Orthogonal Frequency Division MultiplexingOrthogonal Frequency Division Multiplexing(OFDM)(OFDM)(OFDM)(OFDM)
Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat--sensen UniversityUniversity
國立中山大學國立中山大學通訊所通訊所國立中山大學國立中山大學通訊所通訊所
李志鵬李志鵬
July 2010July 2010
Table of ContentsTable of Contentsloz Introduction to OFDM
loz Synchronization
loz Channel Estimation
Introduction to OFDM SystemsIntroduction to OFDM Systems
Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat--sensen UniversityUniversity
OFDM OverviewOFDM Overview
loz OFDMloz Orthogonal Frequency Division Multiplexing
loz Frequency Division Multiplexing (FDM) or multi-tone systems have been employed in military applicationssystems have been employed in military applications since the 196Os
loz OFDM employs multiple carriers overlapping in the f d ifrequency domain
4
OFDM OverviewOFDM Overview
Single carrier (SC) vs multi-carrier (MC)
loz Single carrier data are Multi-carrier data are shared transmitted over only one carrier
among several carriers and simultaneously transmittedFl t f di b iloz Selective fading Flat fading per subcarrier
5
OFDM OverviewOFDM Overview
loz OFDM modulation
loz Featuresloz No intercarrier guard bandsloz Overlapping of bands
S t l ffi iloz Spectral efficiencyloz Easy implementation by IFFTsloz Very sensitive to synchronizationloz Very sensitive to synchronization
6
Applications of OFDM TechnologyApplications of OFDM Technologypp gypp gy
loz Broadband Wired Access AsymmetricDigital Subscriber Loop (ADSL) Digital Multi-tone(DMT)
loz Wireless LANs (IEEE 80211ag IEEE 80211n HIPERLAN-2)
loz Digital Broadcasting (DAB DVB-T DVB-H)loz WiMAX (IEEE 80216 Series) 3GPP Long Term ( ) g
Evolution (3GPP LTE) 4Gloz Wireless Personal Area Network (WPAN) IEEE ( )
80215aMBOAloz Power Line
7
OFDM System ModelOFDM System Modelyy
loz Multi-carrier Block Transmission
frequency
time
8T8
OFDM System ModelOFDM System Modelyy
loz OFDM A block modulation scheme that transmitsa block N source symbols in parallel by using subcarrierssubcarriersloz Sub-carriers are orthogonal in time but overlapped in
frequencyeque cyloz Frequency spacing 1
FFT
fT
Δ =
cos(2 )cos(2 ( ) ) 0FFTTf t f f t dtπ π + Δint 1 10
cos(2 )cos(2 ( ) ) 0f t f f t dtπ π + Δ =int
9
OFDM System ModelOFDM System Model
tNΔ
yy
( )tf02cos π
( )0a
( )0atT
tNΔ1+
( )tf2i
( )
( )0b
T
DataEncoder
tf s Δ=
1 ( ) ( ) njbna +
( )nd SP MUX ( )tDChannelInput
( )tf02sin π
( )tf N 12cos minusπbullbullbull
( )1minusNa
( )1minusNb( )0a1+
( )tf N 12sin minusπ
( )1Nb
tT
( )1a ( )1minusNa1minus
tΔ
Figure 1tNΔ
( )1a ( )1minusNa Figure 1
10
OFDM System ModelOFDM System Model
loz An OFDM system transmitter shown in Figure 1
yy
y gloz The transmitted waveform D(t) can be expressed as
)1( )2sin()()2cos()()(1
0summinus
=
+=N
nnn tfnbtfnatD ππ
where tNffnffn Δ=ΔΔ+=
1 and 0
loz Using a two-dimensional digital modulation format the data
tNΔ
g g symbols d(n) can be represented as a(n) + jb(n)loz a(n) in-phase componentloz b(n) quadrature component
11
OFDM System ModelOFDM System Model
loz The serial data elements spaced by are grouped and tΔ
yy
p y g pused to modulate N carriers Thus they are frequency division multiplexedp
loz The signaling interval is then increased to which tNΔmakes the system less susceptible to channel delay spread impairments
Small-scale fading(Based on multipath time delay spread)
Flat Fading1 BW of signal lt BW of channel
Frequency Selective Fading1 BW of signal gt BW of channel
2 Delay spread lt Symbol preiod 2 Delay spread gt Symbol preiod
12
OrthogonalityOrthogonalityg yg y
loz Consider a set of transmitted carriers as follows
(2) 1 1 0for )( 2 0
minus==⎟⎠⎞
⎜⎝⎛
Δ+
Nnett
tNnfj
n
πψ n
⎨⎧ =minus
intqpab
dtttb for )(
)()( ψψ⎩⎨ Δ=minusne
=int tNabqpdttt
a qp )( and for 0 )()( ψψ
13
OrthogonalityOrthogonality
b tqpjb
intint)(2π
g yg y
dtedtttajbj
b
atN
qpjb
a qp = Δminus
intint )()(
)(2)(2
)(2ψψπ
tNqpjee tN
qpjtN
qpj
Δminusminus
=Δ
minusΔ
minus
)(2
)(2)(2
π
ππ
ee
tNqpjba
tNqpj
tNbqpj
⎟⎟⎞
⎜⎜⎛minus
Δ
minusΔ
minusΔ
minus1
)(2)(1)(2)(2
π
ππ
tNqpj Δminus
⎟⎠
⎜⎝=
)(2πtNabqp Δ=minusne= )( and for 0
14
OrthogonalityOrthogonalityg yg y
tNΔ
15
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFTby using IDFTby using IDFT
loz According to the structure of Tx it must use Ng oscillators That increases the hardware complexity
loz The equivalent method is using IDFT (IFFT)
16
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz In general each carrier can be expressed as
by using IDFTby using IDFT
g p
( ) (6) )()( )(2 ttfjcc
ccetAtS φπ +=
loz We assume that there are N carriers in the OFDM signal Th th t t l l i l S (t) b t d bThen the total complex signal Ss(t) can be represented by
( ) (7))(1)(1
)(2etAtSN
ttfj nn φπ= summinus
+( )
where
(7) )()(
0
0
fnff
etAN
tSn
ns
Δ+=
= sum=
liihffrequency carrier phase amplitude are )( )( and
where 0
fttAfnff
nnn
n
φΔ+
lyrespective carrierth -of n17
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz Then we sample the signal at a sampling frequency tΔ1
by using IDFTby using IDFT
p g p g q y and An(t) and φn(t) becomes
(8))(t = φφ(9) )((8) )(
nn
nn
AtAt
== φφ
( ) (10) 1)(1
0
)(2 0summinus
=
+ΔΔ+=ΔN
n
tkfnfjns
neAN
tkS φπ
loz Then the sampled signal can be expressed as0=nN
( )( ) (11) 1)(1
0
22 0summinus
=
ΔΔ+Δ sdot=ΔN
n
tfnkjtkfjns eeA
NtkS n πφπ
0n
18
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz The inverse discrete Fourier transform (IDFT) is defined
by using IDFTby using IDFT
( )as the following
1 21
NnkjN
πsumminus
(12) )(1)( 2
0
Nnkj
nefnF
Ntkf πsum
=
Δ=Δ
loz Comparing eq(11) and eq(12) the condition must be satisfied in order to make eq(11) an inverse Fouriersatisfied in order to make eq(11) an inverse Fourier transform relationship
1 (13) 1tN
fΔ
=Δ
19
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz If eq(13) is satisfied
by using IDFTby using IDFT
q ( ) loz is the frequency domain signal loz is the time domain signal
( )ntkfjneA φπ +Δ02
)( tkSs Δ gloz is the sub-channel spacingloz is the symbol duration in each sub-channel
)(s
fΔtNΔloz s e sy bo du o e c sub c e
loz This outcome is the same as the result obtained in the
N
loz This outcome is the same as the result obtained in the system of Figure 1 Therefore IDFT can be used to generate an OFDM transmission signalgenerate an OFDM transmission signal
20
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
( )tf02cos π
( )0a
by using IDFTby using IDFT
( )nd ( )tDI( )tf02sin π
( )0a
( )0b
bull
tf s Δ=
1 ( ) ( ) njbna +
( )nd
tNΔffnffn Δ
=Δ+=1 0
( )tDChannelInput
( )tf N 12cos minusπ
( )1minusNa
bullbull
tNΔ
( )tf N 12sin minusπ
( )1minusNb
( )0d
( )nd ( )tDChannelInput
bull bull
( )( )1d
( )2d
tf s Δ=
1 ( ) ( ) njbna +
fNΔffnffn Δ
=Δ+=1 0
ChannelInput
bullbullbullbull
bullbullbullbull
( )1minusNd
21
Frequency Error Results in ICIFrequency Error Results in ICIq yq y
22
Synchronization Error Results in ICISynchronization Error Results in ICIyy
Not Orthogonal Any More
23
Cyclic PrefixCyclic Prefixyy
loz In multipath channel delayed replicas of previous OFDM signal lead to ISI between successive OFDM signals
loz Solution Insert a guard interval between successive OFDMloz Solution Insert a guard interval between successive OFDM signals
24
Cyclic PrefixCyclic Prefix
loz Guard interval leads to intercarrier interference (ICI) in OFDM d d l i
yy
demodulation
loz In DFT interval difference between two subcarriers does not maintain integer number of cycles loss of orthogonality
loz Delayed version of subcarrier 2 causes ICI in the process of d d l ti b i 1demodulating subcarrier 1
25
Cyclic PrefixCyclic Prefixloz Cyclic prefix (CP) A copy of the last part of OFDM signal is
attached to the front of itself
yy
attached to the front of itself
[ ]0d
[ ]1d
[ ]0D[ ]1D
[ ]nd[ ]kD~
symbols dataInput
bullbullbullbull
[ ]1d
[ ]2d
[ ][ ]2D
bullbullbullbullbull
y
bullbullbullbullbull
[ ]1minusNd [ ]1minusND
[ ]gNND minus
bullbullbull
26
Cyclic PrefixCyclic Prefix
loz All delayed replicas of subcarriers always have an
yy
y p yinteger number of cycles within DFT interval no ICI
27
Cyclic PrefixCyclic Prefix
loz Linear convolution vs circular convolution
yy
N
28
Cyclic PrefixCyclic Prefix
loz Channel effect with cyclic prefix
yy
Signal after removed CP
29
Cyclic PrefixCyclic Prefix
loz Time-Domain Explanation
yy
p
30
Cyclic PrefixCyclic Prefixyy
31
Cyclic PrefixCyclic Prefix
loz Spectrum of channel response with length (smaller [ ]nh hL
yy
than )h
gN[ ] nhFFTHk =
loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr
loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=
where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier
[ ] [ ] [ ]N
Notimes
[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce
kk H
X =rArrcomplexity of channel equalizationrdquo
32
Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))
bull bull( )( )nr~Symbols
dataOutput
bullbullbull
bullbullbull( )nr
bullbull
bullbull
33
Cyclic PrefixCyclic Prefix
loz One of the most important reasons to do OFDM is the
yy
pefficient way it deals with multipath delay spread
loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)
34
Bandwidth EfficiencyBandwidth Efficiencyyy
loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )
loz The diagram for bandwidth efficiency of OFDM system is shown below
35
SummarySummaryyy
loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j
the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference
(ISI)loz The main advantage of the OFDM transmission technique is its
high performance even in frequency selective channels
loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems
36
SynchronizationSynchronization
Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg
38 Synchronization2010722
Introduction (13)Introduction (13)( )( )
loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods
loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead
loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the
cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of
computational complexity therefore it may be not be available in short-burst wireless communication
39 Synchronization2010722
Introduction (23)Introduction (23)( )( )
loz Transcevier
40 Synchronization2010722
Introduction (33)Introduction (33)( )( )
loz Synchronization erroryloz Symbol timing offset (TO)
loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)
loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect
41 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg
42 Synchronization2010722
Signal model (14)Signal model (14)g ( )g ( )
loz Time domain sample p1
2
0
1 Nj nk N
n kk
x X eN
πminus
minus= sum
Xk the frequency domain data at the k-th subcarrier0kN =
loz The multipath fading channel
( ) ( ) ( )1L
h t h t δminus
sum
loz Received time domain signal
( ) ( ) ( )0
l ll
h t h tτ τ δ τ τ=
= minussum
loz Received time-domain signal
( ) ( )1
L
n n ly h n l x w nminus
minus= +sum43
( ) ( )0l=sum
Synchronization2010722
Signal model (24)Signal model (24)g ( )g ( )
loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting
12
0
Nj kn N
k nn
Y y e πminus
minus
=
= sum
( )1 1
2
0
N Lj kn N
n l kn l
h n l x e Wπminus minus
minusminus
=
= +sumsum
loz The equation above could be expressed in matrix form
0n l
loz The equation above could be expressed in matrix form
= +Y HX W
44 Synchronization2010722
Signal model (34)Signal model (34)g ( )g ( )
H= = = = +Y Fy FCx FCF X HX W
loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal
y
H=H FCFcirculant matrix leading to be a diagonal matrix
=H FCF
( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0
h h L L hh h
⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥
( )( ) ( )
( ) ( )
1 10 1 0 0i
h Lh L h N L
⎢ ⎥⎢ ⎥minus minus⎢ ⎥
equiv= minus minus⎢ ⎥⎢ ⎥
C( ) ( )
( ) ( )
0 1 1 100
0 0 1 1 0
h L h N L
h N L L h N
⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
45
( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦
Synchronization2010722
Signal model (44)Signal model (44)g ( )g ( )
loz In a time-invariant channel the channel matrix H would be a diagonal matrix
k k k kY H X W= +
( )1 1
2
0 0
N Lj kl N
kn l
H h n l e πminus minus
minus
= =
= sumsum
loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI
k k k k kY H X I Wα= + +k k k k k
( ) ( )1 1 1
2 2 N N L
j m k Nj ml Nk mI X h n l e e ππ
minus minus minusminusminus⎛ ⎞
= ⎜ ⎟⎝ ⎠
sum sum sumICI
46
0 0 0m n lm k= = =ne
⎝ ⎠
Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
47 Synchronization2010722
Impact of timing offset Impact of timing offset p gp g
loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively
( )I f s
Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ
( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr
2 fj k Nk k k kY X H e Wπδminus= +
loz For integer part of timing offset k k k kY X H e W+
I sTδ ( ) 0g IN L δminus minus le leFFT
2 Ij k NY X H e Wπδminus= +
( ) ( ) 2 I sFFT
j f TI sh t T H f e π δδ minusminus rarr
48
k k k kY X H e W= +
Synchronization2010722
Impact of timing offset derivationImpact of timing offset derivationp gp g
49 Synchronization2010722
Discussion of symbol boundaryDiscussion of symbol boundaryy yy y
50 Synchronization2010722
Impact of timing offsetImpact of timing offsetp gp g
loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI
2 Ij k NIk k k k
NY X H e ISI W
Nπδδ minusminus
= + +
loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk
e e
p kloz Phase Shift can be resolved by Differential Encoding
Decoding or carrier recovery with pilots
51 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
52 Synchronization2010722
Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )
53 Synchronization2010722
Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )
( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s
i n t i N N T N T nTy y t e
= + + +=
( ) ( )( ) ( ) ( ) 12
sin
sin
g gI f f
I I
i N N N Nj jf N Ni k i k k
f
Y X H e eN
π ε ε π ε
ε ε
πεπε
+ + minus+
minus minus=⎛ ⎞⎜ ⎟
Faded signal attenuated and rotated by CFO
( )( ) ( ) ( ) ( )11 2
sin
sin g gI f I f
i N N N NN j j l kI f
NN
l k π ε ε π ε επ ε ε + + minusminus + + + minus
⎜ ⎟⎝ ⎠
+ + minussum
( )( )( )
( ) ( )2
0
sin
I f I f
I
j j l kI f N Ni l l
l l k I f
X H e el k
NN
π ε ε π ε ε
ε π ε ε
+ + +
= ne minus
+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠
sum
i k
N
W
⎜ ⎟⎝ ⎠
+ Inter-Carrier Interference (ICI)
54 Synchronization2010722
Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )
loz An easier representation of received signal with only p g yfractional CFO is
1N
( ) ( )1
0
0N
k k k l l kl l k
Y H X C H X C l k Wminus
= ne
= + minus +⎡ ⎤⎣ ⎦sum
where the ICI coefficient is given by
( )( )( ) ( )
sin 1exp 1fl k
C l k j l kπ ε
π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )
( )exp 1sin
f
f
C l k j l kNN l k
N
π επ ε
minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠
55 Synchronization2010722
Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )
loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality
loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift
56 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
57 Synchronization2010722
TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )
loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq
58 Synchronization2010722
TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )
loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below
loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and
rN2 can be expressed by
0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+
( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +
59
( )
Synchronization2010722
TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )
loz Without CFO the first half is identical to the second half (in time order)
loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below
( ) 12 in discretesft T f NT
NTφ π π π ε πε
⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟
⎝ ⎠
helliphellip
sNT⎝ ⎠
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)
60
Length=N2
Synchronization2010722
TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )
loz Received signal without CFOg0
0 0
jr a e θ=jθ 2 0
2 2 0
Nj j
N Nr a e a e
θ θ= =
loz Received signal with CFO0 0
0 0
j jr a e eθ φ= Extra phase rotation due to CFO0 0
2 2 20
2 2 0
N N Nj j jj
N Nr a e e a e e
θ φ φθ= =
due to CFO
Phase difference 2 2 0N N
( )2 02 20 2 0 0
Nj j T fN
r r a e a eφ φ πminus Δ= =
which contains the information about CFO
61
0 2 0 0Nr r a e a e
Synchronization2010722
TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )
loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be
loz Note that d is a time index corresponding to the first( ) ( ) 2 1
20
Nd m d m Nm
P d r rminus
+ + +== sum
loz Note that d is a time index corresponding to the first sample in a window of N samples
loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by
( )22 1
20
Nd m NR d rminus
+ += sumloz A timing metric can be defined as
))(()(
)( 2
2
dRdP
dM =
( ) 20 d m Nm + +=sum
62
))(( dR
Synchronization2010722
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
Table of ContentsTable of Contentsloz Introduction to OFDM
loz Synchronization
loz Channel Estimation
Introduction to OFDM SystemsIntroduction to OFDM Systems
Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat--sensen UniversityUniversity
OFDM OverviewOFDM Overview
loz OFDMloz Orthogonal Frequency Division Multiplexing
loz Frequency Division Multiplexing (FDM) or multi-tone systems have been employed in military applicationssystems have been employed in military applications since the 196Os
loz OFDM employs multiple carriers overlapping in the f d ifrequency domain
4
OFDM OverviewOFDM Overview
Single carrier (SC) vs multi-carrier (MC)
loz Single carrier data are Multi-carrier data are shared transmitted over only one carrier
among several carriers and simultaneously transmittedFl t f di b iloz Selective fading Flat fading per subcarrier
5
OFDM OverviewOFDM Overview
loz OFDM modulation
loz Featuresloz No intercarrier guard bandsloz Overlapping of bands
S t l ffi iloz Spectral efficiencyloz Easy implementation by IFFTsloz Very sensitive to synchronizationloz Very sensitive to synchronization
6
Applications of OFDM TechnologyApplications of OFDM Technologypp gypp gy
loz Broadband Wired Access AsymmetricDigital Subscriber Loop (ADSL) Digital Multi-tone(DMT)
loz Wireless LANs (IEEE 80211ag IEEE 80211n HIPERLAN-2)
loz Digital Broadcasting (DAB DVB-T DVB-H)loz WiMAX (IEEE 80216 Series) 3GPP Long Term ( ) g
Evolution (3GPP LTE) 4Gloz Wireless Personal Area Network (WPAN) IEEE ( )
80215aMBOAloz Power Line
7
OFDM System ModelOFDM System Modelyy
loz Multi-carrier Block Transmission
frequency
time
8T8
OFDM System ModelOFDM System Modelyy
loz OFDM A block modulation scheme that transmitsa block N source symbols in parallel by using subcarrierssubcarriersloz Sub-carriers are orthogonal in time but overlapped in
frequencyeque cyloz Frequency spacing 1
FFT
fT
Δ =
cos(2 )cos(2 ( ) ) 0FFTTf t f f t dtπ π + Δint 1 10
cos(2 )cos(2 ( ) ) 0f t f f t dtπ π + Δ =int
9
OFDM System ModelOFDM System Model
tNΔ
yy
( )tf02cos π
( )0a
( )0atT
tNΔ1+
( )tf2i
( )
( )0b
T
DataEncoder
tf s Δ=
1 ( ) ( ) njbna +
( )nd SP MUX ( )tDChannelInput
( )tf02sin π
( )tf N 12cos minusπbullbullbull
( )1minusNa
( )1minusNb( )0a1+
( )tf N 12sin minusπ
( )1Nb
tT
( )1a ( )1minusNa1minus
tΔ
Figure 1tNΔ
( )1a ( )1minusNa Figure 1
10
OFDM System ModelOFDM System Model
loz An OFDM system transmitter shown in Figure 1
yy
y gloz The transmitted waveform D(t) can be expressed as
)1( )2sin()()2cos()()(1
0summinus
=
+=N
nnn tfnbtfnatD ππ
where tNffnffn Δ=ΔΔ+=
1 and 0
loz Using a two-dimensional digital modulation format the data
tNΔ
g g symbols d(n) can be represented as a(n) + jb(n)loz a(n) in-phase componentloz b(n) quadrature component
11
OFDM System ModelOFDM System Model
loz The serial data elements spaced by are grouped and tΔ
yy
p y g pused to modulate N carriers Thus they are frequency division multiplexedp
loz The signaling interval is then increased to which tNΔmakes the system less susceptible to channel delay spread impairments
Small-scale fading(Based on multipath time delay spread)
Flat Fading1 BW of signal lt BW of channel
Frequency Selective Fading1 BW of signal gt BW of channel
2 Delay spread lt Symbol preiod 2 Delay spread gt Symbol preiod
12
OrthogonalityOrthogonalityg yg y
loz Consider a set of transmitted carriers as follows
(2) 1 1 0for )( 2 0
minus==⎟⎠⎞
⎜⎝⎛
Δ+
Nnett
tNnfj
n
πψ n
⎨⎧ =minus
intqpab
dtttb for )(
)()( ψψ⎩⎨ Δ=minusne
=int tNabqpdttt
a qp )( and for 0 )()( ψψ
13
OrthogonalityOrthogonality
b tqpjb
intint)(2π
g yg y
dtedtttajbj
b
atN
qpjb
a qp = Δminus
intint )()(
)(2)(2
)(2ψψπ
tNqpjee tN
qpjtN
qpj
Δminusminus
=Δ
minusΔ
minus
)(2
)(2)(2
π
ππ
ee
tNqpjba
tNqpj
tNbqpj
⎟⎟⎞
⎜⎜⎛minus
Δ
minusΔ
minusΔ
minus1
)(2)(1)(2)(2
π
ππ
tNqpj Δminus
⎟⎠
⎜⎝=
)(2πtNabqp Δ=minusne= )( and for 0
14
OrthogonalityOrthogonalityg yg y
tNΔ
15
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFTby using IDFTby using IDFT
loz According to the structure of Tx it must use Ng oscillators That increases the hardware complexity
loz The equivalent method is using IDFT (IFFT)
16
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz In general each carrier can be expressed as
by using IDFTby using IDFT
g p
( ) (6) )()( )(2 ttfjcc
ccetAtS φπ +=
loz We assume that there are N carriers in the OFDM signal Th th t t l l i l S (t) b t d bThen the total complex signal Ss(t) can be represented by
( ) (7))(1)(1
)(2etAtSN
ttfj nn φπ= summinus
+( )
where
(7) )()(
0
0
fnff
etAN
tSn
ns
Δ+=
= sum=
liihffrequency carrier phase amplitude are )( )( and
where 0
fttAfnff
nnn
n
φΔ+
lyrespective carrierth -of n17
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz Then we sample the signal at a sampling frequency tΔ1
by using IDFTby using IDFT
p g p g q y and An(t) and φn(t) becomes
(8))(t = φφ(9) )((8) )(
nn
nn
AtAt
== φφ
( ) (10) 1)(1
0
)(2 0summinus
=
+ΔΔ+=ΔN
n
tkfnfjns
neAN
tkS φπ
loz Then the sampled signal can be expressed as0=nN
( )( ) (11) 1)(1
0
22 0summinus
=
ΔΔ+Δ sdot=ΔN
n
tfnkjtkfjns eeA
NtkS n πφπ
0n
18
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz The inverse discrete Fourier transform (IDFT) is defined
by using IDFTby using IDFT
( )as the following
1 21
NnkjN
πsumminus
(12) )(1)( 2
0
Nnkj
nefnF
Ntkf πsum
=
Δ=Δ
loz Comparing eq(11) and eq(12) the condition must be satisfied in order to make eq(11) an inverse Fouriersatisfied in order to make eq(11) an inverse Fourier transform relationship
1 (13) 1tN
fΔ
=Δ
19
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz If eq(13) is satisfied
by using IDFTby using IDFT
q ( ) loz is the frequency domain signal loz is the time domain signal
( )ntkfjneA φπ +Δ02
)( tkSs Δ gloz is the sub-channel spacingloz is the symbol duration in each sub-channel
)(s
fΔtNΔloz s e sy bo du o e c sub c e
loz This outcome is the same as the result obtained in the
N
loz This outcome is the same as the result obtained in the system of Figure 1 Therefore IDFT can be used to generate an OFDM transmission signalgenerate an OFDM transmission signal
20
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
( )tf02cos π
( )0a
by using IDFTby using IDFT
( )nd ( )tDI( )tf02sin π
( )0a
( )0b
bull
tf s Δ=
1 ( ) ( ) njbna +
( )nd
tNΔffnffn Δ
=Δ+=1 0
( )tDChannelInput
( )tf N 12cos minusπ
( )1minusNa
bullbull
tNΔ
( )tf N 12sin minusπ
( )1minusNb
( )0d
( )nd ( )tDChannelInput
bull bull
( )( )1d
( )2d
tf s Δ=
1 ( ) ( ) njbna +
fNΔffnffn Δ
=Δ+=1 0
ChannelInput
bullbullbullbull
bullbullbullbull
( )1minusNd
21
Frequency Error Results in ICIFrequency Error Results in ICIq yq y
22
Synchronization Error Results in ICISynchronization Error Results in ICIyy
Not Orthogonal Any More
23
Cyclic PrefixCyclic Prefixyy
loz In multipath channel delayed replicas of previous OFDM signal lead to ISI between successive OFDM signals
loz Solution Insert a guard interval between successive OFDMloz Solution Insert a guard interval between successive OFDM signals
24
Cyclic PrefixCyclic Prefix
loz Guard interval leads to intercarrier interference (ICI) in OFDM d d l i
yy
demodulation
loz In DFT interval difference between two subcarriers does not maintain integer number of cycles loss of orthogonality
loz Delayed version of subcarrier 2 causes ICI in the process of d d l ti b i 1demodulating subcarrier 1
25
Cyclic PrefixCyclic Prefixloz Cyclic prefix (CP) A copy of the last part of OFDM signal is
attached to the front of itself
yy
attached to the front of itself
[ ]0d
[ ]1d
[ ]0D[ ]1D
[ ]nd[ ]kD~
symbols dataInput
bullbullbullbull
[ ]1d
[ ]2d
[ ][ ]2D
bullbullbullbullbull
y
bullbullbullbullbull
[ ]1minusNd [ ]1minusND
[ ]gNND minus
bullbullbull
26
Cyclic PrefixCyclic Prefix
loz All delayed replicas of subcarriers always have an
yy
y p yinteger number of cycles within DFT interval no ICI
27
Cyclic PrefixCyclic Prefix
loz Linear convolution vs circular convolution
yy
N
28
Cyclic PrefixCyclic Prefix
loz Channel effect with cyclic prefix
yy
Signal after removed CP
29
Cyclic PrefixCyclic Prefix
loz Time-Domain Explanation
yy
p
30
Cyclic PrefixCyclic Prefixyy
31
Cyclic PrefixCyclic Prefix
loz Spectrum of channel response with length (smaller [ ]nh hL
yy
than )h
gN[ ] nhFFTHk =
loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr
loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=
where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier
[ ] [ ] [ ]N
Notimes
[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce
kk H
X =rArrcomplexity of channel equalizationrdquo
32
Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))
bull bull( )( )nr~Symbols
dataOutput
bullbullbull
bullbullbull( )nr
bullbull
bullbull
33
Cyclic PrefixCyclic Prefix
loz One of the most important reasons to do OFDM is the
yy
pefficient way it deals with multipath delay spread
loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)
34
Bandwidth EfficiencyBandwidth Efficiencyyy
loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )
loz The diagram for bandwidth efficiency of OFDM system is shown below
35
SummarySummaryyy
loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j
the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference
(ISI)loz The main advantage of the OFDM transmission technique is its
high performance even in frequency selective channels
loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems
36
SynchronizationSynchronization
Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg
38 Synchronization2010722
Introduction (13)Introduction (13)( )( )
loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods
loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead
loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the
cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of
computational complexity therefore it may be not be available in short-burst wireless communication
39 Synchronization2010722
Introduction (23)Introduction (23)( )( )
loz Transcevier
40 Synchronization2010722
Introduction (33)Introduction (33)( )( )
loz Synchronization erroryloz Symbol timing offset (TO)
loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)
loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect
41 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg
42 Synchronization2010722
Signal model (14)Signal model (14)g ( )g ( )
loz Time domain sample p1
2
0
1 Nj nk N
n kk
x X eN
πminus
minus= sum
Xk the frequency domain data at the k-th subcarrier0kN =
loz The multipath fading channel
( ) ( ) ( )1L
h t h t δminus
sum
loz Received time domain signal
( ) ( ) ( )0
l ll
h t h tτ τ δ τ τ=
= minussum
loz Received time-domain signal
( ) ( )1
L
n n ly h n l x w nminus
minus= +sum43
( ) ( )0l=sum
Synchronization2010722
Signal model (24)Signal model (24)g ( )g ( )
loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting
12
0
Nj kn N
k nn
Y y e πminus
minus
=
= sum
( )1 1
2
0
N Lj kn N
n l kn l
h n l x e Wπminus minus
minusminus
=
= +sumsum
loz The equation above could be expressed in matrix form
0n l
loz The equation above could be expressed in matrix form
= +Y HX W
44 Synchronization2010722
Signal model (34)Signal model (34)g ( )g ( )
H= = = = +Y Fy FCx FCF X HX W
loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal
y
H=H FCFcirculant matrix leading to be a diagonal matrix
=H FCF
( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0
h h L L hh h
⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥
( )( ) ( )
( ) ( )
1 10 1 0 0i
h Lh L h N L
⎢ ⎥⎢ ⎥minus minus⎢ ⎥
equiv= minus minus⎢ ⎥⎢ ⎥
C( ) ( )
( ) ( )
0 1 1 100
0 0 1 1 0
h L h N L
h N L L h N
⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
45
( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦
Synchronization2010722
Signal model (44)Signal model (44)g ( )g ( )
loz In a time-invariant channel the channel matrix H would be a diagonal matrix
k k k kY H X W= +
( )1 1
2
0 0
N Lj kl N
kn l
H h n l e πminus minus
minus
= =
= sumsum
loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI
k k k k kY H X I Wα= + +k k k k k
( ) ( )1 1 1
2 2 N N L
j m k Nj ml Nk mI X h n l e e ππ
minus minus minusminusminus⎛ ⎞
= ⎜ ⎟⎝ ⎠
sum sum sumICI
46
0 0 0m n lm k= = =ne
⎝ ⎠
Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
47 Synchronization2010722
Impact of timing offset Impact of timing offset p gp g
loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively
( )I f s
Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ
( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr
2 fj k Nk k k kY X H e Wπδminus= +
loz For integer part of timing offset k k k kY X H e W+
I sTδ ( ) 0g IN L δminus minus le leFFT
2 Ij k NY X H e Wπδminus= +
( ) ( ) 2 I sFFT
j f TI sh t T H f e π δδ minusminus rarr
48
k k k kY X H e W= +
Synchronization2010722
Impact of timing offset derivationImpact of timing offset derivationp gp g
49 Synchronization2010722
Discussion of symbol boundaryDiscussion of symbol boundaryy yy y
50 Synchronization2010722
Impact of timing offsetImpact of timing offsetp gp g
loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI
2 Ij k NIk k k k
NY X H e ISI W
Nπδδ minusminus
= + +
loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk
e e
p kloz Phase Shift can be resolved by Differential Encoding
Decoding or carrier recovery with pilots
51 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
52 Synchronization2010722
Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )
53 Synchronization2010722
Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )
( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s
i n t i N N T N T nTy y t e
= + + +=
( ) ( )( ) ( ) ( ) 12
sin
sin
g gI f f
I I
i N N N Nj jf N Ni k i k k
f
Y X H e eN
π ε ε π ε
ε ε
πεπε
+ + minus+
minus minus=⎛ ⎞⎜ ⎟
Faded signal attenuated and rotated by CFO
( )( ) ( ) ( ) ( )11 2
sin
sin g gI f I f
i N N N NN j j l kI f
NN
l k π ε ε π ε επ ε ε + + minusminus + + + minus
⎜ ⎟⎝ ⎠
+ + minussum
( )( )( )
( ) ( )2
0
sin
I f I f
I
j j l kI f N Ni l l
l l k I f
X H e el k
NN
π ε ε π ε ε
ε π ε ε
+ + +
= ne minus
+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠
sum
i k
N
W
⎜ ⎟⎝ ⎠
+ Inter-Carrier Interference (ICI)
54 Synchronization2010722
Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )
loz An easier representation of received signal with only p g yfractional CFO is
1N
( ) ( )1
0
0N
k k k l l kl l k
Y H X C H X C l k Wminus
= ne
= + minus +⎡ ⎤⎣ ⎦sum
where the ICI coefficient is given by
( )( )( ) ( )
sin 1exp 1fl k
C l k j l kπ ε
π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )
( )exp 1sin
f
f
C l k j l kNN l k
N
π επ ε
minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠
55 Synchronization2010722
Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )
loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality
loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift
56 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
57 Synchronization2010722
TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )
loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq
58 Synchronization2010722
TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )
loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below
loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and
rN2 can be expressed by
0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+
( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +
59
( )
Synchronization2010722
TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )
loz Without CFO the first half is identical to the second half (in time order)
loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below
( ) 12 in discretesft T f NT
NTφ π π π ε πε
⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟
⎝ ⎠
helliphellip
sNT⎝ ⎠
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)
60
Length=N2
Synchronization2010722
TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )
loz Received signal without CFOg0
0 0
jr a e θ=jθ 2 0
2 2 0
Nj j
N Nr a e a e
θ θ= =
loz Received signal with CFO0 0
0 0
j jr a e eθ φ= Extra phase rotation due to CFO0 0
2 2 20
2 2 0
N N Nj j jj
N Nr a e e a e e
θ φ φθ= =
due to CFO
Phase difference 2 2 0N N
( )2 02 20 2 0 0
Nj j T fN
r r a e a eφ φ πminus Δ= =
which contains the information about CFO
61
0 2 0 0Nr r a e a e
Synchronization2010722
TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )
loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be
loz Note that d is a time index corresponding to the first( ) ( ) 2 1
20
Nd m d m Nm
P d r rminus
+ + +== sum
loz Note that d is a time index corresponding to the first sample in a window of N samples
loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by
( )22 1
20
Nd m NR d rminus
+ += sumloz A timing metric can be defined as
))(()(
)( 2
2
dRdP
dM =
( ) 20 d m Nm + +=sum
62
))(( dR
Synchronization2010722
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
Introduction to OFDM SystemsIntroduction to OFDM Systems
Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat--sensen UniversityUniversity
OFDM OverviewOFDM Overview
loz OFDMloz Orthogonal Frequency Division Multiplexing
loz Frequency Division Multiplexing (FDM) or multi-tone systems have been employed in military applicationssystems have been employed in military applications since the 196Os
loz OFDM employs multiple carriers overlapping in the f d ifrequency domain
4
OFDM OverviewOFDM Overview
Single carrier (SC) vs multi-carrier (MC)
loz Single carrier data are Multi-carrier data are shared transmitted over only one carrier
among several carriers and simultaneously transmittedFl t f di b iloz Selective fading Flat fading per subcarrier
5
OFDM OverviewOFDM Overview
loz OFDM modulation
loz Featuresloz No intercarrier guard bandsloz Overlapping of bands
S t l ffi iloz Spectral efficiencyloz Easy implementation by IFFTsloz Very sensitive to synchronizationloz Very sensitive to synchronization
6
Applications of OFDM TechnologyApplications of OFDM Technologypp gypp gy
loz Broadband Wired Access AsymmetricDigital Subscriber Loop (ADSL) Digital Multi-tone(DMT)
loz Wireless LANs (IEEE 80211ag IEEE 80211n HIPERLAN-2)
loz Digital Broadcasting (DAB DVB-T DVB-H)loz WiMAX (IEEE 80216 Series) 3GPP Long Term ( ) g
Evolution (3GPP LTE) 4Gloz Wireless Personal Area Network (WPAN) IEEE ( )
80215aMBOAloz Power Line
7
OFDM System ModelOFDM System Modelyy
loz Multi-carrier Block Transmission
frequency
time
8T8
OFDM System ModelOFDM System Modelyy
loz OFDM A block modulation scheme that transmitsa block N source symbols in parallel by using subcarrierssubcarriersloz Sub-carriers are orthogonal in time but overlapped in
frequencyeque cyloz Frequency spacing 1
FFT
fT
Δ =
cos(2 )cos(2 ( ) ) 0FFTTf t f f t dtπ π + Δint 1 10
cos(2 )cos(2 ( ) ) 0f t f f t dtπ π + Δ =int
9
OFDM System ModelOFDM System Model
tNΔ
yy
( )tf02cos π
( )0a
( )0atT
tNΔ1+
( )tf2i
( )
( )0b
T
DataEncoder
tf s Δ=
1 ( ) ( ) njbna +
( )nd SP MUX ( )tDChannelInput
( )tf02sin π
( )tf N 12cos minusπbullbullbull
( )1minusNa
( )1minusNb( )0a1+
( )tf N 12sin minusπ
( )1Nb
tT
( )1a ( )1minusNa1minus
tΔ
Figure 1tNΔ
( )1a ( )1minusNa Figure 1
10
OFDM System ModelOFDM System Model
loz An OFDM system transmitter shown in Figure 1
yy
y gloz The transmitted waveform D(t) can be expressed as
)1( )2sin()()2cos()()(1
0summinus
=
+=N
nnn tfnbtfnatD ππ
where tNffnffn Δ=ΔΔ+=
1 and 0
loz Using a two-dimensional digital modulation format the data
tNΔ
g g symbols d(n) can be represented as a(n) + jb(n)loz a(n) in-phase componentloz b(n) quadrature component
11
OFDM System ModelOFDM System Model
loz The serial data elements spaced by are grouped and tΔ
yy
p y g pused to modulate N carriers Thus they are frequency division multiplexedp
loz The signaling interval is then increased to which tNΔmakes the system less susceptible to channel delay spread impairments
Small-scale fading(Based on multipath time delay spread)
Flat Fading1 BW of signal lt BW of channel
Frequency Selective Fading1 BW of signal gt BW of channel
2 Delay spread lt Symbol preiod 2 Delay spread gt Symbol preiod
12
OrthogonalityOrthogonalityg yg y
loz Consider a set of transmitted carriers as follows
(2) 1 1 0for )( 2 0
minus==⎟⎠⎞
⎜⎝⎛
Δ+
Nnett
tNnfj
n
πψ n
⎨⎧ =minus
intqpab
dtttb for )(
)()( ψψ⎩⎨ Δ=minusne
=int tNabqpdttt
a qp )( and for 0 )()( ψψ
13
OrthogonalityOrthogonality
b tqpjb
intint)(2π
g yg y
dtedtttajbj
b
atN
qpjb
a qp = Δminus
intint )()(
)(2)(2
)(2ψψπ
tNqpjee tN
qpjtN
qpj
Δminusminus
=Δ
minusΔ
minus
)(2
)(2)(2
π
ππ
ee
tNqpjba
tNqpj
tNbqpj
⎟⎟⎞
⎜⎜⎛minus
Δ
minusΔ
minusΔ
minus1
)(2)(1)(2)(2
π
ππ
tNqpj Δminus
⎟⎠
⎜⎝=
)(2πtNabqp Δ=minusne= )( and for 0
14
OrthogonalityOrthogonalityg yg y
tNΔ
15
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFTby using IDFTby using IDFT
loz According to the structure of Tx it must use Ng oscillators That increases the hardware complexity
loz The equivalent method is using IDFT (IFFT)
16
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz In general each carrier can be expressed as
by using IDFTby using IDFT
g p
( ) (6) )()( )(2 ttfjcc
ccetAtS φπ +=
loz We assume that there are N carriers in the OFDM signal Th th t t l l i l S (t) b t d bThen the total complex signal Ss(t) can be represented by
( ) (7))(1)(1
)(2etAtSN
ttfj nn φπ= summinus
+( )
where
(7) )()(
0
0
fnff
etAN
tSn
ns
Δ+=
= sum=
liihffrequency carrier phase amplitude are )( )( and
where 0
fttAfnff
nnn
n
φΔ+
lyrespective carrierth -of n17
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz Then we sample the signal at a sampling frequency tΔ1
by using IDFTby using IDFT
p g p g q y and An(t) and φn(t) becomes
(8))(t = φφ(9) )((8) )(
nn
nn
AtAt
== φφ
( ) (10) 1)(1
0
)(2 0summinus
=
+ΔΔ+=ΔN
n
tkfnfjns
neAN
tkS φπ
loz Then the sampled signal can be expressed as0=nN
( )( ) (11) 1)(1
0
22 0summinus
=
ΔΔ+Δ sdot=ΔN
n
tfnkjtkfjns eeA
NtkS n πφπ
0n
18
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz The inverse discrete Fourier transform (IDFT) is defined
by using IDFTby using IDFT
( )as the following
1 21
NnkjN
πsumminus
(12) )(1)( 2
0
Nnkj
nefnF
Ntkf πsum
=
Δ=Δ
loz Comparing eq(11) and eq(12) the condition must be satisfied in order to make eq(11) an inverse Fouriersatisfied in order to make eq(11) an inverse Fourier transform relationship
1 (13) 1tN
fΔ
=Δ
19
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz If eq(13) is satisfied
by using IDFTby using IDFT
q ( ) loz is the frequency domain signal loz is the time domain signal
( )ntkfjneA φπ +Δ02
)( tkSs Δ gloz is the sub-channel spacingloz is the symbol duration in each sub-channel
)(s
fΔtNΔloz s e sy bo du o e c sub c e
loz This outcome is the same as the result obtained in the
N
loz This outcome is the same as the result obtained in the system of Figure 1 Therefore IDFT can be used to generate an OFDM transmission signalgenerate an OFDM transmission signal
20
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
( )tf02cos π
( )0a
by using IDFTby using IDFT
( )nd ( )tDI( )tf02sin π
( )0a
( )0b
bull
tf s Δ=
1 ( ) ( ) njbna +
( )nd
tNΔffnffn Δ
=Δ+=1 0
( )tDChannelInput
( )tf N 12cos minusπ
( )1minusNa
bullbull
tNΔ
( )tf N 12sin minusπ
( )1minusNb
( )0d
( )nd ( )tDChannelInput
bull bull
( )( )1d
( )2d
tf s Δ=
1 ( ) ( ) njbna +
fNΔffnffn Δ
=Δ+=1 0
ChannelInput
bullbullbullbull
bullbullbullbull
( )1minusNd
21
Frequency Error Results in ICIFrequency Error Results in ICIq yq y
22
Synchronization Error Results in ICISynchronization Error Results in ICIyy
Not Orthogonal Any More
23
Cyclic PrefixCyclic Prefixyy
loz In multipath channel delayed replicas of previous OFDM signal lead to ISI between successive OFDM signals
loz Solution Insert a guard interval between successive OFDMloz Solution Insert a guard interval between successive OFDM signals
24
Cyclic PrefixCyclic Prefix
loz Guard interval leads to intercarrier interference (ICI) in OFDM d d l i
yy
demodulation
loz In DFT interval difference between two subcarriers does not maintain integer number of cycles loss of orthogonality
loz Delayed version of subcarrier 2 causes ICI in the process of d d l ti b i 1demodulating subcarrier 1
25
Cyclic PrefixCyclic Prefixloz Cyclic prefix (CP) A copy of the last part of OFDM signal is
attached to the front of itself
yy
attached to the front of itself
[ ]0d
[ ]1d
[ ]0D[ ]1D
[ ]nd[ ]kD~
symbols dataInput
bullbullbullbull
[ ]1d
[ ]2d
[ ][ ]2D
bullbullbullbullbull
y
bullbullbullbullbull
[ ]1minusNd [ ]1minusND
[ ]gNND minus
bullbullbull
26
Cyclic PrefixCyclic Prefix
loz All delayed replicas of subcarriers always have an
yy
y p yinteger number of cycles within DFT interval no ICI
27
Cyclic PrefixCyclic Prefix
loz Linear convolution vs circular convolution
yy
N
28
Cyclic PrefixCyclic Prefix
loz Channel effect with cyclic prefix
yy
Signal after removed CP
29
Cyclic PrefixCyclic Prefix
loz Time-Domain Explanation
yy
p
30
Cyclic PrefixCyclic Prefixyy
31
Cyclic PrefixCyclic Prefix
loz Spectrum of channel response with length (smaller [ ]nh hL
yy
than )h
gN[ ] nhFFTHk =
loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr
loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=
where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier
[ ] [ ] [ ]N
Notimes
[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce
kk H
X =rArrcomplexity of channel equalizationrdquo
32
Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))
bull bull( )( )nr~Symbols
dataOutput
bullbullbull
bullbullbull( )nr
bullbull
bullbull
33
Cyclic PrefixCyclic Prefix
loz One of the most important reasons to do OFDM is the
yy
pefficient way it deals with multipath delay spread
loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)
34
Bandwidth EfficiencyBandwidth Efficiencyyy
loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )
loz The diagram for bandwidth efficiency of OFDM system is shown below
35
SummarySummaryyy
loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j
the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference
(ISI)loz The main advantage of the OFDM transmission technique is its
high performance even in frequency selective channels
loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems
36
SynchronizationSynchronization
Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg
38 Synchronization2010722
Introduction (13)Introduction (13)( )( )
loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods
loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead
loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the
cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of
computational complexity therefore it may be not be available in short-burst wireless communication
39 Synchronization2010722
Introduction (23)Introduction (23)( )( )
loz Transcevier
40 Synchronization2010722
Introduction (33)Introduction (33)( )( )
loz Synchronization erroryloz Symbol timing offset (TO)
loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)
loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect
41 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg
42 Synchronization2010722
Signal model (14)Signal model (14)g ( )g ( )
loz Time domain sample p1
2
0
1 Nj nk N
n kk
x X eN
πminus
minus= sum
Xk the frequency domain data at the k-th subcarrier0kN =
loz The multipath fading channel
( ) ( ) ( )1L
h t h t δminus
sum
loz Received time domain signal
( ) ( ) ( )0
l ll
h t h tτ τ δ τ τ=
= minussum
loz Received time-domain signal
( ) ( )1
L
n n ly h n l x w nminus
minus= +sum43
( ) ( )0l=sum
Synchronization2010722
Signal model (24)Signal model (24)g ( )g ( )
loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting
12
0
Nj kn N
k nn
Y y e πminus
minus
=
= sum
( )1 1
2
0
N Lj kn N
n l kn l
h n l x e Wπminus minus
minusminus
=
= +sumsum
loz The equation above could be expressed in matrix form
0n l
loz The equation above could be expressed in matrix form
= +Y HX W
44 Synchronization2010722
Signal model (34)Signal model (34)g ( )g ( )
H= = = = +Y Fy FCx FCF X HX W
loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal
y
H=H FCFcirculant matrix leading to be a diagonal matrix
=H FCF
( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0
h h L L hh h
⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥
( )( ) ( )
( ) ( )
1 10 1 0 0i
h Lh L h N L
⎢ ⎥⎢ ⎥minus minus⎢ ⎥
equiv= minus minus⎢ ⎥⎢ ⎥
C( ) ( )
( ) ( )
0 1 1 100
0 0 1 1 0
h L h N L
h N L L h N
⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
45
( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦
Synchronization2010722
Signal model (44)Signal model (44)g ( )g ( )
loz In a time-invariant channel the channel matrix H would be a diagonal matrix
k k k kY H X W= +
( )1 1
2
0 0
N Lj kl N
kn l
H h n l e πminus minus
minus
= =
= sumsum
loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI
k k k k kY H X I Wα= + +k k k k k
( ) ( )1 1 1
2 2 N N L
j m k Nj ml Nk mI X h n l e e ππ
minus minus minusminusminus⎛ ⎞
= ⎜ ⎟⎝ ⎠
sum sum sumICI
46
0 0 0m n lm k= = =ne
⎝ ⎠
Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
47 Synchronization2010722
Impact of timing offset Impact of timing offset p gp g
loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively
( )I f s
Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ
( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr
2 fj k Nk k k kY X H e Wπδminus= +
loz For integer part of timing offset k k k kY X H e W+
I sTδ ( ) 0g IN L δminus minus le leFFT
2 Ij k NY X H e Wπδminus= +
( ) ( ) 2 I sFFT
j f TI sh t T H f e π δδ minusminus rarr
48
k k k kY X H e W= +
Synchronization2010722
Impact of timing offset derivationImpact of timing offset derivationp gp g
49 Synchronization2010722
Discussion of symbol boundaryDiscussion of symbol boundaryy yy y
50 Synchronization2010722
Impact of timing offsetImpact of timing offsetp gp g
loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI
2 Ij k NIk k k k
NY X H e ISI W
Nπδδ minusminus
= + +
loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk
e e
p kloz Phase Shift can be resolved by Differential Encoding
Decoding or carrier recovery with pilots
51 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
52 Synchronization2010722
Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )
53 Synchronization2010722
Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )
( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s
i n t i N N T N T nTy y t e
= + + +=
( ) ( )( ) ( ) ( ) 12
sin
sin
g gI f f
I I
i N N N Nj jf N Ni k i k k
f
Y X H e eN
π ε ε π ε
ε ε
πεπε
+ + minus+
minus minus=⎛ ⎞⎜ ⎟
Faded signal attenuated and rotated by CFO
( )( ) ( ) ( ) ( )11 2
sin
sin g gI f I f
i N N N NN j j l kI f
NN
l k π ε ε π ε επ ε ε + + minusminus + + + minus
⎜ ⎟⎝ ⎠
+ + minussum
( )( )( )
( ) ( )2
0
sin
I f I f
I
j j l kI f N Ni l l
l l k I f
X H e el k
NN
π ε ε π ε ε
ε π ε ε
+ + +
= ne minus
+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠
sum
i k
N
W
⎜ ⎟⎝ ⎠
+ Inter-Carrier Interference (ICI)
54 Synchronization2010722
Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )
loz An easier representation of received signal with only p g yfractional CFO is
1N
( ) ( )1
0
0N
k k k l l kl l k
Y H X C H X C l k Wminus
= ne
= + minus +⎡ ⎤⎣ ⎦sum
where the ICI coefficient is given by
( )( )( ) ( )
sin 1exp 1fl k
C l k j l kπ ε
π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )
( )exp 1sin
f
f
C l k j l kNN l k
N
π επ ε
minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠
55 Synchronization2010722
Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )
loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality
loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift
56 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
57 Synchronization2010722
TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )
loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq
58 Synchronization2010722
TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )
loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below
loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and
rN2 can be expressed by
0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+
( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +
59
( )
Synchronization2010722
TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )
loz Without CFO the first half is identical to the second half (in time order)
loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below
( ) 12 in discretesft T f NT
NTφ π π π ε πε
⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟
⎝ ⎠
helliphellip
sNT⎝ ⎠
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)
60
Length=N2
Synchronization2010722
TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )
loz Received signal without CFOg0
0 0
jr a e θ=jθ 2 0
2 2 0
Nj j
N Nr a e a e
θ θ= =
loz Received signal with CFO0 0
0 0
j jr a e eθ φ= Extra phase rotation due to CFO0 0
2 2 20
2 2 0
N N Nj j jj
N Nr a e e a e e
θ φ φθ= =
due to CFO
Phase difference 2 2 0N N
( )2 02 20 2 0 0
Nj j T fN
r r a e a eφ φ πminus Δ= =
which contains the information about CFO
61
0 2 0 0Nr r a e a e
Synchronization2010722
TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )
loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be
loz Note that d is a time index corresponding to the first( ) ( ) 2 1
20
Nd m d m Nm
P d r rminus
+ + +== sum
loz Note that d is a time index corresponding to the first sample in a window of N samples
loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by
( )22 1
20
Nd m NR d rminus
+ += sumloz A timing metric can be defined as
))(()(
)( 2
2
dRdP
dM =
( ) 20 d m Nm + +=sum
62
))(( dR
Synchronization2010722
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
OFDM OverviewOFDM Overview
loz OFDMloz Orthogonal Frequency Division Multiplexing
loz Frequency Division Multiplexing (FDM) or multi-tone systems have been employed in military applicationssystems have been employed in military applications since the 196Os
loz OFDM employs multiple carriers overlapping in the f d ifrequency domain
4
OFDM OverviewOFDM Overview
Single carrier (SC) vs multi-carrier (MC)
loz Single carrier data are Multi-carrier data are shared transmitted over only one carrier
among several carriers and simultaneously transmittedFl t f di b iloz Selective fading Flat fading per subcarrier
5
OFDM OverviewOFDM Overview
loz OFDM modulation
loz Featuresloz No intercarrier guard bandsloz Overlapping of bands
S t l ffi iloz Spectral efficiencyloz Easy implementation by IFFTsloz Very sensitive to synchronizationloz Very sensitive to synchronization
6
Applications of OFDM TechnologyApplications of OFDM Technologypp gypp gy
loz Broadband Wired Access AsymmetricDigital Subscriber Loop (ADSL) Digital Multi-tone(DMT)
loz Wireless LANs (IEEE 80211ag IEEE 80211n HIPERLAN-2)
loz Digital Broadcasting (DAB DVB-T DVB-H)loz WiMAX (IEEE 80216 Series) 3GPP Long Term ( ) g
Evolution (3GPP LTE) 4Gloz Wireless Personal Area Network (WPAN) IEEE ( )
80215aMBOAloz Power Line
7
OFDM System ModelOFDM System Modelyy
loz Multi-carrier Block Transmission
frequency
time
8T8
OFDM System ModelOFDM System Modelyy
loz OFDM A block modulation scheme that transmitsa block N source symbols in parallel by using subcarrierssubcarriersloz Sub-carriers are orthogonal in time but overlapped in
frequencyeque cyloz Frequency spacing 1
FFT
fT
Δ =
cos(2 )cos(2 ( ) ) 0FFTTf t f f t dtπ π + Δint 1 10
cos(2 )cos(2 ( ) ) 0f t f f t dtπ π + Δ =int
9
OFDM System ModelOFDM System Model
tNΔ
yy
( )tf02cos π
( )0a
( )0atT
tNΔ1+
( )tf2i
( )
( )0b
T
DataEncoder
tf s Δ=
1 ( ) ( ) njbna +
( )nd SP MUX ( )tDChannelInput
( )tf02sin π
( )tf N 12cos minusπbullbullbull
( )1minusNa
( )1minusNb( )0a1+
( )tf N 12sin minusπ
( )1Nb
tT
( )1a ( )1minusNa1minus
tΔ
Figure 1tNΔ
( )1a ( )1minusNa Figure 1
10
OFDM System ModelOFDM System Model
loz An OFDM system transmitter shown in Figure 1
yy
y gloz The transmitted waveform D(t) can be expressed as
)1( )2sin()()2cos()()(1
0summinus
=
+=N
nnn tfnbtfnatD ππ
where tNffnffn Δ=ΔΔ+=
1 and 0
loz Using a two-dimensional digital modulation format the data
tNΔ
g g symbols d(n) can be represented as a(n) + jb(n)loz a(n) in-phase componentloz b(n) quadrature component
11
OFDM System ModelOFDM System Model
loz The serial data elements spaced by are grouped and tΔ
yy
p y g pused to modulate N carriers Thus they are frequency division multiplexedp
loz The signaling interval is then increased to which tNΔmakes the system less susceptible to channel delay spread impairments
Small-scale fading(Based on multipath time delay spread)
Flat Fading1 BW of signal lt BW of channel
Frequency Selective Fading1 BW of signal gt BW of channel
2 Delay spread lt Symbol preiod 2 Delay spread gt Symbol preiod
12
OrthogonalityOrthogonalityg yg y
loz Consider a set of transmitted carriers as follows
(2) 1 1 0for )( 2 0
minus==⎟⎠⎞
⎜⎝⎛
Δ+
Nnett
tNnfj
n
πψ n
⎨⎧ =minus
intqpab
dtttb for )(
)()( ψψ⎩⎨ Δ=minusne
=int tNabqpdttt
a qp )( and for 0 )()( ψψ
13
OrthogonalityOrthogonality
b tqpjb
intint)(2π
g yg y
dtedtttajbj
b
atN
qpjb
a qp = Δminus
intint )()(
)(2)(2
)(2ψψπ
tNqpjee tN
qpjtN
qpj
Δminusminus
=Δ
minusΔ
minus
)(2
)(2)(2
π
ππ
ee
tNqpjba
tNqpj
tNbqpj
⎟⎟⎞
⎜⎜⎛minus
Δ
minusΔ
minusΔ
minus1
)(2)(1)(2)(2
π
ππ
tNqpj Δminus
⎟⎠
⎜⎝=
)(2πtNabqp Δ=minusne= )( and for 0
14
OrthogonalityOrthogonalityg yg y
tNΔ
15
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFTby using IDFTby using IDFT
loz According to the structure of Tx it must use Ng oscillators That increases the hardware complexity
loz The equivalent method is using IDFT (IFFT)
16
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz In general each carrier can be expressed as
by using IDFTby using IDFT
g p
( ) (6) )()( )(2 ttfjcc
ccetAtS φπ +=
loz We assume that there are N carriers in the OFDM signal Th th t t l l i l S (t) b t d bThen the total complex signal Ss(t) can be represented by
( ) (7))(1)(1
)(2etAtSN
ttfj nn φπ= summinus
+( )
where
(7) )()(
0
0
fnff
etAN
tSn
ns
Δ+=
= sum=
liihffrequency carrier phase amplitude are )( )( and
where 0
fttAfnff
nnn
n
φΔ+
lyrespective carrierth -of n17
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz Then we sample the signal at a sampling frequency tΔ1
by using IDFTby using IDFT
p g p g q y and An(t) and φn(t) becomes
(8))(t = φφ(9) )((8) )(
nn
nn
AtAt
== φφ
( ) (10) 1)(1
0
)(2 0summinus
=
+ΔΔ+=ΔN
n
tkfnfjns
neAN
tkS φπ
loz Then the sampled signal can be expressed as0=nN
( )( ) (11) 1)(1
0
22 0summinus
=
ΔΔ+Δ sdot=ΔN
n
tfnkjtkfjns eeA
NtkS n πφπ
0n
18
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz The inverse discrete Fourier transform (IDFT) is defined
by using IDFTby using IDFT
( )as the following
1 21
NnkjN
πsumminus
(12) )(1)( 2
0
Nnkj
nefnF
Ntkf πsum
=
Δ=Δ
loz Comparing eq(11) and eq(12) the condition must be satisfied in order to make eq(11) an inverse Fouriersatisfied in order to make eq(11) an inverse Fourier transform relationship
1 (13) 1tN
fΔ
=Δ
19
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz If eq(13) is satisfied
by using IDFTby using IDFT
q ( ) loz is the frequency domain signal loz is the time domain signal
( )ntkfjneA φπ +Δ02
)( tkSs Δ gloz is the sub-channel spacingloz is the symbol duration in each sub-channel
)(s
fΔtNΔloz s e sy bo du o e c sub c e
loz This outcome is the same as the result obtained in the
N
loz This outcome is the same as the result obtained in the system of Figure 1 Therefore IDFT can be used to generate an OFDM transmission signalgenerate an OFDM transmission signal
20
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
( )tf02cos π
( )0a
by using IDFTby using IDFT
( )nd ( )tDI( )tf02sin π
( )0a
( )0b
bull
tf s Δ=
1 ( ) ( ) njbna +
( )nd
tNΔffnffn Δ
=Δ+=1 0
( )tDChannelInput
( )tf N 12cos minusπ
( )1minusNa
bullbull
tNΔ
( )tf N 12sin minusπ
( )1minusNb
( )0d
( )nd ( )tDChannelInput
bull bull
( )( )1d
( )2d
tf s Δ=
1 ( ) ( ) njbna +
fNΔffnffn Δ
=Δ+=1 0
ChannelInput
bullbullbullbull
bullbullbullbull
( )1minusNd
21
Frequency Error Results in ICIFrequency Error Results in ICIq yq y
22
Synchronization Error Results in ICISynchronization Error Results in ICIyy
Not Orthogonal Any More
23
Cyclic PrefixCyclic Prefixyy
loz In multipath channel delayed replicas of previous OFDM signal lead to ISI between successive OFDM signals
loz Solution Insert a guard interval between successive OFDMloz Solution Insert a guard interval between successive OFDM signals
24
Cyclic PrefixCyclic Prefix
loz Guard interval leads to intercarrier interference (ICI) in OFDM d d l i
yy
demodulation
loz In DFT interval difference between two subcarriers does not maintain integer number of cycles loss of orthogonality
loz Delayed version of subcarrier 2 causes ICI in the process of d d l ti b i 1demodulating subcarrier 1
25
Cyclic PrefixCyclic Prefixloz Cyclic prefix (CP) A copy of the last part of OFDM signal is
attached to the front of itself
yy
attached to the front of itself
[ ]0d
[ ]1d
[ ]0D[ ]1D
[ ]nd[ ]kD~
symbols dataInput
bullbullbullbull
[ ]1d
[ ]2d
[ ][ ]2D
bullbullbullbullbull
y
bullbullbullbullbull
[ ]1minusNd [ ]1minusND
[ ]gNND minus
bullbullbull
26
Cyclic PrefixCyclic Prefix
loz All delayed replicas of subcarriers always have an
yy
y p yinteger number of cycles within DFT interval no ICI
27
Cyclic PrefixCyclic Prefix
loz Linear convolution vs circular convolution
yy
N
28
Cyclic PrefixCyclic Prefix
loz Channel effect with cyclic prefix
yy
Signal after removed CP
29
Cyclic PrefixCyclic Prefix
loz Time-Domain Explanation
yy
p
30
Cyclic PrefixCyclic Prefixyy
31
Cyclic PrefixCyclic Prefix
loz Spectrum of channel response with length (smaller [ ]nh hL
yy
than )h
gN[ ] nhFFTHk =
loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr
loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=
where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier
[ ] [ ] [ ]N
Notimes
[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce
kk H
X =rArrcomplexity of channel equalizationrdquo
32
Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))
bull bull( )( )nr~Symbols
dataOutput
bullbullbull
bullbullbull( )nr
bullbull
bullbull
33
Cyclic PrefixCyclic Prefix
loz One of the most important reasons to do OFDM is the
yy
pefficient way it deals with multipath delay spread
loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)
34
Bandwidth EfficiencyBandwidth Efficiencyyy
loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )
loz The diagram for bandwidth efficiency of OFDM system is shown below
35
SummarySummaryyy
loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j
the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference
(ISI)loz The main advantage of the OFDM transmission technique is its
high performance even in frequency selective channels
loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems
36
SynchronizationSynchronization
Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg
38 Synchronization2010722
Introduction (13)Introduction (13)( )( )
loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods
loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead
loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the
cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of
computational complexity therefore it may be not be available in short-burst wireless communication
39 Synchronization2010722
Introduction (23)Introduction (23)( )( )
loz Transcevier
40 Synchronization2010722
Introduction (33)Introduction (33)( )( )
loz Synchronization erroryloz Symbol timing offset (TO)
loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)
loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect
41 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg
42 Synchronization2010722
Signal model (14)Signal model (14)g ( )g ( )
loz Time domain sample p1
2
0
1 Nj nk N
n kk
x X eN
πminus
minus= sum
Xk the frequency domain data at the k-th subcarrier0kN =
loz The multipath fading channel
( ) ( ) ( )1L
h t h t δminus
sum
loz Received time domain signal
( ) ( ) ( )0
l ll
h t h tτ τ δ τ τ=
= minussum
loz Received time-domain signal
( ) ( )1
L
n n ly h n l x w nminus
minus= +sum43
( ) ( )0l=sum
Synchronization2010722
Signal model (24)Signal model (24)g ( )g ( )
loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting
12
0
Nj kn N
k nn
Y y e πminus
minus
=
= sum
( )1 1
2
0
N Lj kn N
n l kn l
h n l x e Wπminus minus
minusminus
=
= +sumsum
loz The equation above could be expressed in matrix form
0n l
loz The equation above could be expressed in matrix form
= +Y HX W
44 Synchronization2010722
Signal model (34)Signal model (34)g ( )g ( )
H= = = = +Y Fy FCx FCF X HX W
loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal
y
H=H FCFcirculant matrix leading to be a diagonal matrix
=H FCF
( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0
h h L L hh h
⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥
( )( ) ( )
( ) ( )
1 10 1 0 0i
h Lh L h N L
⎢ ⎥⎢ ⎥minus minus⎢ ⎥
equiv= minus minus⎢ ⎥⎢ ⎥
C( ) ( )
( ) ( )
0 1 1 100
0 0 1 1 0
h L h N L
h N L L h N
⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
45
( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦
Synchronization2010722
Signal model (44)Signal model (44)g ( )g ( )
loz In a time-invariant channel the channel matrix H would be a diagonal matrix
k k k kY H X W= +
( )1 1
2
0 0
N Lj kl N
kn l
H h n l e πminus minus
minus
= =
= sumsum
loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI
k k k k kY H X I Wα= + +k k k k k
( ) ( )1 1 1
2 2 N N L
j m k Nj ml Nk mI X h n l e e ππ
minus minus minusminusminus⎛ ⎞
= ⎜ ⎟⎝ ⎠
sum sum sumICI
46
0 0 0m n lm k= = =ne
⎝ ⎠
Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
47 Synchronization2010722
Impact of timing offset Impact of timing offset p gp g
loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively
( )I f s
Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ
( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr
2 fj k Nk k k kY X H e Wπδminus= +
loz For integer part of timing offset k k k kY X H e W+
I sTδ ( ) 0g IN L δminus minus le leFFT
2 Ij k NY X H e Wπδminus= +
( ) ( ) 2 I sFFT
j f TI sh t T H f e π δδ minusminus rarr
48
k k k kY X H e W= +
Synchronization2010722
Impact of timing offset derivationImpact of timing offset derivationp gp g
49 Synchronization2010722
Discussion of symbol boundaryDiscussion of symbol boundaryy yy y
50 Synchronization2010722
Impact of timing offsetImpact of timing offsetp gp g
loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI
2 Ij k NIk k k k
NY X H e ISI W
Nπδδ minusminus
= + +
loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk
e e
p kloz Phase Shift can be resolved by Differential Encoding
Decoding or carrier recovery with pilots
51 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
52 Synchronization2010722
Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )
53 Synchronization2010722
Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )
( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s
i n t i N N T N T nTy y t e
= + + +=
( ) ( )( ) ( ) ( ) 12
sin
sin
g gI f f
I I
i N N N Nj jf N Ni k i k k
f
Y X H e eN
π ε ε π ε
ε ε
πεπε
+ + minus+
minus minus=⎛ ⎞⎜ ⎟
Faded signal attenuated and rotated by CFO
( )( ) ( ) ( ) ( )11 2
sin
sin g gI f I f
i N N N NN j j l kI f
NN
l k π ε ε π ε επ ε ε + + minusminus + + + minus
⎜ ⎟⎝ ⎠
+ + minussum
( )( )( )
( ) ( )2
0
sin
I f I f
I
j j l kI f N Ni l l
l l k I f
X H e el k
NN
π ε ε π ε ε
ε π ε ε
+ + +
= ne minus
+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠
sum
i k
N
W
⎜ ⎟⎝ ⎠
+ Inter-Carrier Interference (ICI)
54 Synchronization2010722
Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )
loz An easier representation of received signal with only p g yfractional CFO is
1N
( ) ( )1
0
0N
k k k l l kl l k
Y H X C H X C l k Wminus
= ne
= + minus +⎡ ⎤⎣ ⎦sum
where the ICI coefficient is given by
( )( )( ) ( )
sin 1exp 1fl k
C l k j l kπ ε
π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )
( )exp 1sin
f
f
C l k j l kNN l k
N
π επ ε
minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠
55 Synchronization2010722
Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )
loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality
loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift
56 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
57 Synchronization2010722
TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )
loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq
58 Synchronization2010722
TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )
loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below
loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and
rN2 can be expressed by
0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+
( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +
59
( )
Synchronization2010722
TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )
loz Without CFO the first half is identical to the second half (in time order)
loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below
( ) 12 in discretesft T f NT
NTφ π π π ε πε
⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟
⎝ ⎠
helliphellip
sNT⎝ ⎠
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)
60
Length=N2
Synchronization2010722
TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )
loz Received signal without CFOg0
0 0
jr a e θ=jθ 2 0
2 2 0
Nj j
N Nr a e a e
θ θ= =
loz Received signal with CFO0 0
0 0
j jr a e eθ φ= Extra phase rotation due to CFO0 0
2 2 20
2 2 0
N N Nj j jj
N Nr a e e a e e
θ φ φθ= =
due to CFO
Phase difference 2 2 0N N
( )2 02 20 2 0 0
Nj j T fN
r r a e a eφ φ πminus Δ= =
which contains the information about CFO
61
0 2 0 0Nr r a e a e
Synchronization2010722
TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )
loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be
loz Note that d is a time index corresponding to the first( ) ( ) 2 1
20
Nd m d m Nm
P d r rminus
+ + +== sum
loz Note that d is a time index corresponding to the first sample in a window of N samples
loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by
( )22 1
20
Nd m NR d rminus
+ += sumloz A timing metric can be defined as
))(()(
)( 2
2
dRdP
dM =
( ) 20 d m Nm + +=sum
62
))(( dR
Synchronization2010722
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
OFDM OverviewOFDM Overview
Single carrier (SC) vs multi-carrier (MC)
loz Single carrier data are Multi-carrier data are shared transmitted over only one carrier
among several carriers and simultaneously transmittedFl t f di b iloz Selective fading Flat fading per subcarrier
5
OFDM OverviewOFDM Overview
loz OFDM modulation
loz Featuresloz No intercarrier guard bandsloz Overlapping of bands
S t l ffi iloz Spectral efficiencyloz Easy implementation by IFFTsloz Very sensitive to synchronizationloz Very sensitive to synchronization
6
Applications of OFDM TechnologyApplications of OFDM Technologypp gypp gy
loz Broadband Wired Access AsymmetricDigital Subscriber Loop (ADSL) Digital Multi-tone(DMT)
loz Wireless LANs (IEEE 80211ag IEEE 80211n HIPERLAN-2)
loz Digital Broadcasting (DAB DVB-T DVB-H)loz WiMAX (IEEE 80216 Series) 3GPP Long Term ( ) g
Evolution (3GPP LTE) 4Gloz Wireless Personal Area Network (WPAN) IEEE ( )
80215aMBOAloz Power Line
7
OFDM System ModelOFDM System Modelyy
loz Multi-carrier Block Transmission
frequency
time
8T8
OFDM System ModelOFDM System Modelyy
loz OFDM A block modulation scheme that transmitsa block N source symbols in parallel by using subcarrierssubcarriersloz Sub-carriers are orthogonal in time but overlapped in
frequencyeque cyloz Frequency spacing 1
FFT
fT
Δ =
cos(2 )cos(2 ( ) ) 0FFTTf t f f t dtπ π + Δint 1 10
cos(2 )cos(2 ( ) ) 0f t f f t dtπ π + Δ =int
9
OFDM System ModelOFDM System Model
tNΔ
yy
( )tf02cos π
( )0a
( )0atT
tNΔ1+
( )tf2i
( )
( )0b
T
DataEncoder
tf s Δ=
1 ( ) ( ) njbna +
( )nd SP MUX ( )tDChannelInput
( )tf02sin π
( )tf N 12cos minusπbullbullbull
( )1minusNa
( )1minusNb( )0a1+
( )tf N 12sin minusπ
( )1Nb
tT
( )1a ( )1minusNa1minus
tΔ
Figure 1tNΔ
( )1a ( )1minusNa Figure 1
10
OFDM System ModelOFDM System Model
loz An OFDM system transmitter shown in Figure 1
yy
y gloz The transmitted waveform D(t) can be expressed as
)1( )2sin()()2cos()()(1
0summinus
=
+=N
nnn tfnbtfnatD ππ
where tNffnffn Δ=ΔΔ+=
1 and 0
loz Using a two-dimensional digital modulation format the data
tNΔ
g g symbols d(n) can be represented as a(n) + jb(n)loz a(n) in-phase componentloz b(n) quadrature component
11
OFDM System ModelOFDM System Model
loz The serial data elements spaced by are grouped and tΔ
yy
p y g pused to modulate N carriers Thus they are frequency division multiplexedp
loz The signaling interval is then increased to which tNΔmakes the system less susceptible to channel delay spread impairments
Small-scale fading(Based on multipath time delay spread)
Flat Fading1 BW of signal lt BW of channel
Frequency Selective Fading1 BW of signal gt BW of channel
2 Delay spread lt Symbol preiod 2 Delay spread gt Symbol preiod
12
OrthogonalityOrthogonalityg yg y
loz Consider a set of transmitted carriers as follows
(2) 1 1 0for )( 2 0
minus==⎟⎠⎞
⎜⎝⎛
Δ+
Nnett
tNnfj
n
πψ n
⎨⎧ =minus
intqpab
dtttb for )(
)()( ψψ⎩⎨ Δ=minusne
=int tNabqpdttt
a qp )( and for 0 )()( ψψ
13
OrthogonalityOrthogonality
b tqpjb
intint)(2π
g yg y
dtedtttajbj
b
atN
qpjb
a qp = Δminus
intint )()(
)(2)(2
)(2ψψπ
tNqpjee tN
qpjtN
qpj
Δminusminus
=Δ
minusΔ
minus
)(2
)(2)(2
π
ππ
ee
tNqpjba
tNqpj
tNbqpj
⎟⎟⎞
⎜⎜⎛minus
Δ
minusΔ
minusΔ
minus1
)(2)(1)(2)(2
π
ππ
tNqpj Δminus
⎟⎠
⎜⎝=
)(2πtNabqp Δ=minusne= )( and for 0
14
OrthogonalityOrthogonalityg yg y
tNΔ
15
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFTby using IDFTby using IDFT
loz According to the structure of Tx it must use Ng oscillators That increases the hardware complexity
loz The equivalent method is using IDFT (IFFT)
16
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz In general each carrier can be expressed as
by using IDFTby using IDFT
g p
( ) (6) )()( )(2 ttfjcc
ccetAtS φπ +=
loz We assume that there are N carriers in the OFDM signal Th th t t l l i l S (t) b t d bThen the total complex signal Ss(t) can be represented by
( ) (7))(1)(1
)(2etAtSN
ttfj nn φπ= summinus
+( )
where
(7) )()(
0
0
fnff
etAN
tSn
ns
Δ+=
= sum=
liihffrequency carrier phase amplitude are )( )( and
where 0
fttAfnff
nnn
n
φΔ+
lyrespective carrierth -of n17
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz Then we sample the signal at a sampling frequency tΔ1
by using IDFTby using IDFT
p g p g q y and An(t) and φn(t) becomes
(8))(t = φφ(9) )((8) )(
nn
nn
AtAt
== φφ
( ) (10) 1)(1
0
)(2 0summinus
=
+ΔΔ+=ΔN
n
tkfnfjns
neAN
tkS φπ
loz Then the sampled signal can be expressed as0=nN
( )( ) (11) 1)(1
0
22 0summinus
=
ΔΔ+Δ sdot=ΔN
n
tfnkjtkfjns eeA
NtkS n πφπ
0n
18
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz The inverse discrete Fourier transform (IDFT) is defined
by using IDFTby using IDFT
( )as the following
1 21
NnkjN
πsumminus
(12) )(1)( 2
0
Nnkj
nefnF
Ntkf πsum
=
Δ=Δ
loz Comparing eq(11) and eq(12) the condition must be satisfied in order to make eq(11) an inverse Fouriersatisfied in order to make eq(11) an inverse Fourier transform relationship
1 (13) 1tN
fΔ
=Δ
19
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz If eq(13) is satisfied
by using IDFTby using IDFT
q ( ) loz is the frequency domain signal loz is the time domain signal
( )ntkfjneA φπ +Δ02
)( tkSs Δ gloz is the sub-channel spacingloz is the symbol duration in each sub-channel
)(s
fΔtNΔloz s e sy bo du o e c sub c e
loz This outcome is the same as the result obtained in the
N
loz This outcome is the same as the result obtained in the system of Figure 1 Therefore IDFT can be used to generate an OFDM transmission signalgenerate an OFDM transmission signal
20
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
( )tf02cos π
( )0a
by using IDFTby using IDFT
( )nd ( )tDI( )tf02sin π
( )0a
( )0b
bull
tf s Δ=
1 ( ) ( ) njbna +
( )nd
tNΔffnffn Δ
=Δ+=1 0
( )tDChannelInput
( )tf N 12cos minusπ
( )1minusNa
bullbull
tNΔ
( )tf N 12sin minusπ
( )1minusNb
( )0d
( )nd ( )tDChannelInput
bull bull
( )( )1d
( )2d
tf s Δ=
1 ( ) ( ) njbna +
fNΔffnffn Δ
=Δ+=1 0
ChannelInput
bullbullbullbull
bullbullbullbull
( )1minusNd
21
Frequency Error Results in ICIFrequency Error Results in ICIq yq y
22
Synchronization Error Results in ICISynchronization Error Results in ICIyy
Not Orthogonal Any More
23
Cyclic PrefixCyclic Prefixyy
loz In multipath channel delayed replicas of previous OFDM signal lead to ISI between successive OFDM signals
loz Solution Insert a guard interval between successive OFDMloz Solution Insert a guard interval between successive OFDM signals
24
Cyclic PrefixCyclic Prefix
loz Guard interval leads to intercarrier interference (ICI) in OFDM d d l i
yy
demodulation
loz In DFT interval difference between two subcarriers does not maintain integer number of cycles loss of orthogonality
loz Delayed version of subcarrier 2 causes ICI in the process of d d l ti b i 1demodulating subcarrier 1
25
Cyclic PrefixCyclic Prefixloz Cyclic prefix (CP) A copy of the last part of OFDM signal is
attached to the front of itself
yy
attached to the front of itself
[ ]0d
[ ]1d
[ ]0D[ ]1D
[ ]nd[ ]kD~
symbols dataInput
bullbullbullbull
[ ]1d
[ ]2d
[ ][ ]2D
bullbullbullbullbull
y
bullbullbullbullbull
[ ]1minusNd [ ]1minusND
[ ]gNND minus
bullbullbull
26
Cyclic PrefixCyclic Prefix
loz All delayed replicas of subcarriers always have an
yy
y p yinteger number of cycles within DFT interval no ICI
27
Cyclic PrefixCyclic Prefix
loz Linear convolution vs circular convolution
yy
N
28
Cyclic PrefixCyclic Prefix
loz Channel effect with cyclic prefix
yy
Signal after removed CP
29
Cyclic PrefixCyclic Prefix
loz Time-Domain Explanation
yy
p
30
Cyclic PrefixCyclic Prefixyy
31
Cyclic PrefixCyclic Prefix
loz Spectrum of channel response with length (smaller [ ]nh hL
yy
than )h
gN[ ] nhFFTHk =
loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr
loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=
where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier
[ ] [ ] [ ]N
Notimes
[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce
kk H
X =rArrcomplexity of channel equalizationrdquo
32
Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))
bull bull( )( )nr~Symbols
dataOutput
bullbullbull
bullbullbull( )nr
bullbull
bullbull
33
Cyclic PrefixCyclic Prefix
loz One of the most important reasons to do OFDM is the
yy
pefficient way it deals with multipath delay spread
loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)
34
Bandwidth EfficiencyBandwidth Efficiencyyy
loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )
loz The diagram for bandwidth efficiency of OFDM system is shown below
35
SummarySummaryyy
loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j
the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference
(ISI)loz The main advantage of the OFDM transmission technique is its
high performance even in frequency selective channels
loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems
36
SynchronizationSynchronization
Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg
38 Synchronization2010722
Introduction (13)Introduction (13)( )( )
loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods
loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead
loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the
cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of
computational complexity therefore it may be not be available in short-burst wireless communication
39 Synchronization2010722
Introduction (23)Introduction (23)( )( )
loz Transcevier
40 Synchronization2010722
Introduction (33)Introduction (33)( )( )
loz Synchronization erroryloz Symbol timing offset (TO)
loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)
loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect
41 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg
42 Synchronization2010722
Signal model (14)Signal model (14)g ( )g ( )
loz Time domain sample p1
2
0
1 Nj nk N
n kk
x X eN
πminus
minus= sum
Xk the frequency domain data at the k-th subcarrier0kN =
loz The multipath fading channel
( ) ( ) ( )1L
h t h t δminus
sum
loz Received time domain signal
( ) ( ) ( )0
l ll
h t h tτ τ δ τ τ=
= minussum
loz Received time-domain signal
( ) ( )1
L
n n ly h n l x w nminus
minus= +sum43
( ) ( )0l=sum
Synchronization2010722
Signal model (24)Signal model (24)g ( )g ( )
loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting
12
0
Nj kn N
k nn
Y y e πminus
minus
=
= sum
( )1 1
2
0
N Lj kn N
n l kn l
h n l x e Wπminus minus
minusminus
=
= +sumsum
loz The equation above could be expressed in matrix form
0n l
loz The equation above could be expressed in matrix form
= +Y HX W
44 Synchronization2010722
Signal model (34)Signal model (34)g ( )g ( )
H= = = = +Y Fy FCx FCF X HX W
loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal
y
H=H FCFcirculant matrix leading to be a diagonal matrix
=H FCF
( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0
h h L L hh h
⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥
( )( ) ( )
( ) ( )
1 10 1 0 0i
h Lh L h N L
⎢ ⎥⎢ ⎥minus minus⎢ ⎥
equiv= minus minus⎢ ⎥⎢ ⎥
C( ) ( )
( ) ( )
0 1 1 100
0 0 1 1 0
h L h N L
h N L L h N
⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
45
( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦
Synchronization2010722
Signal model (44)Signal model (44)g ( )g ( )
loz In a time-invariant channel the channel matrix H would be a diagonal matrix
k k k kY H X W= +
( )1 1
2
0 0
N Lj kl N
kn l
H h n l e πminus minus
minus
= =
= sumsum
loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI
k k k k kY H X I Wα= + +k k k k k
( ) ( )1 1 1
2 2 N N L
j m k Nj ml Nk mI X h n l e e ππ
minus minus minusminusminus⎛ ⎞
= ⎜ ⎟⎝ ⎠
sum sum sumICI
46
0 0 0m n lm k= = =ne
⎝ ⎠
Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
47 Synchronization2010722
Impact of timing offset Impact of timing offset p gp g
loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively
( )I f s
Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ
( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr
2 fj k Nk k k kY X H e Wπδminus= +
loz For integer part of timing offset k k k kY X H e W+
I sTδ ( ) 0g IN L δminus minus le leFFT
2 Ij k NY X H e Wπδminus= +
( ) ( ) 2 I sFFT
j f TI sh t T H f e π δδ minusminus rarr
48
k k k kY X H e W= +
Synchronization2010722
Impact of timing offset derivationImpact of timing offset derivationp gp g
49 Synchronization2010722
Discussion of symbol boundaryDiscussion of symbol boundaryy yy y
50 Synchronization2010722
Impact of timing offsetImpact of timing offsetp gp g
loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI
2 Ij k NIk k k k
NY X H e ISI W
Nπδδ minusminus
= + +
loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk
e e
p kloz Phase Shift can be resolved by Differential Encoding
Decoding or carrier recovery with pilots
51 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
52 Synchronization2010722
Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )
53 Synchronization2010722
Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )
( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s
i n t i N N T N T nTy y t e
= + + +=
( ) ( )( ) ( ) ( ) 12
sin
sin
g gI f f
I I
i N N N Nj jf N Ni k i k k
f
Y X H e eN
π ε ε π ε
ε ε
πεπε
+ + minus+
minus minus=⎛ ⎞⎜ ⎟
Faded signal attenuated and rotated by CFO
( )( ) ( ) ( ) ( )11 2
sin
sin g gI f I f
i N N N NN j j l kI f
NN
l k π ε ε π ε επ ε ε + + minusminus + + + minus
⎜ ⎟⎝ ⎠
+ + minussum
( )( )( )
( ) ( )2
0
sin
I f I f
I
j j l kI f N Ni l l
l l k I f
X H e el k
NN
π ε ε π ε ε
ε π ε ε
+ + +
= ne minus
+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠
sum
i k
N
W
⎜ ⎟⎝ ⎠
+ Inter-Carrier Interference (ICI)
54 Synchronization2010722
Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )
loz An easier representation of received signal with only p g yfractional CFO is
1N
( ) ( )1
0
0N
k k k l l kl l k
Y H X C H X C l k Wminus
= ne
= + minus +⎡ ⎤⎣ ⎦sum
where the ICI coefficient is given by
( )( )( ) ( )
sin 1exp 1fl k
C l k j l kπ ε
π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )
( )exp 1sin
f
f
C l k j l kNN l k
N
π επ ε
minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠
55 Synchronization2010722
Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )
loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality
loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift
56 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
57 Synchronization2010722
TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )
loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq
58 Synchronization2010722
TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )
loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below
loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and
rN2 can be expressed by
0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+
( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +
59
( )
Synchronization2010722
TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )
loz Without CFO the first half is identical to the second half (in time order)
loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below
( ) 12 in discretesft T f NT
NTφ π π π ε πε
⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟
⎝ ⎠
helliphellip
sNT⎝ ⎠
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)
60
Length=N2
Synchronization2010722
TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )
loz Received signal without CFOg0
0 0
jr a e θ=jθ 2 0
2 2 0
Nj j
N Nr a e a e
θ θ= =
loz Received signal with CFO0 0
0 0
j jr a e eθ φ= Extra phase rotation due to CFO0 0
2 2 20
2 2 0
N N Nj j jj
N Nr a e e a e e
θ φ φθ= =
due to CFO
Phase difference 2 2 0N N
( )2 02 20 2 0 0
Nj j T fN
r r a e a eφ φ πminus Δ= =
which contains the information about CFO
61
0 2 0 0Nr r a e a e
Synchronization2010722
TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )
loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be
loz Note that d is a time index corresponding to the first( ) ( ) 2 1
20
Nd m d m Nm
P d r rminus
+ + +== sum
loz Note that d is a time index corresponding to the first sample in a window of N samples
loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by
( )22 1
20
Nd m NR d rminus
+ += sumloz A timing metric can be defined as
))(()(
)( 2
2
dRdP
dM =
( ) 20 d m Nm + +=sum
62
))(( dR
Synchronization2010722
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
OFDM OverviewOFDM Overview
loz OFDM modulation
loz Featuresloz No intercarrier guard bandsloz Overlapping of bands
S t l ffi iloz Spectral efficiencyloz Easy implementation by IFFTsloz Very sensitive to synchronizationloz Very sensitive to synchronization
6
Applications of OFDM TechnologyApplications of OFDM Technologypp gypp gy
loz Broadband Wired Access AsymmetricDigital Subscriber Loop (ADSL) Digital Multi-tone(DMT)
loz Wireless LANs (IEEE 80211ag IEEE 80211n HIPERLAN-2)
loz Digital Broadcasting (DAB DVB-T DVB-H)loz WiMAX (IEEE 80216 Series) 3GPP Long Term ( ) g
Evolution (3GPP LTE) 4Gloz Wireless Personal Area Network (WPAN) IEEE ( )
80215aMBOAloz Power Line
7
OFDM System ModelOFDM System Modelyy
loz Multi-carrier Block Transmission
frequency
time
8T8
OFDM System ModelOFDM System Modelyy
loz OFDM A block modulation scheme that transmitsa block N source symbols in parallel by using subcarrierssubcarriersloz Sub-carriers are orthogonal in time but overlapped in
frequencyeque cyloz Frequency spacing 1
FFT
fT
Δ =
cos(2 )cos(2 ( ) ) 0FFTTf t f f t dtπ π + Δint 1 10
cos(2 )cos(2 ( ) ) 0f t f f t dtπ π + Δ =int
9
OFDM System ModelOFDM System Model
tNΔ
yy
( )tf02cos π
( )0a
( )0atT
tNΔ1+
( )tf2i
( )
( )0b
T
DataEncoder
tf s Δ=
1 ( ) ( ) njbna +
( )nd SP MUX ( )tDChannelInput
( )tf02sin π
( )tf N 12cos minusπbullbullbull
( )1minusNa
( )1minusNb( )0a1+
( )tf N 12sin minusπ
( )1Nb
tT
( )1a ( )1minusNa1minus
tΔ
Figure 1tNΔ
( )1a ( )1minusNa Figure 1
10
OFDM System ModelOFDM System Model
loz An OFDM system transmitter shown in Figure 1
yy
y gloz The transmitted waveform D(t) can be expressed as
)1( )2sin()()2cos()()(1
0summinus
=
+=N
nnn tfnbtfnatD ππ
where tNffnffn Δ=ΔΔ+=
1 and 0
loz Using a two-dimensional digital modulation format the data
tNΔ
g g symbols d(n) can be represented as a(n) + jb(n)loz a(n) in-phase componentloz b(n) quadrature component
11
OFDM System ModelOFDM System Model
loz The serial data elements spaced by are grouped and tΔ
yy
p y g pused to modulate N carriers Thus they are frequency division multiplexedp
loz The signaling interval is then increased to which tNΔmakes the system less susceptible to channel delay spread impairments
Small-scale fading(Based on multipath time delay spread)
Flat Fading1 BW of signal lt BW of channel
Frequency Selective Fading1 BW of signal gt BW of channel
2 Delay spread lt Symbol preiod 2 Delay spread gt Symbol preiod
12
OrthogonalityOrthogonalityg yg y
loz Consider a set of transmitted carriers as follows
(2) 1 1 0for )( 2 0
minus==⎟⎠⎞
⎜⎝⎛
Δ+
Nnett
tNnfj
n
πψ n
⎨⎧ =minus
intqpab
dtttb for )(
)()( ψψ⎩⎨ Δ=minusne
=int tNabqpdttt
a qp )( and for 0 )()( ψψ
13
OrthogonalityOrthogonality
b tqpjb
intint)(2π
g yg y
dtedtttajbj
b
atN
qpjb
a qp = Δminus
intint )()(
)(2)(2
)(2ψψπ
tNqpjee tN
qpjtN
qpj
Δminusminus
=Δ
minusΔ
minus
)(2
)(2)(2
π
ππ
ee
tNqpjba
tNqpj
tNbqpj
⎟⎟⎞
⎜⎜⎛minus
Δ
minusΔ
minusΔ
minus1
)(2)(1)(2)(2
π
ππ
tNqpj Δminus
⎟⎠
⎜⎝=
)(2πtNabqp Δ=minusne= )( and for 0
14
OrthogonalityOrthogonalityg yg y
tNΔ
15
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFTby using IDFTby using IDFT
loz According to the structure of Tx it must use Ng oscillators That increases the hardware complexity
loz The equivalent method is using IDFT (IFFT)
16
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz In general each carrier can be expressed as
by using IDFTby using IDFT
g p
( ) (6) )()( )(2 ttfjcc
ccetAtS φπ +=
loz We assume that there are N carriers in the OFDM signal Th th t t l l i l S (t) b t d bThen the total complex signal Ss(t) can be represented by
( ) (7))(1)(1
)(2etAtSN
ttfj nn φπ= summinus
+( )
where
(7) )()(
0
0
fnff
etAN
tSn
ns
Δ+=
= sum=
liihffrequency carrier phase amplitude are )( )( and
where 0
fttAfnff
nnn
n
φΔ+
lyrespective carrierth -of n17
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz Then we sample the signal at a sampling frequency tΔ1
by using IDFTby using IDFT
p g p g q y and An(t) and φn(t) becomes
(8))(t = φφ(9) )((8) )(
nn
nn
AtAt
== φφ
( ) (10) 1)(1
0
)(2 0summinus
=
+ΔΔ+=ΔN
n
tkfnfjns
neAN
tkS φπ
loz Then the sampled signal can be expressed as0=nN
( )( ) (11) 1)(1
0
22 0summinus
=
ΔΔ+Δ sdot=ΔN
n
tfnkjtkfjns eeA
NtkS n πφπ
0n
18
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz The inverse discrete Fourier transform (IDFT) is defined
by using IDFTby using IDFT
( )as the following
1 21
NnkjN
πsumminus
(12) )(1)( 2
0
Nnkj
nefnF
Ntkf πsum
=
Δ=Δ
loz Comparing eq(11) and eq(12) the condition must be satisfied in order to make eq(11) an inverse Fouriersatisfied in order to make eq(11) an inverse Fourier transform relationship
1 (13) 1tN
fΔ
=Δ
19
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz If eq(13) is satisfied
by using IDFTby using IDFT
q ( ) loz is the frequency domain signal loz is the time domain signal
( )ntkfjneA φπ +Δ02
)( tkSs Δ gloz is the sub-channel spacingloz is the symbol duration in each sub-channel
)(s
fΔtNΔloz s e sy bo du o e c sub c e
loz This outcome is the same as the result obtained in the
N
loz This outcome is the same as the result obtained in the system of Figure 1 Therefore IDFT can be used to generate an OFDM transmission signalgenerate an OFDM transmission signal
20
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
( )tf02cos π
( )0a
by using IDFTby using IDFT
( )nd ( )tDI( )tf02sin π
( )0a
( )0b
bull
tf s Δ=
1 ( ) ( ) njbna +
( )nd
tNΔffnffn Δ
=Δ+=1 0
( )tDChannelInput
( )tf N 12cos minusπ
( )1minusNa
bullbull
tNΔ
( )tf N 12sin minusπ
( )1minusNb
( )0d
( )nd ( )tDChannelInput
bull bull
( )( )1d
( )2d
tf s Δ=
1 ( ) ( ) njbna +
fNΔffnffn Δ
=Δ+=1 0
ChannelInput
bullbullbullbull
bullbullbullbull
( )1minusNd
21
Frequency Error Results in ICIFrequency Error Results in ICIq yq y
22
Synchronization Error Results in ICISynchronization Error Results in ICIyy
Not Orthogonal Any More
23
Cyclic PrefixCyclic Prefixyy
loz In multipath channel delayed replicas of previous OFDM signal lead to ISI between successive OFDM signals
loz Solution Insert a guard interval between successive OFDMloz Solution Insert a guard interval between successive OFDM signals
24
Cyclic PrefixCyclic Prefix
loz Guard interval leads to intercarrier interference (ICI) in OFDM d d l i
yy
demodulation
loz In DFT interval difference between two subcarriers does not maintain integer number of cycles loss of orthogonality
loz Delayed version of subcarrier 2 causes ICI in the process of d d l ti b i 1demodulating subcarrier 1
25
Cyclic PrefixCyclic Prefixloz Cyclic prefix (CP) A copy of the last part of OFDM signal is
attached to the front of itself
yy
attached to the front of itself
[ ]0d
[ ]1d
[ ]0D[ ]1D
[ ]nd[ ]kD~
symbols dataInput
bullbullbullbull
[ ]1d
[ ]2d
[ ][ ]2D
bullbullbullbullbull
y
bullbullbullbullbull
[ ]1minusNd [ ]1minusND
[ ]gNND minus
bullbullbull
26
Cyclic PrefixCyclic Prefix
loz All delayed replicas of subcarriers always have an
yy
y p yinteger number of cycles within DFT interval no ICI
27
Cyclic PrefixCyclic Prefix
loz Linear convolution vs circular convolution
yy
N
28
Cyclic PrefixCyclic Prefix
loz Channel effect with cyclic prefix
yy
Signal after removed CP
29
Cyclic PrefixCyclic Prefix
loz Time-Domain Explanation
yy
p
30
Cyclic PrefixCyclic Prefixyy
31
Cyclic PrefixCyclic Prefix
loz Spectrum of channel response with length (smaller [ ]nh hL
yy
than )h
gN[ ] nhFFTHk =
loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr
loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=
where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier
[ ] [ ] [ ]N
Notimes
[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce
kk H
X =rArrcomplexity of channel equalizationrdquo
32
Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))
bull bull( )( )nr~Symbols
dataOutput
bullbullbull
bullbullbull( )nr
bullbull
bullbull
33
Cyclic PrefixCyclic Prefix
loz One of the most important reasons to do OFDM is the
yy
pefficient way it deals with multipath delay spread
loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)
34
Bandwidth EfficiencyBandwidth Efficiencyyy
loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )
loz The diagram for bandwidth efficiency of OFDM system is shown below
35
SummarySummaryyy
loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j
the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference
(ISI)loz The main advantage of the OFDM transmission technique is its
high performance even in frequency selective channels
loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems
36
SynchronizationSynchronization
Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg
38 Synchronization2010722
Introduction (13)Introduction (13)( )( )
loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods
loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead
loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the
cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of
computational complexity therefore it may be not be available in short-burst wireless communication
39 Synchronization2010722
Introduction (23)Introduction (23)( )( )
loz Transcevier
40 Synchronization2010722
Introduction (33)Introduction (33)( )( )
loz Synchronization erroryloz Symbol timing offset (TO)
loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)
loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect
41 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg
42 Synchronization2010722
Signal model (14)Signal model (14)g ( )g ( )
loz Time domain sample p1
2
0
1 Nj nk N
n kk
x X eN
πminus
minus= sum
Xk the frequency domain data at the k-th subcarrier0kN =
loz The multipath fading channel
( ) ( ) ( )1L
h t h t δminus
sum
loz Received time domain signal
( ) ( ) ( )0
l ll
h t h tτ τ δ τ τ=
= minussum
loz Received time-domain signal
( ) ( )1
L
n n ly h n l x w nminus
minus= +sum43
( ) ( )0l=sum
Synchronization2010722
Signal model (24)Signal model (24)g ( )g ( )
loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting
12
0
Nj kn N
k nn
Y y e πminus
minus
=
= sum
( )1 1
2
0
N Lj kn N
n l kn l
h n l x e Wπminus minus
minusminus
=
= +sumsum
loz The equation above could be expressed in matrix form
0n l
loz The equation above could be expressed in matrix form
= +Y HX W
44 Synchronization2010722
Signal model (34)Signal model (34)g ( )g ( )
H= = = = +Y Fy FCx FCF X HX W
loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal
y
H=H FCFcirculant matrix leading to be a diagonal matrix
=H FCF
( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0
h h L L hh h
⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥
( )( ) ( )
( ) ( )
1 10 1 0 0i
h Lh L h N L
⎢ ⎥⎢ ⎥minus minus⎢ ⎥
equiv= minus minus⎢ ⎥⎢ ⎥
C( ) ( )
( ) ( )
0 1 1 100
0 0 1 1 0
h L h N L
h N L L h N
⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
45
( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦
Synchronization2010722
Signal model (44)Signal model (44)g ( )g ( )
loz In a time-invariant channel the channel matrix H would be a diagonal matrix
k k k kY H X W= +
( )1 1
2
0 0
N Lj kl N
kn l
H h n l e πminus minus
minus
= =
= sumsum
loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI
k k k k kY H X I Wα= + +k k k k k
( ) ( )1 1 1
2 2 N N L
j m k Nj ml Nk mI X h n l e e ππ
minus minus minusminusminus⎛ ⎞
= ⎜ ⎟⎝ ⎠
sum sum sumICI
46
0 0 0m n lm k= = =ne
⎝ ⎠
Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
47 Synchronization2010722
Impact of timing offset Impact of timing offset p gp g
loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively
( )I f s
Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ
( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr
2 fj k Nk k k kY X H e Wπδminus= +
loz For integer part of timing offset k k k kY X H e W+
I sTδ ( ) 0g IN L δminus minus le leFFT
2 Ij k NY X H e Wπδminus= +
( ) ( ) 2 I sFFT
j f TI sh t T H f e π δδ minusminus rarr
48
k k k kY X H e W= +
Synchronization2010722
Impact of timing offset derivationImpact of timing offset derivationp gp g
49 Synchronization2010722
Discussion of symbol boundaryDiscussion of symbol boundaryy yy y
50 Synchronization2010722
Impact of timing offsetImpact of timing offsetp gp g
loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI
2 Ij k NIk k k k
NY X H e ISI W
Nπδδ minusminus
= + +
loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk
e e
p kloz Phase Shift can be resolved by Differential Encoding
Decoding or carrier recovery with pilots
51 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
52 Synchronization2010722
Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )
53 Synchronization2010722
Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )
( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s
i n t i N N T N T nTy y t e
= + + +=
( ) ( )( ) ( ) ( ) 12
sin
sin
g gI f f
I I
i N N N Nj jf N Ni k i k k
f
Y X H e eN
π ε ε π ε
ε ε
πεπε
+ + minus+
minus minus=⎛ ⎞⎜ ⎟
Faded signal attenuated and rotated by CFO
( )( ) ( ) ( ) ( )11 2
sin
sin g gI f I f
i N N N NN j j l kI f
NN
l k π ε ε π ε επ ε ε + + minusminus + + + minus
⎜ ⎟⎝ ⎠
+ + minussum
( )( )( )
( ) ( )2
0
sin
I f I f
I
j j l kI f N Ni l l
l l k I f
X H e el k
NN
π ε ε π ε ε
ε π ε ε
+ + +
= ne minus
+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠
sum
i k
N
W
⎜ ⎟⎝ ⎠
+ Inter-Carrier Interference (ICI)
54 Synchronization2010722
Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )
loz An easier representation of received signal with only p g yfractional CFO is
1N
( ) ( )1
0
0N
k k k l l kl l k
Y H X C H X C l k Wminus
= ne
= + minus +⎡ ⎤⎣ ⎦sum
where the ICI coefficient is given by
( )( )( ) ( )
sin 1exp 1fl k
C l k j l kπ ε
π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )
( )exp 1sin
f
f
C l k j l kNN l k
N
π επ ε
minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠
55 Synchronization2010722
Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )
loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality
loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift
56 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
57 Synchronization2010722
TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )
loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq
58 Synchronization2010722
TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )
loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below
loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and
rN2 can be expressed by
0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+
( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +
59
( )
Synchronization2010722
TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )
loz Without CFO the first half is identical to the second half (in time order)
loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below
( ) 12 in discretesft T f NT
NTφ π π π ε πε
⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟
⎝ ⎠
helliphellip
sNT⎝ ⎠
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)
60
Length=N2
Synchronization2010722
TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )
loz Received signal without CFOg0
0 0
jr a e θ=jθ 2 0
2 2 0
Nj j
N Nr a e a e
θ θ= =
loz Received signal with CFO0 0
0 0
j jr a e eθ φ= Extra phase rotation due to CFO0 0
2 2 20
2 2 0
N N Nj j jj
N Nr a e e a e e
θ φ φθ= =
due to CFO
Phase difference 2 2 0N N
( )2 02 20 2 0 0
Nj j T fN
r r a e a eφ φ πminus Δ= =
which contains the information about CFO
61
0 2 0 0Nr r a e a e
Synchronization2010722
TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )
loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be
loz Note that d is a time index corresponding to the first( ) ( ) 2 1
20
Nd m d m Nm
P d r rminus
+ + +== sum
loz Note that d is a time index corresponding to the first sample in a window of N samples
loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by
( )22 1
20
Nd m NR d rminus
+ += sumloz A timing metric can be defined as
))(()(
)( 2
2
dRdP
dM =
( ) 20 d m Nm + +=sum
62
))(( dR
Synchronization2010722
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
Applications of OFDM TechnologyApplications of OFDM Technologypp gypp gy
loz Broadband Wired Access AsymmetricDigital Subscriber Loop (ADSL) Digital Multi-tone(DMT)
loz Wireless LANs (IEEE 80211ag IEEE 80211n HIPERLAN-2)
loz Digital Broadcasting (DAB DVB-T DVB-H)loz WiMAX (IEEE 80216 Series) 3GPP Long Term ( ) g
Evolution (3GPP LTE) 4Gloz Wireless Personal Area Network (WPAN) IEEE ( )
80215aMBOAloz Power Line
7
OFDM System ModelOFDM System Modelyy
loz Multi-carrier Block Transmission
frequency
time
8T8
OFDM System ModelOFDM System Modelyy
loz OFDM A block modulation scheme that transmitsa block N source symbols in parallel by using subcarrierssubcarriersloz Sub-carriers are orthogonal in time but overlapped in
frequencyeque cyloz Frequency spacing 1
FFT
fT
Δ =
cos(2 )cos(2 ( ) ) 0FFTTf t f f t dtπ π + Δint 1 10
cos(2 )cos(2 ( ) ) 0f t f f t dtπ π + Δ =int
9
OFDM System ModelOFDM System Model
tNΔ
yy
( )tf02cos π
( )0a
( )0atT
tNΔ1+
( )tf2i
( )
( )0b
T
DataEncoder
tf s Δ=
1 ( ) ( ) njbna +
( )nd SP MUX ( )tDChannelInput
( )tf02sin π
( )tf N 12cos minusπbullbullbull
( )1minusNa
( )1minusNb( )0a1+
( )tf N 12sin minusπ
( )1Nb
tT
( )1a ( )1minusNa1minus
tΔ
Figure 1tNΔ
( )1a ( )1minusNa Figure 1
10
OFDM System ModelOFDM System Model
loz An OFDM system transmitter shown in Figure 1
yy
y gloz The transmitted waveform D(t) can be expressed as
)1( )2sin()()2cos()()(1
0summinus
=
+=N
nnn tfnbtfnatD ππ
where tNffnffn Δ=ΔΔ+=
1 and 0
loz Using a two-dimensional digital modulation format the data
tNΔ
g g symbols d(n) can be represented as a(n) + jb(n)loz a(n) in-phase componentloz b(n) quadrature component
11
OFDM System ModelOFDM System Model
loz The serial data elements spaced by are grouped and tΔ
yy
p y g pused to modulate N carriers Thus they are frequency division multiplexedp
loz The signaling interval is then increased to which tNΔmakes the system less susceptible to channel delay spread impairments
Small-scale fading(Based on multipath time delay spread)
Flat Fading1 BW of signal lt BW of channel
Frequency Selective Fading1 BW of signal gt BW of channel
2 Delay spread lt Symbol preiod 2 Delay spread gt Symbol preiod
12
OrthogonalityOrthogonalityg yg y
loz Consider a set of transmitted carriers as follows
(2) 1 1 0for )( 2 0
minus==⎟⎠⎞
⎜⎝⎛
Δ+
Nnett
tNnfj
n
πψ n
⎨⎧ =minus
intqpab
dtttb for )(
)()( ψψ⎩⎨ Δ=minusne
=int tNabqpdttt
a qp )( and for 0 )()( ψψ
13
OrthogonalityOrthogonality
b tqpjb
intint)(2π
g yg y
dtedtttajbj
b
atN
qpjb
a qp = Δminus
intint )()(
)(2)(2
)(2ψψπ
tNqpjee tN
qpjtN
qpj
Δminusminus
=Δ
minusΔ
minus
)(2
)(2)(2
π
ππ
ee
tNqpjba
tNqpj
tNbqpj
⎟⎟⎞
⎜⎜⎛minus
Δ
minusΔ
minusΔ
minus1
)(2)(1)(2)(2
π
ππ
tNqpj Δminus
⎟⎠
⎜⎝=
)(2πtNabqp Δ=minusne= )( and for 0
14
OrthogonalityOrthogonalityg yg y
tNΔ
15
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFTby using IDFTby using IDFT
loz According to the structure of Tx it must use Ng oscillators That increases the hardware complexity
loz The equivalent method is using IDFT (IFFT)
16
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz In general each carrier can be expressed as
by using IDFTby using IDFT
g p
( ) (6) )()( )(2 ttfjcc
ccetAtS φπ +=
loz We assume that there are N carriers in the OFDM signal Th th t t l l i l S (t) b t d bThen the total complex signal Ss(t) can be represented by
( ) (7))(1)(1
)(2etAtSN
ttfj nn φπ= summinus
+( )
where
(7) )()(
0
0
fnff
etAN
tSn
ns
Δ+=
= sum=
liihffrequency carrier phase amplitude are )( )( and
where 0
fttAfnff
nnn
n
φΔ+
lyrespective carrierth -of n17
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz Then we sample the signal at a sampling frequency tΔ1
by using IDFTby using IDFT
p g p g q y and An(t) and φn(t) becomes
(8))(t = φφ(9) )((8) )(
nn
nn
AtAt
== φφ
( ) (10) 1)(1
0
)(2 0summinus
=
+ΔΔ+=ΔN
n
tkfnfjns
neAN
tkS φπ
loz Then the sampled signal can be expressed as0=nN
( )( ) (11) 1)(1
0
22 0summinus
=
ΔΔ+Δ sdot=ΔN
n
tfnkjtkfjns eeA
NtkS n πφπ
0n
18
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz The inverse discrete Fourier transform (IDFT) is defined
by using IDFTby using IDFT
( )as the following
1 21
NnkjN
πsumminus
(12) )(1)( 2
0
Nnkj
nefnF
Ntkf πsum
=
Δ=Δ
loz Comparing eq(11) and eq(12) the condition must be satisfied in order to make eq(11) an inverse Fouriersatisfied in order to make eq(11) an inverse Fourier transform relationship
1 (13) 1tN
fΔ
=Δ
19
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz If eq(13) is satisfied
by using IDFTby using IDFT
q ( ) loz is the frequency domain signal loz is the time domain signal
( )ntkfjneA φπ +Δ02
)( tkSs Δ gloz is the sub-channel spacingloz is the symbol duration in each sub-channel
)(s
fΔtNΔloz s e sy bo du o e c sub c e
loz This outcome is the same as the result obtained in the
N
loz This outcome is the same as the result obtained in the system of Figure 1 Therefore IDFT can be used to generate an OFDM transmission signalgenerate an OFDM transmission signal
20
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
( )tf02cos π
( )0a
by using IDFTby using IDFT
( )nd ( )tDI( )tf02sin π
( )0a
( )0b
bull
tf s Δ=
1 ( ) ( ) njbna +
( )nd
tNΔffnffn Δ
=Δ+=1 0
( )tDChannelInput
( )tf N 12cos minusπ
( )1minusNa
bullbull
tNΔ
( )tf N 12sin minusπ
( )1minusNb
( )0d
( )nd ( )tDChannelInput
bull bull
( )( )1d
( )2d
tf s Δ=
1 ( ) ( ) njbna +
fNΔffnffn Δ
=Δ+=1 0
ChannelInput
bullbullbullbull
bullbullbullbull
( )1minusNd
21
Frequency Error Results in ICIFrequency Error Results in ICIq yq y
22
Synchronization Error Results in ICISynchronization Error Results in ICIyy
Not Orthogonal Any More
23
Cyclic PrefixCyclic Prefixyy
loz In multipath channel delayed replicas of previous OFDM signal lead to ISI between successive OFDM signals
loz Solution Insert a guard interval between successive OFDMloz Solution Insert a guard interval between successive OFDM signals
24
Cyclic PrefixCyclic Prefix
loz Guard interval leads to intercarrier interference (ICI) in OFDM d d l i
yy
demodulation
loz In DFT interval difference between two subcarriers does not maintain integer number of cycles loss of orthogonality
loz Delayed version of subcarrier 2 causes ICI in the process of d d l ti b i 1demodulating subcarrier 1
25
Cyclic PrefixCyclic Prefixloz Cyclic prefix (CP) A copy of the last part of OFDM signal is
attached to the front of itself
yy
attached to the front of itself
[ ]0d
[ ]1d
[ ]0D[ ]1D
[ ]nd[ ]kD~
symbols dataInput
bullbullbullbull
[ ]1d
[ ]2d
[ ][ ]2D
bullbullbullbullbull
y
bullbullbullbullbull
[ ]1minusNd [ ]1minusND
[ ]gNND minus
bullbullbull
26
Cyclic PrefixCyclic Prefix
loz All delayed replicas of subcarriers always have an
yy
y p yinteger number of cycles within DFT interval no ICI
27
Cyclic PrefixCyclic Prefix
loz Linear convolution vs circular convolution
yy
N
28
Cyclic PrefixCyclic Prefix
loz Channel effect with cyclic prefix
yy
Signal after removed CP
29
Cyclic PrefixCyclic Prefix
loz Time-Domain Explanation
yy
p
30
Cyclic PrefixCyclic Prefixyy
31
Cyclic PrefixCyclic Prefix
loz Spectrum of channel response with length (smaller [ ]nh hL
yy
than )h
gN[ ] nhFFTHk =
loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr
loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=
where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier
[ ] [ ] [ ]N
Notimes
[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce
kk H
X =rArrcomplexity of channel equalizationrdquo
32
Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))
bull bull( )( )nr~Symbols
dataOutput
bullbullbull
bullbullbull( )nr
bullbull
bullbull
33
Cyclic PrefixCyclic Prefix
loz One of the most important reasons to do OFDM is the
yy
pefficient way it deals with multipath delay spread
loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)
34
Bandwidth EfficiencyBandwidth Efficiencyyy
loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )
loz The diagram for bandwidth efficiency of OFDM system is shown below
35
SummarySummaryyy
loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j
the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference
(ISI)loz The main advantage of the OFDM transmission technique is its
high performance even in frequency selective channels
loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems
36
SynchronizationSynchronization
Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg
38 Synchronization2010722
Introduction (13)Introduction (13)( )( )
loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods
loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead
loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the
cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of
computational complexity therefore it may be not be available in short-burst wireless communication
39 Synchronization2010722
Introduction (23)Introduction (23)( )( )
loz Transcevier
40 Synchronization2010722
Introduction (33)Introduction (33)( )( )
loz Synchronization erroryloz Symbol timing offset (TO)
loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)
loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect
41 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg
42 Synchronization2010722
Signal model (14)Signal model (14)g ( )g ( )
loz Time domain sample p1
2
0
1 Nj nk N
n kk
x X eN
πminus
minus= sum
Xk the frequency domain data at the k-th subcarrier0kN =
loz The multipath fading channel
( ) ( ) ( )1L
h t h t δminus
sum
loz Received time domain signal
( ) ( ) ( )0
l ll
h t h tτ τ δ τ τ=
= minussum
loz Received time-domain signal
( ) ( )1
L
n n ly h n l x w nminus
minus= +sum43
( ) ( )0l=sum
Synchronization2010722
Signal model (24)Signal model (24)g ( )g ( )
loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting
12
0
Nj kn N
k nn
Y y e πminus
minus
=
= sum
( )1 1
2
0
N Lj kn N
n l kn l
h n l x e Wπminus minus
minusminus
=
= +sumsum
loz The equation above could be expressed in matrix form
0n l
loz The equation above could be expressed in matrix form
= +Y HX W
44 Synchronization2010722
Signal model (34)Signal model (34)g ( )g ( )
H= = = = +Y Fy FCx FCF X HX W
loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal
y
H=H FCFcirculant matrix leading to be a diagonal matrix
=H FCF
( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0
h h L L hh h
⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥
( )( ) ( )
( ) ( )
1 10 1 0 0i
h Lh L h N L
⎢ ⎥⎢ ⎥minus minus⎢ ⎥
equiv= minus minus⎢ ⎥⎢ ⎥
C( ) ( )
( ) ( )
0 1 1 100
0 0 1 1 0
h L h N L
h N L L h N
⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
45
( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦
Synchronization2010722
Signal model (44)Signal model (44)g ( )g ( )
loz In a time-invariant channel the channel matrix H would be a diagonal matrix
k k k kY H X W= +
( )1 1
2
0 0
N Lj kl N
kn l
H h n l e πminus minus
minus
= =
= sumsum
loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI
k k k k kY H X I Wα= + +k k k k k
( ) ( )1 1 1
2 2 N N L
j m k Nj ml Nk mI X h n l e e ππ
minus minus minusminusminus⎛ ⎞
= ⎜ ⎟⎝ ⎠
sum sum sumICI
46
0 0 0m n lm k= = =ne
⎝ ⎠
Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
47 Synchronization2010722
Impact of timing offset Impact of timing offset p gp g
loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively
( )I f s
Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ
( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr
2 fj k Nk k k kY X H e Wπδminus= +
loz For integer part of timing offset k k k kY X H e W+
I sTδ ( ) 0g IN L δminus minus le leFFT
2 Ij k NY X H e Wπδminus= +
( ) ( ) 2 I sFFT
j f TI sh t T H f e π δδ minusminus rarr
48
k k k kY X H e W= +
Synchronization2010722
Impact of timing offset derivationImpact of timing offset derivationp gp g
49 Synchronization2010722
Discussion of symbol boundaryDiscussion of symbol boundaryy yy y
50 Synchronization2010722
Impact of timing offsetImpact of timing offsetp gp g
loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI
2 Ij k NIk k k k
NY X H e ISI W
Nπδδ minusminus
= + +
loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk
e e
p kloz Phase Shift can be resolved by Differential Encoding
Decoding or carrier recovery with pilots
51 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
52 Synchronization2010722
Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )
53 Synchronization2010722
Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )
( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s
i n t i N N T N T nTy y t e
= + + +=
( ) ( )( ) ( ) ( ) 12
sin
sin
g gI f f
I I
i N N N Nj jf N Ni k i k k
f
Y X H e eN
π ε ε π ε
ε ε
πεπε
+ + minus+
minus minus=⎛ ⎞⎜ ⎟
Faded signal attenuated and rotated by CFO
( )( ) ( ) ( ) ( )11 2
sin
sin g gI f I f
i N N N NN j j l kI f
NN
l k π ε ε π ε επ ε ε + + minusminus + + + minus
⎜ ⎟⎝ ⎠
+ + minussum
( )( )( )
( ) ( )2
0
sin
I f I f
I
j j l kI f N Ni l l
l l k I f
X H e el k
NN
π ε ε π ε ε
ε π ε ε
+ + +
= ne minus
+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠
sum
i k
N
W
⎜ ⎟⎝ ⎠
+ Inter-Carrier Interference (ICI)
54 Synchronization2010722
Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )
loz An easier representation of received signal with only p g yfractional CFO is
1N
( ) ( )1
0
0N
k k k l l kl l k
Y H X C H X C l k Wminus
= ne
= + minus +⎡ ⎤⎣ ⎦sum
where the ICI coefficient is given by
( )( )( ) ( )
sin 1exp 1fl k
C l k j l kπ ε
π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )
( )exp 1sin
f
f
C l k j l kNN l k
N
π επ ε
minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠
55 Synchronization2010722
Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )
loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality
loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift
56 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
57 Synchronization2010722
TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )
loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq
58 Synchronization2010722
TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )
loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below
loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and
rN2 can be expressed by
0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+
( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +
59
( )
Synchronization2010722
TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )
loz Without CFO the first half is identical to the second half (in time order)
loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below
( ) 12 in discretesft T f NT
NTφ π π π ε πε
⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟
⎝ ⎠
helliphellip
sNT⎝ ⎠
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)
60
Length=N2
Synchronization2010722
TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )
loz Received signal without CFOg0
0 0
jr a e θ=jθ 2 0
2 2 0
Nj j
N Nr a e a e
θ θ= =
loz Received signal with CFO0 0
0 0
j jr a e eθ φ= Extra phase rotation due to CFO0 0
2 2 20
2 2 0
N N Nj j jj
N Nr a e e a e e
θ φ φθ= =
due to CFO
Phase difference 2 2 0N N
( )2 02 20 2 0 0
Nj j T fN
r r a e a eφ φ πminus Δ= =
which contains the information about CFO
61
0 2 0 0Nr r a e a e
Synchronization2010722
TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )
loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be
loz Note that d is a time index corresponding to the first( ) ( ) 2 1
20
Nd m d m Nm
P d r rminus
+ + +== sum
loz Note that d is a time index corresponding to the first sample in a window of N samples
loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by
( )22 1
20
Nd m NR d rminus
+ += sumloz A timing metric can be defined as
))(()(
)( 2
2
dRdP
dM =
( ) 20 d m Nm + +=sum
62
))(( dR
Synchronization2010722
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
OFDM System ModelOFDM System Modelyy
loz Multi-carrier Block Transmission
frequency
time
8T8
OFDM System ModelOFDM System Modelyy
loz OFDM A block modulation scheme that transmitsa block N source symbols in parallel by using subcarrierssubcarriersloz Sub-carriers are orthogonal in time but overlapped in
frequencyeque cyloz Frequency spacing 1
FFT
fT
Δ =
cos(2 )cos(2 ( ) ) 0FFTTf t f f t dtπ π + Δint 1 10
cos(2 )cos(2 ( ) ) 0f t f f t dtπ π + Δ =int
9
OFDM System ModelOFDM System Model
tNΔ
yy
( )tf02cos π
( )0a
( )0atT
tNΔ1+
( )tf2i
( )
( )0b
T
DataEncoder
tf s Δ=
1 ( ) ( ) njbna +
( )nd SP MUX ( )tDChannelInput
( )tf02sin π
( )tf N 12cos minusπbullbullbull
( )1minusNa
( )1minusNb( )0a1+
( )tf N 12sin minusπ
( )1Nb
tT
( )1a ( )1minusNa1minus
tΔ
Figure 1tNΔ
( )1a ( )1minusNa Figure 1
10
OFDM System ModelOFDM System Model
loz An OFDM system transmitter shown in Figure 1
yy
y gloz The transmitted waveform D(t) can be expressed as
)1( )2sin()()2cos()()(1
0summinus
=
+=N
nnn tfnbtfnatD ππ
where tNffnffn Δ=ΔΔ+=
1 and 0
loz Using a two-dimensional digital modulation format the data
tNΔ
g g symbols d(n) can be represented as a(n) + jb(n)loz a(n) in-phase componentloz b(n) quadrature component
11
OFDM System ModelOFDM System Model
loz The serial data elements spaced by are grouped and tΔ
yy
p y g pused to modulate N carriers Thus they are frequency division multiplexedp
loz The signaling interval is then increased to which tNΔmakes the system less susceptible to channel delay spread impairments
Small-scale fading(Based on multipath time delay spread)
Flat Fading1 BW of signal lt BW of channel
Frequency Selective Fading1 BW of signal gt BW of channel
2 Delay spread lt Symbol preiod 2 Delay spread gt Symbol preiod
12
OrthogonalityOrthogonalityg yg y
loz Consider a set of transmitted carriers as follows
(2) 1 1 0for )( 2 0
minus==⎟⎠⎞
⎜⎝⎛
Δ+
Nnett
tNnfj
n
πψ n
⎨⎧ =minus
intqpab
dtttb for )(
)()( ψψ⎩⎨ Δ=minusne
=int tNabqpdttt
a qp )( and for 0 )()( ψψ
13
OrthogonalityOrthogonality
b tqpjb
intint)(2π
g yg y
dtedtttajbj
b
atN
qpjb
a qp = Δminus
intint )()(
)(2)(2
)(2ψψπ
tNqpjee tN
qpjtN
qpj
Δminusminus
=Δ
minusΔ
minus
)(2
)(2)(2
π
ππ
ee
tNqpjba
tNqpj
tNbqpj
⎟⎟⎞
⎜⎜⎛minus
Δ
minusΔ
minusΔ
minus1
)(2)(1)(2)(2
π
ππ
tNqpj Δminus
⎟⎠
⎜⎝=
)(2πtNabqp Δ=minusne= )( and for 0
14
OrthogonalityOrthogonalityg yg y
tNΔ
15
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFTby using IDFTby using IDFT
loz According to the structure of Tx it must use Ng oscillators That increases the hardware complexity
loz The equivalent method is using IDFT (IFFT)
16
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz In general each carrier can be expressed as
by using IDFTby using IDFT
g p
( ) (6) )()( )(2 ttfjcc
ccetAtS φπ +=
loz We assume that there are N carriers in the OFDM signal Th th t t l l i l S (t) b t d bThen the total complex signal Ss(t) can be represented by
( ) (7))(1)(1
)(2etAtSN
ttfj nn φπ= summinus
+( )
where
(7) )()(
0
0
fnff
etAN
tSn
ns
Δ+=
= sum=
liihffrequency carrier phase amplitude are )( )( and
where 0
fttAfnff
nnn
n
φΔ+
lyrespective carrierth -of n17
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz Then we sample the signal at a sampling frequency tΔ1
by using IDFTby using IDFT
p g p g q y and An(t) and φn(t) becomes
(8))(t = φφ(9) )((8) )(
nn
nn
AtAt
== φφ
( ) (10) 1)(1
0
)(2 0summinus
=
+ΔΔ+=ΔN
n
tkfnfjns
neAN
tkS φπ
loz Then the sampled signal can be expressed as0=nN
( )( ) (11) 1)(1
0
22 0summinus
=
ΔΔ+Δ sdot=ΔN
n
tfnkjtkfjns eeA
NtkS n πφπ
0n
18
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz The inverse discrete Fourier transform (IDFT) is defined
by using IDFTby using IDFT
( )as the following
1 21
NnkjN
πsumminus
(12) )(1)( 2
0
Nnkj
nefnF
Ntkf πsum
=
Δ=Δ
loz Comparing eq(11) and eq(12) the condition must be satisfied in order to make eq(11) an inverse Fouriersatisfied in order to make eq(11) an inverse Fourier transform relationship
1 (13) 1tN
fΔ
=Δ
19
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz If eq(13) is satisfied
by using IDFTby using IDFT
q ( ) loz is the frequency domain signal loz is the time domain signal
( )ntkfjneA φπ +Δ02
)( tkSs Δ gloz is the sub-channel spacingloz is the symbol duration in each sub-channel
)(s
fΔtNΔloz s e sy bo du o e c sub c e
loz This outcome is the same as the result obtained in the
N
loz This outcome is the same as the result obtained in the system of Figure 1 Therefore IDFT can be used to generate an OFDM transmission signalgenerate an OFDM transmission signal
20
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
( )tf02cos π
( )0a
by using IDFTby using IDFT
( )nd ( )tDI( )tf02sin π
( )0a
( )0b
bull
tf s Δ=
1 ( ) ( ) njbna +
( )nd
tNΔffnffn Δ
=Δ+=1 0
( )tDChannelInput
( )tf N 12cos minusπ
( )1minusNa
bullbull
tNΔ
( )tf N 12sin minusπ
( )1minusNb
( )0d
( )nd ( )tDChannelInput
bull bull
( )( )1d
( )2d
tf s Δ=
1 ( ) ( ) njbna +
fNΔffnffn Δ
=Δ+=1 0
ChannelInput
bullbullbullbull
bullbullbullbull
( )1minusNd
21
Frequency Error Results in ICIFrequency Error Results in ICIq yq y
22
Synchronization Error Results in ICISynchronization Error Results in ICIyy
Not Orthogonal Any More
23
Cyclic PrefixCyclic Prefixyy
loz In multipath channel delayed replicas of previous OFDM signal lead to ISI between successive OFDM signals
loz Solution Insert a guard interval between successive OFDMloz Solution Insert a guard interval between successive OFDM signals
24
Cyclic PrefixCyclic Prefix
loz Guard interval leads to intercarrier interference (ICI) in OFDM d d l i
yy
demodulation
loz In DFT interval difference between two subcarriers does not maintain integer number of cycles loss of orthogonality
loz Delayed version of subcarrier 2 causes ICI in the process of d d l ti b i 1demodulating subcarrier 1
25
Cyclic PrefixCyclic Prefixloz Cyclic prefix (CP) A copy of the last part of OFDM signal is
attached to the front of itself
yy
attached to the front of itself
[ ]0d
[ ]1d
[ ]0D[ ]1D
[ ]nd[ ]kD~
symbols dataInput
bullbullbullbull
[ ]1d
[ ]2d
[ ][ ]2D
bullbullbullbullbull
y
bullbullbullbullbull
[ ]1minusNd [ ]1minusND
[ ]gNND minus
bullbullbull
26
Cyclic PrefixCyclic Prefix
loz All delayed replicas of subcarriers always have an
yy
y p yinteger number of cycles within DFT interval no ICI
27
Cyclic PrefixCyclic Prefix
loz Linear convolution vs circular convolution
yy
N
28
Cyclic PrefixCyclic Prefix
loz Channel effect with cyclic prefix
yy
Signal after removed CP
29
Cyclic PrefixCyclic Prefix
loz Time-Domain Explanation
yy
p
30
Cyclic PrefixCyclic Prefixyy
31
Cyclic PrefixCyclic Prefix
loz Spectrum of channel response with length (smaller [ ]nh hL
yy
than )h
gN[ ] nhFFTHk =
loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr
loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=
where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier
[ ] [ ] [ ]N
Notimes
[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce
kk H
X =rArrcomplexity of channel equalizationrdquo
32
Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))
bull bull( )( )nr~Symbols
dataOutput
bullbullbull
bullbullbull( )nr
bullbull
bullbull
33
Cyclic PrefixCyclic Prefix
loz One of the most important reasons to do OFDM is the
yy
pefficient way it deals with multipath delay spread
loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)
34
Bandwidth EfficiencyBandwidth Efficiencyyy
loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )
loz The diagram for bandwidth efficiency of OFDM system is shown below
35
SummarySummaryyy
loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j
the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference
(ISI)loz The main advantage of the OFDM transmission technique is its
high performance even in frequency selective channels
loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems
36
SynchronizationSynchronization
Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg
38 Synchronization2010722
Introduction (13)Introduction (13)( )( )
loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods
loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead
loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the
cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of
computational complexity therefore it may be not be available in short-burst wireless communication
39 Synchronization2010722
Introduction (23)Introduction (23)( )( )
loz Transcevier
40 Synchronization2010722
Introduction (33)Introduction (33)( )( )
loz Synchronization erroryloz Symbol timing offset (TO)
loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)
loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect
41 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg
42 Synchronization2010722
Signal model (14)Signal model (14)g ( )g ( )
loz Time domain sample p1
2
0
1 Nj nk N
n kk
x X eN
πminus
minus= sum
Xk the frequency domain data at the k-th subcarrier0kN =
loz The multipath fading channel
( ) ( ) ( )1L
h t h t δminus
sum
loz Received time domain signal
( ) ( ) ( )0
l ll
h t h tτ τ δ τ τ=
= minussum
loz Received time-domain signal
( ) ( )1
L
n n ly h n l x w nminus
minus= +sum43
( ) ( )0l=sum
Synchronization2010722
Signal model (24)Signal model (24)g ( )g ( )
loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting
12
0
Nj kn N
k nn
Y y e πminus
minus
=
= sum
( )1 1
2
0
N Lj kn N
n l kn l
h n l x e Wπminus minus
minusminus
=
= +sumsum
loz The equation above could be expressed in matrix form
0n l
loz The equation above could be expressed in matrix form
= +Y HX W
44 Synchronization2010722
Signal model (34)Signal model (34)g ( )g ( )
H= = = = +Y Fy FCx FCF X HX W
loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal
y
H=H FCFcirculant matrix leading to be a diagonal matrix
=H FCF
( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0
h h L L hh h
⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥
( )( ) ( )
( ) ( )
1 10 1 0 0i
h Lh L h N L
⎢ ⎥⎢ ⎥minus minus⎢ ⎥
equiv= minus minus⎢ ⎥⎢ ⎥
C( ) ( )
( ) ( )
0 1 1 100
0 0 1 1 0
h L h N L
h N L L h N
⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
45
( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦
Synchronization2010722
Signal model (44)Signal model (44)g ( )g ( )
loz In a time-invariant channel the channel matrix H would be a diagonal matrix
k k k kY H X W= +
( )1 1
2
0 0
N Lj kl N
kn l
H h n l e πminus minus
minus
= =
= sumsum
loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI
k k k k kY H X I Wα= + +k k k k k
( ) ( )1 1 1
2 2 N N L
j m k Nj ml Nk mI X h n l e e ππ
minus minus minusminusminus⎛ ⎞
= ⎜ ⎟⎝ ⎠
sum sum sumICI
46
0 0 0m n lm k= = =ne
⎝ ⎠
Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
47 Synchronization2010722
Impact of timing offset Impact of timing offset p gp g
loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively
( )I f s
Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ
( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr
2 fj k Nk k k kY X H e Wπδminus= +
loz For integer part of timing offset k k k kY X H e W+
I sTδ ( ) 0g IN L δminus minus le leFFT
2 Ij k NY X H e Wπδminus= +
( ) ( ) 2 I sFFT
j f TI sh t T H f e π δδ minusminus rarr
48
k k k kY X H e W= +
Synchronization2010722
Impact of timing offset derivationImpact of timing offset derivationp gp g
49 Synchronization2010722
Discussion of symbol boundaryDiscussion of symbol boundaryy yy y
50 Synchronization2010722
Impact of timing offsetImpact of timing offsetp gp g
loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI
2 Ij k NIk k k k
NY X H e ISI W
Nπδδ minusminus
= + +
loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk
e e
p kloz Phase Shift can be resolved by Differential Encoding
Decoding or carrier recovery with pilots
51 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
52 Synchronization2010722
Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )
53 Synchronization2010722
Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )
( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s
i n t i N N T N T nTy y t e
= + + +=
( ) ( )( ) ( ) ( ) 12
sin
sin
g gI f f
I I
i N N N Nj jf N Ni k i k k
f
Y X H e eN
π ε ε π ε
ε ε
πεπε
+ + minus+
minus minus=⎛ ⎞⎜ ⎟
Faded signal attenuated and rotated by CFO
( )( ) ( ) ( ) ( )11 2
sin
sin g gI f I f
i N N N NN j j l kI f
NN
l k π ε ε π ε επ ε ε + + minusminus + + + minus
⎜ ⎟⎝ ⎠
+ + minussum
( )( )( )
( ) ( )2
0
sin
I f I f
I
j j l kI f N Ni l l
l l k I f
X H e el k
NN
π ε ε π ε ε
ε π ε ε
+ + +
= ne minus
+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠
sum
i k
N
W
⎜ ⎟⎝ ⎠
+ Inter-Carrier Interference (ICI)
54 Synchronization2010722
Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )
loz An easier representation of received signal with only p g yfractional CFO is
1N
( ) ( )1
0
0N
k k k l l kl l k
Y H X C H X C l k Wminus
= ne
= + minus +⎡ ⎤⎣ ⎦sum
where the ICI coefficient is given by
( )( )( ) ( )
sin 1exp 1fl k
C l k j l kπ ε
π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )
( )exp 1sin
f
f
C l k j l kNN l k
N
π επ ε
minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠
55 Synchronization2010722
Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )
loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality
loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift
56 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
57 Synchronization2010722
TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )
loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq
58 Synchronization2010722
TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )
loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below
loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and
rN2 can be expressed by
0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+
( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +
59
( )
Synchronization2010722
TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )
loz Without CFO the first half is identical to the second half (in time order)
loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below
( ) 12 in discretesft T f NT
NTφ π π π ε πε
⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟
⎝ ⎠
helliphellip
sNT⎝ ⎠
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)
60
Length=N2
Synchronization2010722
TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )
loz Received signal without CFOg0
0 0
jr a e θ=jθ 2 0
2 2 0
Nj j
N Nr a e a e
θ θ= =
loz Received signal with CFO0 0
0 0
j jr a e eθ φ= Extra phase rotation due to CFO0 0
2 2 20
2 2 0
N N Nj j jj
N Nr a e e a e e
θ φ φθ= =
due to CFO
Phase difference 2 2 0N N
( )2 02 20 2 0 0
Nj j T fN
r r a e a eφ φ πminus Δ= =
which contains the information about CFO
61
0 2 0 0Nr r a e a e
Synchronization2010722
TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )
loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be
loz Note that d is a time index corresponding to the first( ) ( ) 2 1
20
Nd m d m Nm
P d r rminus
+ + +== sum
loz Note that d is a time index corresponding to the first sample in a window of N samples
loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by
( )22 1
20
Nd m NR d rminus
+ += sumloz A timing metric can be defined as
))(()(
)( 2
2
dRdP
dM =
( ) 20 d m Nm + +=sum
62
))(( dR
Synchronization2010722
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
OFDM System ModelOFDM System Modelyy
loz OFDM A block modulation scheme that transmitsa block N source symbols in parallel by using subcarrierssubcarriersloz Sub-carriers are orthogonal in time but overlapped in
frequencyeque cyloz Frequency spacing 1
FFT
fT
Δ =
cos(2 )cos(2 ( ) ) 0FFTTf t f f t dtπ π + Δint 1 10
cos(2 )cos(2 ( ) ) 0f t f f t dtπ π + Δ =int
9
OFDM System ModelOFDM System Model
tNΔ
yy
( )tf02cos π
( )0a
( )0atT
tNΔ1+
( )tf2i
( )
( )0b
T
DataEncoder
tf s Δ=
1 ( ) ( ) njbna +
( )nd SP MUX ( )tDChannelInput
( )tf02sin π
( )tf N 12cos minusπbullbullbull
( )1minusNa
( )1minusNb( )0a1+
( )tf N 12sin minusπ
( )1Nb
tT
( )1a ( )1minusNa1minus
tΔ
Figure 1tNΔ
( )1a ( )1minusNa Figure 1
10
OFDM System ModelOFDM System Model
loz An OFDM system transmitter shown in Figure 1
yy
y gloz The transmitted waveform D(t) can be expressed as
)1( )2sin()()2cos()()(1
0summinus
=
+=N
nnn tfnbtfnatD ππ
where tNffnffn Δ=ΔΔ+=
1 and 0
loz Using a two-dimensional digital modulation format the data
tNΔ
g g symbols d(n) can be represented as a(n) + jb(n)loz a(n) in-phase componentloz b(n) quadrature component
11
OFDM System ModelOFDM System Model
loz The serial data elements spaced by are grouped and tΔ
yy
p y g pused to modulate N carriers Thus they are frequency division multiplexedp
loz The signaling interval is then increased to which tNΔmakes the system less susceptible to channel delay spread impairments
Small-scale fading(Based on multipath time delay spread)
Flat Fading1 BW of signal lt BW of channel
Frequency Selective Fading1 BW of signal gt BW of channel
2 Delay spread lt Symbol preiod 2 Delay spread gt Symbol preiod
12
OrthogonalityOrthogonalityg yg y
loz Consider a set of transmitted carriers as follows
(2) 1 1 0for )( 2 0
minus==⎟⎠⎞
⎜⎝⎛
Δ+
Nnett
tNnfj
n
πψ n
⎨⎧ =minus
intqpab
dtttb for )(
)()( ψψ⎩⎨ Δ=minusne
=int tNabqpdttt
a qp )( and for 0 )()( ψψ
13
OrthogonalityOrthogonality
b tqpjb
intint)(2π
g yg y
dtedtttajbj
b
atN
qpjb
a qp = Δminus
intint )()(
)(2)(2
)(2ψψπ
tNqpjee tN
qpjtN
qpj
Δminusminus
=Δ
minusΔ
minus
)(2
)(2)(2
π
ππ
ee
tNqpjba
tNqpj
tNbqpj
⎟⎟⎞
⎜⎜⎛minus
Δ
minusΔ
minusΔ
minus1
)(2)(1)(2)(2
π
ππ
tNqpj Δminus
⎟⎠
⎜⎝=
)(2πtNabqp Δ=minusne= )( and for 0
14
OrthogonalityOrthogonalityg yg y
tNΔ
15
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFTby using IDFTby using IDFT
loz According to the structure of Tx it must use Ng oscillators That increases the hardware complexity
loz The equivalent method is using IDFT (IFFT)
16
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz In general each carrier can be expressed as
by using IDFTby using IDFT
g p
( ) (6) )()( )(2 ttfjcc
ccetAtS φπ +=
loz We assume that there are N carriers in the OFDM signal Th th t t l l i l S (t) b t d bThen the total complex signal Ss(t) can be represented by
( ) (7))(1)(1
)(2etAtSN
ttfj nn φπ= summinus
+( )
where
(7) )()(
0
0
fnff
etAN
tSn
ns
Δ+=
= sum=
liihffrequency carrier phase amplitude are )( )( and
where 0
fttAfnff
nnn
n
φΔ+
lyrespective carrierth -of n17
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz Then we sample the signal at a sampling frequency tΔ1
by using IDFTby using IDFT
p g p g q y and An(t) and φn(t) becomes
(8))(t = φφ(9) )((8) )(
nn
nn
AtAt
== φφ
( ) (10) 1)(1
0
)(2 0summinus
=
+ΔΔ+=ΔN
n
tkfnfjns
neAN
tkS φπ
loz Then the sampled signal can be expressed as0=nN
( )( ) (11) 1)(1
0
22 0summinus
=
ΔΔ+Δ sdot=ΔN
n
tfnkjtkfjns eeA
NtkS n πφπ
0n
18
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz The inverse discrete Fourier transform (IDFT) is defined
by using IDFTby using IDFT
( )as the following
1 21
NnkjN
πsumminus
(12) )(1)( 2
0
Nnkj
nefnF
Ntkf πsum
=
Δ=Δ
loz Comparing eq(11) and eq(12) the condition must be satisfied in order to make eq(11) an inverse Fouriersatisfied in order to make eq(11) an inverse Fourier transform relationship
1 (13) 1tN
fΔ
=Δ
19
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz If eq(13) is satisfied
by using IDFTby using IDFT
q ( ) loz is the frequency domain signal loz is the time domain signal
( )ntkfjneA φπ +Δ02
)( tkSs Δ gloz is the sub-channel spacingloz is the symbol duration in each sub-channel
)(s
fΔtNΔloz s e sy bo du o e c sub c e
loz This outcome is the same as the result obtained in the
N
loz This outcome is the same as the result obtained in the system of Figure 1 Therefore IDFT can be used to generate an OFDM transmission signalgenerate an OFDM transmission signal
20
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
( )tf02cos π
( )0a
by using IDFTby using IDFT
( )nd ( )tDI( )tf02sin π
( )0a
( )0b
bull
tf s Δ=
1 ( ) ( ) njbna +
( )nd
tNΔffnffn Δ
=Δ+=1 0
( )tDChannelInput
( )tf N 12cos minusπ
( )1minusNa
bullbull
tNΔ
( )tf N 12sin minusπ
( )1minusNb
( )0d
( )nd ( )tDChannelInput
bull bull
( )( )1d
( )2d
tf s Δ=
1 ( ) ( ) njbna +
fNΔffnffn Δ
=Δ+=1 0
ChannelInput
bullbullbullbull
bullbullbullbull
( )1minusNd
21
Frequency Error Results in ICIFrequency Error Results in ICIq yq y
22
Synchronization Error Results in ICISynchronization Error Results in ICIyy
Not Orthogonal Any More
23
Cyclic PrefixCyclic Prefixyy
loz In multipath channel delayed replicas of previous OFDM signal lead to ISI between successive OFDM signals
loz Solution Insert a guard interval between successive OFDMloz Solution Insert a guard interval between successive OFDM signals
24
Cyclic PrefixCyclic Prefix
loz Guard interval leads to intercarrier interference (ICI) in OFDM d d l i
yy
demodulation
loz In DFT interval difference between two subcarriers does not maintain integer number of cycles loss of orthogonality
loz Delayed version of subcarrier 2 causes ICI in the process of d d l ti b i 1demodulating subcarrier 1
25
Cyclic PrefixCyclic Prefixloz Cyclic prefix (CP) A copy of the last part of OFDM signal is
attached to the front of itself
yy
attached to the front of itself
[ ]0d
[ ]1d
[ ]0D[ ]1D
[ ]nd[ ]kD~
symbols dataInput
bullbullbullbull
[ ]1d
[ ]2d
[ ][ ]2D
bullbullbullbullbull
y
bullbullbullbullbull
[ ]1minusNd [ ]1minusND
[ ]gNND minus
bullbullbull
26
Cyclic PrefixCyclic Prefix
loz All delayed replicas of subcarriers always have an
yy
y p yinteger number of cycles within DFT interval no ICI
27
Cyclic PrefixCyclic Prefix
loz Linear convolution vs circular convolution
yy
N
28
Cyclic PrefixCyclic Prefix
loz Channel effect with cyclic prefix
yy
Signal after removed CP
29
Cyclic PrefixCyclic Prefix
loz Time-Domain Explanation
yy
p
30
Cyclic PrefixCyclic Prefixyy
31
Cyclic PrefixCyclic Prefix
loz Spectrum of channel response with length (smaller [ ]nh hL
yy
than )h
gN[ ] nhFFTHk =
loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr
loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=
where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier
[ ] [ ] [ ]N
Notimes
[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce
kk H
X =rArrcomplexity of channel equalizationrdquo
32
Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))
bull bull( )( )nr~Symbols
dataOutput
bullbullbull
bullbullbull( )nr
bullbull
bullbull
33
Cyclic PrefixCyclic Prefix
loz One of the most important reasons to do OFDM is the
yy
pefficient way it deals with multipath delay spread
loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)
34
Bandwidth EfficiencyBandwidth Efficiencyyy
loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )
loz The diagram for bandwidth efficiency of OFDM system is shown below
35
SummarySummaryyy
loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j
the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference
(ISI)loz The main advantage of the OFDM transmission technique is its
high performance even in frequency selective channels
loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems
36
SynchronizationSynchronization
Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg
38 Synchronization2010722
Introduction (13)Introduction (13)( )( )
loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods
loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead
loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the
cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of
computational complexity therefore it may be not be available in short-burst wireless communication
39 Synchronization2010722
Introduction (23)Introduction (23)( )( )
loz Transcevier
40 Synchronization2010722
Introduction (33)Introduction (33)( )( )
loz Synchronization erroryloz Symbol timing offset (TO)
loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)
loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect
41 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg
42 Synchronization2010722
Signal model (14)Signal model (14)g ( )g ( )
loz Time domain sample p1
2
0
1 Nj nk N
n kk
x X eN
πminus
minus= sum
Xk the frequency domain data at the k-th subcarrier0kN =
loz The multipath fading channel
( ) ( ) ( )1L
h t h t δminus
sum
loz Received time domain signal
( ) ( ) ( )0
l ll
h t h tτ τ δ τ τ=
= minussum
loz Received time-domain signal
( ) ( )1
L
n n ly h n l x w nminus
minus= +sum43
( ) ( )0l=sum
Synchronization2010722
Signal model (24)Signal model (24)g ( )g ( )
loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting
12
0
Nj kn N
k nn
Y y e πminus
minus
=
= sum
( )1 1
2
0
N Lj kn N
n l kn l
h n l x e Wπminus minus
minusminus
=
= +sumsum
loz The equation above could be expressed in matrix form
0n l
loz The equation above could be expressed in matrix form
= +Y HX W
44 Synchronization2010722
Signal model (34)Signal model (34)g ( )g ( )
H= = = = +Y Fy FCx FCF X HX W
loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal
y
H=H FCFcirculant matrix leading to be a diagonal matrix
=H FCF
( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0
h h L L hh h
⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥
( )( ) ( )
( ) ( )
1 10 1 0 0i
h Lh L h N L
⎢ ⎥⎢ ⎥minus minus⎢ ⎥
equiv= minus minus⎢ ⎥⎢ ⎥
C( ) ( )
( ) ( )
0 1 1 100
0 0 1 1 0
h L h N L
h N L L h N
⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
45
( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦
Synchronization2010722
Signal model (44)Signal model (44)g ( )g ( )
loz In a time-invariant channel the channel matrix H would be a diagonal matrix
k k k kY H X W= +
( )1 1
2
0 0
N Lj kl N
kn l
H h n l e πminus minus
minus
= =
= sumsum
loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI
k k k k kY H X I Wα= + +k k k k k
( ) ( )1 1 1
2 2 N N L
j m k Nj ml Nk mI X h n l e e ππ
minus minus minusminusminus⎛ ⎞
= ⎜ ⎟⎝ ⎠
sum sum sumICI
46
0 0 0m n lm k= = =ne
⎝ ⎠
Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
47 Synchronization2010722
Impact of timing offset Impact of timing offset p gp g
loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively
( )I f s
Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ
( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr
2 fj k Nk k k kY X H e Wπδminus= +
loz For integer part of timing offset k k k kY X H e W+
I sTδ ( ) 0g IN L δminus minus le leFFT
2 Ij k NY X H e Wπδminus= +
( ) ( ) 2 I sFFT
j f TI sh t T H f e π δδ minusminus rarr
48
k k k kY X H e W= +
Synchronization2010722
Impact of timing offset derivationImpact of timing offset derivationp gp g
49 Synchronization2010722
Discussion of symbol boundaryDiscussion of symbol boundaryy yy y
50 Synchronization2010722
Impact of timing offsetImpact of timing offsetp gp g
loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI
2 Ij k NIk k k k
NY X H e ISI W
Nπδδ minusminus
= + +
loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk
e e
p kloz Phase Shift can be resolved by Differential Encoding
Decoding or carrier recovery with pilots
51 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
52 Synchronization2010722
Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )
53 Synchronization2010722
Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )
( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s
i n t i N N T N T nTy y t e
= + + +=
( ) ( )( ) ( ) ( ) 12
sin
sin
g gI f f
I I
i N N N Nj jf N Ni k i k k
f
Y X H e eN
π ε ε π ε
ε ε
πεπε
+ + minus+
minus minus=⎛ ⎞⎜ ⎟
Faded signal attenuated and rotated by CFO
( )( ) ( ) ( ) ( )11 2
sin
sin g gI f I f
i N N N NN j j l kI f
NN
l k π ε ε π ε επ ε ε + + minusminus + + + minus
⎜ ⎟⎝ ⎠
+ + minussum
( )( )( )
( ) ( )2
0
sin
I f I f
I
j j l kI f N Ni l l
l l k I f
X H e el k
NN
π ε ε π ε ε
ε π ε ε
+ + +
= ne minus
+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠
sum
i k
N
W
⎜ ⎟⎝ ⎠
+ Inter-Carrier Interference (ICI)
54 Synchronization2010722
Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )
loz An easier representation of received signal with only p g yfractional CFO is
1N
( ) ( )1
0
0N
k k k l l kl l k
Y H X C H X C l k Wminus
= ne
= + minus +⎡ ⎤⎣ ⎦sum
where the ICI coefficient is given by
( )( )( ) ( )
sin 1exp 1fl k
C l k j l kπ ε
π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )
( )exp 1sin
f
f
C l k j l kNN l k
N
π επ ε
minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠
55 Synchronization2010722
Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )
loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality
loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift
56 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
57 Synchronization2010722
TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )
loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq
58 Synchronization2010722
TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )
loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below
loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and
rN2 can be expressed by
0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+
( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +
59
( )
Synchronization2010722
TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )
loz Without CFO the first half is identical to the second half (in time order)
loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below
( ) 12 in discretesft T f NT
NTφ π π π ε πε
⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟
⎝ ⎠
helliphellip
sNT⎝ ⎠
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)
60
Length=N2
Synchronization2010722
TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )
loz Received signal without CFOg0
0 0
jr a e θ=jθ 2 0
2 2 0
Nj j
N Nr a e a e
θ θ= =
loz Received signal with CFO0 0
0 0
j jr a e eθ φ= Extra phase rotation due to CFO0 0
2 2 20
2 2 0
N N Nj j jj
N Nr a e e a e e
θ φ φθ= =
due to CFO
Phase difference 2 2 0N N
( )2 02 20 2 0 0
Nj j T fN
r r a e a eφ φ πminus Δ= =
which contains the information about CFO
61
0 2 0 0Nr r a e a e
Synchronization2010722
TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )
loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be
loz Note that d is a time index corresponding to the first( ) ( ) 2 1
20
Nd m d m Nm
P d r rminus
+ + +== sum
loz Note that d is a time index corresponding to the first sample in a window of N samples
loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by
( )22 1
20
Nd m NR d rminus
+ += sumloz A timing metric can be defined as
))(()(
)( 2
2
dRdP
dM =
( ) 20 d m Nm + +=sum
62
))(( dR
Synchronization2010722
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
OFDM System ModelOFDM System Model
tNΔ
yy
( )tf02cos π
( )0a
( )0atT
tNΔ1+
( )tf2i
( )
( )0b
T
DataEncoder
tf s Δ=
1 ( ) ( ) njbna +
( )nd SP MUX ( )tDChannelInput
( )tf02sin π
( )tf N 12cos minusπbullbullbull
( )1minusNa
( )1minusNb( )0a1+
( )tf N 12sin minusπ
( )1Nb
tT
( )1a ( )1minusNa1minus
tΔ
Figure 1tNΔ
( )1a ( )1minusNa Figure 1
10
OFDM System ModelOFDM System Model
loz An OFDM system transmitter shown in Figure 1
yy
y gloz The transmitted waveform D(t) can be expressed as
)1( )2sin()()2cos()()(1
0summinus
=
+=N
nnn tfnbtfnatD ππ
where tNffnffn Δ=ΔΔ+=
1 and 0
loz Using a two-dimensional digital modulation format the data
tNΔ
g g symbols d(n) can be represented as a(n) + jb(n)loz a(n) in-phase componentloz b(n) quadrature component
11
OFDM System ModelOFDM System Model
loz The serial data elements spaced by are grouped and tΔ
yy
p y g pused to modulate N carriers Thus they are frequency division multiplexedp
loz The signaling interval is then increased to which tNΔmakes the system less susceptible to channel delay spread impairments
Small-scale fading(Based on multipath time delay spread)
Flat Fading1 BW of signal lt BW of channel
Frequency Selective Fading1 BW of signal gt BW of channel
2 Delay spread lt Symbol preiod 2 Delay spread gt Symbol preiod
12
OrthogonalityOrthogonalityg yg y
loz Consider a set of transmitted carriers as follows
(2) 1 1 0for )( 2 0
minus==⎟⎠⎞
⎜⎝⎛
Δ+
Nnett
tNnfj
n
πψ n
⎨⎧ =minus
intqpab
dtttb for )(
)()( ψψ⎩⎨ Δ=minusne
=int tNabqpdttt
a qp )( and for 0 )()( ψψ
13
OrthogonalityOrthogonality
b tqpjb
intint)(2π
g yg y
dtedtttajbj
b
atN
qpjb
a qp = Δminus
intint )()(
)(2)(2
)(2ψψπ
tNqpjee tN
qpjtN
qpj
Δminusminus
=Δ
minusΔ
minus
)(2
)(2)(2
π
ππ
ee
tNqpjba
tNqpj
tNbqpj
⎟⎟⎞
⎜⎜⎛minus
Δ
minusΔ
minusΔ
minus1
)(2)(1)(2)(2
π
ππ
tNqpj Δminus
⎟⎠
⎜⎝=
)(2πtNabqp Δ=minusne= )( and for 0
14
OrthogonalityOrthogonalityg yg y
tNΔ
15
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFTby using IDFTby using IDFT
loz According to the structure of Tx it must use Ng oscillators That increases the hardware complexity
loz The equivalent method is using IDFT (IFFT)
16
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz In general each carrier can be expressed as
by using IDFTby using IDFT
g p
( ) (6) )()( )(2 ttfjcc
ccetAtS φπ +=
loz We assume that there are N carriers in the OFDM signal Th th t t l l i l S (t) b t d bThen the total complex signal Ss(t) can be represented by
( ) (7))(1)(1
)(2etAtSN
ttfj nn φπ= summinus
+( )
where
(7) )()(
0
0
fnff
etAN
tSn
ns
Δ+=
= sum=
liihffrequency carrier phase amplitude are )( )( and
where 0
fttAfnff
nnn
n
φΔ+
lyrespective carrierth -of n17
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz Then we sample the signal at a sampling frequency tΔ1
by using IDFTby using IDFT
p g p g q y and An(t) and φn(t) becomes
(8))(t = φφ(9) )((8) )(
nn
nn
AtAt
== φφ
( ) (10) 1)(1
0
)(2 0summinus
=
+ΔΔ+=ΔN
n
tkfnfjns
neAN
tkS φπ
loz Then the sampled signal can be expressed as0=nN
( )( ) (11) 1)(1
0
22 0summinus
=
ΔΔ+Δ sdot=ΔN
n
tfnkjtkfjns eeA
NtkS n πφπ
0n
18
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz The inverse discrete Fourier transform (IDFT) is defined
by using IDFTby using IDFT
( )as the following
1 21
NnkjN
πsumminus
(12) )(1)( 2
0
Nnkj
nefnF
Ntkf πsum
=
Δ=Δ
loz Comparing eq(11) and eq(12) the condition must be satisfied in order to make eq(11) an inverse Fouriersatisfied in order to make eq(11) an inverse Fourier transform relationship
1 (13) 1tN
fΔ
=Δ
19
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz If eq(13) is satisfied
by using IDFTby using IDFT
q ( ) loz is the frequency domain signal loz is the time domain signal
( )ntkfjneA φπ +Δ02
)( tkSs Δ gloz is the sub-channel spacingloz is the symbol duration in each sub-channel
)(s
fΔtNΔloz s e sy bo du o e c sub c e
loz This outcome is the same as the result obtained in the
N
loz This outcome is the same as the result obtained in the system of Figure 1 Therefore IDFT can be used to generate an OFDM transmission signalgenerate an OFDM transmission signal
20
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
( )tf02cos π
( )0a
by using IDFTby using IDFT
( )nd ( )tDI( )tf02sin π
( )0a
( )0b
bull
tf s Δ=
1 ( ) ( ) njbna +
( )nd
tNΔffnffn Δ
=Δ+=1 0
( )tDChannelInput
( )tf N 12cos minusπ
( )1minusNa
bullbull
tNΔ
( )tf N 12sin minusπ
( )1minusNb
( )0d
( )nd ( )tDChannelInput
bull bull
( )( )1d
( )2d
tf s Δ=
1 ( ) ( ) njbna +
fNΔffnffn Δ
=Δ+=1 0
ChannelInput
bullbullbullbull
bullbullbullbull
( )1minusNd
21
Frequency Error Results in ICIFrequency Error Results in ICIq yq y
22
Synchronization Error Results in ICISynchronization Error Results in ICIyy
Not Orthogonal Any More
23
Cyclic PrefixCyclic Prefixyy
loz In multipath channel delayed replicas of previous OFDM signal lead to ISI between successive OFDM signals
loz Solution Insert a guard interval between successive OFDMloz Solution Insert a guard interval between successive OFDM signals
24
Cyclic PrefixCyclic Prefix
loz Guard interval leads to intercarrier interference (ICI) in OFDM d d l i
yy
demodulation
loz In DFT interval difference between two subcarriers does not maintain integer number of cycles loss of orthogonality
loz Delayed version of subcarrier 2 causes ICI in the process of d d l ti b i 1demodulating subcarrier 1
25
Cyclic PrefixCyclic Prefixloz Cyclic prefix (CP) A copy of the last part of OFDM signal is
attached to the front of itself
yy
attached to the front of itself
[ ]0d
[ ]1d
[ ]0D[ ]1D
[ ]nd[ ]kD~
symbols dataInput
bullbullbullbull
[ ]1d
[ ]2d
[ ][ ]2D
bullbullbullbullbull
y
bullbullbullbullbull
[ ]1minusNd [ ]1minusND
[ ]gNND minus
bullbullbull
26
Cyclic PrefixCyclic Prefix
loz All delayed replicas of subcarriers always have an
yy
y p yinteger number of cycles within DFT interval no ICI
27
Cyclic PrefixCyclic Prefix
loz Linear convolution vs circular convolution
yy
N
28
Cyclic PrefixCyclic Prefix
loz Channel effect with cyclic prefix
yy
Signal after removed CP
29
Cyclic PrefixCyclic Prefix
loz Time-Domain Explanation
yy
p
30
Cyclic PrefixCyclic Prefixyy
31
Cyclic PrefixCyclic Prefix
loz Spectrum of channel response with length (smaller [ ]nh hL
yy
than )h
gN[ ] nhFFTHk =
loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr
loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=
where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier
[ ] [ ] [ ]N
Notimes
[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce
kk H
X =rArrcomplexity of channel equalizationrdquo
32
Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))
bull bull( )( )nr~Symbols
dataOutput
bullbullbull
bullbullbull( )nr
bullbull
bullbull
33
Cyclic PrefixCyclic Prefix
loz One of the most important reasons to do OFDM is the
yy
pefficient way it deals with multipath delay spread
loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)
34
Bandwidth EfficiencyBandwidth Efficiencyyy
loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )
loz The diagram for bandwidth efficiency of OFDM system is shown below
35
SummarySummaryyy
loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j
the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference
(ISI)loz The main advantage of the OFDM transmission technique is its
high performance even in frequency selective channels
loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems
36
SynchronizationSynchronization
Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg
38 Synchronization2010722
Introduction (13)Introduction (13)( )( )
loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods
loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead
loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the
cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of
computational complexity therefore it may be not be available in short-burst wireless communication
39 Synchronization2010722
Introduction (23)Introduction (23)( )( )
loz Transcevier
40 Synchronization2010722
Introduction (33)Introduction (33)( )( )
loz Synchronization erroryloz Symbol timing offset (TO)
loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)
loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect
41 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg
42 Synchronization2010722
Signal model (14)Signal model (14)g ( )g ( )
loz Time domain sample p1
2
0
1 Nj nk N
n kk
x X eN
πminus
minus= sum
Xk the frequency domain data at the k-th subcarrier0kN =
loz The multipath fading channel
( ) ( ) ( )1L
h t h t δminus
sum
loz Received time domain signal
( ) ( ) ( )0
l ll
h t h tτ τ δ τ τ=
= minussum
loz Received time-domain signal
( ) ( )1
L
n n ly h n l x w nminus
minus= +sum43
( ) ( )0l=sum
Synchronization2010722
Signal model (24)Signal model (24)g ( )g ( )
loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting
12
0
Nj kn N
k nn
Y y e πminus
minus
=
= sum
( )1 1
2
0
N Lj kn N
n l kn l
h n l x e Wπminus minus
minusminus
=
= +sumsum
loz The equation above could be expressed in matrix form
0n l
loz The equation above could be expressed in matrix form
= +Y HX W
44 Synchronization2010722
Signal model (34)Signal model (34)g ( )g ( )
H= = = = +Y Fy FCx FCF X HX W
loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal
y
H=H FCFcirculant matrix leading to be a diagonal matrix
=H FCF
( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0
h h L L hh h
⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥
( )( ) ( )
( ) ( )
1 10 1 0 0i
h Lh L h N L
⎢ ⎥⎢ ⎥minus minus⎢ ⎥
equiv= minus minus⎢ ⎥⎢ ⎥
C( ) ( )
( ) ( )
0 1 1 100
0 0 1 1 0
h L h N L
h N L L h N
⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
45
( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦
Synchronization2010722
Signal model (44)Signal model (44)g ( )g ( )
loz In a time-invariant channel the channel matrix H would be a diagonal matrix
k k k kY H X W= +
( )1 1
2
0 0
N Lj kl N
kn l
H h n l e πminus minus
minus
= =
= sumsum
loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI
k k k k kY H X I Wα= + +k k k k k
( ) ( )1 1 1
2 2 N N L
j m k Nj ml Nk mI X h n l e e ππ
minus minus minusminusminus⎛ ⎞
= ⎜ ⎟⎝ ⎠
sum sum sumICI
46
0 0 0m n lm k= = =ne
⎝ ⎠
Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
47 Synchronization2010722
Impact of timing offset Impact of timing offset p gp g
loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively
( )I f s
Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ
( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr
2 fj k Nk k k kY X H e Wπδminus= +
loz For integer part of timing offset k k k kY X H e W+
I sTδ ( ) 0g IN L δminus minus le leFFT
2 Ij k NY X H e Wπδminus= +
( ) ( ) 2 I sFFT
j f TI sh t T H f e π δδ minusminus rarr
48
k k k kY X H e W= +
Synchronization2010722
Impact of timing offset derivationImpact of timing offset derivationp gp g
49 Synchronization2010722
Discussion of symbol boundaryDiscussion of symbol boundaryy yy y
50 Synchronization2010722
Impact of timing offsetImpact of timing offsetp gp g
loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI
2 Ij k NIk k k k
NY X H e ISI W
Nπδδ minusminus
= + +
loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk
e e
p kloz Phase Shift can be resolved by Differential Encoding
Decoding or carrier recovery with pilots
51 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
52 Synchronization2010722
Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )
53 Synchronization2010722
Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )
( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s
i n t i N N T N T nTy y t e
= + + +=
( ) ( )( ) ( ) ( ) 12
sin
sin
g gI f f
I I
i N N N Nj jf N Ni k i k k
f
Y X H e eN
π ε ε π ε
ε ε
πεπε
+ + minus+
minus minus=⎛ ⎞⎜ ⎟
Faded signal attenuated and rotated by CFO
( )( ) ( ) ( ) ( )11 2
sin
sin g gI f I f
i N N N NN j j l kI f
NN
l k π ε ε π ε επ ε ε + + minusminus + + + minus
⎜ ⎟⎝ ⎠
+ + minussum
( )( )( )
( ) ( )2
0
sin
I f I f
I
j j l kI f N Ni l l
l l k I f
X H e el k
NN
π ε ε π ε ε
ε π ε ε
+ + +
= ne minus
+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠
sum
i k
N
W
⎜ ⎟⎝ ⎠
+ Inter-Carrier Interference (ICI)
54 Synchronization2010722
Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )
loz An easier representation of received signal with only p g yfractional CFO is
1N
( ) ( )1
0
0N
k k k l l kl l k
Y H X C H X C l k Wminus
= ne
= + minus +⎡ ⎤⎣ ⎦sum
where the ICI coefficient is given by
( )( )( ) ( )
sin 1exp 1fl k
C l k j l kπ ε
π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )
( )exp 1sin
f
f
C l k j l kNN l k
N
π επ ε
minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠
55 Synchronization2010722
Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )
loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality
loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift
56 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
57 Synchronization2010722
TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )
loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq
58 Synchronization2010722
TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )
loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below
loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and
rN2 can be expressed by
0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+
( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +
59
( )
Synchronization2010722
TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )
loz Without CFO the first half is identical to the second half (in time order)
loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below
( ) 12 in discretesft T f NT
NTφ π π π ε πε
⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟
⎝ ⎠
helliphellip
sNT⎝ ⎠
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)
60
Length=N2
Synchronization2010722
TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )
loz Received signal without CFOg0
0 0
jr a e θ=jθ 2 0
2 2 0
Nj j
N Nr a e a e
θ θ= =
loz Received signal with CFO0 0
0 0
j jr a e eθ φ= Extra phase rotation due to CFO0 0
2 2 20
2 2 0
N N Nj j jj
N Nr a e e a e e
θ φ φθ= =
due to CFO
Phase difference 2 2 0N N
( )2 02 20 2 0 0
Nj j T fN
r r a e a eφ φ πminus Δ= =
which contains the information about CFO
61
0 2 0 0Nr r a e a e
Synchronization2010722
TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )
loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be
loz Note that d is a time index corresponding to the first( ) ( ) 2 1
20
Nd m d m Nm
P d r rminus
+ + +== sum
loz Note that d is a time index corresponding to the first sample in a window of N samples
loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by
( )22 1
20
Nd m NR d rminus
+ += sumloz A timing metric can be defined as
))(()(
)( 2
2
dRdP
dM =
( ) 20 d m Nm + +=sum
62
))(( dR
Synchronization2010722
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
OFDM System ModelOFDM System Model
loz An OFDM system transmitter shown in Figure 1
yy
y gloz The transmitted waveform D(t) can be expressed as
)1( )2sin()()2cos()()(1
0summinus
=
+=N
nnn tfnbtfnatD ππ
where tNffnffn Δ=ΔΔ+=
1 and 0
loz Using a two-dimensional digital modulation format the data
tNΔ
g g symbols d(n) can be represented as a(n) + jb(n)loz a(n) in-phase componentloz b(n) quadrature component
11
OFDM System ModelOFDM System Model
loz The serial data elements spaced by are grouped and tΔ
yy
p y g pused to modulate N carriers Thus they are frequency division multiplexedp
loz The signaling interval is then increased to which tNΔmakes the system less susceptible to channel delay spread impairments
Small-scale fading(Based on multipath time delay spread)
Flat Fading1 BW of signal lt BW of channel
Frequency Selective Fading1 BW of signal gt BW of channel
2 Delay spread lt Symbol preiod 2 Delay spread gt Symbol preiod
12
OrthogonalityOrthogonalityg yg y
loz Consider a set of transmitted carriers as follows
(2) 1 1 0for )( 2 0
minus==⎟⎠⎞
⎜⎝⎛
Δ+
Nnett
tNnfj
n
πψ n
⎨⎧ =minus
intqpab
dtttb for )(
)()( ψψ⎩⎨ Δ=minusne
=int tNabqpdttt
a qp )( and for 0 )()( ψψ
13
OrthogonalityOrthogonality
b tqpjb
intint)(2π
g yg y
dtedtttajbj
b
atN
qpjb
a qp = Δminus
intint )()(
)(2)(2
)(2ψψπ
tNqpjee tN
qpjtN
qpj
Δminusminus
=Δ
minusΔ
minus
)(2
)(2)(2
π
ππ
ee
tNqpjba
tNqpj
tNbqpj
⎟⎟⎞
⎜⎜⎛minus
Δ
minusΔ
minusΔ
minus1
)(2)(1)(2)(2
π
ππ
tNqpj Δminus
⎟⎠
⎜⎝=
)(2πtNabqp Δ=minusne= )( and for 0
14
OrthogonalityOrthogonalityg yg y
tNΔ
15
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFTby using IDFTby using IDFT
loz According to the structure of Tx it must use Ng oscillators That increases the hardware complexity
loz The equivalent method is using IDFT (IFFT)
16
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz In general each carrier can be expressed as
by using IDFTby using IDFT
g p
( ) (6) )()( )(2 ttfjcc
ccetAtS φπ +=
loz We assume that there are N carriers in the OFDM signal Th th t t l l i l S (t) b t d bThen the total complex signal Ss(t) can be represented by
( ) (7))(1)(1
)(2etAtSN
ttfj nn φπ= summinus
+( )
where
(7) )()(
0
0
fnff
etAN
tSn
ns
Δ+=
= sum=
liihffrequency carrier phase amplitude are )( )( and
where 0
fttAfnff
nnn
n
φΔ+
lyrespective carrierth -of n17
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz Then we sample the signal at a sampling frequency tΔ1
by using IDFTby using IDFT
p g p g q y and An(t) and φn(t) becomes
(8))(t = φφ(9) )((8) )(
nn
nn
AtAt
== φφ
( ) (10) 1)(1
0
)(2 0summinus
=
+ΔΔ+=ΔN
n
tkfnfjns
neAN
tkS φπ
loz Then the sampled signal can be expressed as0=nN
( )( ) (11) 1)(1
0
22 0summinus
=
ΔΔ+Δ sdot=ΔN
n
tfnkjtkfjns eeA
NtkS n πφπ
0n
18
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz The inverse discrete Fourier transform (IDFT) is defined
by using IDFTby using IDFT
( )as the following
1 21
NnkjN
πsumminus
(12) )(1)( 2
0
Nnkj
nefnF
Ntkf πsum
=
Δ=Δ
loz Comparing eq(11) and eq(12) the condition must be satisfied in order to make eq(11) an inverse Fouriersatisfied in order to make eq(11) an inverse Fourier transform relationship
1 (13) 1tN
fΔ
=Δ
19
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz If eq(13) is satisfied
by using IDFTby using IDFT
q ( ) loz is the frequency domain signal loz is the time domain signal
( )ntkfjneA φπ +Δ02
)( tkSs Δ gloz is the sub-channel spacingloz is the symbol duration in each sub-channel
)(s
fΔtNΔloz s e sy bo du o e c sub c e
loz This outcome is the same as the result obtained in the
N
loz This outcome is the same as the result obtained in the system of Figure 1 Therefore IDFT can be used to generate an OFDM transmission signalgenerate an OFDM transmission signal
20
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
( )tf02cos π
( )0a
by using IDFTby using IDFT
( )nd ( )tDI( )tf02sin π
( )0a
( )0b
bull
tf s Δ=
1 ( ) ( ) njbna +
( )nd
tNΔffnffn Δ
=Δ+=1 0
( )tDChannelInput
( )tf N 12cos minusπ
( )1minusNa
bullbull
tNΔ
( )tf N 12sin minusπ
( )1minusNb
( )0d
( )nd ( )tDChannelInput
bull bull
( )( )1d
( )2d
tf s Δ=
1 ( ) ( ) njbna +
fNΔffnffn Δ
=Δ+=1 0
ChannelInput
bullbullbullbull
bullbullbullbull
( )1minusNd
21
Frequency Error Results in ICIFrequency Error Results in ICIq yq y
22
Synchronization Error Results in ICISynchronization Error Results in ICIyy
Not Orthogonal Any More
23
Cyclic PrefixCyclic Prefixyy
loz In multipath channel delayed replicas of previous OFDM signal lead to ISI between successive OFDM signals
loz Solution Insert a guard interval between successive OFDMloz Solution Insert a guard interval between successive OFDM signals
24
Cyclic PrefixCyclic Prefix
loz Guard interval leads to intercarrier interference (ICI) in OFDM d d l i
yy
demodulation
loz In DFT interval difference between two subcarriers does not maintain integer number of cycles loss of orthogonality
loz Delayed version of subcarrier 2 causes ICI in the process of d d l ti b i 1demodulating subcarrier 1
25
Cyclic PrefixCyclic Prefixloz Cyclic prefix (CP) A copy of the last part of OFDM signal is
attached to the front of itself
yy
attached to the front of itself
[ ]0d
[ ]1d
[ ]0D[ ]1D
[ ]nd[ ]kD~
symbols dataInput
bullbullbullbull
[ ]1d
[ ]2d
[ ][ ]2D
bullbullbullbullbull
y
bullbullbullbullbull
[ ]1minusNd [ ]1minusND
[ ]gNND minus
bullbullbull
26
Cyclic PrefixCyclic Prefix
loz All delayed replicas of subcarriers always have an
yy
y p yinteger number of cycles within DFT interval no ICI
27
Cyclic PrefixCyclic Prefix
loz Linear convolution vs circular convolution
yy
N
28
Cyclic PrefixCyclic Prefix
loz Channel effect with cyclic prefix
yy
Signal after removed CP
29
Cyclic PrefixCyclic Prefix
loz Time-Domain Explanation
yy
p
30
Cyclic PrefixCyclic Prefixyy
31
Cyclic PrefixCyclic Prefix
loz Spectrum of channel response with length (smaller [ ]nh hL
yy
than )h
gN[ ] nhFFTHk =
loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr
loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=
where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier
[ ] [ ] [ ]N
Notimes
[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce
kk H
X =rArrcomplexity of channel equalizationrdquo
32
Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))
bull bull( )( )nr~Symbols
dataOutput
bullbullbull
bullbullbull( )nr
bullbull
bullbull
33
Cyclic PrefixCyclic Prefix
loz One of the most important reasons to do OFDM is the
yy
pefficient way it deals with multipath delay spread
loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)
34
Bandwidth EfficiencyBandwidth Efficiencyyy
loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )
loz The diagram for bandwidth efficiency of OFDM system is shown below
35
SummarySummaryyy
loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j
the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference
(ISI)loz The main advantage of the OFDM transmission technique is its
high performance even in frequency selective channels
loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems
36
SynchronizationSynchronization
Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg
38 Synchronization2010722
Introduction (13)Introduction (13)( )( )
loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods
loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead
loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the
cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of
computational complexity therefore it may be not be available in short-burst wireless communication
39 Synchronization2010722
Introduction (23)Introduction (23)( )( )
loz Transcevier
40 Synchronization2010722
Introduction (33)Introduction (33)( )( )
loz Synchronization erroryloz Symbol timing offset (TO)
loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)
loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect
41 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg
42 Synchronization2010722
Signal model (14)Signal model (14)g ( )g ( )
loz Time domain sample p1
2
0
1 Nj nk N
n kk
x X eN
πminus
minus= sum
Xk the frequency domain data at the k-th subcarrier0kN =
loz The multipath fading channel
( ) ( ) ( )1L
h t h t δminus
sum
loz Received time domain signal
( ) ( ) ( )0
l ll
h t h tτ τ δ τ τ=
= minussum
loz Received time-domain signal
( ) ( )1
L
n n ly h n l x w nminus
minus= +sum43
( ) ( )0l=sum
Synchronization2010722
Signal model (24)Signal model (24)g ( )g ( )
loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting
12
0
Nj kn N
k nn
Y y e πminus
minus
=
= sum
( )1 1
2
0
N Lj kn N
n l kn l
h n l x e Wπminus minus
minusminus
=
= +sumsum
loz The equation above could be expressed in matrix form
0n l
loz The equation above could be expressed in matrix form
= +Y HX W
44 Synchronization2010722
Signal model (34)Signal model (34)g ( )g ( )
H= = = = +Y Fy FCx FCF X HX W
loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal
y
H=H FCFcirculant matrix leading to be a diagonal matrix
=H FCF
( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0
h h L L hh h
⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥
( )( ) ( )
( ) ( )
1 10 1 0 0i
h Lh L h N L
⎢ ⎥⎢ ⎥minus minus⎢ ⎥
equiv= minus minus⎢ ⎥⎢ ⎥
C( ) ( )
( ) ( )
0 1 1 100
0 0 1 1 0
h L h N L
h N L L h N
⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
45
( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦
Synchronization2010722
Signal model (44)Signal model (44)g ( )g ( )
loz In a time-invariant channel the channel matrix H would be a diagonal matrix
k k k kY H X W= +
( )1 1
2
0 0
N Lj kl N
kn l
H h n l e πminus minus
minus
= =
= sumsum
loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI
k k k k kY H X I Wα= + +k k k k k
( ) ( )1 1 1
2 2 N N L
j m k Nj ml Nk mI X h n l e e ππ
minus minus minusminusminus⎛ ⎞
= ⎜ ⎟⎝ ⎠
sum sum sumICI
46
0 0 0m n lm k= = =ne
⎝ ⎠
Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
47 Synchronization2010722
Impact of timing offset Impact of timing offset p gp g
loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively
( )I f s
Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ
( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr
2 fj k Nk k k kY X H e Wπδminus= +
loz For integer part of timing offset k k k kY X H e W+
I sTδ ( ) 0g IN L δminus minus le leFFT
2 Ij k NY X H e Wπδminus= +
( ) ( ) 2 I sFFT
j f TI sh t T H f e π δδ minusminus rarr
48
k k k kY X H e W= +
Synchronization2010722
Impact of timing offset derivationImpact of timing offset derivationp gp g
49 Synchronization2010722
Discussion of symbol boundaryDiscussion of symbol boundaryy yy y
50 Synchronization2010722
Impact of timing offsetImpact of timing offsetp gp g
loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI
2 Ij k NIk k k k
NY X H e ISI W
Nπδδ minusminus
= + +
loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk
e e
p kloz Phase Shift can be resolved by Differential Encoding
Decoding or carrier recovery with pilots
51 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
52 Synchronization2010722
Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )
53 Synchronization2010722
Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )
( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s
i n t i N N T N T nTy y t e
= + + +=
( ) ( )( ) ( ) ( ) 12
sin
sin
g gI f f
I I
i N N N Nj jf N Ni k i k k
f
Y X H e eN
π ε ε π ε
ε ε
πεπε
+ + minus+
minus minus=⎛ ⎞⎜ ⎟
Faded signal attenuated and rotated by CFO
( )( ) ( ) ( ) ( )11 2
sin
sin g gI f I f
i N N N NN j j l kI f
NN
l k π ε ε π ε επ ε ε + + minusminus + + + minus
⎜ ⎟⎝ ⎠
+ + minussum
( )( )( )
( ) ( )2
0
sin
I f I f
I
j j l kI f N Ni l l
l l k I f
X H e el k
NN
π ε ε π ε ε
ε π ε ε
+ + +
= ne minus
+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠
sum
i k
N
W
⎜ ⎟⎝ ⎠
+ Inter-Carrier Interference (ICI)
54 Synchronization2010722
Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )
loz An easier representation of received signal with only p g yfractional CFO is
1N
( ) ( )1
0
0N
k k k l l kl l k
Y H X C H X C l k Wminus
= ne
= + minus +⎡ ⎤⎣ ⎦sum
where the ICI coefficient is given by
( )( )( ) ( )
sin 1exp 1fl k
C l k j l kπ ε
π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )
( )exp 1sin
f
f
C l k j l kNN l k
N
π επ ε
minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠
55 Synchronization2010722
Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )
loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality
loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift
56 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
57 Synchronization2010722
TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )
loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq
58 Synchronization2010722
TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )
loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below
loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and
rN2 can be expressed by
0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+
( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +
59
( )
Synchronization2010722
TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )
loz Without CFO the first half is identical to the second half (in time order)
loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below
( ) 12 in discretesft T f NT
NTφ π π π ε πε
⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟
⎝ ⎠
helliphellip
sNT⎝ ⎠
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)
60
Length=N2
Synchronization2010722
TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )
loz Received signal without CFOg0
0 0
jr a e θ=jθ 2 0
2 2 0
Nj j
N Nr a e a e
θ θ= =
loz Received signal with CFO0 0
0 0
j jr a e eθ φ= Extra phase rotation due to CFO0 0
2 2 20
2 2 0
N N Nj j jj
N Nr a e e a e e
θ φ φθ= =
due to CFO
Phase difference 2 2 0N N
( )2 02 20 2 0 0
Nj j T fN
r r a e a eφ φ πminus Δ= =
which contains the information about CFO
61
0 2 0 0Nr r a e a e
Synchronization2010722
TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )
loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be
loz Note that d is a time index corresponding to the first( ) ( ) 2 1
20
Nd m d m Nm
P d r rminus
+ + +== sum
loz Note that d is a time index corresponding to the first sample in a window of N samples
loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by
( )22 1
20
Nd m NR d rminus
+ += sumloz A timing metric can be defined as
))(()(
)( 2
2
dRdP
dM =
( ) 20 d m Nm + +=sum
62
))(( dR
Synchronization2010722
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
OFDM System ModelOFDM System Model
loz The serial data elements spaced by are grouped and tΔ
yy
p y g pused to modulate N carriers Thus they are frequency division multiplexedp
loz The signaling interval is then increased to which tNΔmakes the system less susceptible to channel delay spread impairments
Small-scale fading(Based on multipath time delay spread)
Flat Fading1 BW of signal lt BW of channel
Frequency Selective Fading1 BW of signal gt BW of channel
2 Delay spread lt Symbol preiod 2 Delay spread gt Symbol preiod
12
OrthogonalityOrthogonalityg yg y
loz Consider a set of transmitted carriers as follows
(2) 1 1 0for )( 2 0
minus==⎟⎠⎞
⎜⎝⎛
Δ+
Nnett
tNnfj
n
πψ n
⎨⎧ =minus
intqpab
dtttb for )(
)()( ψψ⎩⎨ Δ=minusne
=int tNabqpdttt
a qp )( and for 0 )()( ψψ
13
OrthogonalityOrthogonality
b tqpjb
intint)(2π
g yg y
dtedtttajbj
b
atN
qpjb
a qp = Δminus
intint )()(
)(2)(2
)(2ψψπ
tNqpjee tN
qpjtN
qpj
Δminusminus
=Δ
minusΔ
minus
)(2
)(2)(2
π
ππ
ee
tNqpjba
tNqpj
tNbqpj
⎟⎟⎞
⎜⎜⎛minus
Δ
minusΔ
minusΔ
minus1
)(2)(1)(2)(2
π
ππ
tNqpj Δminus
⎟⎠
⎜⎝=
)(2πtNabqp Δ=minusne= )( and for 0
14
OrthogonalityOrthogonalityg yg y
tNΔ
15
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFTby using IDFTby using IDFT
loz According to the structure of Tx it must use Ng oscillators That increases the hardware complexity
loz The equivalent method is using IDFT (IFFT)
16
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz In general each carrier can be expressed as
by using IDFTby using IDFT
g p
( ) (6) )()( )(2 ttfjcc
ccetAtS φπ +=
loz We assume that there are N carriers in the OFDM signal Th th t t l l i l S (t) b t d bThen the total complex signal Ss(t) can be represented by
( ) (7))(1)(1
)(2etAtSN
ttfj nn φπ= summinus
+( )
where
(7) )()(
0
0
fnff
etAN
tSn
ns
Δ+=
= sum=
liihffrequency carrier phase amplitude are )( )( and
where 0
fttAfnff
nnn
n
φΔ+
lyrespective carrierth -of n17
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz Then we sample the signal at a sampling frequency tΔ1
by using IDFTby using IDFT
p g p g q y and An(t) and φn(t) becomes
(8))(t = φφ(9) )((8) )(
nn
nn
AtAt
== φφ
( ) (10) 1)(1
0
)(2 0summinus
=
+ΔΔ+=ΔN
n
tkfnfjns
neAN
tkS φπ
loz Then the sampled signal can be expressed as0=nN
( )( ) (11) 1)(1
0
22 0summinus
=
ΔΔ+Δ sdot=ΔN
n
tfnkjtkfjns eeA
NtkS n πφπ
0n
18
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz The inverse discrete Fourier transform (IDFT) is defined
by using IDFTby using IDFT
( )as the following
1 21
NnkjN
πsumminus
(12) )(1)( 2
0
Nnkj
nefnF
Ntkf πsum
=
Δ=Δ
loz Comparing eq(11) and eq(12) the condition must be satisfied in order to make eq(11) an inverse Fouriersatisfied in order to make eq(11) an inverse Fourier transform relationship
1 (13) 1tN
fΔ
=Δ
19
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz If eq(13) is satisfied
by using IDFTby using IDFT
q ( ) loz is the frequency domain signal loz is the time domain signal
( )ntkfjneA φπ +Δ02
)( tkSs Δ gloz is the sub-channel spacingloz is the symbol duration in each sub-channel
)(s
fΔtNΔloz s e sy bo du o e c sub c e
loz This outcome is the same as the result obtained in the
N
loz This outcome is the same as the result obtained in the system of Figure 1 Therefore IDFT can be used to generate an OFDM transmission signalgenerate an OFDM transmission signal
20
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
( )tf02cos π
( )0a
by using IDFTby using IDFT
( )nd ( )tDI( )tf02sin π
( )0a
( )0b
bull
tf s Δ=
1 ( ) ( ) njbna +
( )nd
tNΔffnffn Δ
=Δ+=1 0
( )tDChannelInput
( )tf N 12cos minusπ
( )1minusNa
bullbull
tNΔ
( )tf N 12sin minusπ
( )1minusNb
( )0d
( )nd ( )tDChannelInput
bull bull
( )( )1d
( )2d
tf s Δ=
1 ( ) ( ) njbna +
fNΔffnffn Δ
=Δ+=1 0
ChannelInput
bullbullbullbull
bullbullbullbull
( )1minusNd
21
Frequency Error Results in ICIFrequency Error Results in ICIq yq y
22
Synchronization Error Results in ICISynchronization Error Results in ICIyy
Not Orthogonal Any More
23
Cyclic PrefixCyclic Prefixyy
loz In multipath channel delayed replicas of previous OFDM signal lead to ISI between successive OFDM signals
loz Solution Insert a guard interval between successive OFDMloz Solution Insert a guard interval between successive OFDM signals
24
Cyclic PrefixCyclic Prefix
loz Guard interval leads to intercarrier interference (ICI) in OFDM d d l i
yy
demodulation
loz In DFT interval difference between two subcarriers does not maintain integer number of cycles loss of orthogonality
loz Delayed version of subcarrier 2 causes ICI in the process of d d l ti b i 1demodulating subcarrier 1
25
Cyclic PrefixCyclic Prefixloz Cyclic prefix (CP) A copy of the last part of OFDM signal is
attached to the front of itself
yy
attached to the front of itself
[ ]0d
[ ]1d
[ ]0D[ ]1D
[ ]nd[ ]kD~
symbols dataInput
bullbullbullbull
[ ]1d
[ ]2d
[ ][ ]2D
bullbullbullbullbull
y
bullbullbullbullbull
[ ]1minusNd [ ]1minusND
[ ]gNND minus
bullbullbull
26
Cyclic PrefixCyclic Prefix
loz All delayed replicas of subcarriers always have an
yy
y p yinteger number of cycles within DFT interval no ICI
27
Cyclic PrefixCyclic Prefix
loz Linear convolution vs circular convolution
yy
N
28
Cyclic PrefixCyclic Prefix
loz Channel effect with cyclic prefix
yy
Signal after removed CP
29
Cyclic PrefixCyclic Prefix
loz Time-Domain Explanation
yy
p
30
Cyclic PrefixCyclic Prefixyy
31
Cyclic PrefixCyclic Prefix
loz Spectrum of channel response with length (smaller [ ]nh hL
yy
than )h
gN[ ] nhFFTHk =
loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr
loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=
where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier
[ ] [ ] [ ]N
Notimes
[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce
kk H
X =rArrcomplexity of channel equalizationrdquo
32
Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))
bull bull( )( )nr~Symbols
dataOutput
bullbullbull
bullbullbull( )nr
bullbull
bullbull
33
Cyclic PrefixCyclic Prefix
loz One of the most important reasons to do OFDM is the
yy
pefficient way it deals with multipath delay spread
loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)
34
Bandwidth EfficiencyBandwidth Efficiencyyy
loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )
loz The diagram for bandwidth efficiency of OFDM system is shown below
35
SummarySummaryyy
loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j
the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference
(ISI)loz The main advantage of the OFDM transmission technique is its
high performance even in frequency selective channels
loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems
36
SynchronizationSynchronization
Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg
38 Synchronization2010722
Introduction (13)Introduction (13)( )( )
loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods
loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead
loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the
cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of
computational complexity therefore it may be not be available in short-burst wireless communication
39 Synchronization2010722
Introduction (23)Introduction (23)( )( )
loz Transcevier
40 Synchronization2010722
Introduction (33)Introduction (33)( )( )
loz Synchronization erroryloz Symbol timing offset (TO)
loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)
loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect
41 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg
42 Synchronization2010722
Signal model (14)Signal model (14)g ( )g ( )
loz Time domain sample p1
2
0
1 Nj nk N
n kk
x X eN
πminus
minus= sum
Xk the frequency domain data at the k-th subcarrier0kN =
loz The multipath fading channel
( ) ( ) ( )1L
h t h t δminus
sum
loz Received time domain signal
( ) ( ) ( )0
l ll
h t h tτ τ δ τ τ=
= minussum
loz Received time-domain signal
( ) ( )1
L
n n ly h n l x w nminus
minus= +sum43
( ) ( )0l=sum
Synchronization2010722
Signal model (24)Signal model (24)g ( )g ( )
loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting
12
0
Nj kn N
k nn
Y y e πminus
minus
=
= sum
( )1 1
2
0
N Lj kn N
n l kn l
h n l x e Wπminus minus
minusminus
=
= +sumsum
loz The equation above could be expressed in matrix form
0n l
loz The equation above could be expressed in matrix form
= +Y HX W
44 Synchronization2010722
Signal model (34)Signal model (34)g ( )g ( )
H= = = = +Y Fy FCx FCF X HX W
loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal
y
H=H FCFcirculant matrix leading to be a diagonal matrix
=H FCF
( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0
h h L L hh h
⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥
( )( ) ( )
( ) ( )
1 10 1 0 0i
h Lh L h N L
⎢ ⎥⎢ ⎥minus minus⎢ ⎥
equiv= minus minus⎢ ⎥⎢ ⎥
C( ) ( )
( ) ( )
0 1 1 100
0 0 1 1 0
h L h N L
h N L L h N
⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
45
( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦
Synchronization2010722
Signal model (44)Signal model (44)g ( )g ( )
loz In a time-invariant channel the channel matrix H would be a diagonal matrix
k k k kY H X W= +
( )1 1
2
0 0
N Lj kl N
kn l
H h n l e πminus minus
minus
= =
= sumsum
loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI
k k k k kY H X I Wα= + +k k k k k
( ) ( )1 1 1
2 2 N N L
j m k Nj ml Nk mI X h n l e e ππ
minus minus minusminusminus⎛ ⎞
= ⎜ ⎟⎝ ⎠
sum sum sumICI
46
0 0 0m n lm k= = =ne
⎝ ⎠
Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
47 Synchronization2010722
Impact of timing offset Impact of timing offset p gp g
loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively
( )I f s
Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ
( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr
2 fj k Nk k k kY X H e Wπδminus= +
loz For integer part of timing offset k k k kY X H e W+
I sTδ ( ) 0g IN L δminus minus le leFFT
2 Ij k NY X H e Wπδminus= +
( ) ( ) 2 I sFFT
j f TI sh t T H f e π δδ minusminus rarr
48
k k k kY X H e W= +
Synchronization2010722
Impact of timing offset derivationImpact of timing offset derivationp gp g
49 Synchronization2010722
Discussion of symbol boundaryDiscussion of symbol boundaryy yy y
50 Synchronization2010722
Impact of timing offsetImpact of timing offsetp gp g
loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI
2 Ij k NIk k k k
NY X H e ISI W
Nπδδ minusminus
= + +
loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk
e e
p kloz Phase Shift can be resolved by Differential Encoding
Decoding or carrier recovery with pilots
51 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
52 Synchronization2010722
Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )
53 Synchronization2010722
Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )
( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s
i n t i N N T N T nTy y t e
= + + +=
( ) ( )( ) ( ) ( ) 12
sin
sin
g gI f f
I I
i N N N Nj jf N Ni k i k k
f
Y X H e eN
π ε ε π ε
ε ε
πεπε
+ + minus+
minus minus=⎛ ⎞⎜ ⎟
Faded signal attenuated and rotated by CFO
( )( ) ( ) ( ) ( )11 2
sin
sin g gI f I f
i N N N NN j j l kI f
NN
l k π ε ε π ε επ ε ε + + minusminus + + + minus
⎜ ⎟⎝ ⎠
+ + minussum
( )( )( )
( ) ( )2
0
sin
I f I f
I
j j l kI f N Ni l l
l l k I f
X H e el k
NN
π ε ε π ε ε
ε π ε ε
+ + +
= ne minus
+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠
sum
i k
N
W
⎜ ⎟⎝ ⎠
+ Inter-Carrier Interference (ICI)
54 Synchronization2010722
Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )
loz An easier representation of received signal with only p g yfractional CFO is
1N
( ) ( )1
0
0N
k k k l l kl l k
Y H X C H X C l k Wminus
= ne
= + minus +⎡ ⎤⎣ ⎦sum
where the ICI coefficient is given by
( )( )( ) ( )
sin 1exp 1fl k
C l k j l kπ ε
π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )
( )exp 1sin
f
f
C l k j l kNN l k
N
π επ ε
minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠
55 Synchronization2010722
Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )
loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality
loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift
56 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
57 Synchronization2010722
TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )
loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq
58 Synchronization2010722
TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )
loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below
loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and
rN2 can be expressed by
0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+
( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +
59
( )
Synchronization2010722
TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )
loz Without CFO the first half is identical to the second half (in time order)
loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below
( ) 12 in discretesft T f NT
NTφ π π π ε πε
⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟
⎝ ⎠
helliphellip
sNT⎝ ⎠
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)
60
Length=N2
Synchronization2010722
TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )
loz Received signal without CFOg0
0 0
jr a e θ=jθ 2 0
2 2 0
Nj j
N Nr a e a e
θ θ= =
loz Received signal with CFO0 0
0 0
j jr a e eθ φ= Extra phase rotation due to CFO0 0
2 2 20
2 2 0
N N Nj j jj
N Nr a e e a e e
θ φ φθ= =
due to CFO
Phase difference 2 2 0N N
( )2 02 20 2 0 0
Nj j T fN
r r a e a eφ φ πminus Δ= =
which contains the information about CFO
61
0 2 0 0Nr r a e a e
Synchronization2010722
TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )
loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be
loz Note that d is a time index corresponding to the first( ) ( ) 2 1
20
Nd m d m Nm
P d r rminus
+ + +== sum
loz Note that d is a time index corresponding to the first sample in a window of N samples
loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by
( )22 1
20
Nd m NR d rminus
+ += sumloz A timing metric can be defined as
))(()(
)( 2
2
dRdP
dM =
( ) 20 d m Nm + +=sum
62
))(( dR
Synchronization2010722
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
OrthogonalityOrthogonalityg yg y
loz Consider a set of transmitted carriers as follows
(2) 1 1 0for )( 2 0
minus==⎟⎠⎞
⎜⎝⎛
Δ+
Nnett
tNnfj
n
πψ n
⎨⎧ =minus
intqpab
dtttb for )(
)()( ψψ⎩⎨ Δ=minusne
=int tNabqpdttt
a qp )( and for 0 )()( ψψ
13
OrthogonalityOrthogonality
b tqpjb
intint)(2π
g yg y
dtedtttajbj
b
atN
qpjb
a qp = Δminus
intint )()(
)(2)(2
)(2ψψπ
tNqpjee tN
qpjtN
qpj
Δminusminus
=Δ
minusΔ
minus
)(2
)(2)(2
π
ππ
ee
tNqpjba
tNqpj
tNbqpj
⎟⎟⎞
⎜⎜⎛minus
Δ
minusΔ
minusΔ
minus1
)(2)(1)(2)(2
π
ππ
tNqpj Δminus
⎟⎠
⎜⎝=
)(2πtNabqp Δ=minusne= )( and for 0
14
OrthogonalityOrthogonalityg yg y
tNΔ
15
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFTby using IDFTby using IDFT
loz According to the structure of Tx it must use Ng oscillators That increases the hardware complexity
loz The equivalent method is using IDFT (IFFT)
16
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz In general each carrier can be expressed as
by using IDFTby using IDFT
g p
( ) (6) )()( )(2 ttfjcc
ccetAtS φπ +=
loz We assume that there are N carriers in the OFDM signal Th th t t l l i l S (t) b t d bThen the total complex signal Ss(t) can be represented by
( ) (7))(1)(1
)(2etAtSN
ttfj nn φπ= summinus
+( )
where
(7) )()(
0
0
fnff
etAN
tSn
ns
Δ+=
= sum=
liihffrequency carrier phase amplitude are )( )( and
where 0
fttAfnff
nnn
n
φΔ+
lyrespective carrierth -of n17
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz Then we sample the signal at a sampling frequency tΔ1
by using IDFTby using IDFT
p g p g q y and An(t) and φn(t) becomes
(8))(t = φφ(9) )((8) )(
nn
nn
AtAt
== φφ
( ) (10) 1)(1
0
)(2 0summinus
=
+ΔΔ+=ΔN
n
tkfnfjns
neAN
tkS φπ
loz Then the sampled signal can be expressed as0=nN
( )( ) (11) 1)(1
0
22 0summinus
=
ΔΔ+Δ sdot=ΔN
n
tfnkjtkfjns eeA
NtkS n πφπ
0n
18
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz The inverse discrete Fourier transform (IDFT) is defined
by using IDFTby using IDFT
( )as the following
1 21
NnkjN
πsumminus
(12) )(1)( 2
0
Nnkj
nefnF
Ntkf πsum
=
Δ=Δ
loz Comparing eq(11) and eq(12) the condition must be satisfied in order to make eq(11) an inverse Fouriersatisfied in order to make eq(11) an inverse Fourier transform relationship
1 (13) 1tN
fΔ
=Δ
19
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz If eq(13) is satisfied
by using IDFTby using IDFT
q ( ) loz is the frequency domain signal loz is the time domain signal
( )ntkfjneA φπ +Δ02
)( tkSs Δ gloz is the sub-channel spacingloz is the symbol duration in each sub-channel
)(s
fΔtNΔloz s e sy bo du o e c sub c e
loz This outcome is the same as the result obtained in the
N
loz This outcome is the same as the result obtained in the system of Figure 1 Therefore IDFT can be used to generate an OFDM transmission signalgenerate an OFDM transmission signal
20
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
( )tf02cos π
( )0a
by using IDFTby using IDFT
( )nd ( )tDI( )tf02sin π
( )0a
( )0b
bull
tf s Δ=
1 ( ) ( ) njbna +
( )nd
tNΔffnffn Δ
=Δ+=1 0
( )tDChannelInput
( )tf N 12cos minusπ
( )1minusNa
bullbull
tNΔ
( )tf N 12sin minusπ
( )1minusNb
( )0d
( )nd ( )tDChannelInput
bull bull
( )( )1d
( )2d
tf s Δ=
1 ( ) ( ) njbna +
fNΔffnffn Δ
=Δ+=1 0
ChannelInput
bullbullbullbull
bullbullbullbull
( )1minusNd
21
Frequency Error Results in ICIFrequency Error Results in ICIq yq y
22
Synchronization Error Results in ICISynchronization Error Results in ICIyy
Not Orthogonal Any More
23
Cyclic PrefixCyclic Prefixyy
loz In multipath channel delayed replicas of previous OFDM signal lead to ISI between successive OFDM signals
loz Solution Insert a guard interval between successive OFDMloz Solution Insert a guard interval between successive OFDM signals
24
Cyclic PrefixCyclic Prefix
loz Guard interval leads to intercarrier interference (ICI) in OFDM d d l i
yy
demodulation
loz In DFT interval difference between two subcarriers does not maintain integer number of cycles loss of orthogonality
loz Delayed version of subcarrier 2 causes ICI in the process of d d l ti b i 1demodulating subcarrier 1
25
Cyclic PrefixCyclic Prefixloz Cyclic prefix (CP) A copy of the last part of OFDM signal is
attached to the front of itself
yy
attached to the front of itself
[ ]0d
[ ]1d
[ ]0D[ ]1D
[ ]nd[ ]kD~
symbols dataInput
bullbullbullbull
[ ]1d
[ ]2d
[ ][ ]2D
bullbullbullbullbull
y
bullbullbullbullbull
[ ]1minusNd [ ]1minusND
[ ]gNND minus
bullbullbull
26
Cyclic PrefixCyclic Prefix
loz All delayed replicas of subcarriers always have an
yy
y p yinteger number of cycles within DFT interval no ICI
27
Cyclic PrefixCyclic Prefix
loz Linear convolution vs circular convolution
yy
N
28
Cyclic PrefixCyclic Prefix
loz Channel effect with cyclic prefix
yy
Signal after removed CP
29
Cyclic PrefixCyclic Prefix
loz Time-Domain Explanation
yy
p
30
Cyclic PrefixCyclic Prefixyy
31
Cyclic PrefixCyclic Prefix
loz Spectrum of channel response with length (smaller [ ]nh hL
yy
than )h
gN[ ] nhFFTHk =
loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr
loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=
where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier
[ ] [ ] [ ]N
Notimes
[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce
kk H
X =rArrcomplexity of channel equalizationrdquo
32
Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))
bull bull( )( )nr~Symbols
dataOutput
bullbullbull
bullbullbull( )nr
bullbull
bullbull
33
Cyclic PrefixCyclic Prefix
loz One of the most important reasons to do OFDM is the
yy
pefficient way it deals with multipath delay spread
loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)
34
Bandwidth EfficiencyBandwidth Efficiencyyy
loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )
loz The diagram for bandwidth efficiency of OFDM system is shown below
35
SummarySummaryyy
loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j
the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference
(ISI)loz The main advantage of the OFDM transmission technique is its
high performance even in frequency selective channels
loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems
36
SynchronizationSynchronization
Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg
38 Synchronization2010722
Introduction (13)Introduction (13)( )( )
loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods
loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead
loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the
cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of
computational complexity therefore it may be not be available in short-burst wireless communication
39 Synchronization2010722
Introduction (23)Introduction (23)( )( )
loz Transcevier
40 Synchronization2010722
Introduction (33)Introduction (33)( )( )
loz Synchronization erroryloz Symbol timing offset (TO)
loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)
loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect
41 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg
42 Synchronization2010722
Signal model (14)Signal model (14)g ( )g ( )
loz Time domain sample p1
2
0
1 Nj nk N
n kk
x X eN
πminus
minus= sum
Xk the frequency domain data at the k-th subcarrier0kN =
loz The multipath fading channel
( ) ( ) ( )1L
h t h t δminus
sum
loz Received time domain signal
( ) ( ) ( )0
l ll
h t h tτ τ δ τ τ=
= minussum
loz Received time-domain signal
( ) ( )1
L
n n ly h n l x w nminus
minus= +sum43
( ) ( )0l=sum
Synchronization2010722
Signal model (24)Signal model (24)g ( )g ( )
loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting
12
0
Nj kn N
k nn
Y y e πminus
minus
=
= sum
( )1 1
2
0
N Lj kn N
n l kn l
h n l x e Wπminus minus
minusminus
=
= +sumsum
loz The equation above could be expressed in matrix form
0n l
loz The equation above could be expressed in matrix form
= +Y HX W
44 Synchronization2010722
Signal model (34)Signal model (34)g ( )g ( )
H= = = = +Y Fy FCx FCF X HX W
loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal
y
H=H FCFcirculant matrix leading to be a diagonal matrix
=H FCF
( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0
h h L L hh h
⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥
( )( ) ( )
( ) ( )
1 10 1 0 0i
h Lh L h N L
⎢ ⎥⎢ ⎥minus minus⎢ ⎥
equiv= minus minus⎢ ⎥⎢ ⎥
C( ) ( )
( ) ( )
0 1 1 100
0 0 1 1 0
h L h N L
h N L L h N
⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
45
( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦
Synchronization2010722
Signal model (44)Signal model (44)g ( )g ( )
loz In a time-invariant channel the channel matrix H would be a diagonal matrix
k k k kY H X W= +
( )1 1
2
0 0
N Lj kl N
kn l
H h n l e πminus minus
minus
= =
= sumsum
loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI
k k k k kY H X I Wα= + +k k k k k
( ) ( )1 1 1
2 2 N N L
j m k Nj ml Nk mI X h n l e e ππ
minus minus minusminusminus⎛ ⎞
= ⎜ ⎟⎝ ⎠
sum sum sumICI
46
0 0 0m n lm k= = =ne
⎝ ⎠
Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
47 Synchronization2010722
Impact of timing offset Impact of timing offset p gp g
loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively
( )I f s
Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ
( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr
2 fj k Nk k k kY X H e Wπδminus= +
loz For integer part of timing offset k k k kY X H e W+
I sTδ ( ) 0g IN L δminus minus le leFFT
2 Ij k NY X H e Wπδminus= +
( ) ( ) 2 I sFFT
j f TI sh t T H f e π δδ minusminus rarr
48
k k k kY X H e W= +
Synchronization2010722
Impact of timing offset derivationImpact of timing offset derivationp gp g
49 Synchronization2010722
Discussion of symbol boundaryDiscussion of symbol boundaryy yy y
50 Synchronization2010722
Impact of timing offsetImpact of timing offsetp gp g
loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI
2 Ij k NIk k k k
NY X H e ISI W
Nπδδ minusminus
= + +
loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk
e e
p kloz Phase Shift can be resolved by Differential Encoding
Decoding or carrier recovery with pilots
51 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
52 Synchronization2010722
Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )
53 Synchronization2010722
Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )
( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s
i n t i N N T N T nTy y t e
= + + +=
( ) ( )( ) ( ) ( ) 12
sin
sin
g gI f f
I I
i N N N Nj jf N Ni k i k k
f
Y X H e eN
π ε ε π ε
ε ε
πεπε
+ + minus+
minus minus=⎛ ⎞⎜ ⎟
Faded signal attenuated and rotated by CFO
( )( ) ( ) ( ) ( )11 2
sin
sin g gI f I f
i N N N NN j j l kI f
NN
l k π ε ε π ε επ ε ε + + minusminus + + + minus
⎜ ⎟⎝ ⎠
+ + minussum
( )( )( )
( ) ( )2
0
sin
I f I f
I
j j l kI f N Ni l l
l l k I f
X H e el k
NN
π ε ε π ε ε
ε π ε ε
+ + +
= ne minus
+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠
sum
i k
N
W
⎜ ⎟⎝ ⎠
+ Inter-Carrier Interference (ICI)
54 Synchronization2010722
Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )
loz An easier representation of received signal with only p g yfractional CFO is
1N
( ) ( )1
0
0N
k k k l l kl l k
Y H X C H X C l k Wminus
= ne
= + minus +⎡ ⎤⎣ ⎦sum
where the ICI coefficient is given by
( )( )( ) ( )
sin 1exp 1fl k
C l k j l kπ ε
π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )
( )exp 1sin
f
f
C l k j l kNN l k
N
π επ ε
minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠
55 Synchronization2010722
Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )
loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality
loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift
56 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
57 Synchronization2010722
TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )
loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq
58 Synchronization2010722
TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )
loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below
loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and
rN2 can be expressed by
0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+
( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +
59
( )
Synchronization2010722
TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )
loz Without CFO the first half is identical to the second half (in time order)
loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below
( ) 12 in discretesft T f NT
NTφ π π π ε πε
⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟
⎝ ⎠
helliphellip
sNT⎝ ⎠
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)
60
Length=N2
Synchronization2010722
TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )
loz Received signal without CFOg0
0 0
jr a e θ=jθ 2 0
2 2 0
Nj j
N Nr a e a e
θ θ= =
loz Received signal with CFO0 0
0 0
j jr a e eθ φ= Extra phase rotation due to CFO0 0
2 2 20
2 2 0
N N Nj j jj
N Nr a e e a e e
θ φ φθ= =
due to CFO
Phase difference 2 2 0N N
( )2 02 20 2 0 0
Nj j T fN
r r a e a eφ φ πminus Δ= =
which contains the information about CFO
61
0 2 0 0Nr r a e a e
Synchronization2010722
TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )
loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be
loz Note that d is a time index corresponding to the first( ) ( ) 2 1
20
Nd m d m Nm
P d r rminus
+ + +== sum
loz Note that d is a time index corresponding to the first sample in a window of N samples
loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by
( )22 1
20
Nd m NR d rminus
+ += sumloz A timing metric can be defined as
))(()(
)( 2
2
dRdP
dM =
( ) 20 d m Nm + +=sum
62
))(( dR
Synchronization2010722
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
OrthogonalityOrthogonality
b tqpjb
intint)(2π
g yg y
dtedtttajbj
b
atN
qpjb
a qp = Δminus
intint )()(
)(2)(2
)(2ψψπ
tNqpjee tN
qpjtN
qpj
Δminusminus
=Δ
minusΔ
minus
)(2
)(2)(2
π
ππ
ee
tNqpjba
tNqpj
tNbqpj
⎟⎟⎞
⎜⎜⎛minus
Δ
minusΔ
minusΔ
minus1
)(2)(1)(2)(2
π
ππ
tNqpj Δminus
⎟⎠
⎜⎝=
)(2πtNabqp Δ=minusne= )( and for 0
14
OrthogonalityOrthogonalityg yg y
tNΔ
15
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFTby using IDFTby using IDFT
loz According to the structure of Tx it must use Ng oscillators That increases the hardware complexity
loz The equivalent method is using IDFT (IFFT)
16
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz In general each carrier can be expressed as
by using IDFTby using IDFT
g p
( ) (6) )()( )(2 ttfjcc
ccetAtS φπ +=
loz We assume that there are N carriers in the OFDM signal Th th t t l l i l S (t) b t d bThen the total complex signal Ss(t) can be represented by
( ) (7))(1)(1
)(2etAtSN
ttfj nn φπ= summinus
+( )
where
(7) )()(
0
0
fnff
etAN
tSn
ns
Δ+=
= sum=
liihffrequency carrier phase amplitude are )( )( and
where 0
fttAfnff
nnn
n
φΔ+
lyrespective carrierth -of n17
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz Then we sample the signal at a sampling frequency tΔ1
by using IDFTby using IDFT
p g p g q y and An(t) and φn(t) becomes
(8))(t = φφ(9) )((8) )(
nn
nn
AtAt
== φφ
( ) (10) 1)(1
0
)(2 0summinus
=
+ΔΔ+=ΔN
n
tkfnfjns
neAN
tkS φπ
loz Then the sampled signal can be expressed as0=nN
( )( ) (11) 1)(1
0
22 0summinus
=
ΔΔ+Δ sdot=ΔN
n
tfnkjtkfjns eeA
NtkS n πφπ
0n
18
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz The inverse discrete Fourier transform (IDFT) is defined
by using IDFTby using IDFT
( )as the following
1 21
NnkjN
πsumminus
(12) )(1)( 2
0
Nnkj
nefnF
Ntkf πsum
=
Δ=Δ
loz Comparing eq(11) and eq(12) the condition must be satisfied in order to make eq(11) an inverse Fouriersatisfied in order to make eq(11) an inverse Fourier transform relationship
1 (13) 1tN
fΔ
=Δ
19
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz If eq(13) is satisfied
by using IDFTby using IDFT
q ( ) loz is the frequency domain signal loz is the time domain signal
( )ntkfjneA φπ +Δ02
)( tkSs Δ gloz is the sub-channel spacingloz is the symbol duration in each sub-channel
)(s
fΔtNΔloz s e sy bo du o e c sub c e
loz This outcome is the same as the result obtained in the
N
loz This outcome is the same as the result obtained in the system of Figure 1 Therefore IDFT can be used to generate an OFDM transmission signalgenerate an OFDM transmission signal
20
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
( )tf02cos π
( )0a
by using IDFTby using IDFT
( )nd ( )tDI( )tf02sin π
( )0a
( )0b
bull
tf s Δ=
1 ( ) ( ) njbna +
( )nd
tNΔffnffn Δ
=Δ+=1 0
( )tDChannelInput
( )tf N 12cos minusπ
( )1minusNa
bullbull
tNΔ
( )tf N 12sin minusπ
( )1minusNb
( )0d
( )nd ( )tDChannelInput
bull bull
( )( )1d
( )2d
tf s Δ=
1 ( ) ( ) njbna +
fNΔffnffn Δ
=Δ+=1 0
ChannelInput
bullbullbullbull
bullbullbullbull
( )1minusNd
21
Frequency Error Results in ICIFrequency Error Results in ICIq yq y
22
Synchronization Error Results in ICISynchronization Error Results in ICIyy
Not Orthogonal Any More
23
Cyclic PrefixCyclic Prefixyy
loz In multipath channel delayed replicas of previous OFDM signal lead to ISI between successive OFDM signals
loz Solution Insert a guard interval between successive OFDMloz Solution Insert a guard interval between successive OFDM signals
24
Cyclic PrefixCyclic Prefix
loz Guard interval leads to intercarrier interference (ICI) in OFDM d d l i
yy
demodulation
loz In DFT interval difference between two subcarriers does not maintain integer number of cycles loss of orthogonality
loz Delayed version of subcarrier 2 causes ICI in the process of d d l ti b i 1demodulating subcarrier 1
25
Cyclic PrefixCyclic Prefixloz Cyclic prefix (CP) A copy of the last part of OFDM signal is
attached to the front of itself
yy
attached to the front of itself
[ ]0d
[ ]1d
[ ]0D[ ]1D
[ ]nd[ ]kD~
symbols dataInput
bullbullbullbull
[ ]1d
[ ]2d
[ ][ ]2D
bullbullbullbullbull
y
bullbullbullbullbull
[ ]1minusNd [ ]1minusND
[ ]gNND minus
bullbullbull
26
Cyclic PrefixCyclic Prefix
loz All delayed replicas of subcarriers always have an
yy
y p yinteger number of cycles within DFT interval no ICI
27
Cyclic PrefixCyclic Prefix
loz Linear convolution vs circular convolution
yy
N
28
Cyclic PrefixCyclic Prefix
loz Channel effect with cyclic prefix
yy
Signal after removed CP
29
Cyclic PrefixCyclic Prefix
loz Time-Domain Explanation
yy
p
30
Cyclic PrefixCyclic Prefixyy
31
Cyclic PrefixCyclic Prefix
loz Spectrum of channel response with length (smaller [ ]nh hL
yy
than )h
gN[ ] nhFFTHk =
loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr
loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=
where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier
[ ] [ ] [ ]N
Notimes
[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce
kk H
X =rArrcomplexity of channel equalizationrdquo
32
Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))
bull bull( )( )nr~Symbols
dataOutput
bullbullbull
bullbullbull( )nr
bullbull
bullbull
33
Cyclic PrefixCyclic Prefix
loz One of the most important reasons to do OFDM is the
yy
pefficient way it deals with multipath delay spread
loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)
34
Bandwidth EfficiencyBandwidth Efficiencyyy
loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )
loz The diagram for bandwidth efficiency of OFDM system is shown below
35
SummarySummaryyy
loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j
the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference
(ISI)loz The main advantage of the OFDM transmission technique is its
high performance even in frequency selective channels
loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems
36
SynchronizationSynchronization
Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg
38 Synchronization2010722
Introduction (13)Introduction (13)( )( )
loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods
loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead
loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the
cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of
computational complexity therefore it may be not be available in short-burst wireless communication
39 Synchronization2010722
Introduction (23)Introduction (23)( )( )
loz Transcevier
40 Synchronization2010722
Introduction (33)Introduction (33)( )( )
loz Synchronization erroryloz Symbol timing offset (TO)
loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)
loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect
41 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg
42 Synchronization2010722
Signal model (14)Signal model (14)g ( )g ( )
loz Time domain sample p1
2
0
1 Nj nk N
n kk
x X eN
πminus
minus= sum
Xk the frequency domain data at the k-th subcarrier0kN =
loz The multipath fading channel
( ) ( ) ( )1L
h t h t δminus
sum
loz Received time domain signal
( ) ( ) ( )0
l ll
h t h tτ τ δ τ τ=
= minussum
loz Received time-domain signal
( ) ( )1
L
n n ly h n l x w nminus
minus= +sum43
( ) ( )0l=sum
Synchronization2010722
Signal model (24)Signal model (24)g ( )g ( )
loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting
12
0
Nj kn N
k nn
Y y e πminus
minus
=
= sum
( )1 1
2
0
N Lj kn N
n l kn l
h n l x e Wπminus minus
minusminus
=
= +sumsum
loz The equation above could be expressed in matrix form
0n l
loz The equation above could be expressed in matrix form
= +Y HX W
44 Synchronization2010722
Signal model (34)Signal model (34)g ( )g ( )
H= = = = +Y Fy FCx FCF X HX W
loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal
y
H=H FCFcirculant matrix leading to be a diagonal matrix
=H FCF
( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0
h h L L hh h
⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥
( )( ) ( )
( ) ( )
1 10 1 0 0i
h Lh L h N L
⎢ ⎥⎢ ⎥minus minus⎢ ⎥
equiv= minus minus⎢ ⎥⎢ ⎥
C( ) ( )
( ) ( )
0 1 1 100
0 0 1 1 0
h L h N L
h N L L h N
⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
45
( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦
Synchronization2010722
Signal model (44)Signal model (44)g ( )g ( )
loz In a time-invariant channel the channel matrix H would be a diagonal matrix
k k k kY H X W= +
( )1 1
2
0 0
N Lj kl N
kn l
H h n l e πminus minus
minus
= =
= sumsum
loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI
k k k k kY H X I Wα= + +k k k k k
( ) ( )1 1 1
2 2 N N L
j m k Nj ml Nk mI X h n l e e ππ
minus minus minusminusminus⎛ ⎞
= ⎜ ⎟⎝ ⎠
sum sum sumICI
46
0 0 0m n lm k= = =ne
⎝ ⎠
Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
47 Synchronization2010722
Impact of timing offset Impact of timing offset p gp g
loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively
( )I f s
Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ
( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr
2 fj k Nk k k kY X H e Wπδminus= +
loz For integer part of timing offset k k k kY X H e W+
I sTδ ( ) 0g IN L δminus minus le leFFT
2 Ij k NY X H e Wπδminus= +
( ) ( ) 2 I sFFT
j f TI sh t T H f e π δδ minusminus rarr
48
k k k kY X H e W= +
Synchronization2010722
Impact of timing offset derivationImpact of timing offset derivationp gp g
49 Synchronization2010722
Discussion of symbol boundaryDiscussion of symbol boundaryy yy y
50 Synchronization2010722
Impact of timing offsetImpact of timing offsetp gp g
loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI
2 Ij k NIk k k k
NY X H e ISI W
Nπδδ minusminus
= + +
loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk
e e
p kloz Phase Shift can be resolved by Differential Encoding
Decoding or carrier recovery with pilots
51 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
52 Synchronization2010722
Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )
53 Synchronization2010722
Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )
( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s
i n t i N N T N T nTy y t e
= + + +=
( ) ( )( ) ( ) ( ) 12
sin
sin
g gI f f
I I
i N N N Nj jf N Ni k i k k
f
Y X H e eN
π ε ε π ε
ε ε
πεπε
+ + minus+
minus minus=⎛ ⎞⎜ ⎟
Faded signal attenuated and rotated by CFO
( )( ) ( ) ( ) ( )11 2
sin
sin g gI f I f
i N N N NN j j l kI f
NN
l k π ε ε π ε επ ε ε + + minusminus + + + minus
⎜ ⎟⎝ ⎠
+ + minussum
( )( )( )
( ) ( )2
0
sin
I f I f
I
j j l kI f N Ni l l
l l k I f
X H e el k
NN
π ε ε π ε ε
ε π ε ε
+ + +
= ne minus
+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠
sum
i k
N
W
⎜ ⎟⎝ ⎠
+ Inter-Carrier Interference (ICI)
54 Synchronization2010722
Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )
loz An easier representation of received signal with only p g yfractional CFO is
1N
( ) ( )1
0
0N
k k k l l kl l k
Y H X C H X C l k Wminus
= ne
= + minus +⎡ ⎤⎣ ⎦sum
where the ICI coefficient is given by
( )( )( ) ( )
sin 1exp 1fl k
C l k j l kπ ε
π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )
( )exp 1sin
f
f
C l k j l kNN l k
N
π επ ε
minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠
55 Synchronization2010722
Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )
loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality
loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift
56 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
57 Synchronization2010722
TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )
loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq
58 Synchronization2010722
TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )
loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below
loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and
rN2 can be expressed by
0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+
( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +
59
( )
Synchronization2010722
TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )
loz Without CFO the first half is identical to the second half (in time order)
loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below
( ) 12 in discretesft T f NT
NTφ π π π ε πε
⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟
⎝ ⎠
helliphellip
sNT⎝ ⎠
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)
60
Length=N2
Synchronization2010722
TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )
loz Received signal without CFOg0
0 0
jr a e θ=jθ 2 0
2 2 0
Nj j
N Nr a e a e
θ θ= =
loz Received signal with CFO0 0
0 0
j jr a e eθ φ= Extra phase rotation due to CFO0 0
2 2 20
2 2 0
N N Nj j jj
N Nr a e e a e e
θ φ φθ= =
due to CFO
Phase difference 2 2 0N N
( )2 02 20 2 0 0
Nj j T fN
r r a e a eφ φ πminus Δ= =
which contains the information about CFO
61
0 2 0 0Nr r a e a e
Synchronization2010722
TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )
loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be
loz Note that d is a time index corresponding to the first( ) ( ) 2 1
20
Nd m d m Nm
P d r rminus
+ + +== sum
loz Note that d is a time index corresponding to the first sample in a window of N samples
loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by
( )22 1
20
Nd m NR d rminus
+ += sumloz A timing metric can be defined as
))(()(
)( 2
2
dRdP
dM =
( ) 20 d m Nm + +=sum
62
))(( dR
Synchronization2010722
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
OrthogonalityOrthogonalityg yg y
tNΔ
15
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFTby using IDFTby using IDFT
loz According to the structure of Tx it must use Ng oscillators That increases the hardware complexity
loz The equivalent method is using IDFT (IFFT)
16
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz In general each carrier can be expressed as
by using IDFTby using IDFT
g p
( ) (6) )()( )(2 ttfjcc
ccetAtS φπ +=
loz We assume that there are N carriers in the OFDM signal Th th t t l l i l S (t) b t d bThen the total complex signal Ss(t) can be represented by
( ) (7))(1)(1
)(2etAtSN
ttfj nn φπ= summinus
+( )
where
(7) )()(
0
0
fnff
etAN
tSn
ns
Δ+=
= sum=
liihffrequency carrier phase amplitude are )( )( and
where 0
fttAfnff
nnn
n
φΔ+
lyrespective carrierth -of n17
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz Then we sample the signal at a sampling frequency tΔ1
by using IDFTby using IDFT
p g p g q y and An(t) and φn(t) becomes
(8))(t = φφ(9) )((8) )(
nn
nn
AtAt
== φφ
( ) (10) 1)(1
0
)(2 0summinus
=
+ΔΔ+=ΔN
n
tkfnfjns
neAN
tkS φπ
loz Then the sampled signal can be expressed as0=nN
( )( ) (11) 1)(1
0
22 0summinus
=
ΔΔ+Δ sdot=ΔN
n
tfnkjtkfjns eeA
NtkS n πφπ
0n
18
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz The inverse discrete Fourier transform (IDFT) is defined
by using IDFTby using IDFT
( )as the following
1 21
NnkjN
πsumminus
(12) )(1)( 2
0
Nnkj
nefnF
Ntkf πsum
=
Δ=Δ
loz Comparing eq(11) and eq(12) the condition must be satisfied in order to make eq(11) an inverse Fouriersatisfied in order to make eq(11) an inverse Fourier transform relationship
1 (13) 1tN
fΔ
=Δ
19
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz If eq(13) is satisfied
by using IDFTby using IDFT
q ( ) loz is the frequency domain signal loz is the time domain signal
( )ntkfjneA φπ +Δ02
)( tkSs Δ gloz is the sub-channel spacingloz is the symbol duration in each sub-channel
)(s
fΔtNΔloz s e sy bo du o e c sub c e
loz This outcome is the same as the result obtained in the
N
loz This outcome is the same as the result obtained in the system of Figure 1 Therefore IDFT can be used to generate an OFDM transmission signalgenerate an OFDM transmission signal
20
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
( )tf02cos π
( )0a
by using IDFTby using IDFT
( )nd ( )tDI( )tf02sin π
( )0a
( )0b
bull
tf s Δ=
1 ( ) ( ) njbna +
( )nd
tNΔffnffn Δ
=Δ+=1 0
( )tDChannelInput
( )tf N 12cos minusπ
( )1minusNa
bullbull
tNΔ
( )tf N 12sin minusπ
( )1minusNb
( )0d
( )nd ( )tDChannelInput
bull bull
( )( )1d
( )2d
tf s Δ=
1 ( ) ( ) njbna +
fNΔffnffn Δ
=Δ+=1 0
ChannelInput
bullbullbullbull
bullbullbullbull
( )1minusNd
21
Frequency Error Results in ICIFrequency Error Results in ICIq yq y
22
Synchronization Error Results in ICISynchronization Error Results in ICIyy
Not Orthogonal Any More
23
Cyclic PrefixCyclic Prefixyy
loz In multipath channel delayed replicas of previous OFDM signal lead to ISI between successive OFDM signals
loz Solution Insert a guard interval between successive OFDMloz Solution Insert a guard interval between successive OFDM signals
24
Cyclic PrefixCyclic Prefix
loz Guard interval leads to intercarrier interference (ICI) in OFDM d d l i
yy
demodulation
loz In DFT interval difference between two subcarriers does not maintain integer number of cycles loss of orthogonality
loz Delayed version of subcarrier 2 causes ICI in the process of d d l ti b i 1demodulating subcarrier 1
25
Cyclic PrefixCyclic Prefixloz Cyclic prefix (CP) A copy of the last part of OFDM signal is
attached to the front of itself
yy
attached to the front of itself
[ ]0d
[ ]1d
[ ]0D[ ]1D
[ ]nd[ ]kD~
symbols dataInput
bullbullbullbull
[ ]1d
[ ]2d
[ ][ ]2D
bullbullbullbullbull
y
bullbullbullbullbull
[ ]1minusNd [ ]1minusND
[ ]gNND minus
bullbullbull
26
Cyclic PrefixCyclic Prefix
loz All delayed replicas of subcarriers always have an
yy
y p yinteger number of cycles within DFT interval no ICI
27
Cyclic PrefixCyclic Prefix
loz Linear convolution vs circular convolution
yy
N
28
Cyclic PrefixCyclic Prefix
loz Channel effect with cyclic prefix
yy
Signal after removed CP
29
Cyclic PrefixCyclic Prefix
loz Time-Domain Explanation
yy
p
30
Cyclic PrefixCyclic Prefixyy
31
Cyclic PrefixCyclic Prefix
loz Spectrum of channel response with length (smaller [ ]nh hL
yy
than )h
gN[ ] nhFFTHk =
loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr
loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=
where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier
[ ] [ ] [ ]N
Notimes
[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce
kk H
X =rArrcomplexity of channel equalizationrdquo
32
Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))
bull bull( )( )nr~Symbols
dataOutput
bullbullbull
bullbullbull( )nr
bullbull
bullbull
33
Cyclic PrefixCyclic Prefix
loz One of the most important reasons to do OFDM is the
yy
pefficient way it deals with multipath delay spread
loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)
34
Bandwidth EfficiencyBandwidth Efficiencyyy
loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )
loz The diagram for bandwidth efficiency of OFDM system is shown below
35
SummarySummaryyy
loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j
the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference
(ISI)loz The main advantage of the OFDM transmission technique is its
high performance even in frequency selective channels
loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems
36
SynchronizationSynchronization
Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg
38 Synchronization2010722
Introduction (13)Introduction (13)( )( )
loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods
loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead
loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the
cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of
computational complexity therefore it may be not be available in short-burst wireless communication
39 Synchronization2010722
Introduction (23)Introduction (23)( )( )
loz Transcevier
40 Synchronization2010722
Introduction (33)Introduction (33)( )( )
loz Synchronization erroryloz Symbol timing offset (TO)
loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)
loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect
41 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg
42 Synchronization2010722
Signal model (14)Signal model (14)g ( )g ( )
loz Time domain sample p1
2
0
1 Nj nk N
n kk
x X eN
πminus
minus= sum
Xk the frequency domain data at the k-th subcarrier0kN =
loz The multipath fading channel
( ) ( ) ( )1L
h t h t δminus
sum
loz Received time domain signal
( ) ( ) ( )0
l ll
h t h tτ τ δ τ τ=
= minussum
loz Received time-domain signal
( ) ( )1
L
n n ly h n l x w nminus
minus= +sum43
( ) ( )0l=sum
Synchronization2010722
Signal model (24)Signal model (24)g ( )g ( )
loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting
12
0
Nj kn N
k nn
Y y e πminus
minus
=
= sum
( )1 1
2
0
N Lj kn N
n l kn l
h n l x e Wπminus minus
minusminus
=
= +sumsum
loz The equation above could be expressed in matrix form
0n l
loz The equation above could be expressed in matrix form
= +Y HX W
44 Synchronization2010722
Signal model (34)Signal model (34)g ( )g ( )
H= = = = +Y Fy FCx FCF X HX W
loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal
y
H=H FCFcirculant matrix leading to be a diagonal matrix
=H FCF
( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0
h h L L hh h
⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥
( )( ) ( )
( ) ( )
1 10 1 0 0i
h Lh L h N L
⎢ ⎥⎢ ⎥minus minus⎢ ⎥
equiv= minus minus⎢ ⎥⎢ ⎥
C( ) ( )
( ) ( )
0 1 1 100
0 0 1 1 0
h L h N L
h N L L h N
⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
45
( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦
Synchronization2010722
Signal model (44)Signal model (44)g ( )g ( )
loz In a time-invariant channel the channel matrix H would be a diagonal matrix
k k k kY H X W= +
( )1 1
2
0 0
N Lj kl N
kn l
H h n l e πminus minus
minus
= =
= sumsum
loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI
k k k k kY H X I Wα= + +k k k k k
( ) ( )1 1 1
2 2 N N L
j m k Nj ml Nk mI X h n l e e ππ
minus minus minusminusminus⎛ ⎞
= ⎜ ⎟⎝ ⎠
sum sum sumICI
46
0 0 0m n lm k= = =ne
⎝ ⎠
Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
47 Synchronization2010722
Impact of timing offset Impact of timing offset p gp g
loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively
( )I f s
Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ
( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr
2 fj k Nk k k kY X H e Wπδminus= +
loz For integer part of timing offset k k k kY X H e W+
I sTδ ( ) 0g IN L δminus minus le leFFT
2 Ij k NY X H e Wπδminus= +
( ) ( ) 2 I sFFT
j f TI sh t T H f e π δδ minusminus rarr
48
k k k kY X H e W= +
Synchronization2010722
Impact of timing offset derivationImpact of timing offset derivationp gp g
49 Synchronization2010722
Discussion of symbol boundaryDiscussion of symbol boundaryy yy y
50 Synchronization2010722
Impact of timing offsetImpact of timing offsetp gp g
loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI
2 Ij k NIk k k k
NY X H e ISI W
Nπδδ minusminus
= + +
loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk
e e
p kloz Phase Shift can be resolved by Differential Encoding
Decoding or carrier recovery with pilots
51 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
52 Synchronization2010722
Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )
53 Synchronization2010722
Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )
( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s
i n t i N N T N T nTy y t e
= + + +=
( ) ( )( ) ( ) ( ) 12
sin
sin
g gI f f
I I
i N N N Nj jf N Ni k i k k
f
Y X H e eN
π ε ε π ε
ε ε
πεπε
+ + minus+
minus minus=⎛ ⎞⎜ ⎟
Faded signal attenuated and rotated by CFO
( )( ) ( ) ( ) ( )11 2
sin
sin g gI f I f
i N N N NN j j l kI f
NN
l k π ε ε π ε επ ε ε + + minusminus + + + minus
⎜ ⎟⎝ ⎠
+ + minussum
( )( )( )
( ) ( )2
0
sin
I f I f
I
j j l kI f N Ni l l
l l k I f
X H e el k
NN
π ε ε π ε ε
ε π ε ε
+ + +
= ne minus
+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠
sum
i k
N
W
⎜ ⎟⎝ ⎠
+ Inter-Carrier Interference (ICI)
54 Synchronization2010722
Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )
loz An easier representation of received signal with only p g yfractional CFO is
1N
( ) ( )1
0
0N
k k k l l kl l k
Y H X C H X C l k Wminus
= ne
= + minus +⎡ ⎤⎣ ⎦sum
where the ICI coefficient is given by
( )( )( ) ( )
sin 1exp 1fl k
C l k j l kπ ε
π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )
( )exp 1sin
f
f
C l k j l kNN l k
N
π επ ε
minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠
55 Synchronization2010722
Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )
loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality
loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift
56 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
57 Synchronization2010722
TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )
loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq
58 Synchronization2010722
TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )
loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below
loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and
rN2 can be expressed by
0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+
( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +
59
( )
Synchronization2010722
TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )
loz Without CFO the first half is identical to the second half (in time order)
loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below
( ) 12 in discretesft T f NT
NTφ π π π ε πε
⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟
⎝ ⎠
helliphellip
sNT⎝ ⎠
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)
60
Length=N2
Synchronization2010722
TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )
loz Received signal without CFOg0
0 0
jr a e θ=jθ 2 0
2 2 0
Nj j
N Nr a e a e
θ θ= =
loz Received signal with CFO0 0
0 0
j jr a e eθ φ= Extra phase rotation due to CFO0 0
2 2 20
2 2 0
N N Nj j jj
N Nr a e e a e e
θ φ φθ= =
due to CFO
Phase difference 2 2 0N N
( )2 02 20 2 0 0
Nj j T fN
r r a e a eφ φ πminus Δ= =
which contains the information about CFO
61
0 2 0 0Nr r a e a e
Synchronization2010722
TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )
loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be
loz Note that d is a time index corresponding to the first( ) ( ) 2 1
20
Nd m d m Nm
P d r rminus
+ + +== sum
loz Note that d is a time index corresponding to the first sample in a window of N samples
loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by
( )22 1
20
Nd m NR d rminus
+ += sumloz A timing metric can be defined as
))(()(
)( 2
2
dRdP
dM =
( ) 20 d m Nm + +=sum
62
))(( dR
Synchronization2010722
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFTby using IDFTby using IDFT
loz According to the structure of Tx it must use Ng oscillators That increases the hardware complexity
loz The equivalent method is using IDFT (IFFT)
16
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz In general each carrier can be expressed as
by using IDFTby using IDFT
g p
( ) (6) )()( )(2 ttfjcc
ccetAtS φπ +=
loz We assume that there are N carriers in the OFDM signal Th th t t l l i l S (t) b t d bThen the total complex signal Ss(t) can be represented by
( ) (7))(1)(1
)(2etAtSN
ttfj nn φπ= summinus
+( )
where
(7) )()(
0
0
fnff
etAN
tSn
ns
Δ+=
= sum=
liihffrequency carrier phase amplitude are )( )( and
where 0
fttAfnff
nnn
n
φΔ+
lyrespective carrierth -of n17
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz Then we sample the signal at a sampling frequency tΔ1
by using IDFTby using IDFT
p g p g q y and An(t) and φn(t) becomes
(8))(t = φφ(9) )((8) )(
nn
nn
AtAt
== φφ
( ) (10) 1)(1
0
)(2 0summinus
=
+ΔΔ+=ΔN
n
tkfnfjns
neAN
tkS φπ
loz Then the sampled signal can be expressed as0=nN
( )( ) (11) 1)(1
0
22 0summinus
=
ΔΔ+Δ sdot=ΔN
n
tfnkjtkfjns eeA
NtkS n πφπ
0n
18
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz The inverse discrete Fourier transform (IDFT) is defined
by using IDFTby using IDFT
( )as the following
1 21
NnkjN
πsumminus
(12) )(1)( 2
0
Nnkj
nefnF
Ntkf πsum
=
Δ=Δ
loz Comparing eq(11) and eq(12) the condition must be satisfied in order to make eq(11) an inverse Fouriersatisfied in order to make eq(11) an inverse Fourier transform relationship
1 (13) 1tN
fΔ
=Δ
19
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz If eq(13) is satisfied
by using IDFTby using IDFT
q ( ) loz is the frequency domain signal loz is the time domain signal
( )ntkfjneA φπ +Δ02
)( tkSs Δ gloz is the sub-channel spacingloz is the symbol duration in each sub-channel
)(s
fΔtNΔloz s e sy bo du o e c sub c e
loz This outcome is the same as the result obtained in the
N
loz This outcome is the same as the result obtained in the system of Figure 1 Therefore IDFT can be used to generate an OFDM transmission signalgenerate an OFDM transmission signal
20
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
( )tf02cos π
( )0a
by using IDFTby using IDFT
( )nd ( )tDI( )tf02sin π
( )0a
( )0b
bull
tf s Δ=
1 ( ) ( ) njbna +
( )nd
tNΔffnffn Δ
=Δ+=1 0
( )tDChannelInput
( )tf N 12cos minusπ
( )1minusNa
bullbull
tNΔ
( )tf N 12sin minusπ
( )1minusNb
( )0d
( )nd ( )tDChannelInput
bull bull
( )( )1d
( )2d
tf s Δ=
1 ( ) ( ) njbna +
fNΔffnffn Δ
=Δ+=1 0
ChannelInput
bullbullbullbull
bullbullbullbull
( )1minusNd
21
Frequency Error Results in ICIFrequency Error Results in ICIq yq y
22
Synchronization Error Results in ICISynchronization Error Results in ICIyy
Not Orthogonal Any More
23
Cyclic PrefixCyclic Prefixyy
loz In multipath channel delayed replicas of previous OFDM signal lead to ISI between successive OFDM signals
loz Solution Insert a guard interval between successive OFDMloz Solution Insert a guard interval between successive OFDM signals
24
Cyclic PrefixCyclic Prefix
loz Guard interval leads to intercarrier interference (ICI) in OFDM d d l i
yy
demodulation
loz In DFT interval difference between two subcarriers does not maintain integer number of cycles loss of orthogonality
loz Delayed version of subcarrier 2 causes ICI in the process of d d l ti b i 1demodulating subcarrier 1
25
Cyclic PrefixCyclic Prefixloz Cyclic prefix (CP) A copy of the last part of OFDM signal is
attached to the front of itself
yy
attached to the front of itself
[ ]0d
[ ]1d
[ ]0D[ ]1D
[ ]nd[ ]kD~
symbols dataInput
bullbullbullbull
[ ]1d
[ ]2d
[ ][ ]2D
bullbullbullbullbull
y
bullbullbullbullbull
[ ]1minusNd [ ]1minusND
[ ]gNND minus
bullbullbull
26
Cyclic PrefixCyclic Prefix
loz All delayed replicas of subcarriers always have an
yy
y p yinteger number of cycles within DFT interval no ICI
27
Cyclic PrefixCyclic Prefix
loz Linear convolution vs circular convolution
yy
N
28
Cyclic PrefixCyclic Prefix
loz Channel effect with cyclic prefix
yy
Signal after removed CP
29
Cyclic PrefixCyclic Prefix
loz Time-Domain Explanation
yy
p
30
Cyclic PrefixCyclic Prefixyy
31
Cyclic PrefixCyclic Prefix
loz Spectrum of channel response with length (smaller [ ]nh hL
yy
than )h
gN[ ] nhFFTHk =
loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr
loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=
where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier
[ ] [ ] [ ]N
Notimes
[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce
kk H
X =rArrcomplexity of channel equalizationrdquo
32
Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))
bull bull( )( )nr~Symbols
dataOutput
bullbullbull
bullbullbull( )nr
bullbull
bullbull
33
Cyclic PrefixCyclic Prefix
loz One of the most important reasons to do OFDM is the
yy
pefficient way it deals with multipath delay spread
loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)
34
Bandwidth EfficiencyBandwidth Efficiencyyy
loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )
loz The diagram for bandwidth efficiency of OFDM system is shown below
35
SummarySummaryyy
loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j
the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference
(ISI)loz The main advantage of the OFDM transmission technique is its
high performance even in frequency selective channels
loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems
36
SynchronizationSynchronization
Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg
38 Synchronization2010722
Introduction (13)Introduction (13)( )( )
loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods
loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead
loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the
cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of
computational complexity therefore it may be not be available in short-burst wireless communication
39 Synchronization2010722
Introduction (23)Introduction (23)( )( )
loz Transcevier
40 Synchronization2010722
Introduction (33)Introduction (33)( )( )
loz Synchronization erroryloz Symbol timing offset (TO)
loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)
loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect
41 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg
42 Synchronization2010722
Signal model (14)Signal model (14)g ( )g ( )
loz Time domain sample p1
2
0
1 Nj nk N
n kk
x X eN
πminus
minus= sum
Xk the frequency domain data at the k-th subcarrier0kN =
loz The multipath fading channel
( ) ( ) ( )1L
h t h t δminus
sum
loz Received time domain signal
( ) ( ) ( )0
l ll
h t h tτ τ δ τ τ=
= minussum
loz Received time-domain signal
( ) ( )1
L
n n ly h n l x w nminus
minus= +sum43
( ) ( )0l=sum
Synchronization2010722
Signal model (24)Signal model (24)g ( )g ( )
loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting
12
0
Nj kn N
k nn
Y y e πminus
minus
=
= sum
( )1 1
2
0
N Lj kn N
n l kn l
h n l x e Wπminus minus
minusminus
=
= +sumsum
loz The equation above could be expressed in matrix form
0n l
loz The equation above could be expressed in matrix form
= +Y HX W
44 Synchronization2010722
Signal model (34)Signal model (34)g ( )g ( )
H= = = = +Y Fy FCx FCF X HX W
loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal
y
H=H FCFcirculant matrix leading to be a diagonal matrix
=H FCF
( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0
h h L L hh h
⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥
( )( ) ( )
( ) ( )
1 10 1 0 0i
h Lh L h N L
⎢ ⎥⎢ ⎥minus minus⎢ ⎥
equiv= minus minus⎢ ⎥⎢ ⎥
C( ) ( )
( ) ( )
0 1 1 100
0 0 1 1 0
h L h N L
h N L L h N
⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
45
( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦
Synchronization2010722
Signal model (44)Signal model (44)g ( )g ( )
loz In a time-invariant channel the channel matrix H would be a diagonal matrix
k k k kY H X W= +
( )1 1
2
0 0
N Lj kl N
kn l
H h n l e πminus minus
minus
= =
= sumsum
loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI
k k k k kY H X I Wα= + +k k k k k
( ) ( )1 1 1
2 2 N N L
j m k Nj ml Nk mI X h n l e e ππ
minus minus minusminusminus⎛ ⎞
= ⎜ ⎟⎝ ⎠
sum sum sumICI
46
0 0 0m n lm k= = =ne
⎝ ⎠
Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
47 Synchronization2010722
Impact of timing offset Impact of timing offset p gp g
loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively
( )I f s
Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ
( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr
2 fj k Nk k k kY X H e Wπδminus= +
loz For integer part of timing offset k k k kY X H e W+
I sTδ ( ) 0g IN L δminus minus le leFFT
2 Ij k NY X H e Wπδminus= +
( ) ( ) 2 I sFFT
j f TI sh t T H f e π δδ minusminus rarr
48
k k k kY X H e W= +
Synchronization2010722
Impact of timing offset derivationImpact of timing offset derivationp gp g
49 Synchronization2010722
Discussion of symbol boundaryDiscussion of symbol boundaryy yy y
50 Synchronization2010722
Impact of timing offsetImpact of timing offsetp gp g
loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI
2 Ij k NIk k k k
NY X H e ISI W
Nπδδ minusminus
= + +
loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk
e e
p kloz Phase Shift can be resolved by Differential Encoding
Decoding or carrier recovery with pilots
51 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
52 Synchronization2010722
Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )
53 Synchronization2010722
Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )
( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s
i n t i N N T N T nTy y t e
= + + +=
( ) ( )( ) ( ) ( ) 12
sin
sin
g gI f f
I I
i N N N Nj jf N Ni k i k k
f
Y X H e eN
π ε ε π ε
ε ε
πεπε
+ + minus+
minus minus=⎛ ⎞⎜ ⎟
Faded signal attenuated and rotated by CFO
( )( ) ( ) ( ) ( )11 2
sin
sin g gI f I f
i N N N NN j j l kI f
NN
l k π ε ε π ε επ ε ε + + minusminus + + + minus
⎜ ⎟⎝ ⎠
+ + minussum
( )( )( )
( ) ( )2
0
sin
I f I f
I
j j l kI f N Ni l l
l l k I f
X H e el k
NN
π ε ε π ε ε
ε π ε ε
+ + +
= ne minus
+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠
sum
i k
N
W
⎜ ⎟⎝ ⎠
+ Inter-Carrier Interference (ICI)
54 Synchronization2010722
Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )
loz An easier representation of received signal with only p g yfractional CFO is
1N
( ) ( )1
0
0N
k k k l l kl l k
Y H X C H X C l k Wminus
= ne
= + minus +⎡ ⎤⎣ ⎦sum
where the ICI coefficient is given by
( )( )( ) ( )
sin 1exp 1fl k
C l k j l kπ ε
π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )
( )exp 1sin
f
f
C l k j l kNN l k
N
π επ ε
minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠
55 Synchronization2010722
Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )
loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality
loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift
56 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
57 Synchronization2010722
TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )
loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq
58 Synchronization2010722
TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )
loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below
loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and
rN2 can be expressed by
0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+
( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +
59
( )
Synchronization2010722
TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )
loz Without CFO the first half is identical to the second half (in time order)
loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below
( ) 12 in discretesft T f NT
NTφ π π π ε πε
⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟
⎝ ⎠
helliphellip
sNT⎝ ⎠
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)
60
Length=N2
Synchronization2010722
TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )
loz Received signal without CFOg0
0 0
jr a e θ=jθ 2 0
2 2 0
Nj j
N Nr a e a e
θ θ= =
loz Received signal with CFO0 0
0 0
j jr a e eθ φ= Extra phase rotation due to CFO0 0
2 2 20
2 2 0
N N Nj j jj
N Nr a e e a e e
θ φ φθ= =
due to CFO
Phase difference 2 2 0N N
( )2 02 20 2 0 0
Nj j T fN
r r a e a eφ φ πminus Δ= =
which contains the information about CFO
61
0 2 0 0Nr r a e a e
Synchronization2010722
TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )
loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be
loz Note that d is a time index corresponding to the first( ) ( ) 2 1
20
Nd m d m Nm
P d r rminus
+ + +== sum
loz Note that d is a time index corresponding to the first sample in a window of N samples
loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by
( )22 1
20
Nd m NR d rminus
+ += sumloz A timing metric can be defined as
))(()(
)( 2
2
dRdP
dM =
( ) 20 d m Nm + +=sum
62
))(( dR
Synchronization2010722
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz In general each carrier can be expressed as
by using IDFTby using IDFT
g p
( ) (6) )()( )(2 ttfjcc
ccetAtS φπ +=
loz We assume that there are N carriers in the OFDM signal Th th t t l l i l S (t) b t d bThen the total complex signal Ss(t) can be represented by
( ) (7))(1)(1
)(2etAtSN
ttfj nn φπ= summinus
+( )
where
(7) )()(
0
0
fnff
etAN
tSn
ns
Δ+=
= sum=
liihffrequency carrier phase amplitude are )( )( and
where 0
fttAfnff
nnn
n
φΔ+
lyrespective carrierth -of n17
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz Then we sample the signal at a sampling frequency tΔ1
by using IDFTby using IDFT
p g p g q y and An(t) and φn(t) becomes
(8))(t = φφ(9) )((8) )(
nn
nn
AtAt
== φφ
( ) (10) 1)(1
0
)(2 0summinus
=
+ΔΔ+=ΔN
n
tkfnfjns
neAN
tkS φπ
loz Then the sampled signal can be expressed as0=nN
( )( ) (11) 1)(1
0
22 0summinus
=
ΔΔ+Δ sdot=ΔN
n
tfnkjtkfjns eeA
NtkS n πφπ
0n
18
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz The inverse discrete Fourier transform (IDFT) is defined
by using IDFTby using IDFT
( )as the following
1 21
NnkjN
πsumminus
(12) )(1)( 2
0
Nnkj
nefnF
Ntkf πsum
=
Δ=Δ
loz Comparing eq(11) and eq(12) the condition must be satisfied in order to make eq(11) an inverse Fouriersatisfied in order to make eq(11) an inverse Fourier transform relationship
1 (13) 1tN
fΔ
=Δ
19
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz If eq(13) is satisfied
by using IDFTby using IDFT
q ( ) loz is the frequency domain signal loz is the time domain signal
( )ntkfjneA φπ +Δ02
)( tkSs Δ gloz is the sub-channel spacingloz is the symbol duration in each sub-channel
)(s
fΔtNΔloz s e sy bo du o e c sub c e
loz This outcome is the same as the result obtained in the
N
loz This outcome is the same as the result obtained in the system of Figure 1 Therefore IDFT can be used to generate an OFDM transmission signalgenerate an OFDM transmission signal
20
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
( )tf02cos π
( )0a
by using IDFTby using IDFT
( )nd ( )tDI( )tf02sin π
( )0a
( )0b
bull
tf s Δ=
1 ( ) ( ) njbna +
( )nd
tNΔffnffn Δ
=Δ+=1 0
( )tDChannelInput
( )tf N 12cos minusπ
( )1minusNa
bullbull
tNΔ
( )tf N 12sin minusπ
( )1minusNb
( )0d
( )nd ( )tDChannelInput
bull bull
( )( )1d
( )2d
tf s Δ=
1 ( ) ( ) njbna +
fNΔffnffn Δ
=Δ+=1 0
ChannelInput
bullbullbullbull
bullbullbullbull
( )1minusNd
21
Frequency Error Results in ICIFrequency Error Results in ICIq yq y
22
Synchronization Error Results in ICISynchronization Error Results in ICIyy
Not Orthogonal Any More
23
Cyclic PrefixCyclic Prefixyy
loz In multipath channel delayed replicas of previous OFDM signal lead to ISI between successive OFDM signals
loz Solution Insert a guard interval between successive OFDMloz Solution Insert a guard interval between successive OFDM signals
24
Cyclic PrefixCyclic Prefix
loz Guard interval leads to intercarrier interference (ICI) in OFDM d d l i
yy
demodulation
loz In DFT interval difference between two subcarriers does not maintain integer number of cycles loss of orthogonality
loz Delayed version of subcarrier 2 causes ICI in the process of d d l ti b i 1demodulating subcarrier 1
25
Cyclic PrefixCyclic Prefixloz Cyclic prefix (CP) A copy of the last part of OFDM signal is
attached to the front of itself
yy
attached to the front of itself
[ ]0d
[ ]1d
[ ]0D[ ]1D
[ ]nd[ ]kD~
symbols dataInput
bullbullbullbull
[ ]1d
[ ]2d
[ ][ ]2D
bullbullbullbullbull
y
bullbullbullbullbull
[ ]1minusNd [ ]1minusND
[ ]gNND minus
bullbullbull
26
Cyclic PrefixCyclic Prefix
loz All delayed replicas of subcarriers always have an
yy
y p yinteger number of cycles within DFT interval no ICI
27
Cyclic PrefixCyclic Prefix
loz Linear convolution vs circular convolution
yy
N
28
Cyclic PrefixCyclic Prefix
loz Channel effect with cyclic prefix
yy
Signal after removed CP
29
Cyclic PrefixCyclic Prefix
loz Time-Domain Explanation
yy
p
30
Cyclic PrefixCyclic Prefixyy
31
Cyclic PrefixCyclic Prefix
loz Spectrum of channel response with length (smaller [ ]nh hL
yy
than )h
gN[ ] nhFFTHk =
loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr
loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=
where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier
[ ] [ ] [ ]N
Notimes
[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce
kk H
X =rArrcomplexity of channel equalizationrdquo
32
Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))
bull bull( )( )nr~Symbols
dataOutput
bullbullbull
bullbullbull( )nr
bullbull
bullbull
33
Cyclic PrefixCyclic Prefix
loz One of the most important reasons to do OFDM is the
yy
pefficient way it deals with multipath delay spread
loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)
34
Bandwidth EfficiencyBandwidth Efficiencyyy
loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )
loz The diagram for bandwidth efficiency of OFDM system is shown below
35
SummarySummaryyy
loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j
the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference
(ISI)loz The main advantage of the OFDM transmission technique is its
high performance even in frequency selective channels
loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems
36
SynchronizationSynchronization
Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg
38 Synchronization2010722
Introduction (13)Introduction (13)( )( )
loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods
loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead
loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the
cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of
computational complexity therefore it may be not be available in short-burst wireless communication
39 Synchronization2010722
Introduction (23)Introduction (23)( )( )
loz Transcevier
40 Synchronization2010722
Introduction (33)Introduction (33)( )( )
loz Synchronization erroryloz Symbol timing offset (TO)
loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)
loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect
41 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg
42 Synchronization2010722
Signal model (14)Signal model (14)g ( )g ( )
loz Time domain sample p1
2
0
1 Nj nk N
n kk
x X eN
πminus
minus= sum
Xk the frequency domain data at the k-th subcarrier0kN =
loz The multipath fading channel
( ) ( ) ( )1L
h t h t δminus
sum
loz Received time domain signal
( ) ( ) ( )0
l ll
h t h tτ τ δ τ τ=
= minussum
loz Received time-domain signal
( ) ( )1
L
n n ly h n l x w nminus
minus= +sum43
( ) ( )0l=sum
Synchronization2010722
Signal model (24)Signal model (24)g ( )g ( )
loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting
12
0
Nj kn N
k nn
Y y e πminus
minus
=
= sum
( )1 1
2
0
N Lj kn N
n l kn l
h n l x e Wπminus minus
minusminus
=
= +sumsum
loz The equation above could be expressed in matrix form
0n l
loz The equation above could be expressed in matrix form
= +Y HX W
44 Synchronization2010722
Signal model (34)Signal model (34)g ( )g ( )
H= = = = +Y Fy FCx FCF X HX W
loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal
y
H=H FCFcirculant matrix leading to be a diagonal matrix
=H FCF
( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0
h h L L hh h
⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥
( )( ) ( )
( ) ( )
1 10 1 0 0i
h Lh L h N L
⎢ ⎥⎢ ⎥minus minus⎢ ⎥
equiv= minus minus⎢ ⎥⎢ ⎥
C( ) ( )
( ) ( )
0 1 1 100
0 0 1 1 0
h L h N L
h N L L h N
⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
45
( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦
Synchronization2010722
Signal model (44)Signal model (44)g ( )g ( )
loz In a time-invariant channel the channel matrix H would be a diagonal matrix
k k k kY H X W= +
( )1 1
2
0 0
N Lj kl N
kn l
H h n l e πminus minus
minus
= =
= sumsum
loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI
k k k k kY H X I Wα= + +k k k k k
( ) ( )1 1 1
2 2 N N L
j m k Nj ml Nk mI X h n l e e ππ
minus minus minusminusminus⎛ ⎞
= ⎜ ⎟⎝ ⎠
sum sum sumICI
46
0 0 0m n lm k= = =ne
⎝ ⎠
Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
47 Synchronization2010722
Impact of timing offset Impact of timing offset p gp g
loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively
( )I f s
Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ
( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr
2 fj k Nk k k kY X H e Wπδminus= +
loz For integer part of timing offset k k k kY X H e W+
I sTδ ( ) 0g IN L δminus minus le leFFT
2 Ij k NY X H e Wπδminus= +
( ) ( ) 2 I sFFT
j f TI sh t T H f e π δδ minusminus rarr
48
k k k kY X H e W= +
Synchronization2010722
Impact of timing offset derivationImpact of timing offset derivationp gp g
49 Synchronization2010722
Discussion of symbol boundaryDiscussion of symbol boundaryy yy y
50 Synchronization2010722
Impact of timing offsetImpact of timing offsetp gp g
loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI
2 Ij k NIk k k k
NY X H e ISI W
Nπδδ minusminus
= + +
loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk
e e
p kloz Phase Shift can be resolved by Differential Encoding
Decoding or carrier recovery with pilots
51 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
52 Synchronization2010722
Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )
53 Synchronization2010722
Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )
( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s
i n t i N N T N T nTy y t e
= + + +=
( ) ( )( ) ( ) ( ) 12
sin
sin
g gI f f
I I
i N N N Nj jf N Ni k i k k
f
Y X H e eN
π ε ε π ε
ε ε
πεπε
+ + minus+
minus minus=⎛ ⎞⎜ ⎟
Faded signal attenuated and rotated by CFO
( )( ) ( ) ( ) ( )11 2
sin
sin g gI f I f
i N N N NN j j l kI f
NN
l k π ε ε π ε επ ε ε + + minusminus + + + minus
⎜ ⎟⎝ ⎠
+ + minussum
( )( )( )
( ) ( )2
0
sin
I f I f
I
j j l kI f N Ni l l
l l k I f
X H e el k
NN
π ε ε π ε ε
ε π ε ε
+ + +
= ne minus
+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠
sum
i k
N
W
⎜ ⎟⎝ ⎠
+ Inter-Carrier Interference (ICI)
54 Synchronization2010722
Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )
loz An easier representation of received signal with only p g yfractional CFO is
1N
( ) ( )1
0
0N
k k k l l kl l k
Y H X C H X C l k Wminus
= ne
= + minus +⎡ ⎤⎣ ⎦sum
where the ICI coefficient is given by
( )( )( ) ( )
sin 1exp 1fl k
C l k j l kπ ε
π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )
( )exp 1sin
f
f
C l k j l kNN l k
N
π επ ε
minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠
55 Synchronization2010722
Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )
loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality
loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift
56 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
57 Synchronization2010722
TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )
loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq
58 Synchronization2010722
TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )
loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below
loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and
rN2 can be expressed by
0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+
( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +
59
( )
Synchronization2010722
TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )
loz Without CFO the first half is identical to the second half (in time order)
loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below
( ) 12 in discretesft T f NT
NTφ π π π ε πε
⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟
⎝ ⎠
helliphellip
sNT⎝ ⎠
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)
60
Length=N2
Synchronization2010722
TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )
loz Received signal without CFOg0
0 0
jr a e θ=jθ 2 0
2 2 0
Nj j
N Nr a e a e
θ θ= =
loz Received signal with CFO0 0
0 0
j jr a e eθ φ= Extra phase rotation due to CFO0 0
2 2 20
2 2 0
N N Nj j jj
N Nr a e e a e e
θ φ φθ= =
due to CFO
Phase difference 2 2 0N N
( )2 02 20 2 0 0
Nj j T fN
r r a e a eφ φ πminus Δ= =
which contains the information about CFO
61
0 2 0 0Nr r a e a e
Synchronization2010722
TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )
loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be
loz Note that d is a time index corresponding to the first( ) ( ) 2 1
20
Nd m d m Nm
P d r rminus
+ + +== sum
loz Note that d is a time index corresponding to the first sample in a window of N samples
loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by
( )22 1
20
Nd m NR d rminus
+ += sumloz A timing metric can be defined as
))(()(
)( 2
2
dRdP
dM =
( ) 20 d m Nm + +=sum
62
))(( dR
Synchronization2010722
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz Then we sample the signal at a sampling frequency tΔ1
by using IDFTby using IDFT
p g p g q y and An(t) and φn(t) becomes
(8))(t = φφ(9) )((8) )(
nn
nn
AtAt
== φφ
( ) (10) 1)(1
0
)(2 0summinus
=
+ΔΔ+=ΔN
n
tkfnfjns
neAN
tkS φπ
loz Then the sampled signal can be expressed as0=nN
( )( ) (11) 1)(1
0
22 0summinus
=
ΔΔ+Δ sdot=ΔN
n
tfnkjtkfjns eeA
NtkS n πφπ
0n
18
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz The inverse discrete Fourier transform (IDFT) is defined
by using IDFTby using IDFT
( )as the following
1 21
NnkjN
πsumminus
(12) )(1)( 2
0
Nnkj
nefnF
Ntkf πsum
=
Δ=Δ
loz Comparing eq(11) and eq(12) the condition must be satisfied in order to make eq(11) an inverse Fouriersatisfied in order to make eq(11) an inverse Fourier transform relationship
1 (13) 1tN
fΔ
=Δ
19
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz If eq(13) is satisfied
by using IDFTby using IDFT
q ( ) loz is the frequency domain signal loz is the time domain signal
( )ntkfjneA φπ +Δ02
)( tkSs Δ gloz is the sub-channel spacingloz is the symbol duration in each sub-channel
)(s
fΔtNΔloz s e sy bo du o e c sub c e
loz This outcome is the same as the result obtained in the
N
loz This outcome is the same as the result obtained in the system of Figure 1 Therefore IDFT can be used to generate an OFDM transmission signalgenerate an OFDM transmission signal
20
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
( )tf02cos π
( )0a
by using IDFTby using IDFT
( )nd ( )tDI( )tf02sin π
( )0a
( )0b
bull
tf s Δ=
1 ( ) ( ) njbna +
( )nd
tNΔffnffn Δ
=Δ+=1 0
( )tDChannelInput
( )tf N 12cos minusπ
( )1minusNa
bullbull
tNΔ
( )tf N 12sin minusπ
( )1minusNb
( )0d
( )nd ( )tDChannelInput
bull bull
( )( )1d
( )2d
tf s Δ=
1 ( ) ( ) njbna +
fNΔffnffn Δ
=Δ+=1 0
ChannelInput
bullbullbullbull
bullbullbullbull
( )1minusNd
21
Frequency Error Results in ICIFrequency Error Results in ICIq yq y
22
Synchronization Error Results in ICISynchronization Error Results in ICIyy
Not Orthogonal Any More
23
Cyclic PrefixCyclic Prefixyy
loz In multipath channel delayed replicas of previous OFDM signal lead to ISI between successive OFDM signals
loz Solution Insert a guard interval between successive OFDMloz Solution Insert a guard interval between successive OFDM signals
24
Cyclic PrefixCyclic Prefix
loz Guard interval leads to intercarrier interference (ICI) in OFDM d d l i
yy
demodulation
loz In DFT interval difference between two subcarriers does not maintain integer number of cycles loss of orthogonality
loz Delayed version of subcarrier 2 causes ICI in the process of d d l ti b i 1demodulating subcarrier 1
25
Cyclic PrefixCyclic Prefixloz Cyclic prefix (CP) A copy of the last part of OFDM signal is
attached to the front of itself
yy
attached to the front of itself
[ ]0d
[ ]1d
[ ]0D[ ]1D
[ ]nd[ ]kD~
symbols dataInput
bullbullbullbull
[ ]1d
[ ]2d
[ ][ ]2D
bullbullbullbullbull
y
bullbullbullbullbull
[ ]1minusNd [ ]1minusND
[ ]gNND minus
bullbullbull
26
Cyclic PrefixCyclic Prefix
loz All delayed replicas of subcarriers always have an
yy
y p yinteger number of cycles within DFT interval no ICI
27
Cyclic PrefixCyclic Prefix
loz Linear convolution vs circular convolution
yy
N
28
Cyclic PrefixCyclic Prefix
loz Channel effect with cyclic prefix
yy
Signal after removed CP
29
Cyclic PrefixCyclic Prefix
loz Time-Domain Explanation
yy
p
30
Cyclic PrefixCyclic Prefixyy
31
Cyclic PrefixCyclic Prefix
loz Spectrum of channel response with length (smaller [ ]nh hL
yy
than )h
gN[ ] nhFFTHk =
loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr
loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=
where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier
[ ] [ ] [ ]N
Notimes
[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce
kk H
X =rArrcomplexity of channel equalizationrdquo
32
Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))
bull bull( )( )nr~Symbols
dataOutput
bullbullbull
bullbullbull( )nr
bullbull
bullbull
33
Cyclic PrefixCyclic Prefix
loz One of the most important reasons to do OFDM is the
yy
pefficient way it deals with multipath delay spread
loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)
34
Bandwidth EfficiencyBandwidth Efficiencyyy
loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )
loz The diagram for bandwidth efficiency of OFDM system is shown below
35
SummarySummaryyy
loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j
the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference
(ISI)loz The main advantage of the OFDM transmission technique is its
high performance even in frequency selective channels
loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems
36
SynchronizationSynchronization
Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg
38 Synchronization2010722
Introduction (13)Introduction (13)( )( )
loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods
loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead
loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the
cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of
computational complexity therefore it may be not be available in short-burst wireless communication
39 Synchronization2010722
Introduction (23)Introduction (23)( )( )
loz Transcevier
40 Synchronization2010722
Introduction (33)Introduction (33)( )( )
loz Synchronization erroryloz Symbol timing offset (TO)
loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)
loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect
41 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg
42 Synchronization2010722
Signal model (14)Signal model (14)g ( )g ( )
loz Time domain sample p1
2
0
1 Nj nk N
n kk
x X eN
πminus
minus= sum
Xk the frequency domain data at the k-th subcarrier0kN =
loz The multipath fading channel
( ) ( ) ( )1L
h t h t δminus
sum
loz Received time domain signal
( ) ( ) ( )0
l ll
h t h tτ τ δ τ τ=
= minussum
loz Received time-domain signal
( ) ( )1
L
n n ly h n l x w nminus
minus= +sum43
( ) ( )0l=sum
Synchronization2010722
Signal model (24)Signal model (24)g ( )g ( )
loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting
12
0
Nj kn N
k nn
Y y e πminus
minus
=
= sum
( )1 1
2
0
N Lj kn N
n l kn l
h n l x e Wπminus minus
minusminus
=
= +sumsum
loz The equation above could be expressed in matrix form
0n l
loz The equation above could be expressed in matrix form
= +Y HX W
44 Synchronization2010722
Signal model (34)Signal model (34)g ( )g ( )
H= = = = +Y Fy FCx FCF X HX W
loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal
y
H=H FCFcirculant matrix leading to be a diagonal matrix
=H FCF
( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0
h h L L hh h
⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥
( )( ) ( )
( ) ( )
1 10 1 0 0i
h Lh L h N L
⎢ ⎥⎢ ⎥minus minus⎢ ⎥
equiv= minus minus⎢ ⎥⎢ ⎥
C( ) ( )
( ) ( )
0 1 1 100
0 0 1 1 0
h L h N L
h N L L h N
⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
45
( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦
Synchronization2010722
Signal model (44)Signal model (44)g ( )g ( )
loz In a time-invariant channel the channel matrix H would be a diagonal matrix
k k k kY H X W= +
( )1 1
2
0 0
N Lj kl N
kn l
H h n l e πminus minus
minus
= =
= sumsum
loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI
k k k k kY H X I Wα= + +k k k k k
( ) ( )1 1 1
2 2 N N L
j m k Nj ml Nk mI X h n l e e ππ
minus minus minusminusminus⎛ ⎞
= ⎜ ⎟⎝ ⎠
sum sum sumICI
46
0 0 0m n lm k= = =ne
⎝ ⎠
Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
47 Synchronization2010722
Impact of timing offset Impact of timing offset p gp g
loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively
( )I f s
Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ
( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr
2 fj k Nk k k kY X H e Wπδminus= +
loz For integer part of timing offset k k k kY X H e W+
I sTδ ( ) 0g IN L δminus minus le leFFT
2 Ij k NY X H e Wπδminus= +
( ) ( ) 2 I sFFT
j f TI sh t T H f e π δδ minusminus rarr
48
k k k kY X H e W= +
Synchronization2010722
Impact of timing offset derivationImpact of timing offset derivationp gp g
49 Synchronization2010722
Discussion of symbol boundaryDiscussion of symbol boundaryy yy y
50 Synchronization2010722
Impact of timing offsetImpact of timing offsetp gp g
loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI
2 Ij k NIk k k k
NY X H e ISI W
Nπδδ minusminus
= + +
loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk
e e
p kloz Phase Shift can be resolved by Differential Encoding
Decoding or carrier recovery with pilots
51 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
52 Synchronization2010722
Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )
53 Synchronization2010722
Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )
( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s
i n t i N N T N T nTy y t e
= + + +=
( ) ( )( ) ( ) ( ) 12
sin
sin
g gI f f
I I
i N N N Nj jf N Ni k i k k
f
Y X H e eN
π ε ε π ε
ε ε
πεπε
+ + minus+
minus minus=⎛ ⎞⎜ ⎟
Faded signal attenuated and rotated by CFO
( )( ) ( ) ( ) ( )11 2
sin
sin g gI f I f
i N N N NN j j l kI f
NN
l k π ε ε π ε επ ε ε + + minusminus + + + minus
⎜ ⎟⎝ ⎠
+ + minussum
( )( )( )
( ) ( )2
0
sin
I f I f
I
j j l kI f N Ni l l
l l k I f
X H e el k
NN
π ε ε π ε ε
ε π ε ε
+ + +
= ne minus
+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠
sum
i k
N
W
⎜ ⎟⎝ ⎠
+ Inter-Carrier Interference (ICI)
54 Synchronization2010722
Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )
loz An easier representation of received signal with only p g yfractional CFO is
1N
( ) ( )1
0
0N
k k k l l kl l k
Y H X C H X C l k Wminus
= ne
= + minus +⎡ ⎤⎣ ⎦sum
where the ICI coefficient is given by
( )( )( ) ( )
sin 1exp 1fl k
C l k j l kπ ε
π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )
( )exp 1sin
f
f
C l k j l kNN l k
N
π επ ε
minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠
55 Synchronization2010722
Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )
loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality
loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift
56 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
57 Synchronization2010722
TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )
loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq
58 Synchronization2010722
TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )
loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below
loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and
rN2 can be expressed by
0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+
( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +
59
( )
Synchronization2010722
TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )
loz Without CFO the first half is identical to the second half (in time order)
loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below
( ) 12 in discretesft T f NT
NTφ π π π ε πε
⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟
⎝ ⎠
helliphellip
sNT⎝ ⎠
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)
60
Length=N2
Synchronization2010722
TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )
loz Received signal without CFOg0
0 0
jr a e θ=jθ 2 0
2 2 0
Nj j
N Nr a e a e
θ θ= =
loz Received signal with CFO0 0
0 0
j jr a e eθ φ= Extra phase rotation due to CFO0 0
2 2 20
2 2 0
N N Nj j jj
N Nr a e e a e e
θ φ φθ= =
due to CFO
Phase difference 2 2 0N N
( )2 02 20 2 0 0
Nj j T fN
r r a e a eφ φ πminus Δ= =
which contains the information about CFO
61
0 2 0 0Nr r a e a e
Synchronization2010722
TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )
loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be
loz Note that d is a time index corresponding to the first( ) ( ) 2 1
20
Nd m d m Nm
P d r rminus
+ + +== sum
loz Note that d is a time index corresponding to the first sample in a window of N samples
loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by
( )22 1
20
Nd m NR d rminus
+ += sumloz A timing metric can be defined as
))(()(
)( 2
2
dRdP
dM =
( ) 20 d m Nm + +=sum
62
))(( dR
Synchronization2010722
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz The inverse discrete Fourier transform (IDFT) is defined
by using IDFTby using IDFT
( )as the following
1 21
NnkjN
πsumminus
(12) )(1)( 2
0
Nnkj
nefnF
Ntkf πsum
=
Δ=Δ
loz Comparing eq(11) and eq(12) the condition must be satisfied in order to make eq(11) an inverse Fouriersatisfied in order to make eq(11) an inverse Fourier transform relationship
1 (13) 1tN
fΔ
=Δ
19
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz If eq(13) is satisfied
by using IDFTby using IDFT
q ( ) loz is the frequency domain signal loz is the time domain signal
( )ntkfjneA φπ +Δ02
)( tkSs Δ gloz is the sub-channel spacingloz is the symbol duration in each sub-channel
)(s
fΔtNΔloz s e sy bo du o e c sub c e
loz This outcome is the same as the result obtained in the
N
loz This outcome is the same as the result obtained in the system of Figure 1 Therefore IDFT can be used to generate an OFDM transmission signalgenerate an OFDM transmission signal
20
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
( )tf02cos π
( )0a
by using IDFTby using IDFT
( )nd ( )tDI( )tf02sin π
( )0a
( )0b
bull
tf s Δ=
1 ( ) ( ) njbna +
( )nd
tNΔffnffn Δ
=Δ+=1 0
( )tDChannelInput
( )tf N 12cos minusπ
( )1minusNa
bullbull
tNΔ
( )tf N 12sin minusπ
( )1minusNb
( )0d
( )nd ( )tDChannelInput
bull bull
( )( )1d
( )2d
tf s Δ=
1 ( ) ( ) njbna +
fNΔffnffn Δ
=Δ+=1 0
ChannelInput
bullbullbullbull
bullbullbullbull
( )1minusNd
21
Frequency Error Results in ICIFrequency Error Results in ICIq yq y
22
Synchronization Error Results in ICISynchronization Error Results in ICIyy
Not Orthogonal Any More
23
Cyclic PrefixCyclic Prefixyy
loz In multipath channel delayed replicas of previous OFDM signal lead to ISI between successive OFDM signals
loz Solution Insert a guard interval between successive OFDMloz Solution Insert a guard interval between successive OFDM signals
24
Cyclic PrefixCyclic Prefix
loz Guard interval leads to intercarrier interference (ICI) in OFDM d d l i
yy
demodulation
loz In DFT interval difference between two subcarriers does not maintain integer number of cycles loss of orthogonality
loz Delayed version of subcarrier 2 causes ICI in the process of d d l ti b i 1demodulating subcarrier 1
25
Cyclic PrefixCyclic Prefixloz Cyclic prefix (CP) A copy of the last part of OFDM signal is
attached to the front of itself
yy
attached to the front of itself
[ ]0d
[ ]1d
[ ]0D[ ]1D
[ ]nd[ ]kD~
symbols dataInput
bullbullbullbull
[ ]1d
[ ]2d
[ ][ ]2D
bullbullbullbullbull
y
bullbullbullbullbull
[ ]1minusNd [ ]1minusND
[ ]gNND minus
bullbullbull
26
Cyclic PrefixCyclic Prefix
loz All delayed replicas of subcarriers always have an
yy
y p yinteger number of cycles within DFT interval no ICI
27
Cyclic PrefixCyclic Prefix
loz Linear convolution vs circular convolution
yy
N
28
Cyclic PrefixCyclic Prefix
loz Channel effect with cyclic prefix
yy
Signal after removed CP
29
Cyclic PrefixCyclic Prefix
loz Time-Domain Explanation
yy
p
30
Cyclic PrefixCyclic Prefixyy
31
Cyclic PrefixCyclic Prefix
loz Spectrum of channel response with length (smaller [ ]nh hL
yy
than )h
gN[ ] nhFFTHk =
loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr
loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=
where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier
[ ] [ ] [ ]N
Notimes
[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce
kk H
X =rArrcomplexity of channel equalizationrdquo
32
Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))
bull bull( )( )nr~Symbols
dataOutput
bullbullbull
bullbullbull( )nr
bullbull
bullbull
33
Cyclic PrefixCyclic Prefix
loz One of the most important reasons to do OFDM is the
yy
pefficient way it deals with multipath delay spread
loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)
34
Bandwidth EfficiencyBandwidth Efficiencyyy
loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )
loz The diagram for bandwidth efficiency of OFDM system is shown below
35
SummarySummaryyy
loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j
the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference
(ISI)loz The main advantage of the OFDM transmission technique is its
high performance even in frequency selective channels
loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems
36
SynchronizationSynchronization
Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg
38 Synchronization2010722
Introduction (13)Introduction (13)( )( )
loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods
loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead
loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the
cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of
computational complexity therefore it may be not be available in short-burst wireless communication
39 Synchronization2010722
Introduction (23)Introduction (23)( )( )
loz Transcevier
40 Synchronization2010722
Introduction (33)Introduction (33)( )( )
loz Synchronization erroryloz Symbol timing offset (TO)
loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)
loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect
41 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg
42 Synchronization2010722
Signal model (14)Signal model (14)g ( )g ( )
loz Time domain sample p1
2
0
1 Nj nk N
n kk
x X eN
πminus
minus= sum
Xk the frequency domain data at the k-th subcarrier0kN =
loz The multipath fading channel
( ) ( ) ( )1L
h t h t δminus
sum
loz Received time domain signal
( ) ( ) ( )0
l ll
h t h tτ τ δ τ τ=
= minussum
loz Received time-domain signal
( ) ( )1
L
n n ly h n l x w nminus
minus= +sum43
( ) ( )0l=sum
Synchronization2010722
Signal model (24)Signal model (24)g ( )g ( )
loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting
12
0
Nj kn N
k nn
Y y e πminus
minus
=
= sum
( )1 1
2
0
N Lj kn N
n l kn l
h n l x e Wπminus minus
minusminus
=
= +sumsum
loz The equation above could be expressed in matrix form
0n l
loz The equation above could be expressed in matrix form
= +Y HX W
44 Synchronization2010722
Signal model (34)Signal model (34)g ( )g ( )
H= = = = +Y Fy FCx FCF X HX W
loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal
y
H=H FCFcirculant matrix leading to be a diagonal matrix
=H FCF
( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0
h h L L hh h
⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥
( )( ) ( )
( ) ( )
1 10 1 0 0i
h Lh L h N L
⎢ ⎥⎢ ⎥minus minus⎢ ⎥
equiv= minus minus⎢ ⎥⎢ ⎥
C( ) ( )
( ) ( )
0 1 1 100
0 0 1 1 0
h L h N L
h N L L h N
⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
45
( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦
Synchronization2010722
Signal model (44)Signal model (44)g ( )g ( )
loz In a time-invariant channel the channel matrix H would be a diagonal matrix
k k k kY H X W= +
( )1 1
2
0 0
N Lj kl N
kn l
H h n l e πminus minus
minus
= =
= sumsum
loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI
k k k k kY H X I Wα= + +k k k k k
( ) ( )1 1 1
2 2 N N L
j m k Nj ml Nk mI X h n l e e ππ
minus minus minusminusminus⎛ ⎞
= ⎜ ⎟⎝ ⎠
sum sum sumICI
46
0 0 0m n lm k= = =ne
⎝ ⎠
Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
47 Synchronization2010722
Impact of timing offset Impact of timing offset p gp g
loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively
( )I f s
Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ
( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr
2 fj k Nk k k kY X H e Wπδminus= +
loz For integer part of timing offset k k k kY X H e W+
I sTδ ( ) 0g IN L δminus minus le leFFT
2 Ij k NY X H e Wπδminus= +
( ) ( ) 2 I sFFT
j f TI sh t T H f e π δδ minusminus rarr
48
k k k kY X H e W= +
Synchronization2010722
Impact of timing offset derivationImpact of timing offset derivationp gp g
49 Synchronization2010722
Discussion of symbol boundaryDiscussion of symbol boundaryy yy y
50 Synchronization2010722
Impact of timing offsetImpact of timing offsetp gp g
loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI
2 Ij k NIk k k k
NY X H e ISI W
Nπδδ minusminus
= + +
loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk
e e
p kloz Phase Shift can be resolved by Differential Encoding
Decoding or carrier recovery with pilots
51 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
52 Synchronization2010722
Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )
53 Synchronization2010722
Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )
( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s
i n t i N N T N T nTy y t e
= + + +=
( ) ( )( ) ( ) ( ) 12
sin
sin
g gI f f
I I
i N N N Nj jf N Ni k i k k
f
Y X H e eN
π ε ε π ε
ε ε
πεπε
+ + minus+
minus minus=⎛ ⎞⎜ ⎟
Faded signal attenuated and rotated by CFO
( )( ) ( ) ( ) ( )11 2
sin
sin g gI f I f
i N N N NN j j l kI f
NN
l k π ε ε π ε επ ε ε + + minusminus + + + minus
⎜ ⎟⎝ ⎠
+ + minussum
( )( )( )
( ) ( )2
0
sin
I f I f
I
j j l kI f N Ni l l
l l k I f
X H e el k
NN
π ε ε π ε ε
ε π ε ε
+ + +
= ne minus
+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠
sum
i k
N
W
⎜ ⎟⎝ ⎠
+ Inter-Carrier Interference (ICI)
54 Synchronization2010722
Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )
loz An easier representation of received signal with only p g yfractional CFO is
1N
( ) ( )1
0
0N
k k k l l kl l k
Y H X C H X C l k Wminus
= ne
= + minus +⎡ ⎤⎣ ⎦sum
where the ICI coefficient is given by
( )( )( ) ( )
sin 1exp 1fl k
C l k j l kπ ε
π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )
( )exp 1sin
f
f
C l k j l kNN l k
N
π επ ε
minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠
55 Synchronization2010722
Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )
loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality
loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift
56 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
57 Synchronization2010722
TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )
loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq
58 Synchronization2010722
TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )
loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below
loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and
rN2 can be expressed by
0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+
( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +
59
( )
Synchronization2010722
TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )
loz Without CFO the first half is identical to the second half (in time order)
loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below
( ) 12 in discretesft T f NT
NTφ π π π ε πε
⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟
⎝ ⎠
helliphellip
sNT⎝ ⎠
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)
60
Length=N2
Synchronization2010722
TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )
loz Received signal without CFOg0
0 0
jr a e θ=jθ 2 0
2 2 0
Nj j
N Nr a e a e
θ θ= =
loz Received signal with CFO0 0
0 0
j jr a e eθ φ= Extra phase rotation due to CFO0 0
2 2 20
2 2 0
N N Nj j jj
N Nr a e e a e e
θ φ φθ= =
due to CFO
Phase difference 2 2 0N N
( )2 02 20 2 0 0
Nj j T fN
r r a e a eφ φ πminus Δ= =
which contains the information about CFO
61
0 2 0 0Nr r a e a e
Synchronization2010722
TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )
loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be
loz Note that d is a time index corresponding to the first( ) ( ) 2 1
20
Nd m d m Nm
P d r rminus
+ + +== sum
loz Note that d is a time index corresponding to the first sample in a window of N samples
loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by
( )22 1
20
Nd m NR d rminus
+ += sumloz A timing metric can be defined as
))(()(
)( 2
2
dRdP
dM =
( ) 20 d m Nm + +=sum
62
))(( dR
Synchronization2010722
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
loz If eq(13) is satisfied
by using IDFTby using IDFT
q ( ) loz is the frequency domain signal loz is the time domain signal
( )ntkfjneA φπ +Δ02
)( tkSs Δ gloz is the sub-channel spacingloz is the symbol duration in each sub-channel
)(s
fΔtNΔloz s e sy bo du o e c sub c e
loz This outcome is the same as the result obtained in the
N
loz This outcome is the same as the result obtained in the system of Figure 1 Therefore IDFT can be used to generate an OFDM transmission signalgenerate an OFDM transmission signal
20
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
( )tf02cos π
( )0a
by using IDFTby using IDFT
( )nd ( )tDI( )tf02sin π
( )0a
( )0b
bull
tf s Δ=
1 ( ) ( ) njbna +
( )nd
tNΔffnffn Δ
=Δ+=1 0
( )tDChannelInput
( )tf N 12cos minusπ
( )1minusNa
bullbull
tNΔ
( )tf N 12sin minusπ
( )1minusNb
( )0d
( )nd ( )tDChannelInput
bull bull
( )( )1d
( )2d
tf s Δ=
1 ( ) ( ) njbna +
fNΔffnffn Δ
=Δ+=1 0
ChannelInput
bullbullbullbull
bullbullbullbull
( )1minusNd
21
Frequency Error Results in ICIFrequency Error Results in ICIq yq y
22
Synchronization Error Results in ICISynchronization Error Results in ICIyy
Not Orthogonal Any More
23
Cyclic PrefixCyclic Prefixyy
loz In multipath channel delayed replicas of previous OFDM signal lead to ISI between successive OFDM signals
loz Solution Insert a guard interval between successive OFDMloz Solution Insert a guard interval between successive OFDM signals
24
Cyclic PrefixCyclic Prefix
loz Guard interval leads to intercarrier interference (ICI) in OFDM d d l i
yy
demodulation
loz In DFT interval difference between two subcarriers does not maintain integer number of cycles loss of orthogonality
loz Delayed version of subcarrier 2 causes ICI in the process of d d l ti b i 1demodulating subcarrier 1
25
Cyclic PrefixCyclic Prefixloz Cyclic prefix (CP) A copy of the last part of OFDM signal is
attached to the front of itself
yy
attached to the front of itself
[ ]0d
[ ]1d
[ ]0D[ ]1D
[ ]nd[ ]kD~
symbols dataInput
bullbullbullbull
[ ]1d
[ ]2d
[ ][ ]2D
bullbullbullbullbull
y
bullbullbullbullbull
[ ]1minusNd [ ]1minusND
[ ]gNND minus
bullbullbull
26
Cyclic PrefixCyclic Prefix
loz All delayed replicas of subcarriers always have an
yy
y p yinteger number of cycles within DFT interval no ICI
27
Cyclic PrefixCyclic Prefix
loz Linear convolution vs circular convolution
yy
N
28
Cyclic PrefixCyclic Prefix
loz Channel effect with cyclic prefix
yy
Signal after removed CP
29
Cyclic PrefixCyclic Prefix
loz Time-Domain Explanation
yy
p
30
Cyclic PrefixCyclic Prefixyy
31
Cyclic PrefixCyclic Prefix
loz Spectrum of channel response with length (smaller [ ]nh hL
yy
than )h
gN[ ] nhFFTHk =
loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr
loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=
where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier
[ ] [ ] [ ]N
Notimes
[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce
kk H
X =rArrcomplexity of channel equalizationrdquo
32
Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))
bull bull( )( )nr~Symbols
dataOutput
bullbullbull
bullbullbull( )nr
bullbull
bullbull
33
Cyclic PrefixCyclic Prefix
loz One of the most important reasons to do OFDM is the
yy
pefficient way it deals with multipath delay spread
loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)
34
Bandwidth EfficiencyBandwidth Efficiencyyy
loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )
loz The diagram for bandwidth efficiency of OFDM system is shown below
35
SummarySummaryyy
loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j
the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference
(ISI)loz The main advantage of the OFDM transmission technique is its
high performance even in frequency selective channels
loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems
36
SynchronizationSynchronization
Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg
38 Synchronization2010722
Introduction (13)Introduction (13)( )( )
loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods
loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead
loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the
cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of
computational complexity therefore it may be not be available in short-burst wireless communication
39 Synchronization2010722
Introduction (23)Introduction (23)( )( )
loz Transcevier
40 Synchronization2010722
Introduction (33)Introduction (33)( )( )
loz Synchronization erroryloz Symbol timing offset (TO)
loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)
loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect
41 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg
42 Synchronization2010722
Signal model (14)Signal model (14)g ( )g ( )
loz Time domain sample p1
2
0
1 Nj nk N
n kk
x X eN
πminus
minus= sum
Xk the frequency domain data at the k-th subcarrier0kN =
loz The multipath fading channel
( ) ( ) ( )1L
h t h t δminus
sum
loz Received time domain signal
( ) ( ) ( )0
l ll
h t h tτ τ δ τ τ=
= minussum
loz Received time-domain signal
( ) ( )1
L
n n ly h n l x w nminus
minus= +sum43
( ) ( )0l=sum
Synchronization2010722
Signal model (24)Signal model (24)g ( )g ( )
loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting
12
0
Nj kn N
k nn
Y y e πminus
minus
=
= sum
( )1 1
2
0
N Lj kn N
n l kn l
h n l x e Wπminus minus
minusminus
=
= +sumsum
loz The equation above could be expressed in matrix form
0n l
loz The equation above could be expressed in matrix form
= +Y HX W
44 Synchronization2010722
Signal model (34)Signal model (34)g ( )g ( )
H= = = = +Y Fy FCx FCF X HX W
loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal
y
H=H FCFcirculant matrix leading to be a diagonal matrix
=H FCF
( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0
h h L L hh h
⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥
( )( ) ( )
( ) ( )
1 10 1 0 0i
h Lh L h N L
⎢ ⎥⎢ ⎥minus minus⎢ ⎥
equiv= minus minus⎢ ⎥⎢ ⎥
C( ) ( )
( ) ( )
0 1 1 100
0 0 1 1 0
h L h N L
h N L L h N
⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
45
( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦
Synchronization2010722
Signal model (44)Signal model (44)g ( )g ( )
loz In a time-invariant channel the channel matrix H would be a diagonal matrix
k k k kY H X W= +
( )1 1
2
0 0
N Lj kl N
kn l
H h n l e πminus minus
minus
= =
= sumsum
loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI
k k k k kY H X I Wα= + +k k k k k
( ) ( )1 1 1
2 2 N N L
j m k Nj ml Nk mI X h n l e e ππ
minus minus minusminusminus⎛ ⎞
= ⎜ ⎟⎝ ⎠
sum sum sumICI
46
0 0 0m n lm k= = =ne
⎝ ⎠
Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
47 Synchronization2010722
Impact of timing offset Impact of timing offset p gp g
loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively
( )I f s
Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ
( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr
2 fj k Nk k k kY X H e Wπδminus= +
loz For integer part of timing offset k k k kY X H e W+
I sTδ ( ) 0g IN L δminus minus le leFFT
2 Ij k NY X H e Wπδminus= +
( ) ( ) 2 I sFFT
j f TI sh t T H f e π δδ minusminus rarr
48
k k k kY X H e W= +
Synchronization2010722
Impact of timing offset derivationImpact of timing offset derivationp gp g
49 Synchronization2010722
Discussion of symbol boundaryDiscussion of symbol boundaryy yy y
50 Synchronization2010722
Impact of timing offsetImpact of timing offsetp gp g
loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI
2 Ij k NIk k k k
NY X H e ISI W
Nπδδ minusminus
= + +
loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk
e e
p kloz Phase Shift can be resolved by Differential Encoding
Decoding or carrier recovery with pilots
51 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
52 Synchronization2010722
Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )
53 Synchronization2010722
Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )
( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s
i n t i N N T N T nTy y t e
= + + +=
( ) ( )( ) ( ) ( ) 12
sin
sin
g gI f f
I I
i N N N Nj jf N Ni k i k k
f
Y X H e eN
π ε ε π ε
ε ε
πεπε
+ + minus+
minus minus=⎛ ⎞⎜ ⎟
Faded signal attenuated and rotated by CFO
( )( ) ( ) ( ) ( )11 2
sin
sin g gI f I f
i N N N NN j j l kI f
NN
l k π ε ε π ε επ ε ε + + minusminus + + + minus
⎜ ⎟⎝ ⎠
+ + minussum
( )( )( )
( ) ( )2
0
sin
I f I f
I
j j l kI f N Ni l l
l l k I f
X H e el k
NN
π ε ε π ε ε
ε π ε ε
+ + +
= ne minus
+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠
sum
i k
N
W
⎜ ⎟⎝ ⎠
+ Inter-Carrier Interference (ICI)
54 Synchronization2010722
Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )
loz An easier representation of received signal with only p g yfractional CFO is
1N
( ) ( )1
0
0N
k k k l l kl l k
Y H X C H X C l k Wminus
= ne
= + minus +⎡ ⎤⎣ ⎦sum
where the ICI coefficient is given by
( )( )( ) ( )
sin 1exp 1fl k
C l k j l kπ ε
π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )
( )exp 1sin
f
f
C l k j l kNN l k
N
π επ ε
minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠
55 Synchronization2010722
Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )
loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality
loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift
56 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
57 Synchronization2010722
TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )
loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq
58 Synchronization2010722
TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )
loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below
loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and
rN2 can be expressed by
0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+
( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +
59
( )
Synchronization2010722
TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )
loz Without CFO the first half is identical to the second half (in time order)
loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below
( ) 12 in discretesft T f NT
NTφ π π π ε πε
⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟
⎝ ⎠
helliphellip
sNT⎝ ⎠
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)
60
Length=N2
Synchronization2010722
TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )
loz Received signal without CFOg0
0 0
jr a e θ=jθ 2 0
2 2 0
Nj j
N Nr a e a e
θ θ= =
loz Received signal with CFO0 0
0 0
j jr a e eθ φ= Extra phase rotation due to CFO0 0
2 2 20
2 2 0
N N Nj j jj
N Nr a e e a e e
θ φ φθ= =
due to CFO
Phase difference 2 2 0N N
( )2 02 20 2 0 0
Nj j T fN
r r a e a eφ φ πminus Δ= =
which contains the information about CFO
61
0 2 0 0Nr r a e a e
Synchronization2010722
TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )
loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be
loz Note that d is a time index corresponding to the first( ) ( ) 2 1
20
Nd m d m Nm
P d r rminus
+ + +== sum
loz Note that d is a time index corresponding to the first sample in a window of N samples
loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by
( )22 1
20
Nd m NR d rminus
+ += sumloz A timing metric can be defined as
))(()(
)( 2
2
dRdP
dM =
( ) 20 d m Nm + +=sum
62
))(( dR
Synchronization2010722
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
MultiMulti--carrier Equivalent Implementation carrier Equivalent Implementation by using IDFTby using IDFT
( )tf02cos π
( )0a
by using IDFTby using IDFT
( )nd ( )tDI( )tf02sin π
( )0a
( )0b
bull
tf s Δ=
1 ( ) ( ) njbna +
( )nd
tNΔffnffn Δ
=Δ+=1 0
( )tDChannelInput
( )tf N 12cos minusπ
( )1minusNa
bullbull
tNΔ
( )tf N 12sin minusπ
( )1minusNb
( )0d
( )nd ( )tDChannelInput
bull bull
( )( )1d
( )2d
tf s Δ=
1 ( ) ( ) njbna +
fNΔffnffn Δ
=Δ+=1 0
ChannelInput
bullbullbullbull
bullbullbullbull
( )1minusNd
21
Frequency Error Results in ICIFrequency Error Results in ICIq yq y
22
Synchronization Error Results in ICISynchronization Error Results in ICIyy
Not Orthogonal Any More
23
Cyclic PrefixCyclic Prefixyy
loz In multipath channel delayed replicas of previous OFDM signal lead to ISI between successive OFDM signals
loz Solution Insert a guard interval between successive OFDMloz Solution Insert a guard interval between successive OFDM signals
24
Cyclic PrefixCyclic Prefix
loz Guard interval leads to intercarrier interference (ICI) in OFDM d d l i
yy
demodulation
loz In DFT interval difference between two subcarriers does not maintain integer number of cycles loss of orthogonality
loz Delayed version of subcarrier 2 causes ICI in the process of d d l ti b i 1demodulating subcarrier 1
25
Cyclic PrefixCyclic Prefixloz Cyclic prefix (CP) A copy of the last part of OFDM signal is
attached to the front of itself
yy
attached to the front of itself
[ ]0d
[ ]1d
[ ]0D[ ]1D
[ ]nd[ ]kD~
symbols dataInput
bullbullbullbull
[ ]1d
[ ]2d
[ ][ ]2D
bullbullbullbullbull
y
bullbullbullbullbull
[ ]1minusNd [ ]1minusND
[ ]gNND minus
bullbullbull
26
Cyclic PrefixCyclic Prefix
loz All delayed replicas of subcarriers always have an
yy
y p yinteger number of cycles within DFT interval no ICI
27
Cyclic PrefixCyclic Prefix
loz Linear convolution vs circular convolution
yy
N
28
Cyclic PrefixCyclic Prefix
loz Channel effect with cyclic prefix
yy
Signal after removed CP
29
Cyclic PrefixCyclic Prefix
loz Time-Domain Explanation
yy
p
30
Cyclic PrefixCyclic Prefixyy
31
Cyclic PrefixCyclic Prefix
loz Spectrum of channel response with length (smaller [ ]nh hL
yy
than )h
gN[ ] nhFFTHk =
loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr
loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=
where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier
[ ] [ ] [ ]N
Notimes
[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce
kk H
X =rArrcomplexity of channel equalizationrdquo
32
Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))
bull bull( )( )nr~Symbols
dataOutput
bullbullbull
bullbullbull( )nr
bullbull
bullbull
33
Cyclic PrefixCyclic Prefix
loz One of the most important reasons to do OFDM is the
yy
pefficient way it deals with multipath delay spread
loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)
34
Bandwidth EfficiencyBandwidth Efficiencyyy
loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )
loz The diagram for bandwidth efficiency of OFDM system is shown below
35
SummarySummaryyy
loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j
the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference
(ISI)loz The main advantage of the OFDM transmission technique is its
high performance even in frequency selective channels
loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems
36
SynchronizationSynchronization
Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg
38 Synchronization2010722
Introduction (13)Introduction (13)( )( )
loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods
loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead
loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the
cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of
computational complexity therefore it may be not be available in short-burst wireless communication
39 Synchronization2010722
Introduction (23)Introduction (23)( )( )
loz Transcevier
40 Synchronization2010722
Introduction (33)Introduction (33)( )( )
loz Synchronization erroryloz Symbol timing offset (TO)
loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)
loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect
41 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg
42 Synchronization2010722
Signal model (14)Signal model (14)g ( )g ( )
loz Time domain sample p1
2
0
1 Nj nk N
n kk
x X eN
πminus
minus= sum
Xk the frequency domain data at the k-th subcarrier0kN =
loz The multipath fading channel
( ) ( ) ( )1L
h t h t δminus
sum
loz Received time domain signal
( ) ( ) ( )0
l ll
h t h tτ τ δ τ τ=
= minussum
loz Received time-domain signal
( ) ( )1
L
n n ly h n l x w nminus
minus= +sum43
( ) ( )0l=sum
Synchronization2010722
Signal model (24)Signal model (24)g ( )g ( )
loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting
12
0
Nj kn N
k nn
Y y e πminus
minus
=
= sum
( )1 1
2
0
N Lj kn N
n l kn l
h n l x e Wπminus minus
minusminus
=
= +sumsum
loz The equation above could be expressed in matrix form
0n l
loz The equation above could be expressed in matrix form
= +Y HX W
44 Synchronization2010722
Signal model (34)Signal model (34)g ( )g ( )
H= = = = +Y Fy FCx FCF X HX W
loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal
y
H=H FCFcirculant matrix leading to be a diagonal matrix
=H FCF
( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0
h h L L hh h
⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥
( )( ) ( )
( ) ( )
1 10 1 0 0i
h Lh L h N L
⎢ ⎥⎢ ⎥minus minus⎢ ⎥
equiv= minus minus⎢ ⎥⎢ ⎥
C( ) ( )
( ) ( )
0 1 1 100
0 0 1 1 0
h L h N L
h N L L h N
⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
45
( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦
Synchronization2010722
Signal model (44)Signal model (44)g ( )g ( )
loz In a time-invariant channel the channel matrix H would be a diagonal matrix
k k k kY H X W= +
( )1 1
2
0 0
N Lj kl N
kn l
H h n l e πminus minus
minus
= =
= sumsum
loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI
k k k k kY H X I Wα= + +k k k k k
( ) ( )1 1 1
2 2 N N L
j m k Nj ml Nk mI X h n l e e ππ
minus minus minusminusminus⎛ ⎞
= ⎜ ⎟⎝ ⎠
sum sum sumICI
46
0 0 0m n lm k= = =ne
⎝ ⎠
Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
47 Synchronization2010722
Impact of timing offset Impact of timing offset p gp g
loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively
( )I f s
Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ
( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr
2 fj k Nk k k kY X H e Wπδminus= +
loz For integer part of timing offset k k k kY X H e W+
I sTδ ( ) 0g IN L δminus minus le leFFT
2 Ij k NY X H e Wπδminus= +
( ) ( ) 2 I sFFT
j f TI sh t T H f e π δδ minusminus rarr
48
k k k kY X H e W= +
Synchronization2010722
Impact of timing offset derivationImpact of timing offset derivationp gp g
49 Synchronization2010722
Discussion of symbol boundaryDiscussion of symbol boundaryy yy y
50 Synchronization2010722
Impact of timing offsetImpact of timing offsetp gp g
loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI
2 Ij k NIk k k k
NY X H e ISI W
Nπδδ minusminus
= + +
loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk
e e
p kloz Phase Shift can be resolved by Differential Encoding
Decoding or carrier recovery with pilots
51 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
52 Synchronization2010722
Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )
53 Synchronization2010722
Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )
( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s
i n t i N N T N T nTy y t e
= + + +=
( ) ( )( ) ( ) ( ) 12
sin
sin
g gI f f
I I
i N N N Nj jf N Ni k i k k
f
Y X H e eN
π ε ε π ε
ε ε
πεπε
+ + minus+
minus minus=⎛ ⎞⎜ ⎟
Faded signal attenuated and rotated by CFO
( )( ) ( ) ( ) ( )11 2
sin
sin g gI f I f
i N N N NN j j l kI f
NN
l k π ε ε π ε επ ε ε + + minusminus + + + minus
⎜ ⎟⎝ ⎠
+ + minussum
( )( )( )
( ) ( )2
0
sin
I f I f
I
j j l kI f N Ni l l
l l k I f
X H e el k
NN
π ε ε π ε ε
ε π ε ε
+ + +
= ne minus
+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠
sum
i k
N
W
⎜ ⎟⎝ ⎠
+ Inter-Carrier Interference (ICI)
54 Synchronization2010722
Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )
loz An easier representation of received signal with only p g yfractional CFO is
1N
( ) ( )1
0
0N
k k k l l kl l k
Y H X C H X C l k Wminus
= ne
= + minus +⎡ ⎤⎣ ⎦sum
where the ICI coefficient is given by
( )( )( ) ( )
sin 1exp 1fl k
C l k j l kπ ε
π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )
( )exp 1sin
f
f
C l k j l kNN l k
N
π επ ε
minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠
55 Synchronization2010722
Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )
loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality
loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift
56 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
57 Synchronization2010722
TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )
loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq
58 Synchronization2010722
TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )
loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below
loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and
rN2 can be expressed by
0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+
( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +
59
( )
Synchronization2010722
TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )
loz Without CFO the first half is identical to the second half (in time order)
loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below
( ) 12 in discretesft T f NT
NTφ π π π ε πε
⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟
⎝ ⎠
helliphellip
sNT⎝ ⎠
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)
60
Length=N2
Synchronization2010722
TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )
loz Received signal without CFOg0
0 0
jr a e θ=jθ 2 0
2 2 0
Nj j
N Nr a e a e
θ θ= =
loz Received signal with CFO0 0
0 0
j jr a e eθ φ= Extra phase rotation due to CFO0 0
2 2 20
2 2 0
N N Nj j jj
N Nr a e e a e e
θ φ φθ= =
due to CFO
Phase difference 2 2 0N N
( )2 02 20 2 0 0
Nj j T fN
r r a e a eφ φ πminus Δ= =
which contains the information about CFO
61
0 2 0 0Nr r a e a e
Synchronization2010722
TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )
loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be
loz Note that d is a time index corresponding to the first( ) ( ) 2 1
20
Nd m d m Nm
P d r rminus
+ + +== sum
loz Note that d is a time index corresponding to the first sample in a window of N samples
loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by
( )22 1
20
Nd m NR d rminus
+ += sumloz A timing metric can be defined as
))(()(
)( 2
2
dRdP
dM =
( ) 20 d m Nm + +=sum
62
))(( dR
Synchronization2010722
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
Frequency Error Results in ICIFrequency Error Results in ICIq yq y
22
Synchronization Error Results in ICISynchronization Error Results in ICIyy
Not Orthogonal Any More
23
Cyclic PrefixCyclic Prefixyy
loz In multipath channel delayed replicas of previous OFDM signal lead to ISI between successive OFDM signals
loz Solution Insert a guard interval between successive OFDMloz Solution Insert a guard interval between successive OFDM signals
24
Cyclic PrefixCyclic Prefix
loz Guard interval leads to intercarrier interference (ICI) in OFDM d d l i
yy
demodulation
loz In DFT interval difference between two subcarriers does not maintain integer number of cycles loss of orthogonality
loz Delayed version of subcarrier 2 causes ICI in the process of d d l ti b i 1demodulating subcarrier 1
25
Cyclic PrefixCyclic Prefixloz Cyclic prefix (CP) A copy of the last part of OFDM signal is
attached to the front of itself
yy
attached to the front of itself
[ ]0d
[ ]1d
[ ]0D[ ]1D
[ ]nd[ ]kD~
symbols dataInput
bullbullbullbull
[ ]1d
[ ]2d
[ ][ ]2D
bullbullbullbullbull
y
bullbullbullbullbull
[ ]1minusNd [ ]1minusND
[ ]gNND minus
bullbullbull
26
Cyclic PrefixCyclic Prefix
loz All delayed replicas of subcarriers always have an
yy
y p yinteger number of cycles within DFT interval no ICI
27
Cyclic PrefixCyclic Prefix
loz Linear convolution vs circular convolution
yy
N
28
Cyclic PrefixCyclic Prefix
loz Channel effect with cyclic prefix
yy
Signal after removed CP
29
Cyclic PrefixCyclic Prefix
loz Time-Domain Explanation
yy
p
30
Cyclic PrefixCyclic Prefixyy
31
Cyclic PrefixCyclic Prefix
loz Spectrum of channel response with length (smaller [ ]nh hL
yy
than )h
gN[ ] nhFFTHk =
loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr
loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=
where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier
[ ] [ ] [ ]N
Notimes
[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce
kk H
X =rArrcomplexity of channel equalizationrdquo
32
Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))
bull bull( )( )nr~Symbols
dataOutput
bullbullbull
bullbullbull( )nr
bullbull
bullbull
33
Cyclic PrefixCyclic Prefix
loz One of the most important reasons to do OFDM is the
yy
pefficient way it deals with multipath delay spread
loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)
34
Bandwidth EfficiencyBandwidth Efficiencyyy
loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )
loz The diagram for bandwidth efficiency of OFDM system is shown below
35
SummarySummaryyy
loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j
the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference
(ISI)loz The main advantage of the OFDM transmission technique is its
high performance even in frequency selective channels
loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems
36
SynchronizationSynchronization
Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg
38 Synchronization2010722
Introduction (13)Introduction (13)( )( )
loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods
loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead
loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the
cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of
computational complexity therefore it may be not be available in short-burst wireless communication
39 Synchronization2010722
Introduction (23)Introduction (23)( )( )
loz Transcevier
40 Synchronization2010722
Introduction (33)Introduction (33)( )( )
loz Synchronization erroryloz Symbol timing offset (TO)
loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)
loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect
41 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg
42 Synchronization2010722
Signal model (14)Signal model (14)g ( )g ( )
loz Time domain sample p1
2
0
1 Nj nk N
n kk
x X eN
πminus
minus= sum
Xk the frequency domain data at the k-th subcarrier0kN =
loz The multipath fading channel
( ) ( ) ( )1L
h t h t δminus
sum
loz Received time domain signal
( ) ( ) ( )0
l ll
h t h tτ τ δ τ τ=
= minussum
loz Received time-domain signal
( ) ( )1
L
n n ly h n l x w nminus
minus= +sum43
( ) ( )0l=sum
Synchronization2010722
Signal model (24)Signal model (24)g ( )g ( )
loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting
12
0
Nj kn N
k nn
Y y e πminus
minus
=
= sum
( )1 1
2
0
N Lj kn N
n l kn l
h n l x e Wπminus minus
minusminus
=
= +sumsum
loz The equation above could be expressed in matrix form
0n l
loz The equation above could be expressed in matrix form
= +Y HX W
44 Synchronization2010722
Signal model (34)Signal model (34)g ( )g ( )
H= = = = +Y Fy FCx FCF X HX W
loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal
y
H=H FCFcirculant matrix leading to be a diagonal matrix
=H FCF
( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0
h h L L hh h
⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥
( )( ) ( )
( ) ( )
1 10 1 0 0i
h Lh L h N L
⎢ ⎥⎢ ⎥minus minus⎢ ⎥
equiv= minus minus⎢ ⎥⎢ ⎥
C( ) ( )
( ) ( )
0 1 1 100
0 0 1 1 0
h L h N L
h N L L h N
⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
45
( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦
Synchronization2010722
Signal model (44)Signal model (44)g ( )g ( )
loz In a time-invariant channel the channel matrix H would be a diagonal matrix
k k k kY H X W= +
( )1 1
2
0 0
N Lj kl N
kn l
H h n l e πminus minus
minus
= =
= sumsum
loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI
k k k k kY H X I Wα= + +k k k k k
( ) ( )1 1 1
2 2 N N L
j m k Nj ml Nk mI X h n l e e ππ
minus minus minusminusminus⎛ ⎞
= ⎜ ⎟⎝ ⎠
sum sum sumICI
46
0 0 0m n lm k= = =ne
⎝ ⎠
Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
47 Synchronization2010722
Impact of timing offset Impact of timing offset p gp g
loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively
( )I f s
Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ
( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr
2 fj k Nk k k kY X H e Wπδminus= +
loz For integer part of timing offset k k k kY X H e W+
I sTδ ( ) 0g IN L δminus minus le leFFT
2 Ij k NY X H e Wπδminus= +
( ) ( ) 2 I sFFT
j f TI sh t T H f e π δδ minusminus rarr
48
k k k kY X H e W= +
Synchronization2010722
Impact of timing offset derivationImpact of timing offset derivationp gp g
49 Synchronization2010722
Discussion of symbol boundaryDiscussion of symbol boundaryy yy y
50 Synchronization2010722
Impact of timing offsetImpact of timing offsetp gp g
loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI
2 Ij k NIk k k k
NY X H e ISI W
Nπδδ minusminus
= + +
loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk
e e
p kloz Phase Shift can be resolved by Differential Encoding
Decoding or carrier recovery with pilots
51 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
52 Synchronization2010722
Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )
53 Synchronization2010722
Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )
( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s
i n t i N N T N T nTy y t e
= + + +=
( ) ( )( ) ( ) ( ) 12
sin
sin
g gI f f
I I
i N N N Nj jf N Ni k i k k
f
Y X H e eN
π ε ε π ε
ε ε
πεπε
+ + minus+
minus minus=⎛ ⎞⎜ ⎟
Faded signal attenuated and rotated by CFO
( )( ) ( ) ( ) ( )11 2
sin
sin g gI f I f
i N N N NN j j l kI f
NN
l k π ε ε π ε επ ε ε + + minusminus + + + minus
⎜ ⎟⎝ ⎠
+ + minussum
( )( )( )
( ) ( )2
0
sin
I f I f
I
j j l kI f N Ni l l
l l k I f
X H e el k
NN
π ε ε π ε ε
ε π ε ε
+ + +
= ne minus
+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠
sum
i k
N
W
⎜ ⎟⎝ ⎠
+ Inter-Carrier Interference (ICI)
54 Synchronization2010722
Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )
loz An easier representation of received signal with only p g yfractional CFO is
1N
( ) ( )1
0
0N
k k k l l kl l k
Y H X C H X C l k Wminus
= ne
= + minus +⎡ ⎤⎣ ⎦sum
where the ICI coefficient is given by
( )( )( ) ( )
sin 1exp 1fl k
C l k j l kπ ε
π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )
( )exp 1sin
f
f
C l k j l kNN l k
N
π επ ε
minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠
55 Synchronization2010722
Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )
loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality
loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift
56 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
57 Synchronization2010722
TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )
loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq
58 Synchronization2010722
TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )
loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below
loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and
rN2 can be expressed by
0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+
( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +
59
( )
Synchronization2010722
TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )
loz Without CFO the first half is identical to the second half (in time order)
loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below
( ) 12 in discretesft T f NT
NTφ π π π ε πε
⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟
⎝ ⎠
helliphellip
sNT⎝ ⎠
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)
60
Length=N2
Synchronization2010722
TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )
loz Received signal without CFOg0
0 0
jr a e θ=jθ 2 0
2 2 0
Nj j
N Nr a e a e
θ θ= =
loz Received signal with CFO0 0
0 0
j jr a e eθ φ= Extra phase rotation due to CFO0 0
2 2 20
2 2 0
N N Nj j jj
N Nr a e e a e e
θ φ φθ= =
due to CFO
Phase difference 2 2 0N N
( )2 02 20 2 0 0
Nj j T fN
r r a e a eφ φ πminus Δ= =
which contains the information about CFO
61
0 2 0 0Nr r a e a e
Synchronization2010722
TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )
loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be
loz Note that d is a time index corresponding to the first( ) ( ) 2 1
20
Nd m d m Nm
P d r rminus
+ + +== sum
loz Note that d is a time index corresponding to the first sample in a window of N samples
loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by
( )22 1
20
Nd m NR d rminus
+ += sumloz A timing metric can be defined as
))(()(
)( 2
2
dRdP
dM =
( ) 20 d m Nm + +=sum
62
))(( dR
Synchronization2010722
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
Synchronization Error Results in ICISynchronization Error Results in ICIyy
Not Orthogonal Any More
23
Cyclic PrefixCyclic Prefixyy
loz In multipath channel delayed replicas of previous OFDM signal lead to ISI between successive OFDM signals
loz Solution Insert a guard interval between successive OFDMloz Solution Insert a guard interval between successive OFDM signals
24
Cyclic PrefixCyclic Prefix
loz Guard interval leads to intercarrier interference (ICI) in OFDM d d l i
yy
demodulation
loz In DFT interval difference between two subcarriers does not maintain integer number of cycles loss of orthogonality
loz Delayed version of subcarrier 2 causes ICI in the process of d d l ti b i 1demodulating subcarrier 1
25
Cyclic PrefixCyclic Prefixloz Cyclic prefix (CP) A copy of the last part of OFDM signal is
attached to the front of itself
yy
attached to the front of itself
[ ]0d
[ ]1d
[ ]0D[ ]1D
[ ]nd[ ]kD~
symbols dataInput
bullbullbullbull
[ ]1d
[ ]2d
[ ][ ]2D
bullbullbullbullbull
y
bullbullbullbullbull
[ ]1minusNd [ ]1minusND
[ ]gNND minus
bullbullbull
26
Cyclic PrefixCyclic Prefix
loz All delayed replicas of subcarriers always have an
yy
y p yinteger number of cycles within DFT interval no ICI
27
Cyclic PrefixCyclic Prefix
loz Linear convolution vs circular convolution
yy
N
28
Cyclic PrefixCyclic Prefix
loz Channel effect with cyclic prefix
yy
Signal after removed CP
29
Cyclic PrefixCyclic Prefix
loz Time-Domain Explanation
yy
p
30
Cyclic PrefixCyclic Prefixyy
31
Cyclic PrefixCyclic Prefix
loz Spectrum of channel response with length (smaller [ ]nh hL
yy
than )h
gN[ ] nhFFTHk =
loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr
loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=
where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier
[ ] [ ] [ ]N
Notimes
[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce
kk H
X =rArrcomplexity of channel equalizationrdquo
32
Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))
bull bull( )( )nr~Symbols
dataOutput
bullbullbull
bullbullbull( )nr
bullbull
bullbull
33
Cyclic PrefixCyclic Prefix
loz One of the most important reasons to do OFDM is the
yy
pefficient way it deals with multipath delay spread
loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)
34
Bandwidth EfficiencyBandwidth Efficiencyyy
loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )
loz The diagram for bandwidth efficiency of OFDM system is shown below
35
SummarySummaryyy
loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j
the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference
(ISI)loz The main advantage of the OFDM transmission technique is its
high performance even in frequency selective channels
loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems
36
SynchronizationSynchronization
Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg
38 Synchronization2010722
Introduction (13)Introduction (13)( )( )
loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods
loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead
loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the
cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of
computational complexity therefore it may be not be available in short-burst wireless communication
39 Synchronization2010722
Introduction (23)Introduction (23)( )( )
loz Transcevier
40 Synchronization2010722
Introduction (33)Introduction (33)( )( )
loz Synchronization erroryloz Symbol timing offset (TO)
loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)
loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect
41 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg
42 Synchronization2010722
Signal model (14)Signal model (14)g ( )g ( )
loz Time domain sample p1
2
0
1 Nj nk N
n kk
x X eN
πminus
minus= sum
Xk the frequency domain data at the k-th subcarrier0kN =
loz The multipath fading channel
( ) ( ) ( )1L
h t h t δminus
sum
loz Received time domain signal
( ) ( ) ( )0
l ll
h t h tτ τ δ τ τ=
= minussum
loz Received time-domain signal
( ) ( )1
L
n n ly h n l x w nminus
minus= +sum43
( ) ( )0l=sum
Synchronization2010722
Signal model (24)Signal model (24)g ( )g ( )
loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting
12
0
Nj kn N
k nn
Y y e πminus
minus
=
= sum
( )1 1
2
0
N Lj kn N
n l kn l
h n l x e Wπminus minus
minusminus
=
= +sumsum
loz The equation above could be expressed in matrix form
0n l
loz The equation above could be expressed in matrix form
= +Y HX W
44 Synchronization2010722
Signal model (34)Signal model (34)g ( )g ( )
H= = = = +Y Fy FCx FCF X HX W
loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal
y
H=H FCFcirculant matrix leading to be a diagonal matrix
=H FCF
( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0
h h L L hh h
⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥
( )( ) ( )
( ) ( )
1 10 1 0 0i
h Lh L h N L
⎢ ⎥⎢ ⎥minus minus⎢ ⎥
equiv= minus minus⎢ ⎥⎢ ⎥
C( ) ( )
( ) ( )
0 1 1 100
0 0 1 1 0
h L h N L
h N L L h N
⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
45
( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦
Synchronization2010722
Signal model (44)Signal model (44)g ( )g ( )
loz In a time-invariant channel the channel matrix H would be a diagonal matrix
k k k kY H X W= +
( )1 1
2
0 0
N Lj kl N
kn l
H h n l e πminus minus
minus
= =
= sumsum
loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI
k k k k kY H X I Wα= + +k k k k k
( ) ( )1 1 1
2 2 N N L
j m k Nj ml Nk mI X h n l e e ππ
minus minus minusminusminus⎛ ⎞
= ⎜ ⎟⎝ ⎠
sum sum sumICI
46
0 0 0m n lm k= = =ne
⎝ ⎠
Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
47 Synchronization2010722
Impact of timing offset Impact of timing offset p gp g
loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively
( )I f s
Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ
( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr
2 fj k Nk k k kY X H e Wπδminus= +
loz For integer part of timing offset k k k kY X H e W+
I sTδ ( ) 0g IN L δminus minus le leFFT
2 Ij k NY X H e Wπδminus= +
( ) ( ) 2 I sFFT
j f TI sh t T H f e π δδ minusminus rarr
48
k k k kY X H e W= +
Synchronization2010722
Impact of timing offset derivationImpact of timing offset derivationp gp g
49 Synchronization2010722
Discussion of symbol boundaryDiscussion of symbol boundaryy yy y
50 Synchronization2010722
Impact of timing offsetImpact of timing offsetp gp g
loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI
2 Ij k NIk k k k
NY X H e ISI W
Nπδδ minusminus
= + +
loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk
e e
p kloz Phase Shift can be resolved by Differential Encoding
Decoding or carrier recovery with pilots
51 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
52 Synchronization2010722
Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )
53 Synchronization2010722
Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )
( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s
i n t i N N T N T nTy y t e
= + + +=
( ) ( )( ) ( ) ( ) 12
sin
sin
g gI f f
I I
i N N N Nj jf N Ni k i k k
f
Y X H e eN
π ε ε π ε
ε ε
πεπε
+ + minus+
minus minus=⎛ ⎞⎜ ⎟
Faded signal attenuated and rotated by CFO
( )( ) ( ) ( ) ( )11 2
sin
sin g gI f I f
i N N N NN j j l kI f
NN
l k π ε ε π ε επ ε ε + + minusminus + + + minus
⎜ ⎟⎝ ⎠
+ + minussum
( )( )( )
( ) ( )2
0
sin
I f I f
I
j j l kI f N Ni l l
l l k I f
X H e el k
NN
π ε ε π ε ε
ε π ε ε
+ + +
= ne minus
+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠
sum
i k
N
W
⎜ ⎟⎝ ⎠
+ Inter-Carrier Interference (ICI)
54 Synchronization2010722
Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )
loz An easier representation of received signal with only p g yfractional CFO is
1N
( ) ( )1
0
0N
k k k l l kl l k
Y H X C H X C l k Wminus
= ne
= + minus +⎡ ⎤⎣ ⎦sum
where the ICI coefficient is given by
( )( )( ) ( )
sin 1exp 1fl k
C l k j l kπ ε
π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )
( )exp 1sin
f
f
C l k j l kNN l k
N
π επ ε
minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠
55 Synchronization2010722
Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )
loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality
loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift
56 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
57 Synchronization2010722
TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )
loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq
58 Synchronization2010722
TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )
loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below
loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and
rN2 can be expressed by
0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+
( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +
59
( )
Synchronization2010722
TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )
loz Without CFO the first half is identical to the second half (in time order)
loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below
( ) 12 in discretesft T f NT
NTφ π π π ε πε
⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟
⎝ ⎠
helliphellip
sNT⎝ ⎠
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)
60
Length=N2
Synchronization2010722
TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )
loz Received signal without CFOg0
0 0
jr a e θ=jθ 2 0
2 2 0
Nj j
N Nr a e a e
θ θ= =
loz Received signal with CFO0 0
0 0
j jr a e eθ φ= Extra phase rotation due to CFO0 0
2 2 20
2 2 0
N N Nj j jj
N Nr a e e a e e
θ φ φθ= =
due to CFO
Phase difference 2 2 0N N
( )2 02 20 2 0 0
Nj j T fN
r r a e a eφ φ πminus Δ= =
which contains the information about CFO
61
0 2 0 0Nr r a e a e
Synchronization2010722
TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )
loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be
loz Note that d is a time index corresponding to the first( ) ( ) 2 1
20
Nd m d m Nm
P d r rminus
+ + +== sum
loz Note that d is a time index corresponding to the first sample in a window of N samples
loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by
( )22 1
20
Nd m NR d rminus
+ += sumloz A timing metric can be defined as
))(()(
)( 2
2
dRdP
dM =
( ) 20 d m Nm + +=sum
62
))(( dR
Synchronization2010722
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
Cyclic PrefixCyclic Prefixyy
loz In multipath channel delayed replicas of previous OFDM signal lead to ISI between successive OFDM signals
loz Solution Insert a guard interval between successive OFDMloz Solution Insert a guard interval between successive OFDM signals
24
Cyclic PrefixCyclic Prefix
loz Guard interval leads to intercarrier interference (ICI) in OFDM d d l i
yy
demodulation
loz In DFT interval difference between two subcarriers does not maintain integer number of cycles loss of orthogonality
loz Delayed version of subcarrier 2 causes ICI in the process of d d l ti b i 1demodulating subcarrier 1
25
Cyclic PrefixCyclic Prefixloz Cyclic prefix (CP) A copy of the last part of OFDM signal is
attached to the front of itself
yy
attached to the front of itself
[ ]0d
[ ]1d
[ ]0D[ ]1D
[ ]nd[ ]kD~
symbols dataInput
bullbullbullbull
[ ]1d
[ ]2d
[ ][ ]2D
bullbullbullbullbull
y
bullbullbullbullbull
[ ]1minusNd [ ]1minusND
[ ]gNND minus
bullbullbull
26
Cyclic PrefixCyclic Prefix
loz All delayed replicas of subcarriers always have an
yy
y p yinteger number of cycles within DFT interval no ICI
27
Cyclic PrefixCyclic Prefix
loz Linear convolution vs circular convolution
yy
N
28
Cyclic PrefixCyclic Prefix
loz Channel effect with cyclic prefix
yy
Signal after removed CP
29
Cyclic PrefixCyclic Prefix
loz Time-Domain Explanation
yy
p
30
Cyclic PrefixCyclic Prefixyy
31
Cyclic PrefixCyclic Prefix
loz Spectrum of channel response with length (smaller [ ]nh hL
yy
than )h
gN[ ] nhFFTHk =
loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr
loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=
where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier
[ ] [ ] [ ]N
Notimes
[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce
kk H
X =rArrcomplexity of channel equalizationrdquo
32
Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))
bull bull( )( )nr~Symbols
dataOutput
bullbullbull
bullbullbull( )nr
bullbull
bullbull
33
Cyclic PrefixCyclic Prefix
loz One of the most important reasons to do OFDM is the
yy
pefficient way it deals with multipath delay spread
loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)
34
Bandwidth EfficiencyBandwidth Efficiencyyy
loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )
loz The diagram for bandwidth efficiency of OFDM system is shown below
35
SummarySummaryyy
loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j
the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference
(ISI)loz The main advantage of the OFDM transmission technique is its
high performance even in frequency selective channels
loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems
36
SynchronizationSynchronization
Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg
38 Synchronization2010722
Introduction (13)Introduction (13)( )( )
loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods
loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead
loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the
cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of
computational complexity therefore it may be not be available in short-burst wireless communication
39 Synchronization2010722
Introduction (23)Introduction (23)( )( )
loz Transcevier
40 Synchronization2010722
Introduction (33)Introduction (33)( )( )
loz Synchronization erroryloz Symbol timing offset (TO)
loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)
loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect
41 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg
42 Synchronization2010722
Signal model (14)Signal model (14)g ( )g ( )
loz Time domain sample p1
2
0
1 Nj nk N
n kk
x X eN
πminus
minus= sum
Xk the frequency domain data at the k-th subcarrier0kN =
loz The multipath fading channel
( ) ( ) ( )1L
h t h t δminus
sum
loz Received time domain signal
( ) ( ) ( )0
l ll
h t h tτ τ δ τ τ=
= minussum
loz Received time-domain signal
( ) ( )1
L
n n ly h n l x w nminus
minus= +sum43
( ) ( )0l=sum
Synchronization2010722
Signal model (24)Signal model (24)g ( )g ( )
loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting
12
0
Nj kn N
k nn
Y y e πminus
minus
=
= sum
( )1 1
2
0
N Lj kn N
n l kn l
h n l x e Wπminus minus
minusminus
=
= +sumsum
loz The equation above could be expressed in matrix form
0n l
loz The equation above could be expressed in matrix form
= +Y HX W
44 Synchronization2010722
Signal model (34)Signal model (34)g ( )g ( )
H= = = = +Y Fy FCx FCF X HX W
loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal
y
H=H FCFcirculant matrix leading to be a diagonal matrix
=H FCF
( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0
h h L L hh h
⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥
( )( ) ( )
( ) ( )
1 10 1 0 0i
h Lh L h N L
⎢ ⎥⎢ ⎥minus minus⎢ ⎥
equiv= minus minus⎢ ⎥⎢ ⎥
C( ) ( )
( ) ( )
0 1 1 100
0 0 1 1 0
h L h N L
h N L L h N
⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
45
( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦
Synchronization2010722
Signal model (44)Signal model (44)g ( )g ( )
loz In a time-invariant channel the channel matrix H would be a diagonal matrix
k k k kY H X W= +
( )1 1
2
0 0
N Lj kl N
kn l
H h n l e πminus minus
minus
= =
= sumsum
loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI
k k k k kY H X I Wα= + +k k k k k
( ) ( )1 1 1
2 2 N N L
j m k Nj ml Nk mI X h n l e e ππ
minus minus minusminusminus⎛ ⎞
= ⎜ ⎟⎝ ⎠
sum sum sumICI
46
0 0 0m n lm k= = =ne
⎝ ⎠
Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
47 Synchronization2010722
Impact of timing offset Impact of timing offset p gp g
loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively
( )I f s
Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ
( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr
2 fj k Nk k k kY X H e Wπδminus= +
loz For integer part of timing offset k k k kY X H e W+
I sTδ ( ) 0g IN L δminus minus le leFFT
2 Ij k NY X H e Wπδminus= +
( ) ( ) 2 I sFFT
j f TI sh t T H f e π δδ minusminus rarr
48
k k k kY X H e W= +
Synchronization2010722
Impact of timing offset derivationImpact of timing offset derivationp gp g
49 Synchronization2010722
Discussion of symbol boundaryDiscussion of symbol boundaryy yy y
50 Synchronization2010722
Impact of timing offsetImpact of timing offsetp gp g
loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI
2 Ij k NIk k k k
NY X H e ISI W
Nπδδ minusminus
= + +
loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk
e e
p kloz Phase Shift can be resolved by Differential Encoding
Decoding or carrier recovery with pilots
51 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
52 Synchronization2010722
Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )
53 Synchronization2010722
Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )
( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s
i n t i N N T N T nTy y t e
= + + +=
( ) ( )( ) ( ) ( ) 12
sin
sin
g gI f f
I I
i N N N Nj jf N Ni k i k k
f
Y X H e eN
π ε ε π ε
ε ε
πεπε
+ + minus+
minus minus=⎛ ⎞⎜ ⎟
Faded signal attenuated and rotated by CFO
( )( ) ( ) ( ) ( )11 2
sin
sin g gI f I f
i N N N NN j j l kI f
NN
l k π ε ε π ε επ ε ε + + minusminus + + + minus
⎜ ⎟⎝ ⎠
+ + minussum
( )( )( )
( ) ( )2
0
sin
I f I f
I
j j l kI f N Ni l l
l l k I f
X H e el k
NN
π ε ε π ε ε
ε π ε ε
+ + +
= ne minus
+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠
sum
i k
N
W
⎜ ⎟⎝ ⎠
+ Inter-Carrier Interference (ICI)
54 Synchronization2010722
Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )
loz An easier representation of received signal with only p g yfractional CFO is
1N
( ) ( )1
0
0N
k k k l l kl l k
Y H X C H X C l k Wminus
= ne
= + minus +⎡ ⎤⎣ ⎦sum
where the ICI coefficient is given by
( )( )( ) ( )
sin 1exp 1fl k
C l k j l kπ ε
π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )
( )exp 1sin
f
f
C l k j l kNN l k
N
π επ ε
minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠
55 Synchronization2010722
Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )
loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality
loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift
56 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
57 Synchronization2010722
TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )
loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq
58 Synchronization2010722
TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )
loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below
loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and
rN2 can be expressed by
0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+
( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +
59
( )
Synchronization2010722
TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )
loz Without CFO the first half is identical to the second half (in time order)
loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below
( ) 12 in discretesft T f NT
NTφ π π π ε πε
⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟
⎝ ⎠
helliphellip
sNT⎝ ⎠
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)
60
Length=N2
Synchronization2010722
TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )
loz Received signal without CFOg0
0 0
jr a e θ=jθ 2 0
2 2 0
Nj j
N Nr a e a e
θ θ= =
loz Received signal with CFO0 0
0 0
j jr a e eθ φ= Extra phase rotation due to CFO0 0
2 2 20
2 2 0
N N Nj j jj
N Nr a e e a e e
θ φ φθ= =
due to CFO
Phase difference 2 2 0N N
( )2 02 20 2 0 0
Nj j T fN
r r a e a eφ φ πminus Δ= =
which contains the information about CFO
61
0 2 0 0Nr r a e a e
Synchronization2010722
TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )
loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be
loz Note that d is a time index corresponding to the first( ) ( ) 2 1
20
Nd m d m Nm
P d r rminus
+ + +== sum
loz Note that d is a time index corresponding to the first sample in a window of N samples
loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by
( )22 1
20
Nd m NR d rminus
+ += sumloz A timing metric can be defined as
))(()(
)( 2
2
dRdP
dM =
( ) 20 d m Nm + +=sum
62
))(( dR
Synchronization2010722
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
Cyclic PrefixCyclic Prefix
loz Guard interval leads to intercarrier interference (ICI) in OFDM d d l i
yy
demodulation
loz In DFT interval difference between two subcarriers does not maintain integer number of cycles loss of orthogonality
loz Delayed version of subcarrier 2 causes ICI in the process of d d l ti b i 1demodulating subcarrier 1
25
Cyclic PrefixCyclic Prefixloz Cyclic prefix (CP) A copy of the last part of OFDM signal is
attached to the front of itself
yy
attached to the front of itself
[ ]0d
[ ]1d
[ ]0D[ ]1D
[ ]nd[ ]kD~
symbols dataInput
bullbullbullbull
[ ]1d
[ ]2d
[ ][ ]2D
bullbullbullbullbull
y
bullbullbullbullbull
[ ]1minusNd [ ]1minusND
[ ]gNND minus
bullbullbull
26
Cyclic PrefixCyclic Prefix
loz All delayed replicas of subcarriers always have an
yy
y p yinteger number of cycles within DFT interval no ICI
27
Cyclic PrefixCyclic Prefix
loz Linear convolution vs circular convolution
yy
N
28
Cyclic PrefixCyclic Prefix
loz Channel effect with cyclic prefix
yy
Signal after removed CP
29
Cyclic PrefixCyclic Prefix
loz Time-Domain Explanation
yy
p
30
Cyclic PrefixCyclic Prefixyy
31
Cyclic PrefixCyclic Prefix
loz Spectrum of channel response with length (smaller [ ]nh hL
yy
than )h
gN[ ] nhFFTHk =
loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr
loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=
where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier
[ ] [ ] [ ]N
Notimes
[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce
kk H
X =rArrcomplexity of channel equalizationrdquo
32
Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))
bull bull( )( )nr~Symbols
dataOutput
bullbullbull
bullbullbull( )nr
bullbull
bullbull
33
Cyclic PrefixCyclic Prefix
loz One of the most important reasons to do OFDM is the
yy
pefficient way it deals with multipath delay spread
loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)
34
Bandwidth EfficiencyBandwidth Efficiencyyy
loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )
loz The diagram for bandwidth efficiency of OFDM system is shown below
35
SummarySummaryyy
loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j
the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference
(ISI)loz The main advantage of the OFDM transmission technique is its
high performance even in frequency selective channels
loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems
36
SynchronizationSynchronization
Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg
38 Synchronization2010722
Introduction (13)Introduction (13)( )( )
loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods
loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead
loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the
cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of
computational complexity therefore it may be not be available in short-burst wireless communication
39 Synchronization2010722
Introduction (23)Introduction (23)( )( )
loz Transcevier
40 Synchronization2010722
Introduction (33)Introduction (33)( )( )
loz Synchronization erroryloz Symbol timing offset (TO)
loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)
loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect
41 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg
42 Synchronization2010722
Signal model (14)Signal model (14)g ( )g ( )
loz Time domain sample p1
2
0
1 Nj nk N
n kk
x X eN
πminus
minus= sum
Xk the frequency domain data at the k-th subcarrier0kN =
loz The multipath fading channel
( ) ( ) ( )1L
h t h t δminus
sum
loz Received time domain signal
( ) ( ) ( )0
l ll
h t h tτ τ δ τ τ=
= minussum
loz Received time-domain signal
( ) ( )1
L
n n ly h n l x w nminus
minus= +sum43
( ) ( )0l=sum
Synchronization2010722
Signal model (24)Signal model (24)g ( )g ( )
loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting
12
0
Nj kn N
k nn
Y y e πminus
minus
=
= sum
( )1 1
2
0
N Lj kn N
n l kn l
h n l x e Wπminus minus
minusminus
=
= +sumsum
loz The equation above could be expressed in matrix form
0n l
loz The equation above could be expressed in matrix form
= +Y HX W
44 Synchronization2010722
Signal model (34)Signal model (34)g ( )g ( )
H= = = = +Y Fy FCx FCF X HX W
loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal
y
H=H FCFcirculant matrix leading to be a diagonal matrix
=H FCF
( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0
h h L L hh h
⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥
( )( ) ( )
( ) ( )
1 10 1 0 0i
h Lh L h N L
⎢ ⎥⎢ ⎥minus minus⎢ ⎥
equiv= minus minus⎢ ⎥⎢ ⎥
C( ) ( )
( ) ( )
0 1 1 100
0 0 1 1 0
h L h N L
h N L L h N
⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
45
( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦
Synchronization2010722
Signal model (44)Signal model (44)g ( )g ( )
loz In a time-invariant channel the channel matrix H would be a diagonal matrix
k k k kY H X W= +
( )1 1
2
0 0
N Lj kl N
kn l
H h n l e πminus minus
minus
= =
= sumsum
loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI
k k k k kY H X I Wα= + +k k k k k
( ) ( )1 1 1
2 2 N N L
j m k Nj ml Nk mI X h n l e e ππ
minus minus minusminusminus⎛ ⎞
= ⎜ ⎟⎝ ⎠
sum sum sumICI
46
0 0 0m n lm k= = =ne
⎝ ⎠
Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
47 Synchronization2010722
Impact of timing offset Impact of timing offset p gp g
loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively
( )I f s
Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ
( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr
2 fj k Nk k k kY X H e Wπδminus= +
loz For integer part of timing offset k k k kY X H e W+
I sTδ ( ) 0g IN L δminus minus le leFFT
2 Ij k NY X H e Wπδminus= +
( ) ( ) 2 I sFFT
j f TI sh t T H f e π δδ minusminus rarr
48
k k k kY X H e W= +
Synchronization2010722
Impact of timing offset derivationImpact of timing offset derivationp gp g
49 Synchronization2010722
Discussion of symbol boundaryDiscussion of symbol boundaryy yy y
50 Synchronization2010722
Impact of timing offsetImpact of timing offsetp gp g
loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI
2 Ij k NIk k k k
NY X H e ISI W
Nπδδ minusminus
= + +
loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk
e e
p kloz Phase Shift can be resolved by Differential Encoding
Decoding or carrier recovery with pilots
51 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
52 Synchronization2010722
Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )
53 Synchronization2010722
Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )
( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s
i n t i N N T N T nTy y t e
= + + +=
( ) ( )( ) ( ) ( ) 12
sin
sin
g gI f f
I I
i N N N Nj jf N Ni k i k k
f
Y X H e eN
π ε ε π ε
ε ε
πεπε
+ + minus+
minus minus=⎛ ⎞⎜ ⎟
Faded signal attenuated and rotated by CFO
( )( ) ( ) ( ) ( )11 2
sin
sin g gI f I f
i N N N NN j j l kI f
NN
l k π ε ε π ε επ ε ε + + minusminus + + + minus
⎜ ⎟⎝ ⎠
+ + minussum
( )( )( )
( ) ( )2
0
sin
I f I f
I
j j l kI f N Ni l l
l l k I f
X H e el k
NN
π ε ε π ε ε
ε π ε ε
+ + +
= ne minus
+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠
sum
i k
N
W
⎜ ⎟⎝ ⎠
+ Inter-Carrier Interference (ICI)
54 Synchronization2010722
Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )
loz An easier representation of received signal with only p g yfractional CFO is
1N
( ) ( )1
0
0N
k k k l l kl l k
Y H X C H X C l k Wminus
= ne
= + minus +⎡ ⎤⎣ ⎦sum
where the ICI coefficient is given by
( )( )( ) ( )
sin 1exp 1fl k
C l k j l kπ ε
π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )
( )exp 1sin
f
f
C l k j l kNN l k
N
π επ ε
minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠
55 Synchronization2010722
Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )
loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality
loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift
56 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
57 Synchronization2010722
TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )
loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq
58 Synchronization2010722
TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )
loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below
loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and
rN2 can be expressed by
0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+
( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +
59
( )
Synchronization2010722
TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )
loz Without CFO the first half is identical to the second half (in time order)
loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below
( ) 12 in discretesft T f NT
NTφ π π π ε πε
⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟
⎝ ⎠
helliphellip
sNT⎝ ⎠
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)
60
Length=N2
Synchronization2010722
TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )
loz Received signal without CFOg0
0 0
jr a e θ=jθ 2 0
2 2 0
Nj j
N Nr a e a e
θ θ= =
loz Received signal with CFO0 0
0 0
j jr a e eθ φ= Extra phase rotation due to CFO0 0
2 2 20
2 2 0
N N Nj j jj
N Nr a e e a e e
θ φ φθ= =
due to CFO
Phase difference 2 2 0N N
( )2 02 20 2 0 0
Nj j T fN
r r a e a eφ φ πminus Δ= =
which contains the information about CFO
61
0 2 0 0Nr r a e a e
Synchronization2010722
TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )
loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be
loz Note that d is a time index corresponding to the first( ) ( ) 2 1
20
Nd m d m Nm
P d r rminus
+ + +== sum
loz Note that d is a time index corresponding to the first sample in a window of N samples
loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by
( )22 1
20
Nd m NR d rminus
+ += sumloz A timing metric can be defined as
))(()(
)( 2
2
dRdP
dM =
( ) 20 d m Nm + +=sum
62
))(( dR
Synchronization2010722
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
Cyclic PrefixCyclic Prefixloz Cyclic prefix (CP) A copy of the last part of OFDM signal is
attached to the front of itself
yy
attached to the front of itself
[ ]0d
[ ]1d
[ ]0D[ ]1D
[ ]nd[ ]kD~
symbols dataInput
bullbullbullbull
[ ]1d
[ ]2d
[ ][ ]2D
bullbullbullbullbull
y
bullbullbullbullbull
[ ]1minusNd [ ]1minusND
[ ]gNND minus
bullbullbull
26
Cyclic PrefixCyclic Prefix
loz All delayed replicas of subcarriers always have an
yy
y p yinteger number of cycles within DFT interval no ICI
27
Cyclic PrefixCyclic Prefix
loz Linear convolution vs circular convolution
yy
N
28
Cyclic PrefixCyclic Prefix
loz Channel effect with cyclic prefix
yy
Signal after removed CP
29
Cyclic PrefixCyclic Prefix
loz Time-Domain Explanation
yy
p
30
Cyclic PrefixCyclic Prefixyy
31
Cyclic PrefixCyclic Prefix
loz Spectrum of channel response with length (smaller [ ]nh hL
yy
than )h
gN[ ] nhFFTHk =
loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr
loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=
where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier
[ ] [ ] [ ]N
Notimes
[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce
kk H
X =rArrcomplexity of channel equalizationrdquo
32
Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))
bull bull( )( )nr~Symbols
dataOutput
bullbullbull
bullbullbull( )nr
bullbull
bullbull
33
Cyclic PrefixCyclic Prefix
loz One of the most important reasons to do OFDM is the
yy
pefficient way it deals with multipath delay spread
loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)
34
Bandwidth EfficiencyBandwidth Efficiencyyy
loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )
loz The diagram for bandwidth efficiency of OFDM system is shown below
35
SummarySummaryyy
loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j
the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference
(ISI)loz The main advantage of the OFDM transmission technique is its
high performance even in frequency selective channels
loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems
36
SynchronizationSynchronization
Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg
38 Synchronization2010722
Introduction (13)Introduction (13)( )( )
loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods
loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead
loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the
cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of
computational complexity therefore it may be not be available in short-burst wireless communication
39 Synchronization2010722
Introduction (23)Introduction (23)( )( )
loz Transcevier
40 Synchronization2010722
Introduction (33)Introduction (33)( )( )
loz Synchronization erroryloz Symbol timing offset (TO)
loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)
loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect
41 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg
42 Synchronization2010722
Signal model (14)Signal model (14)g ( )g ( )
loz Time domain sample p1
2
0
1 Nj nk N
n kk
x X eN
πminus
minus= sum
Xk the frequency domain data at the k-th subcarrier0kN =
loz The multipath fading channel
( ) ( ) ( )1L
h t h t δminus
sum
loz Received time domain signal
( ) ( ) ( )0
l ll
h t h tτ τ δ τ τ=
= minussum
loz Received time-domain signal
( ) ( )1
L
n n ly h n l x w nminus
minus= +sum43
( ) ( )0l=sum
Synchronization2010722
Signal model (24)Signal model (24)g ( )g ( )
loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting
12
0
Nj kn N
k nn
Y y e πminus
minus
=
= sum
( )1 1
2
0
N Lj kn N
n l kn l
h n l x e Wπminus minus
minusminus
=
= +sumsum
loz The equation above could be expressed in matrix form
0n l
loz The equation above could be expressed in matrix form
= +Y HX W
44 Synchronization2010722
Signal model (34)Signal model (34)g ( )g ( )
H= = = = +Y Fy FCx FCF X HX W
loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal
y
H=H FCFcirculant matrix leading to be a diagonal matrix
=H FCF
( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0
h h L L hh h
⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥
( )( ) ( )
( ) ( )
1 10 1 0 0i
h Lh L h N L
⎢ ⎥⎢ ⎥minus minus⎢ ⎥
equiv= minus minus⎢ ⎥⎢ ⎥
C( ) ( )
( ) ( )
0 1 1 100
0 0 1 1 0
h L h N L
h N L L h N
⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
45
( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦
Synchronization2010722
Signal model (44)Signal model (44)g ( )g ( )
loz In a time-invariant channel the channel matrix H would be a diagonal matrix
k k k kY H X W= +
( )1 1
2
0 0
N Lj kl N
kn l
H h n l e πminus minus
minus
= =
= sumsum
loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI
k k k k kY H X I Wα= + +k k k k k
( ) ( )1 1 1
2 2 N N L
j m k Nj ml Nk mI X h n l e e ππ
minus minus minusminusminus⎛ ⎞
= ⎜ ⎟⎝ ⎠
sum sum sumICI
46
0 0 0m n lm k= = =ne
⎝ ⎠
Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
47 Synchronization2010722
Impact of timing offset Impact of timing offset p gp g
loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively
( )I f s
Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ
( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr
2 fj k Nk k k kY X H e Wπδminus= +
loz For integer part of timing offset k k k kY X H e W+
I sTδ ( ) 0g IN L δminus minus le leFFT
2 Ij k NY X H e Wπδminus= +
( ) ( ) 2 I sFFT
j f TI sh t T H f e π δδ minusminus rarr
48
k k k kY X H e W= +
Synchronization2010722
Impact of timing offset derivationImpact of timing offset derivationp gp g
49 Synchronization2010722
Discussion of symbol boundaryDiscussion of symbol boundaryy yy y
50 Synchronization2010722
Impact of timing offsetImpact of timing offsetp gp g
loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI
2 Ij k NIk k k k
NY X H e ISI W
Nπδδ minusminus
= + +
loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk
e e
p kloz Phase Shift can be resolved by Differential Encoding
Decoding or carrier recovery with pilots
51 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
52 Synchronization2010722
Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )
53 Synchronization2010722
Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )
( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s
i n t i N N T N T nTy y t e
= + + +=
( ) ( )( ) ( ) ( ) 12
sin
sin
g gI f f
I I
i N N N Nj jf N Ni k i k k
f
Y X H e eN
π ε ε π ε
ε ε
πεπε
+ + minus+
minus minus=⎛ ⎞⎜ ⎟
Faded signal attenuated and rotated by CFO
( )( ) ( ) ( ) ( )11 2
sin
sin g gI f I f
i N N N NN j j l kI f
NN
l k π ε ε π ε επ ε ε + + minusminus + + + minus
⎜ ⎟⎝ ⎠
+ + minussum
( )( )( )
( ) ( )2
0
sin
I f I f
I
j j l kI f N Ni l l
l l k I f
X H e el k
NN
π ε ε π ε ε
ε π ε ε
+ + +
= ne minus
+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠
sum
i k
N
W
⎜ ⎟⎝ ⎠
+ Inter-Carrier Interference (ICI)
54 Synchronization2010722
Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )
loz An easier representation of received signal with only p g yfractional CFO is
1N
( ) ( )1
0
0N
k k k l l kl l k
Y H X C H X C l k Wminus
= ne
= + minus +⎡ ⎤⎣ ⎦sum
where the ICI coefficient is given by
( )( )( ) ( )
sin 1exp 1fl k
C l k j l kπ ε
π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )
( )exp 1sin
f
f
C l k j l kNN l k
N
π επ ε
minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠
55 Synchronization2010722
Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )
loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality
loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift
56 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
57 Synchronization2010722
TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )
loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq
58 Synchronization2010722
TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )
loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below
loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and
rN2 can be expressed by
0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+
( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +
59
( )
Synchronization2010722
TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )
loz Without CFO the first half is identical to the second half (in time order)
loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below
( ) 12 in discretesft T f NT
NTφ π π π ε πε
⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟
⎝ ⎠
helliphellip
sNT⎝ ⎠
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)
60
Length=N2
Synchronization2010722
TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )
loz Received signal without CFOg0
0 0
jr a e θ=jθ 2 0
2 2 0
Nj j
N Nr a e a e
θ θ= =
loz Received signal with CFO0 0
0 0
j jr a e eθ φ= Extra phase rotation due to CFO0 0
2 2 20
2 2 0
N N Nj j jj
N Nr a e e a e e
θ φ φθ= =
due to CFO
Phase difference 2 2 0N N
( )2 02 20 2 0 0
Nj j T fN
r r a e a eφ φ πminus Δ= =
which contains the information about CFO
61
0 2 0 0Nr r a e a e
Synchronization2010722
TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )
loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be
loz Note that d is a time index corresponding to the first( ) ( ) 2 1
20
Nd m d m Nm
P d r rminus
+ + +== sum
loz Note that d is a time index corresponding to the first sample in a window of N samples
loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by
( )22 1
20
Nd m NR d rminus
+ += sumloz A timing metric can be defined as
))(()(
)( 2
2
dRdP
dM =
( ) 20 d m Nm + +=sum
62
))(( dR
Synchronization2010722
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
Cyclic PrefixCyclic Prefix
loz All delayed replicas of subcarriers always have an
yy
y p yinteger number of cycles within DFT interval no ICI
27
Cyclic PrefixCyclic Prefix
loz Linear convolution vs circular convolution
yy
N
28
Cyclic PrefixCyclic Prefix
loz Channel effect with cyclic prefix
yy
Signal after removed CP
29
Cyclic PrefixCyclic Prefix
loz Time-Domain Explanation
yy
p
30
Cyclic PrefixCyclic Prefixyy
31
Cyclic PrefixCyclic Prefix
loz Spectrum of channel response with length (smaller [ ]nh hL
yy
than )h
gN[ ] nhFFTHk =
loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr
loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=
where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier
[ ] [ ] [ ]N
Notimes
[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce
kk H
X =rArrcomplexity of channel equalizationrdquo
32
Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))
bull bull( )( )nr~Symbols
dataOutput
bullbullbull
bullbullbull( )nr
bullbull
bullbull
33
Cyclic PrefixCyclic Prefix
loz One of the most important reasons to do OFDM is the
yy
pefficient way it deals with multipath delay spread
loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)
34
Bandwidth EfficiencyBandwidth Efficiencyyy
loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )
loz The diagram for bandwidth efficiency of OFDM system is shown below
35
SummarySummaryyy
loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j
the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference
(ISI)loz The main advantage of the OFDM transmission technique is its
high performance even in frequency selective channels
loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems
36
SynchronizationSynchronization
Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg
38 Synchronization2010722
Introduction (13)Introduction (13)( )( )
loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods
loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead
loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the
cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of
computational complexity therefore it may be not be available in short-burst wireless communication
39 Synchronization2010722
Introduction (23)Introduction (23)( )( )
loz Transcevier
40 Synchronization2010722
Introduction (33)Introduction (33)( )( )
loz Synchronization erroryloz Symbol timing offset (TO)
loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)
loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect
41 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg
42 Synchronization2010722
Signal model (14)Signal model (14)g ( )g ( )
loz Time domain sample p1
2
0
1 Nj nk N
n kk
x X eN
πminus
minus= sum
Xk the frequency domain data at the k-th subcarrier0kN =
loz The multipath fading channel
( ) ( ) ( )1L
h t h t δminus
sum
loz Received time domain signal
( ) ( ) ( )0
l ll
h t h tτ τ δ τ τ=
= minussum
loz Received time-domain signal
( ) ( )1
L
n n ly h n l x w nminus
minus= +sum43
( ) ( )0l=sum
Synchronization2010722
Signal model (24)Signal model (24)g ( )g ( )
loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting
12
0
Nj kn N
k nn
Y y e πminus
minus
=
= sum
( )1 1
2
0
N Lj kn N
n l kn l
h n l x e Wπminus minus
minusminus
=
= +sumsum
loz The equation above could be expressed in matrix form
0n l
loz The equation above could be expressed in matrix form
= +Y HX W
44 Synchronization2010722
Signal model (34)Signal model (34)g ( )g ( )
H= = = = +Y Fy FCx FCF X HX W
loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal
y
H=H FCFcirculant matrix leading to be a diagonal matrix
=H FCF
( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0
h h L L hh h
⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥
( )( ) ( )
( ) ( )
1 10 1 0 0i
h Lh L h N L
⎢ ⎥⎢ ⎥minus minus⎢ ⎥
equiv= minus minus⎢ ⎥⎢ ⎥
C( ) ( )
( ) ( )
0 1 1 100
0 0 1 1 0
h L h N L
h N L L h N
⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
45
( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦
Synchronization2010722
Signal model (44)Signal model (44)g ( )g ( )
loz In a time-invariant channel the channel matrix H would be a diagonal matrix
k k k kY H X W= +
( )1 1
2
0 0
N Lj kl N
kn l
H h n l e πminus minus
minus
= =
= sumsum
loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI
k k k k kY H X I Wα= + +k k k k k
( ) ( )1 1 1
2 2 N N L
j m k Nj ml Nk mI X h n l e e ππ
minus minus minusminusminus⎛ ⎞
= ⎜ ⎟⎝ ⎠
sum sum sumICI
46
0 0 0m n lm k= = =ne
⎝ ⎠
Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
47 Synchronization2010722
Impact of timing offset Impact of timing offset p gp g
loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively
( )I f s
Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ
( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr
2 fj k Nk k k kY X H e Wπδminus= +
loz For integer part of timing offset k k k kY X H e W+
I sTδ ( ) 0g IN L δminus minus le leFFT
2 Ij k NY X H e Wπδminus= +
( ) ( ) 2 I sFFT
j f TI sh t T H f e π δδ minusminus rarr
48
k k k kY X H e W= +
Synchronization2010722
Impact of timing offset derivationImpact of timing offset derivationp gp g
49 Synchronization2010722
Discussion of symbol boundaryDiscussion of symbol boundaryy yy y
50 Synchronization2010722
Impact of timing offsetImpact of timing offsetp gp g
loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI
2 Ij k NIk k k k
NY X H e ISI W
Nπδδ minusminus
= + +
loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk
e e
p kloz Phase Shift can be resolved by Differential Encoding
Decoding or carrier recovery with pilots
51 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
52 Synchronization2010722
Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )
53 Synchronization2010722
Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )
( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s
i n t i N N T N T nTy y t e
= + + +=
( ) ( )( ) ( ) ( ) 12
sin
sin
g gI f f
I I
i N N N Nj jf N Ni k i k k
f
Y X H e eN
π ε ε π ε
ε ε
πεπε
+ + minus+
minus minus=⎛ ⎞⎜ ⎟
Faded signal attenuated and rotated by CFO
( )( ) ( ) ( ) ( )11 2
sin
sin g gI f I f
i N N N NN j j l kI f
NN
l k π ε ε π ε επ ε ε + + minusminus + + + minus
⎜ ⎟⎝ ⎠
+ + minussum
( )( )( )
( ) ( )2
0
sin
I f I f
I
j j l kI f N Ni l l
l l k I f
X H e el k
NN
π ε ε π ε ε
ε π ε ε
+ + +
= ne minus
+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠
sum
i k
N
W
⎜ ⎟⎝ ⎠
+ Inter-Carrier Interference (ICI)
54 Synchronization2010722
Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )
loz An easier representation of received signal with only p g yfractional CFO is
1N
( ) ( )1
0
0N
k k k l l kl l k
Y H X C H X C l k Wminus
= ne
= + minus +⎡ ⎤⎣ ⎦sum
where the ICI coefficient is given by
( )( )( ) ( )
sin 1exp 1fl k
C l k j l kπ ε
π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )
( )exp 1sin
f
f
C l k j l kNN l k
N
π επ ε
minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠
55 Synchronization2010722
Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )
loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality
loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift
56 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
57 Synchronization2010722
TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )
loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq
58 Synchronization2010722
TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )
loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below
loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and
rN2 can be expressed by
0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+
( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +
59
( )
Synchronization2010722
TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )
loz Without CFO the first half is identical to the second half (in time order)
loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below
( ) 12 in discretesft T f NT
NTφ π π π ε πε
⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟
⎝ ⎠
helliphellip
sNT⎝ ⎠
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)
60
Length=N2
Synchronization2010722
TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )
loz Received signal without CFOg0
0 0
jr a e θ=jθ 2 0
2 2 0
Nj j
N Nr a e a e
θ θ= =
loz Received signal with CFO0 0
0 0
j jr a e eθ φ= Extra phase rotation due to CFO0 0
2 2 20
2 2 0
N N Nj j jj
N Nr a e e a e e
θ φ φθ= =
due to CFO
Phase difference 2 2 0N N
( )2 02 20 2 0 0
Nj j T fN
r r a e a eφ φ πminus Δ= =
which contains the information about CFO
61
0 2 0 0Nr r a e a e
Synchronization2010722
TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )
loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be
loz Note that d is a time index corresponding to the first( ) ( ) 2 1
20
Nd m d m Nm
P d r rminus
+ + +== sum
loz Note that d is a time index corresponding to the first sample in a window of N samples
loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by
( )22 1
20
Nd m NR d rminus
+ += sumloz A timing metric can be defined as
))(()(
)( 2
2
dRdP
dM =
( ) 20 d m Nm + +=sum
62
))(( dR
Synchronization2010722
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
Cyclic PrefixCyclic Prefix
loz Linear convolution vs circular convolution
yy
N
28
Cyclic PrefixCyclic Prefix
loz Channel effect with cyclic prefix
yy
Signal after removed CP
29
Cyclic PrefixCyclic Prefix
loz Time-Domain Explanation
yy
p
30
Cyclic PrefixCyclic Prefixyy
31
Cyclic PrefixCyclic Prefix
loz Spectrum of channel response with length (smaller [ ]nh hL
yy
than )h
gN[ ] nhFFTHk =
loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr
loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=
where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier
[ ] [ ] [ ]N
Notimes
[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce
kk H
X =rArrcomplexity of channel equalizationrdquo
32
Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))
bull bull( )( )nr~Symbols
dataOutput
bullbullbull
bullbullbull( )nr
bullbull
bullbull
33
Cyclic PrefixCyclic Prefix
loz One of the most important reasons to do OFDM is the
yy
pefficient way it deals with multipath delay spread
loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)
34
Bandwidth EfficiencyBandwidth Efficiencyyy
loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )
loz The diagram for bandwidth efficiency of OFDM system is shown below
35
SummarySummaryyy
loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j
the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference
(ISI)loz The main advantage of the OFDM transmission technique is its
high performance even in frequency selective channels
loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems
36
SynchronizationSynchronization
Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg
38 Synchronization2010722
Introduction (13)Introduction (13)( )( )
loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods
loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead
loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the
cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of
computational complexity therefore it may be not be available in short-burst wireless communication
39 Synchronization2010722
Introduction (23)Introduction (23)( )( )
loz Transcevier
40 Synchronization2010722
Introduction (33)Introduction (33)( )( )
loz Synchronization erroryloz Symbol timing offset (TO)
loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)
loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect
41 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg
42 Synchronization2010722
Signal model (14)Signal model (14)g ( )g ( )
loz Time domain sample p1
2
0
1 Nj nk N
n kk
x X eN
πminus
minus= sum
Xk the frequency domain data at the k-th subcarrier0kN =
loz The multipath fading channel
( ) ( ) ( )1L
h t h t δminus
sum
loz Received time domain signal
( ) ( ) ( )0
l ll
h t h tτ τ δ τ τ=
= minussum
loz Received time-domain signal
( ) ( )1
L
n n ly h n l x w nminus
minus= +sum43
( ) ( )0l=sum
Synchronization2010722
Signal model (24)Signal model (24)g ( )g ( )
loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting
12
0
Nj kn N
k nn
Y y e πminus
minus
=
= sum
( )1 1
2
0
N Lj kn N
n l kn l
h n l x e Wπminus minus
minusminus
=
= +sumsum
loz The equation above could be expressed in matrix form
0n l
loz The equation above could be expressed in matrix form
= +Y HX W
44 Synchronization2010722
Signal model (34)Signal model (34)g ( )g ( )
H= = = = +Y Fy FCx FCF X HX W
loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal
y
H=H FCFcirculant matrix leading to be a diagonal matrix
=H FCF
( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0
h h L L hh h
⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥
( )( ) ( )
( ) ( )
1 10 1 0 0i
h Lh L h N L
⎢ ⎥⎢ ⎥minus minus⎢ ⎥
equiv= minus minus⎢ ⎥⎢ ⎥
C( ) ( )
( ) ( )
0 1 1 100
0 0 1 1 0
h L h N L
h N L L h N
⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
45
( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦
Synchronization2010722
Signal model (44)Signal model (44)g ( )g ( )
loz In a time-invariant channel the channel matrix H would be a diagonal matrix
k k k kY H X W= +
( )1 1
2
0 0
N Lj kl N
kn l
H h n l e πminus minus
minus
= =
= sumsum
loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI
k k k k kY H X I Wα= + +k k k k k
( ) ( )1 1 1
2 2 N N L
j m k Nj ml Nk mI X h n l e e ππ
minus minus minusminusminus⎛ ⎞
= ⎜ ⎟⎝ ⎠
sum sum sumICI
46
0 0 0m n lm k= = =ne
⎝ ⎠
Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
47 Synchronization2010722
Impact of timing offset Impact of timing offset p gp g
loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively
( )I f s
Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ
( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr
2 fj k Nk k k kY X H e Wπδminus= +
loz For integer part of timing offset k k k kY X H e W+
I sTδ ( ) 0g IN L δminus minus le leFFT
2 Ij k NY X H e Wπδminus= +
( ) ( ) 2 I sFFT
j f TI sh t T H f e π δδ minusminus rarr
48
k k k kY X H e W= +
Synchronization2010722
Impact of timing offset derivationImpact of timing offset derivationp gp g
49 Synchronization2010722
Discussion of symbol boundaryDiscussion of symbol boundaryy yy y
50 Synchronization2010722
Impact of timing offsetImpact of timing offsetp gp g
loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI
2 Ij k NIk k k k
NY X H e ISI W
Nπδδ minusminus
= + +
loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk
e e
p kloz Phase Shift can be resolved by Differential Encoding
Decoding or carrier recovery with pilots
51 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
52 Synchronization2010722
Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )
53 Synchronization2010722
Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )
( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s
i n t i N N T N T nTy y t e
= + + +=
( ) ( )( ) ( ) ( ) 12
sin
sin
g gI f f
I I
i N N N Nj jf N Ni k i k k
f
Y X H e eN
π ε ε π ε
ε ε
πεπε
+ + minus+
minus minus=⎛ ⎞⎜ ⎟
Faded signal attenuated and rotated by CFO
( )( ) ( ) ( ) ( )11 2
sin
sin g gI f I f
i N N N NN j j l kI f
NN
l k π ε ε π ε επ ε ε + + minusminus + + + minus
⎜ ⎟⎝ ⎠
+ + minussum
( )( )( )
( ) ( )2
0
sin
I f I f
I
j j l kI f N Ni l l
l l k I f
X H e el k
NN
π ε ε π ε ε
ε π ε ε
+ + +
= ne minus
+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠
sum
i k
N
W
⎜ ⎟⎝ ⎠
+ Inter-Carrier Interference (ICI)
54 Synchronization2010722
Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )
loz An easier representation of received signal with only p g yfractional CFO is
1N
( ) ( )1
0
0N
k k k l l kl l k
Y H X C H X C l k Wminus
= ne
= + minus +⎡ ⎤⎣ ⎦sum
where the ICI coefficient is given by
( )( )( ) ( )
sin 1exp 1fl k
C l k j l kπ ε
π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )
( )exp 1sin
f
f
C l k j l kNN l k
N
π επ ε
minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠
55 Synchronization2010722
Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )
loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality
loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift
56 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
57 Synchronization2010722
TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )
loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq
58 Synchronization2010722
TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )
loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below
loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and
rN2 can be expressed by
0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+
( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +
59
( )
Synchronization2010722
TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )
loz Without CFO the first half is identical to the second half (in time order)
loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below
( ) 12 in discretesft T f NT
NTφ π π π ε πε
⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟
⎝ ⎠
helliphellip
sNT⎝ ⎠
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)
60
Length=N2
Synchronization2010722
TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )
loz Received signal without CFOg0
0 0
jr a e θ=jθ 2 0
2 2 0
Nj j
N Nr a e a e
θ θ= =
loz Received signal with CFO0 0
0 0
j jr a e eθ φ= Extra phase rotation due to CFO0 0
2 2 20
2 2 0
N N Nj j jj
N Nr a e e a e e
θ φ φθ= =
due to CFO
Phase difference 2 2 0N N
( )2 02 20 2 0 0
Nj j T fN
r r a e a eφ φ πminus Δ= =
which contains the information about CFO
61
0 2 0 0Nr r a e a e
Synchronization2010722
TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )
loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be
loz Note that d is a time index corresponding to the first( ) ( ) 2 1
20
Nd m d m Nm
P d r rminus
+ + +== sum
loz Note that d is a time index corresponding to the first sample in a window of N samples
loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by
( )22 1
20
Nd m NR d rminus
+ += sumloz A timing metric can be defined as
))(()(
)( 2
2
dRdP
dM =
( ) 20 d m Nm + +=sum
62
))(( dR
Synchronization2010722
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
Cyclic PrefixCyclic Prefix
loz Channel effect with cyclic prefix
yy
Signal after removed CP
29
Cyclic PrefixCyclic Prefix
loz Time-Domain Explanation
yy
p
30
Cyclic PrefixCyclic Prefixyy
31
Cyclic PrefixCyclic Prefix
loz Spectrum of channel response with length (smaller [ ]nh hL
yy
than )h
gN[ ] nhFFTHk =
loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr
loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=
where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier
[ ] [ ] [ ]N
Notimes
[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce
kk H
X =rArrcomplexity of channel equalizationrdquo
32
Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))
bull bull( )( )nr~Symbols
dataOutput
bullbullbull
bullbullbull( )nr
bullbull
bullbull
33
Cyclic PrefixCyclic Prefix
loz One of the most important reasons to do OFDM is the
yy
pefficient way it deals with multipath delay spread
loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)
34
Bandwidth EfficiencyBandwidth Efficiencyyy
loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )
loz The diagram for bandwidth efficiency of OFDM system is shown below
35
SummarySummaryyy
loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j
the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference
(ISI)loz The main advantage of the OFDM transmission technique is its
high performance even in frequency selective channels
loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems
36
SynchronizationSynchronization
Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg
38 Synchronization2010722
Introduction (13)Introduction (13)( )( )
loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods
loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead
loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the
cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of
computational complexity therefore it may be not be available in short-burst wireless communication
39 Synchronization2010722
Introduction (23)Introduction (23)( )( )
loz Transcevier
40 Synchronization2010722
Introduction (33)Introduction (33)( )( )
loz Synchronization erroryloz Symbol timing offset (TO)
loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)
loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect
41 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg
42 Synchronization2010722
Signal model (14)Signal model (14)g ( )g ( )
loz Time domain sample p1
2
0
1 Nj nk N
n kk
x X eN
πminus
minus= sum
Xk the frequency domain data at the k-th subcarrier0kN =
loz The multipath fading channel
( ) ( ) ( )1L
h t h t δminus
sum
loz Received time domain signal
( ) ( ) ( )0
l ll
h t h tτ τ δ τ τ=
= minussum
loz Received time-domain signal
( ) ( )1
L
n n ly h n l x w nminus
minus= +sum43
( ) ( )0l=sum
Synchronization2010722
Signal model (24)Signal model (24)g ( )g ( )
loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting
12
0
Nj kn N
k nn
Y y e πminus
minus
=
= sum
( )1 1
2
0
N Lj kn N
n l kn l
h n l x e Wπminus minus
minusminus
=
= +sumsum
loz The equation above could be expressed in matrix form
0n l
loz The equation above could be expressed in matrix form
= +Y HX W
44 Synchronization2010722
Signal model (34)Signal model (34)g ( )g ( )
H= = = = +Y Fy FCx FCF X HX W
loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal
y
H=H FCFcirculant matrix leading to be a diagonal matrix
=H FCF
( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0
h h L L hh h
⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥
( )( ) ( )
( ) ( )
1 10 1 0 0i
h Lh L h N L
⎢ ⎥⎢ ⎥minus minus⎢ ⎥
equiv= minus minus⎢ ⎥⎢ ⎥
C( ) ( )
( ) ( )
0 1 1 100
0 0 1 1 0
h L h N L
h N L L h N
⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
45
( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦
Synchronization2010722
Signal model (44)Signal model (44)g ( )g ( )
loz In a time-invariant channel the channel matrix H would be a diagonal matrix
k k k kY H X W= +
( )1 1
2
0 0
N Lj kl N
kn l
H h n l e πminus minus
minus
= =
= sumsum
loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI
k k k k kY H X I Wα= + +k k k k k
( ) ( )1 1 1
2 2 N N L
j m k Nj ml Nk mI X h n l e e ππ
minus minus minusminusminus⎛ ⎞
= ⎜ ⎟⎝ ⎠
sum sum sumICI
46
0 0 0m n lm k= = =ne
⎝ ⎠
Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
47 Synchronization2010722
Impact of timing offset Impact of timing offset p gp g
loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively
( )I f s
Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ
( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr
2 fj k Nk k k kY X H e Wπδminus= +
loz For integer part of timing offset k k k kY X H e W+
I sTδ ( ) 0g IN L δminus minus le leFFT
2 Ij k NY X H e Wπδminus= +
( ) ( ) 2 I sFFT
j f TI sh t T H f e π δδ minusminus rarr
48
k k k kY X H e W= +
Synchronization2010722
Impact of timing offset derivationImpact of timing offset derivationp gp g
49 Synchronization2010722
Discussion of symbol boundaryDiscussion of symbol boundaryy yy y
50 Synchronization2010722
Impact of timing offsetImpact of timing offsetp gp g
loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI
2 Ij k NIk k k k
NY X H e ISI W
Nπδδ minusminus
= + +
loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk
e e
p kloz Phase Shift can be resolved by Differential Encoding
Decoding or carrier recovery with pilots
51 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
52 Synchronization2010722
Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )
53 Synchronization2010722
Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )
( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s
i n t i N N T N T nTy y t e
= + + +=
( ) ( )( ) ( ) ( ) 12
sin
sin
g gI f f
I I
i N N N Nj jf N Ni k i k k
f
Y X H e eN
π ε ε π ε
ε ε
πεπε
+ + minus+
minus minus=⎛ ⎞⎜ ⎟
Faded signal attenuated and rotated by CFO
( )( ) ( ) ( ) ( )11 2
sin
sin g gI f I f
i N N N NN j j l kI f
NN
l k π ε ε π ε επ ε ε + + minusminus + + + minus
⎜ ⎟⎝ ⎠
+ + minussum
( )( )( )
( ) ( )2
0
sin
I f I f
I
j j l kI f N Ni l l
l l k I f
X H e el k
NN
π ε ε π ε ε
ε π ε ε
+ + +
= ne minus
+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠
sum
i k
N
W
⎜ ⎟⎝ ⎠
+ Inter-Carrier Interference (ICI)
54 Synchronization2010722
Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )
loz An easier representation of received signal with only p g yfractional CFO is
1N
( ) ( )1
0
0N
k k k l l kl l k
Y H X C H X C l k Wminus
= ne
= + minus +⎡ ⎤⎣ ⎦sum
where the ICI coefficient is given by
( )( )( ) ( )
sin 1exp 1fl k
C l k j l kπ ε
π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )
( )exp 1sin
f
f
C l k j l kNN l k
N
π επ ε
minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠
55 Synchronization2010722
Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )
loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality
loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift
56 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
57 Synchronization2010722
TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )
loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq
58 Synchronization2010722
TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )
loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below
loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and
rN2 can be expressed by
0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+
( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +
59
( )
Synchronization2010722
TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )
loz Without CFO the first half is identical to the second half (in time order)
loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below
( ) 12 in discretesft T f NT
NTφ π π π ε πε
⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟
⎝ ⎠
helliphellip
sNT⎝ ⎠
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)
60
Length=N2
Synchronization2010722
TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )
loz Received signal without CFOg0
0 0
jr a e θ=jθ 2 0
2 2 0
Nj j
N Nr a e a e
θ θ= =
loz Received signal with CFO0 0
0 0
j jr a e eθ φ= Extra phase rotation due to CFO0 0
2 2 20
2 2 0
N N Nj j jj
N Nr a e e a e e
θ φ φθ= =
due to CFO
Phase difference 2 2 0N N
( )2 02 20 2 0 0
Nj j T fN
r r a e a eφ φ πminus Δ= =
which contains the information about CFO
61
0 2 0 0Nr r a e a e
Synchronization2010722
TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )
loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be
loz Note that d is a time index corresponding to the first( ) ( ) 2 1
20
Nd m d m Nm
P d r rminus
+ + +== sum
loz Note that d is a time index corresponding to the first sample in a window of N samples
loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by
( )22 1
20
Nd m NR d rminus
+ += sumloz A timing metric can be defined as
))(()(
)( 2
2
dRdP
dM =
( ) 20 d m Nm + +=sum
62
))(( dR
Synchronization2010722
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
Cyclic PrefixCyclic Prefix
loz Time-Domain Explanation
yy
p
30
Cyclic PrefixCyclic Prefixyy
31
Cyclic PrefixCyclic Prefix
loz Spectrum of channel response with length (smaller [ ]nh hL
yy
than )h
gN[ ] nhFFTHk =
loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr
loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=
where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier
[ ] [ ] [ ]N
Notimes
[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce
kk H
X =rArrcomplexity of channel equalizationrdquo
32
Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))
bull bull( )( )nr~Symbols
dataOutput
bullbullbull
bullbullbull( )nr
bullbull
bullbull
33
Cyclic PrefixCyclic Prefix
loz One of the most important reasons to do OFDM is the
yy
pefficient way it deals with multipath delay spread
loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)
34
Bandwidth EfficiencyBandwidth Efficiencyyy
loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )
loz The diagram for bandwidth efficiency of OFDM system is shown below
35
SummarySummaryyy
loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j
the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference
(ISI)loz The main advantage of the OFDM transmission technique is its
high performance even in frequency selective channels
loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems
36
SynchronizationSynchronization
Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg
38 Synchronization2010722
Introduction (13)Introduction (13)( )( )
loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods
loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead
loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the
cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of
computational complexity therefore it may be not be available in short-burst wireless communication
39 Synchronization2010722
Introduction (23)Introduction (23)( )( )
loz Transcevier
40 Synchronization2010722
Introduction (33)Introduction (33)( )( )
loz Synchronization erroryloz Symbol timing offset (TO)
loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)
loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect
41 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg
42 Synchronization2010722
Signal model (14)Signal model (14)g ( )g ( )
loz Time domain sample p1
2
0
1 Nj nk N
n kk
x X eN
πminus
minus= sum
Xk the frequency domain data at the k-th subcarrier0kN =
loz The multipath fading channel
( ) ( ) ( )1L
h t h t δminus
sum
loz Received time domain signal
( ) ( ) ( )0
l ll
h t h tτ τ δ τ τ=
= minussum
loz Received time-domain signal
( ) ( )1
L
n n ly h n l x w nminus
minus= +sum43
( ) ( )0l=sum
Synchronization2010722
Signal model (24)Signal model (24)g ( )g ( )
loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting
12
0
Nj kn N
k nn
Y y e πminus
minus
=
= sum
( )1 1
2
0
N Lj kn N
n l kn l
h n l x e Wπminus minus
minusminus
=
= +sumsum
loz The equation above could be expressed in matrix form
0n l
loz The equation above could be expressed in matrix form
= +Y HX W
44 Synchronization2010722
Signal model (34)Signal model (34)g ( )g ( )
H= = = = +Y Fy FCx FCF X HX W
loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal
y
H=H FCFcirculant matrix leading to be a diagonal matrix
=H FCF
( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0
h h L L hh h
⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥
( )( ) ( )
( ) ( )
1 10 1 0 0i
h Lh L h N L
⎢ ⎥⎢ ⎥minus minus⎢ ⎥
equiv= minus minus⎢ ⎥⎢ ⎥
C( ) ( )
( ) ( )
0 1 1 100
0 0 1 1 0
h L h N L
h N L L h N
⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
45
( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦
Synchronization2010722
Signal model (44)Signal model (44)g ( )g ( )
loz In a time-invariant channel the channel matrix H would be a diagonal matrix
k k k kY H X W= +
( )1 1
2
0 0
N Lj kl N
kn l
H h n l e πminus minus
minus
= =
= sumsum
loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI
k k k k kY H X I Wα= + +k k k k k
( ) ( )1 1 1
2 2 N N L
j m k Nj ml Nk mI X h n l e e ππ
minus minus minusminusminus⎛ ⎞
= ⎜ ⎟⎝ ⎠
sum sum sumICI
46
0 0 0m n lm k= = =ne
⎝ ⎠
Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
47 Synchronization2010722
Impact of timing offset Impact of timing offset p gp g
loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively
( )I f s
Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ
( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr
2 fj k Nk k k kY X H e Wπδminus= +
loz For integer part of timing offset k k k kY X H e W+
I sTδ ( ) 0g IN L δminus minus le leFFT
2 Ij k NY X H e Wπδminus= +
( ) ( ) 2 I sFFT
j f TI sh t T H f e π δδ minusminus rarr
48
k k k kY X H e W= +
Synchronization2010722
Impact of timing offset derivationImpact of timing offset derivationp gp g
49 Synchronization2010722
Discussion of symbol boundaryDiscussion of symbol boundaryy yy y
50 Synchronization2010722
Impact of timing offsetImpact of timing offsetp gp g
loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI
2 Ij k NIk k k k
NY X H e ISI W
Nπδδ minusminus
= + +
loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk
e e
p kloz Phase Shift can be resolved by Differential Encoding
Decoding or carrier recovery with pilots
51 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
52 Synchronization2010722
Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )
53 Synchronization2010722
Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )
( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s
i n t i N N T N T nTy y t e
= + + +=
( ) ( )( ) ( ) ( ) 12
sin
sin
g gI f f
I I
i N N N Nj jf N Ni k i k k
f
Y X H e eN
π ε ε π ε
ε ε
πεπε
+ + minus+
minus minus=⎛ ⎞⎜ ⎟
Faded signal attenuated and rotated by CFO
( )( ) ( ) ( ) ( )11 2
sin
sin g gI f I f
i N N N NN j j l kI f
NN
l k π ε ε π ε επ ε ε + + minusminus + + + minus
⎜ ⎟⎝ ⎠
+ + minussum
( )( )( )
( ) ( )2
0
sin
I f I f
I
j j l kI f N Ni l l
l l k I f
X H e el k
NN
π ε ε π ε ε
ε π ε ε
+ + +
= ne minus
+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠
sum
i k
N
W
⎜ ⎟⎝ ⎠
+ Inter-Carrier Interference (ICI)
54 Synchronization2010722
Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )
loz An easier representation of received signal with only p g yfractional CFO is
1N
( ) ( )1
0
0N
k k k l l kl l k
Y H X C H X C l k Wminus
= ne
= + minus +⎡ ⎤⎣ ⎦sum
where the ICI coefficient is given by
( )( )( ) ( )
sin 1exp 1fl k
C l k j l kπ ε
π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )
( )exp 1sin
f
f
C l k j l kNN l k
N
π επ ε
minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠
55 Synchronization2010722
Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )
loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality
loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift
56 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
57 Synchronization2010722
TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )
loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq
58 Synchronization2010722
TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )
loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below
loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and
rN2 can be expressed by
0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+
( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +
59
( )
Synchronization2010722
TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )
loz Without CFO the first half is identical to the second half (in time order)
loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below
( ) 12 in discretesft T f NT
NTφ π π π ε πε
⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟
⎝ ⎠
helliphellip
sNT⎝ ⎠
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)
60
Length=N2
Synchronization2010722
TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )
loz Received signal without CFOg0
0 0
jr a e θ=jθ 2 0
2 2 0
Nj j
N Nr a e a e
θ θ= =
loz Received signal with CFO0 0
0 0
j jr a e eθ φ= Extra phase rotation due to CFO0 0
2 2 20
2 2 0
N N Nj j jj
N Nr a e e a e e
θ φ φθ= =
due to CFO
Phase difference 2 2 0N N
( )2 02 20 2 0 0
Nj j T fN
r r a e a eφ φ πminus Δ= =
which contains the information about CFO
61
0 2 0 0Nr r a e a e
Synchronization2010722
TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )
loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be
loz Note that d is a time index corresponding to the first( ) ( ) 2 1
20
Nd m d m Nm
P d r rminus
+ + +== sum
loz Note that d is a time index corresponding to the first sample in a window of N samples
loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by
( )22 1
20
Nd m NR d rminus
+ += sumloz A timing metric can be defined as
))(()(
)( 2
2
dRdP
dM =
( ) 20 d m Nm + +=sum
62
))(( dR
Synchronization2010722
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
Cyclic PrefixCyclic Prefixyy
31
Cyclic PrefixCyclic Prefix
loz Spectrum of channel response with length (smaller [ ]nh hL
yy
than )h
gN[ ] nhFFTHk =
loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr
loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=
where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier
[ ] [ ] [ ]N
Notimes
[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce
kk H
X =rArrcomplexity of channel equalizationrdquo
32
Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))
bull bull( )( )nr~Symbols
dataOutput
bullbullbull
bullbullbull( )nr
bullbull
bullbull
33
Cyclic PrefixCyclic Prefix
loz One of the most important reasons to do OFDM is the
yy
pefficient way it deals with multipath delay spread
loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)
34
Bandwidth EfficiencyBandwidth Efficiencyyy
loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )
loz The diagram for bandwidth efficiency of OFDM system is shown below
35
SummarySummaryyy
loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j
the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference
(ISI)loz The main advantage of the OFDM transmission technique is its
high performance even in frequency selective channels
loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems
36
SynchronizationSynchronization
Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg
38 Synchronization2010722
Introduction (13)Introduction (13)( )( )
loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods
loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead
loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the
cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of
computational complexity therefore it may be not be available in short-burst wireless communication
39 Synchronization2010722
Introduction (23)Introduction (23)( )( )
loz Transcevier
40 Synchronization2010722
Introduction (33)Introduction (33)( )( )
loz Synchronization erroryloz Symbol timing offset (TO)
loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)
loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect
41 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg
42 Synchronization2010722
Signal model (14)Signal model (14)g ( )g ( )
loz Time domain sample p1
2
0
1 Nj nk N
n kk
x X eN
πminus
minus= sum
Xk the frequency domain data at the k-th subcarrier0kN =
loz The multipath fading channel
( ) ( ) ( )1L
h t h t δminus
sum
loz Received time domain signal
( ) ( ) ( )0
l ll
h t h tτ τ δ τ τ=
= minussum
loz Received time-domain signal
( ) ( )1
L
n n ly h n l x w nminus
minus= +sum43
( ) ( )0l=sum
Synchronization2010722
Signal model (24)Signal model (24)g ( )g ( )
loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting
12
0
Nj kn N
k nn
Y y e πminus
minus
=
= sum
( )1 1
2
0
N Lj kn N
n l kn l
h n l x e Wπminus minus
minusminus
=
= +sumsum
loz The equation above could be expressed in matrix form
0n l
loz The equation above could be expressed in matrix form
= +Y HX W
44 Synchronization2010722
Signal model (34)Signal model (34)g ( )g ( )
H= = = = +Y Fy FCx FCF X HX W
loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal
y
H=H FCFcirculant matrix leading to be a diagonal matrix
=H FCF
( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0
h h L L hh h
⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥
( )( ) ( )
( ) ( )
1 10 1 0 0i
h Lh L h N L
⎢ ⎥⎢ ⎥minus minus⎢ ⎥
equiv= minus minus⎢ ⎥⎢ ⎥
C( ) ( )
( ) ( )
0 1 1 100
0 0 1 1 0
h L h N L
h N L L h N
⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
45
( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦
Synchronization2010722
Signal model (44)Signal model (44)g ( )g ( )
loz In a time-invariant channel the channel matrix H would be a diagonal matrix
k k k kY H X W= +
( )1 1
2
0 0
N Lj kl N
kn l
H h n l e πminus minus
minus
= =
= sumsum
loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI
k k k k kY H X I Wα= + +k k k k k
( ) ( )1 1 1
2 2 N N L
j m k Nj ml Nk mI X h n l e e ππ
minus minus minusminusminus⎛ ⎞
= ⎜ ⎟⎝ ⎠
sum sum sumICI
46
0 0 0m n lm k= = =ne
⎝ ⎠
Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
47 Synchronization2010722
Impact of timing offset Impact of timing offset p gp g
loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively
( )I f s
Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ
( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr
2 fj k Nk k k kY X H e Wπδminus= +
loz For integer part of timing offset k k k kY X H e W+
I sTδ ( ) 0g IN L δminus minus le leFFT
2 Ij k NY X H e Wπδminus= +
( ) ( ) 2 I sFFT
j f TI sh t T H f e π δδ minusminus rarr
48
k k k kY X H e W= +
Synchronization2010722
Impact of timing offset derivationImpact of timing offset derivationp gp g
49 Synchronization2010722
Discussion of symbol boundaryDiscussion of symbol boundaryy yy y
50 Synchronization2010722
Impact of timing offsetImpact of timing offsetp gp g
loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI
2 Ij k NIk k k k
NY X H e ISI W
Nπδδ minusminus
= + +
loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk
e e
p kloz Phase Shift can be resolved by Differential Encoding
Decoding or carrier recovery with pilots
51 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
52 Synchronization2010722
Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )
53 Synchronization2010722
Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )
( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s
i n t i N N T N T nTy y t e
= + + +=
( ) ( )( ) ( ) ( ) 12
sin
sin
g gI f f
I I
i N N N Nj jf N Ni k i k k
f
Y X H e eN
π ε ε π ε
ε ε
πεπε
+ + minus+
minus minus=⎛ ⎞⎜ ⎟
Faded signal attenuated and rotated by CFO
( )( ) ( ) ( ) ( )11 2
sin
sin g gI f I f
i N N N NN j j l kI f
NN
l k π ε ε π ε επ ε ε + + minusminus + + + minus
⎜ ⎟⎝ ⎠
+ + minussum
( )( )( )
( ) ( )2
0
sin
I f I f
I
j j l kI f N Ni l l
l l k I f
X H e el k
NN
π ε ε π ε ε
ε π ε ε
+ + +
= ne minus
+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠
sum
i k
N
W
⎜ ⎟⎝ ⎠
+ Inter-Carrier Interference (ICI)
54 Synchronization2010722
Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )
loz An easier representation of received signal with only p g yfractional CFO is
1N
( ) ( )1
0
0N
k k k l l kl l k
Y H X C H X C l k Wminus
= ne
= + minus +⎡ ⎤⎣ ⎦sum
where the ICI coefficient is given by
( )( )( ) ( )
sin 1exp 1fl k
C l k j l kπ ε
π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )
( )exp 1sin
f
f
C l k j l kNN l k
N
π επ ε
minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠
55 Synchronization2010722
Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )
loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality
loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift
56 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
57 Synchronization2010722
TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )
loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq
58 Synchronization2010722
TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )
loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below
loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and
rN2 can be expressed by
0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+
( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +
59
( )
Synchronization2010722
TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )
loz Without CFO the first half is identical to the second half (in time order)
loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below
( ) 12 in discretesft T f NT
NTφ π π π ε πε
⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟
⎝ ⎠
helliphellip
sNT⎝ ⎠
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)
60
Length=N2
Synchronization2010722
TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )
loz Received signal without CFOg0
0 0
jr a e θ=jθ 2 0
2 2 0
Nj j
N Nr a e a e
θ θ= =
loz Received signal with CFO0 0
0 0
j jr a e eθ φ= Extra phase rotation due to CFO0 0
2 2 20
2 2 0
N N Nj j jj
N Nr a e e a e e
θ φ φθ= =
due to CFO
Phase difference 2 2 0N N
( )2 02 20 2 0 0
Nj j T fN
r r a e a eφ φ πminus Δ= =
which contains the information about CFO
61
0 2 0 0Nr r a e a e
Synchronization2010722
TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )
loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be
loz Note that d is a time index corresponding to the first( ) ( ) 2 1
20
Nd m d m Nm
P d r rminus
+ + +== sum
loz Note that d is a time index corresponding to the first sample in a window of N samples
loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by
( )22 1
20
Nd m NR d rminus
+ += sumloz A timing metric can be defined as
))(()(
)( 2
2
dRdP
dM =
( ) 20 d m Nm + +=sum
62
))(( dR
Synchronization2010722
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
Cyclic PrefixCyclic Prefix
loz Spectrum of channel response with length (smaller [ ]nh hL
yy
than )h
gN[ ] nhFFTHk =
loz Received complete OFDM signal[ ] [ ] [ ] 20 ~~ minus++leleotimes= hg LNNnnhnDnr
loz Received useful part [ ]nr[ ] [ ] [ ]nhnDnr Notimes=
where is N-point circular convolution (due to CP)loz Received symbol at k-th subcarrier
[ ] [ ] [ ]N
Notimes
[ ] [ ] [ ] kkNk HXnhnDFFTnrFFTY =otimes== kYX =rArr ldquoUseful property for OFDM system to reduce
kk H
X =rArrcomplexity of channel equalizationrdquo
32
Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))
bull bull( )( )nr~Symbols
dataOutput
bullbullbull
bullbullbull( )nr
bullbull
bullbull
33
Cyclic PrefixCyclic Prefix
loz One of the most important reasons to do OFDM is the
yy
pefficient way it deals with multipath delay spread
loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)
34
Bandwidth EfficiencyBandwidth Efficiencyyy
loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )
loz The diagram for bandwidth efficiency of OFDM system is shown below
35
SummarySummaryyy
loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j
the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference
(ISI)loz The main advantage of the OFDM transmission technique is its
high performance even in frequency selective channels
loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems
36
SynchronizationSynchronization
Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg
38 Synchronization2010722
Introduction (13)Introduction (13)( )( )
loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods
loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead
loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the
cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of
computational complexity therefore it may be not be available in short-burst wireless communication
39 Synchronization2010722
Introduction (23)Introduction (23)( )( )
loz Transcevier
40 Synchronization2010722
Introduction (33)Introduction (33)( )( )
loz Synchronization erroryloz Symbol timing offset (TO)
loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)
loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect
41 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg
42 Synchronization2010722
Signal model (14)Signal model (14)g ( )g ( )
loz Time domain sample p1
2
0
1 Nj nk N
n kk
x X eN
πminus
minus= sum
Xk the frequency domain data at the k-th subcarrier0kN =
loz The multipath fading channel
( ) ( ) ( )1L
h t h t δminus
sum
loz Received time domain signal
( ) ( ) ( )0
l ll
h t h tτ τ δ τ τ=
= minussum
loz Received time-domain signal
( ) ( )1
L
n n ly h n l x w nminus
minus= +sum43
( ) ( )0l=sum
Synchronization2010722
Signal model (24)Signal model (24)g ( )g ( )
loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting
12
0
Nj kn N
k nn
Y y e πminus
minus
=
= sum
( )1 1
2
0
N Lj kn N
n l kn l
h n l x e Wπminus minus
minusminus
=
= +sumsum
loz The equation above could be expressed in matrix form
0n l
loz The equation above could be expressed in matrix form
= +Y HX W
44 Synchronization2010722
Signal model (34)Signal model (34)g ( )g ( )
H= = = = +Y Fy FCx FCF X HX W
loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal
y
H=H FCFcirculant matrix leading to be a diagonal matrix
=H FCF
( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0
h h L L hh h
⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥
( )( ) ( )
( ) ( )
1 10 1 0 0i
h Lh L h N L
⎢ ⎥⎢ ⎥minus minus⎢ ⎥
equiv= minus minus⎢ ⎥⎢ ⎥
C( ) ( )
( ) ( )
0 1 1 100
0 0 1 1 0
h L h N L
h N L L h N
⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
45
( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦
Synchronization2010722
Signal model (44)Signal model (44)g ( )g ( )
loz In a time-invariant channel the channel matrix H would be a diagonal matrix
k k k kY H X W= +
( )1 1
2
0 0
N Lj kl N
kn l
H h n l e πminus minus
minus
= =
= sumsum
loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI
k k k k kY H X I Wα= + +k k k k k
( ) ( )1 1 1
2 2 N N L
j m k Nj ml Nk mI X h n l e e ππ
minus minus minusminusminus⎛ ⎞
= ⎜ ⎟⎝ ⎠
sum sum sumICI
46
0 0 0m n lm k= = =ne
⎝ ⎠
Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
47 Synchronization2010722
Impact of timing offset Impact of timing offset p gp g
loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively
( )I f s
Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ
( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr
2 fj k Nk k k kY X H e Wπδminus= +
loz For integer part of timing offset k k k kY X H e W+
I sTδ ( ) 0g IN L δminus minus le leFFT
2 Ij k NY X H e Wπδminus= +
( ) ( ) 2 I sFFT
j f TI sh t T H f e π δδ minusminus rarr
48
k k k kY X H e W= +
Synchronization2010722
Impact of timing offset derivationImpact of timing offset derivationp gp g
49 Synchronization2010722
Discussion of symbol boundaryDiscussion of symbol boundaryy yy y
50 Synchronization2010722
Impact of timing offsetImpact of timing offsetp gp g
loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI
2 Ij k NIk k k k
NY X H e ISI W
Nπδδ minusminus
= + +
loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk
e e
p kloz Phase Shift can be resolved by Differential Encoding
Decoding or carrier recovery with pilots
51 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
52 Synchronization2010722
Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )
53 Synchronization2010722
Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )
( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s
i n t i N N T N T nTy y t e
= + + +=
( ) ( )( ) ( ) ( ) 12
sin
sin
g gI f f
I I
i N N N Nj jf N Ni k i k k
f
Y X H e eN
π ε ε π ε
ε ε
πεπε
+ + minus+
minus minus=⎛ ⎞⎜ ⎟
Faded signal attenuated and rotated by CFO
( )( ) ( ) ( ) ( )11 2
sin
sin g gI f I f
i N N N NN j j l kI f
NN
l k π ε ε π ε επ ε ε + + minusminus + + + minus
⎜ ⎟⎝ ⎠
+ + minussum
( )( )( )
( ) ( )2
0
sin
I f I f
I
j j l kI f N Ni l l
l l k I f
X H e el k
NN
π ε ε π ε ε
ε π ε ε
+ + +
= ne minus
+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠
sum
i k
N
W
⎜ ⎟⎝ ⎠
+ Inter-Carrier Interference (ICI)
54 Synchronization2010722
Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )
loz An easier representation of received signal with only p g yfractional CFO is
1N
( ) ( )1
0
0N
k k k l l kl l k
Y H X C H X C l k Wminus
= ne
= + minus +⎡ ⎤⎣ ⎦sum
where the ICI coefficient is given by
( )( )( ) ( )
sin 1exp 1fl k
C l k j l kπ ε
π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )
( )exp 1sin
f
f
C l k j l kNN l k
N
π επ ε
minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠
55 Synchronization2010722
Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )
loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality
loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift
56 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
57 Synchronization2010722
TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )
loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq
58 Synchronization2010722
TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )
loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below
loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and
rN2 can be expressed by
0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+
( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +
59
( )
Synchronization2010722
TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )
loz Without CFO the first half is identical to the second half (in time order)
loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below
( ) 12 in discretesft T f NT
NTφ π π π ε πε
⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟
⎝ ⎠
helliphellip
sNT⎝ ⎠
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)
60
Length=N2
Synchronization2010722
TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )
loz Received signal without CFOg0
0 0
jr a e θ=jθ 2 0
2 2 0
Nj j
N Nr a e a e
θ θ= =
loz Received signal with CFO0 0
0 0
j jr a e eθ φ= Extra phase rotation due to CFO0 0
2 2 20
2 2 0
N N Nj j jj
N Nr a e e a e e
θ φ φθ= =
due to CFO
Phase difference 2 2 0N N
( )2 02 20 2 0 0
Nj j T fN
r r a e a eφ φ πminus Δ= =
which contains the information about CFO
61
0 2 0 0Nr r a e a e
Synchronization2010722
TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )
loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be
loz Note that d is a time index corresponding to the first( ) ( ) 2 1
20
Nd m d m Nm
P d r rminus
+ + +== sum
loz Note that d is a time index corresponding to the first sample in a window of N samples
loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by
( )22 1
20
Nd m NR d rminus
+ += sumloz A timing metric can be defined as
))(()(
)( 2
2
dRdP
dM =
( ) 20 d m Nm + +=sum
62
))(( dR
Synchronization2010722
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
Cyclic Cyclic Prefix (OFDM Prefix (OFDM Receiver)Receiver)yy (( ))
bull bull( )( )nr~Symbols
dataOutput
bullbullbull
bullbullbull( )nr
bullbull
bullbull
33
Cyclic PrefixCyclic Prefix
loz One of the most important reasons to do OFDM is the
yy
pefficient way it deals with multipath delay spread
loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)
34
Bandwidth EfficiencyBandwidth Efficiencyyy
loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )
loz The diagram for bandwidth efficiency of OFDM system is shown below
35
SummarySummaryyy
loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j
the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference
(ISI)loz The main advantage of the OFDM transmission technique is its
high performance even in frequency selective channels
loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems
36
SynchronizationSynchronization
Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg
38 Synchronization2010722
Introduction (13)Introduction (13)( )( )
loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods
loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead
loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the
cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of
computational complexity therefore it may be not be available in short-burst wireless communication
39 Synchronization2010722
Introduction (23)Introduction (23)( )( )
loz Transcevier
40 Synchronization2010722
Introduction (33)Introduction (33)( )( )
loz Synchronization erroryloz Symbol timing offset (TO)
loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)
loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect
41 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg
42 Synchronization2010722
Signal model (14)Signal model (14)g ( )g ( )
loz Time domain sample p1
2
0
1 Nj nk N
n kk
x X eN
πminus
minus= sum
Xk the frequency domain data at the k-th subcarrier0kN =
loz The multipath fading channel
( ) ( ) ( )1L
h t h t δminus
sum
loz Received time domain signal
( ) ( ) ( )0
l ll
h t h tτ τ δ τ τ=
= minussum
loz Received time-domain signal
( ) ( )1
L
n n ly h n l x w nminus
minus= +sum43
( ) ( )0l=sum
Synchronization2010722
Signal model (24)Signal model (24)g ( )g ( )
loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting
12
0
Nj kn N
k nn
Y y e πminus
minus
=
= sum
( )1 1
2
0
N Lj kn N
n l kn l
h n l x e Wπminus minus
minusminus
=
= +sumsum
loz The equation above could be expressed in matrix form
0n l
loz The equation above could be expressed in matrix form
= +Y HX W
44 Synchronization2010722
Signal model (34)Signal model (34)g ( )g ( )
H= = = = +Y Fy FCx FCF X HX W
loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal
y
H=H FCFcirculant matrix leading to be a diagonal matrix
=H FCF
( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0
h h L L hh h
⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥
( )( ) ( )
( ) ( )
1 10 1 0 0i
h Lh L h N L
⎢ ⎥⎢ ⎥minus minus⎢ ⎥
equiv= minus minus⎢ ⎥⎢ ⎥
C( ) ( )
( ) ( )
0 1 1 100
0 0 1 1 0
h L h N L
h N L L h N
⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
45
( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦
Synchronization2010722
Signal model (44)Signal model (44)g ( )g ( )
loz In a time-invariant channel the channel matrix H would be a diagonal matrix
k k k kY H X W= +
( )1 1
2
0 0
N Lj kl N
kn l
H h n l e πminus minus
minus
= =
= sumsum
loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI
k k k k kY H X I Wα= + +k k k k k
( ) ( )1 1 1
2 2 N N L
j m k Nj ml Nk mI X h n l e e ππ
minus minus minusminusminus⎛ ⎞
= ⎜ ⎟⎝ ⎠
sum sum sumICI
46
0 0 0m n lm k= = =ne
⎝ ⎠
Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
47 Synchronization2010722
Impact of timing offset Impact of timing offset p gp g
loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively
( )I f s
Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ
( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr
2 fj k Nk k k kY X H e Wπδminus= +
loz For integer part of timing offset k k k kY X H e W+
I sTδ ( ) 0g IN L δminus minus le leFFT
2 Ij k NY X H e Wπδminus= +
( ) ( ) 2 I sFFT
j f TI sh t T H f e π δδ minusminus rarr
48
k k k kY X H e W= +
Synchronization2010722
Impact of timing offset derivationImpact of timing offset derivationp gp g
49 Synchronization2010722
Discussion of symbol boundaryDiscussion of symbol boundaryy yy y
50 Synchronization2010722
Impact of timing offsetImpact of timing offsetp gp g
loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI
2 Ij k NIk k k k
NY X H e ISI W
Nπδδ minusminus
= + +
loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk
e e
p kloz Phase Shift can be resolved by Differential Encoding
Decoding or carrier recovery with pilots
51 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
52 Synchronization2010722
Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )
53 Synchronization2010722
Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )
( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s
i n t i N N T N T nTy y t e
= + + +=
( ) ( )( ) ( ) ( ) 12
sin
sin
g gI f f
I I
i N N N Nj jf N Ni k i k k
f
Y X H e eN
π ε ε π ε
ε ε
πεπε
+ + minus+
minus minus=⎛ ⎞⎜ ⎟
Faded signal attenuated and rotated by CFO
( )( ) ( ) ( ) ( )11 2
sin
sin g gI f I f
i N N N NN j j l kI f
NN
l k π ε ε π ε επ ε ε + + minusminus + + + minus
⎜ ⎟⎝ ⎠
+ + minussum
( )( )( )
( ) ( )2
0
sin
I f I f
I
j j l kI f N Ni l l
l l k I f
X H e el k
NN
π ε ε π ε ε
ε π ε ε
+ + +
= ne minus
+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠
sum
i k
N
W
⎜ ⎟⎝ ⎠
+ Inter-Carrier Interference (ICI)
54 Synchronization2010722
Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )
loz An easier representation of received signal with only p g yfractional CFO is
1N
( ) ( )1
0
0N
k k k l l kl l k
Y H X C H X C l k Wminus
= ne
= + minus +⎡ ⎤⎣ ⎦sum
where the ICI coefficient is given by
( )( )( ) ( )
sin 1exp 1fl k
C l k j l kπ ε
π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )
( )exp 1sin
f
f
C l k j l kNN l k
N
π επ ε
minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠
55 Synchronization2010722
Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )
loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality
loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift
56 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
57 Synchronization2010722
TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )
loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq
58 Synchronization2010722
TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )
loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below
loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and
rN2 can be expressed by
0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+
( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +
59
( )
Synchronization2010722
TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )
loz Without CFO the first half is identical to the second half (in time order)
loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below
( ) 12 in discretesft T f NT
NTφ π π π ε πε
⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟
⎝ ⎠
helliphellip
sNT⎝ ⎠
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)
60
Length=N2
Synchronization2010722
TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )
loz Received signal without CFOg0
0 0
jr a e θ=jθ 2 0
2 2 0
Nj j
N Nr a e a e
θ θ= =
loz Received signal with CFO0 0
0 0
j jr a e eθ φ= Extra phase rotation due to CFO0 0
2 2 20
2 2 0
N N Nj j jj
N Nr a e e a e e
θ φ φθ= =
due to CFO
Phase difference 2 2 0N N
( )2 02 20 2 0 0
Nj j T fN
r r a e a eφ φ πminus Δ= =
which contains the information about CFO
61
0 2 0 0Nr r a e a e
Synchronization2010722
TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )
loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be
loz Note that d is a time index corresponding to the first( ) ( ) 2 1
20
Nd m d m Nm
P d r rminus
+ + +== sum
loz Note that d is a time index corresponding to the first sample in a window of N samples
loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by
( )22 1
20
Nd m NR d rminus
+ += sumloz A timing metric can be defined as
))(()(
)( 2
2
dRdP
dM =
( ) 20 d m Nm + +=sum
62
))(( dR
Synchronization2010722
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
Cyclic PrefixCyclic Prefix
loz One of the most important reasons to do OFDM is the
yy
pefficient way it deals with multipath delay spread
loz To eliminate inter-symbol interference (ISI) almostloz To eliminate inter symbol interference (ISI) almost completely a guard time is introduced for each OFDM symbolsymbol(The guard time is chosen larger than the delay spread)
34
Bandwidth EfficiencyBandwidth Efficiencyyy
loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )
loz The diagram for bandwidth efficiency of OFDM system is shown below
35
SummarySummaryyy
loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j
the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference
(ISI)loz The main advantage of the OFDM transmission technique is its
high performance even in frequency selective channels
loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems
36
SynchronizationSynchronization
Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg
38 Synchronization2010722
Introduction (13)Introduction (13)( )( )
loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods
loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead
loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the
cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of
computational complexity therefore it may be not be available in short-burst wireless communication
39 Synchronization2010722
Introduction (23)Introduction (23)( )( )
loz Transcevier
40 Synchronization2010722
Introduction (33)Introduction (33)( )( )
loz Synchronization erroryloz Symbol timing offset (TO)
loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)
loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect
41 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg
42 Synchronization2010722
Signal model (14)Signal model (14)g ( )g ( )
loz Time domain sample p1
2
0
1 Nj nk N
n kk
x X eN
πminus
minus= sum
Xk the frequency domain data at the k-th subcarrier0kN =
loz The multipath fading channel
( ) ( ) ( )1L
h t h t δminus
sum
loz Received time domain signal
( ) ( ) ( )0
l ll
h t h tτ τ δ τ τ=
= minussum
loz Received time-domain signal
( ) ( )1
L
n n ly h n l x w nminus
minus= +sum43
( ) ( )0l=sum
Synchronization2010722
Signal model (24)Signal model (24)g ( )g ( )
loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting
12
0
Nj kn N
k nn
Y y e πminus
minus
=
= sum
( )1 1
2
0
N Lj kn N
n l kn l
h n l x e Wπminus minus
minusminus
=
= +sumsum
loz The equation above could be expressed in matrix form
0n l
loz The equation above could be expressed in matrix form
= +Y HX W
44 Synchronization2010722
Signal model (34)Signal model (34)g ( )g ( )
H= = = = +Y Fy FCx FCF X HX W
loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal
y
H=H FCFcirculant matrix leading to be a diagonal matrix
=H FCF
( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0
h h L L hh h
⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥
( )( ) ( )
( ) ( )
1 10 1 0 0i
h Lh L h N L
⎢ ⎥⎢ ⎥minus minus⎢ ⎥
equiv= minus minus⎢ ⎥⎢ ⎥
C( ) ( )
( ) ( )
0 1 1 100
0 0 1 1 0
h L h N L
h N L L h N
⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
45
( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦
Synchronization2010722
Signal model (44)Signal model (44)g ( )g ( )
loz In a time-invariant channel the channel matrix H would be a diagonal matrix
k k k kY H X W= +
( )1 1
2
0 0
N Lj kl N
kn l
H h n l e πminus minus
minus
= =
= sumsum
loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI
k k k k kY H X I Wα= + +k k k k k
( ) ( )1 1 1
2 2 N N L
j m k Nj ml Nk mI X h n l e e ππ
minus minus minusminusminus⎛ ⎞
= ⎜ ⎟⎝ ⎠
sum sum sumICI
46
0 0 0m n lm k= = =ne
⎝ ⎠
Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
47 Synchronization2010722
Impact of timing offset Impact of timing offset p gp g
loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively
( )I f s
Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ
( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr
2 fj k Nk k k kY X H e Wπδminus= +
loz For integer part of timing offset k k k kY X H e W+
I sTδ ( ) 0g IN L δminus minus le leFFT
2 Ij k NY X H e Wπδminus= +
( ) ( ) 2 I sFFT
j f TI sh t T H f e π δδ minusminus rarr
48
k k k kY X H e W= +
Synchronization2010722
Impact of timing offset derivationImpact of timing offset derivationp gp g
49 Synchronization2010722
Discussion of symbol boundaryDiscussion of symbol boundaryy yy y
50 Synchronization2010722
Impact of timing offsetImpact of timing offsetp gp g
loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI
2 Ij k NIk k k k
NY X H e ISI W
Nπδδ minusminus
= + +
loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk
e e
p kloz Phase Shift can be resolved by Differential Encoding
Decoding or carrier recovery with pilots
51 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
52 Synchronization2010722
Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )
53 Synchronization2010722
Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )
( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s
i n t i N N T N T nTy y t e
= + + +=
( ) ( )( ) ( ) ( ) 12
sin
sin
g gI f f
I I
i N N N Nj jf N Ni k i k k
f
Y X H e eN
π ε ε π ε
ε ε
πεπε
+ + minus+
minus minus=⎛ ⎞⎜ ⎟
Faded signal attenuated and rotated by CFO
( )( ) ( ) ( ) ( )11 2
sin
sin g gI f I f
i N N N NN j j l kI f
NN
l k π ε ε π ε επ ε ε + + minusminus + + + minus
⎜ ⎟⎝ ⎠
+ + minussum
( )( )( )
( ) ( )2
0
sin
I f I f
I
j j l kI f N Ni l l
l l k I f
X H e el k
NN
π ε ε π ε ε
ε π ε ε
+ + +
= ne minus
+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠
sum
i k
N
W
⎜ ⎟⎝ ⎠
+ Inter-Carrier Interference (ICI)
54 Synchronization2010722
Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )
loz An easier representation of received signal with only p g yfractional CFO is
1N
( ) ( )1
0
0N
k k k l l kl l k
Y H X C H X C l k Wminus
= ne
= + minus +⎡ ⎤⎣ ⎦sum
where the ICI coefficient is given by
( )( )( ) ( )
sin 1exp 1fl k
C l k j l kπ ε
π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )
( )exp 1sin
f
f
C l k j l kNN l k
N
π επ ε
minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠
55 Synchronization2010722
Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )
loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality
loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift
56 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
57 Synchronization2010722
TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )
loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq
58 Synchronization2010722
TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )
loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below
loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and
rN2 can be expressed by
0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+
( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +
59
( )
Synchronization2010722
TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )
loz Without CFO the first half is identical to the second half (in time order)
loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below
( ) 12 in discretesft T f NT
NTφ π π π ε πε
⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟
⎝ ⎠
helliphellip
sNT⎝ ⎠
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)
60
Length=N2
Synchronization2010722
TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )
loz Received signal without CFOg0
0 0
jr a e θ=jθ 2 0
2 2 0
Nj j
N Nr a e a e
θ θ= =
loz Received signal with CFO0 0
0 0
j jr a e eθ φ= Extra phase rotation due to CFO0 0
2 2 20
2 2 0
N N Nj j jj
N Nr a e e a e e
θ φ φθ= =
due to CFO
Phase difference 2 2 0N N
( )2 02 20 2 0 0
Nj j T fN
r r a e a eφ φ πminus Δ= =
which contains the information about CFO
61
0 2 0 0Nr r a e a e
Synchronization2010722
TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )
loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be
loz Note that d is a time index corresponding to the first( ) ( ) 2 1
20
Nd m d m Nm
P d r rminus
+ + +== sum
loz Note that d is a time index corresponding to the first sample in a window of N samples
loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by
( )22 1
20
Nd m NR d rminus
+ += sumloz A timing metric can be defined as
))(()(
)( 2
2
dRdP
dM =
( ) 20 d m Nm + +=sum
62
))(( dR
Synchronization2010722
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
Bandwidth EfficiencyBandwidth Efficiencyyy
loz In a classical parallel system the channel is divided into p y N non-overlapping sub-channels to avoid inter-carrier interference (ICI)( )
loz The diagram for bandwidth efficiency of OFDM system is shown below
35
SummarySummaryyy
loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j
the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference
(ISI)loz The main advantage of the OFDM transmission technique is its
high performance even in frequency selective channels
loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems
36
SynchronizationSynchronization
Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg
38 Synchronization2010722
Introduction (13)Introduction (13)( )( )
loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods
loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead
loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the
cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of
computational complexity therefore it may be not be available in short-burst wireless communication
39 Synchronization2010722
Introduction (23)Introduction (23)( )( )
loz Transcevier
40 Synchronization2010722
Introduction (33)Introduction (33)( )( )
loz Synchronization erroryloz Symbol timing offset (TO)
loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)
loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect
41 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg
42 Synchronization2010722
Signal model (14)Signal model (14)g ( )g ( )
loz Time domain sample p1
2
0
1 Nj nk N
n kk
x X eN
πminus
minus= sum
Xk the frequency domain data at the k-th subcarrier0kN =
loz The multipath fading channel
( ) ( ) ( )1L
h t h t δminus
sum
loz Received time domain signal
( ) ( ) ( )0
l ll
h t h tτ τ δ τ τ=
= minussum
loz Received time-domain signal
( ) ( )1
L
n n ly h n l x w nminus
minus= +sum43
( ) ( )0l=sum
Synchronization2010722
Signal model (24)Signal model (24)g ( )g ( )
loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting
12
0
Nj kn N
k nn
Y y e πminus
minus
=
= sum
( )1 1
2
0
N Lj kn N
n l kn l
h n l x e Wπminus minus
minusminus
=
= +sumsum
loz The equation above could be expressed in matrix form
0n l
loz The equation above could be expressed in matrix form
= +Y HX W
44 Synchronization2010722
Signal model (34)Signal model (34)g ( )g ( )
H= = = = +Y Fy FCx FCF X HX W
loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal
y
H=H FCFcirculant matrix leading to be a diagonal matrix
=H FCF
( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0
h h L L hh h
⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥
( )( ) ( )
( ) ( )
1 10 1 0 0i
h Lh L h N L
⎢ ⎥⎢ ⎥minus minus⎢ ⎥
equiv= minus minus⎢ ⎥⎢ ⎥
C( ) ( )
( ) ( )
0 1 1 100
0 0 1 1 0
h L h N L
h N L L h N
⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
45
( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦
Synchronization2010722
Signal model (44)Signal model (44)g ( )g ( )
loz In a time-invariant channel the channel matrix H would be a diagonal matrix
k k k kY H X W= +
( )1 1
2
0 0
N Lj kl N
kn l
H h n l e πminus minus
minus
= =
= sumsum
loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI
k k k k kY H X I Wα= + +k k k k k
( ) ( )1 1 1
2 2 N N L
j m k Nj ml Nk mI X h n l e e ππ
minus minus minusminusminus⎛ ⎞
= ⎜ ⎟⎝ ⎠
sum sum sumICI
46
0 0 0m n lm k= = =ne
⎝ ⎠
Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
47 Synchronization2010722
Impact of timing offset Impact of timing offset p gp g
loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively
( )I f s
Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ
( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr
2 fj k Nk k k kY X H e Wπδminus= +
loz For integer part of timing offset k k k kY X H e W+
I sTδ ( ) 0g IN L δminus minus le leFFT
2 Ij k NY X H e Wπδminus= +
( ) ( ) 2 I sFFT
j f TI sh t T H f e π δδ minusminus rarr
48
k k k kY X H e W= +
Synchronization2010722
Impact of timing offset derivationImpact of timing offset derivationp gp g
49 Synchronization2010722
Discussion of symbol boundaryDiscussion of symbol boundaryy yy y
50 Synchronization2010722
Impact of timing offsetImpact of timing offsetp gp g
loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI
2 Ij k NIk k k k
NY X H e ISI W
Nπδδ minusminus
= + +
loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk
e e
p kloz Phase Shift can be resolved by Differential Encoding
Decoding or carrier recovery with pilots
51 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
52 Synchronization2010722
Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )
53 Synchronization2010722
Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )
( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s
i n t i N N T N T nTy y t e
= + + +=
( ) ( )( ) ( ) ( ) 12
sin
sin
g gI f f
I I
i N N N Nj jf N Ni k i k k
f
Y X H e eN
π ε ε π ε
ε ε
πεπε
+ + minus+
minus minus=⎛ ⎞⎜ ⎟
Faded signal attenuated and rotated by CFO
( )( ) ( ) ( ) ( )11 2
sin
sin g gI f I f
i N N N NN j j l kI f
NN
l k π ε ε π ε επ ε ε + + minusminus + + + minus
⎜ ⎟⎝ ⎠
+ + minussum
( )( )( )
( ) ( )2
0
sin
I f I f
I
j j l kI f N Ni l l
l l k I f
X H e el k
NN
π ε ε π ε ε
ε π ε ε
+ + +
= ne minus
+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠
sum
i k
N
W
⎜ ⎟⎝ ⎠
+ Inter-Carrier Interference (ICI)
54 Synchronization2010722
Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )
loz An easier representation of received signal with only p g yfractional CFO is
1N
( ) ( )1
0
0N
k k k l l kl l k
Y H X C H X C l k Wminus
= ne
= + minus +⎡ ⎤⎣ ⎦sum
where the ICI coefficient is given by
( )( )( ) ( )
sin 1exp 1fl k
C l k j l kπ ε
π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )
( )exp 1sin
f
f
C l k j l kNN l k
N
π επ ε
minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠
55 Synchronization2010722
Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )
loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality
loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift
56 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
57 Synchronization2010722
TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )
loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq
58 Synchronization2010722
TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )
loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below
loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and
rN2 can be expressed by
0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+
( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +
59
( )
Synchronization2010722
TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )
loz Without CFO the first half is identical to the second half (in time order)
loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below
( ) 12 in discretesft T f NT
NTφ π π π ε πε
⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟
⎝ ⎠
helliphellip
sNT⎝ ⎠
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)
60
Length=N2
Synchronization2010722
TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )
loz Received signal without CFOg0
0 0
jr a e θ=jθ 2 0
2 2 0
Nj j
N Nr a e a e
θ θ= =
loz Received signal with CFO0 0
0 0
j jr a e eθ φ= Extra phase rotation due to CFO0 0
2 2 20
2 2 0
N N Nj j jj
N Nr a e e a e e
θ φ φθ= =
due to CFO
Phase difference 2 2 0N N
( )2 02 20 2 0 0
Nj j T fN
r r a e a eφ φ πminus Δ= =
which contains the information about CFO
61
0 2 0 0Nr r a e a e
Synchronization2010722
TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )
loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be
loz Note that d is a time index corresponding to the first( ) ( ) 2 1
20
Nd m d m Nm
P d r rminus
+ + +== sum
loz Note that d is a time index corresponding to the first sample in a window of N samples
loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by
( )22 1
20
Nd m NR d rminus
+ += sumloz A timing metric can be defined as
))(()(
)( 2
2
dRdP
dM =
( ) 20 d m Nm + +=sum
62
))(( dR
Synchronization2010722
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
SummarySummaryyy
loz The advantage of the FFT-based OFDM system g yloz The use of IFFTFFT can reduce the computation complexityloz The orthogonality between the adjacent sub-carriers will make g y j
the use of transmission bandwidth more efficientloz The guard interval is used to resist the inter-symbol interference
(ISI)loz The main advantage of the OFDM transmission technique is its
high performance even in frequency selective channels
loz The drawbacks of the OFDM system loz It is highly vulnerable to synchronization errorsloz Peak to Average Power Ratio (PAPR) problems
36
SynchronizationSynchronization
Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg
38 Synchronization2010722
Introduction (13)Introduction (13)( )( )
loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods
loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead
loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the
cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of
computational complexity therefore it may be not be available in short-burst wireless communication
39 Synchronization2010722
Introduction (23)Introduction (23)( )( )
loz Transcevier
40 Synchronization2010722
Introduction (33)Introduction (33)( )( )
loz Synchronization erroryloz Symbol timing offset (TO)
loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)
loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect
41 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg
42 Synchronization2010722
Signal model (14)Signal model (14)g ( )g ( )
loz Time domain sample p1
2
0
1 Nj nk N
n kk
x X eN
πminus
minus= sum
Xk the frequency domain data at the k-th subcarrier0kN =
loz The multipath fading channel
( ) ( ) ( )1L
h t h t δminus
sum
loz Received time domain signal
( ) ( ) ( )0
l ll
h t h tτ τ δ τ τ=
= minussum
loz Received time-domain signal
( ) ( )1
L
n n ly h n l x w nminus
minus= +sum43
( ) ( )0l=sum
Synchronization2010722
Signal model (24)Signal model (24)g ( )g ( )
loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting
12
0
Nj kn N
k nn
Y y e πminus
minus
=
= sum
( )1 1
2
0
N Lj kn N
n l kn l
h n l x e Wπminus minus
minusminus
=
= +sumsum
loz The equation above could be expressed in matrix form
0n l
loz The equation above could be expressed in matrix form
= +Y HX W
44 Synchronization2010722
Signal model (34)Signal model (34)g ( )g ( )
H= = = = +Y Fy FCx FCF X HX W
loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal
y
H=H FCFcirculant matrix leading to be a diagonal matrix
=H FCF
( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0
h h L L hh h
⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥
( )( ) ( )
( ) ( )
1 10 1 0 0i
h Lh L h N L
⎢ ⎥⎢ ⎥minus minus⎢ ⎥
equiv= minus minus⎢ ⎥⎢ ⎥
C( ) ( )
( ) ( )
0 1 1 100
0 0 1 1 0
h L h N L
h N L L h N
⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
45
( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦
Synchronization2010722
Signal model (44)Signal model (44)g ( )g ( )
loz In a time-invariant channel the channel matrix H would be a diagonal matrix
k k k kY H X W= +
( )1 1
2
0 0
N Lj kl N
kn l
H h n l e πminus minus
minus
= =
= sumsum
loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI
k k k k kY H X I Wα= + +k k k k k
( ) ( )1 1 1
2 2 N N L
j m k Nj ml Nk mI X h n l e e ππ
minus minus minusminusminus⎛ ⎞
= ⎜ ⎟⎝ ⎠
sum sum sumICI
46
0 0 0m n lm k= = =ne
⎝ ⎠
Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
47 Synchronization2010722
Impact of timing offset Impact of timing offset p gp g
loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively
( )I f s
Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ
( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr
2 fj k Nk k k kY X H e Wπδminus= +
loz For integer part of timing offset k k k kY X H e W+
I sTδ ( ) 0g IN L δminus minus le leFFT
2 Ij k NY X H e Wπδminus= +
( ) ( ) 2 I sFFT
j f TI sh t T H f e π δδ minusminus rarr
48
k k k kY X H e W= +
Synchronization2010722
Impact of timing offset derivationImpact of timing offset derivationp gp g
49 Synchronization2010722
Discussion of symbol boundaryDiscussion of symbol boundaryy yy y
50 Synchronization2010722
Impact of timing offsetImpact of timing offsetp gp g
loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI
2 Ij k NIk k k k
NY X H e ISI W
Nπδδ minusminus
= + +
loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk
e e
p kloz Phase Shift can be resolved by Differential Encoding
Decoding or carrier recovery with pilots
51 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
52 Synchronization2010722
Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )
53 Synchronization2010722
Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )
( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s
i n t i N N T N T nTy y t e
= + + +=
( ) ( )( ) ( ) ( ) 12
sin
sin
g gI f f
I I
i N N N Nj jf N Ni k i k k
f
Y X H e eN
π ε ε π ε
ε ε
πεπε
+ + minus+
minus minus=⎛ ⎞⎜ ⎟
Faded signal attenuated and rotated by CFO
( )( ) ( ) ( ) ( )11 2
sin
sin g gI f I f
i N N N NN j j l kI f
NN
l k π ε ε π ε επ ε ε + + minusminus + + + minus
⎜ ⎟⎝ ⎠
+ + minussum
( )( )( )
( ) ( )2
0
sin
I f I f
I
j j l kI f N Ni l l
l l k I f
X H e el k
NN
π ε ε π ε ε
ε π ε ε
+ + +
= ne minus
+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠
sum
i k
N
W
⎜ ⎟⎝ ⎠
+ Inter-Carrier Interference (ICI)
54 Synchronization2010722
Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )
loz An easier representation of received signal with only p g yfractional CFO is
1N
( ) ( )1
0
0N
k k k l l kl l k
Y H X C H X C l k Wminus
= ne
= + minus +⎡ ⎤⎣ ⎦sum
where the ICI coefficient is given by
( )( )( ) ( )
sin 1exp 1fl k
C l k j l kπ ε
π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )
( )exp 1sin
f
f
C l k j l kNN l k
N
π επ ε
minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠
55 Synchronization2010722
Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )
loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality
loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift
56 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
57 Synchronization2010722
TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )
loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq
58 Synchronization2010722
TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )
loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below
loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and
rN2 can be expressed by
0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+
( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +
59
( )
Synchronization2010722
TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )
loz Without CFO the first half is identical to the second half (in time order)
loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below
( ) 12 in discretesft T f NT
NTφ π π π ε πε
⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟
⎝ ⎠
helliphellip
sNT⎝ ⎠
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)
60
Length=N2
Synchronization2010722
TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )
loz Received signal without CFOg0
0 0
jr a e θ=jθ 2 0
2 2 0
Nj j
N Nr a e a e
θ θ= =
loz Received signal with CFO0 0
0 0
j jr a e eθ φ= Extra phase rotation due to CFO0 0
2 2 20
2 2 0
N N Nj j jj
N Nr a e e a e e
θ φ φθ= =
due to CFO
Phase difference 2 2 0N N
( )2 02 20 2 0 0
Nj j T fN
r r a e a eφ φ πminus Δ= =
which contains the information about CFO
61
0 2 0 0Nr r a e a e
Synchronization2010722
TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )
loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be
loz Note that d is a time index corresponding to the first( ) ( ) 2 1
20
Nd m d m Nm
P d r rminus
+ + +== sum
loz Note that d is a time index corresponding to the first sample in a window of N samples
loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by
( )22 1
20
Nd m NR d rminus
+ += sumloz A timing metric can be defined as
))(()(
)( 2
2
dRdP
dM =
( ) 20 d m Nm + +=sum
62
))(( dR
Synchronization2010722
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
SynchronizationSynchronization
Wireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg
38 Synchronization2010722
Introduction (13)Introduction (13)( )( )
loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods
loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead
loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the
cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of
computational complexity therefore it may be not be available in short-burst wireless communication
39 Synchronization2010722
Introduction (23)Introduction (23)( )( )
loz Transcevier
40 Synchronization2010722
Introduction (33)Introduction (33)( )( )
loz Synchronization erroryloz Symbol timing offset (TO)
loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)
loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect
41 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg
42 Synchronization2010722
Signal model (14)Signal model (14)g ( )g ( )
loz Time domain sample p1
2
0
1 Nj nk N
n kk
x X eN
πminus
minus= sum
Xk the frequency domain data at the k-th subcarrier0kN =
loz The multipath fading channel
( ) ( ) ( )1L
h t h t δminus
sum
loz Received time domain signal
( ) ( ) ( )0
l ll
h t h tτ τ δ τ τ=
= minussum
loz Received time-domain signal
( ) ( )1
L
n n ly h n l x w nminus
minus= +sum43
( ) ( )0l=sum
Synchronization2010722
Signal model (24)Signal model (24)g ( )g ( )
loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting
12
0
Nj kn N
k nn
Y y e πminus
minus
=
= sum
( )1 1
2
0
N Lj kn N
n l kn l
h n l x e Wπminus minus
minusminus
=
= +sumsum
loz The equation above could be expressed in matrix form
0n l
loz The equation above could be expressed in matrix form
= +Y HX W
44 Synchronization2010722
Signal model (34)Signal model (34)g ( )g ( )
H= = = = +Y Fy FCx FCF X HX W
loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal
y
H=H FCFcirculant matrix leading to be a diagonal matrix
=H FCF
( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0
h h L L hh h
⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥
( )( ) ( )
( ) ( )
1 10 1 0 0i
h Lh L h N L
⎢ ⎥⎢ ⎥minus minus⎢ ⎥
equiv= minus minus⎢ ⎥⎢ ⎥
C( ) ( )
( ) ( )
0 1 1 100
0 0 1 1 0
h L h N L
h N L L h N
⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
45
( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦
Synchronization2010722
Signal model (44)Signal model (44)g ( )g ( )
loz In a time-invariant channel the channel matrix H would be a diagonal matrix
k k k kY H X W= +
( )1 1
2
0 0
N Lj kl N
kn l
H h n l e πminus minus
minus
= =
= sumsum
loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI
k k k k kY H X I Wα= + +k k k k k
( ) ( )1 1 1
2 2 N N L
j m k Nj ml Nk mI X h n l e e ππ
minus minus minusminusminus⎛ ⎞
= ⎜ ⎟⎝ ⎠
sum sum sumICI
46
0 0 0m n lm k= = =ne
⎝ ⎠
Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
47 Synchronization2010722
Impact of timing offset Impact of timing offset p gp g
loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively
( )I f s
Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ
( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr
2 fj k Nk k k kY X H e Wπδminus= +
loz For integer part of timing offset k k k kY X H e W+
I sTδ ( ) 0g IN L δminus minus le leFFT
2 Ij k NY X H e Wπδminus= +
( ) ( ) 2 I sFFT
j f TI sh t T H f e π δδ minusminus rarr
48
k k k kY X H e W= +
Synchronization2010722
Impact of timing offset derivationImpact of timing offset derivationp gp g
49 Synchronization2010722
Discussion of symbol boundaryDiscussion of symbol boundaryy yy y
50 Synchronization2010722
Impact of timing offsetImpact of timing offsetp gp g
loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI
2 Ij k NIk k k k
NY X H e ISI W
Nπδδ minusminus
= + +
loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk
e e
p kloz Phase Shift can be resolved by Differential Encoding
Decoding or carrier recovery with pilots
51 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
52 Synchronization2010722
Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )
53 Synchronization2010722
Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )
( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s
i n t i N N T N T nTy y t e
= + + +=
( ) ( )( ) ( ) ( ) 12
sin
sin
g gI f f
I I
i N N N Nj jf N Ni k i k k
f
Y X H e eN
π ε ε π ε
ε ε
πεπε
+ + minus+
minus minus=⎛ ⎞⎜ ⎟
Faded signal attenuated and rotated by CFO
( )( ) ( ) ( ) ( )11 2
sin
sin g gI f I f
i N N N NN j j l kI f
NN
l k π ε ε π ε επ ε ε + + minusminus + + + minus
⎜ ⎟⎝ ⎠
+ + minussum
( )( )( )
( ) ( )2
0
sin
I f I f
I
j j l kI f N Ni l l
l l k I f
X H e el k
NN
π ε ε π ε ε
ε π ε ε
+ + +
= ne minus
+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠
sum
i k
N
W
⎜ ⎟⎝ ⎠
+ Inter-Carrier Interference (ICI)
54 Synchronization2010722
Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )
loz An easier representation of received signal with only p g yfractional CFO is
1N
( ) ( )1
0
0N
k k k l l kl l k
Y H X C H X C l k Wminus
= ne
= + minus +⎡ ⎤⎣ ⎦sum
where the ICI coefficient is given by
( )( )( ) ( )
sin 1exp 1fl k
C l k j l kπ ε
π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )
( )exp 1sin
f
f
C l k j l kNN l k
N
π επ ε
minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠
55 Synchronization2010722
Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )
loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality
loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift
56 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
57 Synchronization2010722
TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )
loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq
58 Synchronization2010722
TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )
loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below
loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and
rN2 can be expressed by
0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+
( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +
59
( )
Synchronization2010722
TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )
loz Without CFO the first half is identical to the second half (in time order)
loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below
( ) 12 in discretesft T f NT
NTφ π π π ε πε
⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟
⎝ ⎠
helliphellip
sNT⎝ ⎠
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)
60
Length=N2
Synchronization2010722
TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )
loz Received signal without CFOg0
0 0
jr a e θ=jθ 2 0
2 2 0
Nj j
N Nr a e a e
θ θ= =
loz Received signal with CFO0 0
0 0
j jr a e eθ φ= Extra phase rotation due to CFO0 0
2 2 20
2 2 0
N N Nj j jj
N Nr a e e a e e
θ φ φθ= =
due to CFO
Phase difference 2 2 0N N
( )2 02 20 2 0 0
Nj j T fN
r r a e a eφ φ πminus Δ= =
which contains the information about CFO
61
0 2 0 0Nr r a e a e
Synchronization2010722
TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )
loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be
loz Note that d is a time index corresponding to the first( ) ( ) 2 1
20
Nd m d m Nm
P d r rminus
+ + +== sum
loz Note that d is a time index corresponding to the first sample in a window of N samples
loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by
( )22 1
20
Nd m NR d rminus
+ += sumloz A timing metric can be defined as
))(()(
)( 2
2
dRdP
dM =
( ) 20 d m Nm + +=sum
62
))(( dR
Synchronization2010722
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offset (TO)loz Impact of timing offset (TO)loz Impact of carrier frequency offset (CFO)loz TO estimation algorithmloz CFO estimation algorithmg
38 Synchronization2010722
Introduction (13)Introduction (13)( )( )
loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods
loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead
loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the
cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of
computational complexity therefore it may be not be available in short-burst wireless communication
39 Synchronization2010722
Introduction (23)Introduction (23)( )( )
loz Transcevier
40 Synchronization2010722
Introduction (33)Introduction (33)( )( )
loz Synchronization erroryloz Symbol timing offset (TO)
loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)
loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect
41 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg
42 Synchronization2010722
Signal model (14)Signal model (14)g ( )g ( )
loz Time domain sample p1
2
0
1 Nj nk N
n kk
x X eN
πminus
minus= sum
Xk the frequency domain data at the k-th subcarrier0kN =
loz The multipath fading channel
( ) ( ) ( )1L
h t h t δminus
sum
loz Received time domain signal
( ) ( ) ( )0
l ll
h t h tτ τ δ τ τ=
= minussum
loz Received time-domain signal
( ) ( )1
L
n n ly h n l x w nminus
minus= +sum43
( ) ( )0l=sum
Synchronization2010722
Signal model (24)Signal model (24)g ( )g ( )
loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting
12
0
Nj kn N
k nn
Y y e πminus
minus
=
= sum
( )1 1
2
0
N Lj kn N
n l kn l
h n l x e Wπminus minus
minusminus
=
= +sumsum
loz The equation above could be expressed in matrix form
0n l
loz The equation above could be expressed in matrix form
= +Y HX W
44 Synchronization2010722
Signal model (34)Signal model (34)g ( )g ( )
H= = = = +Y Fy FCx FCF X HX W
loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal
y
H=H FCFcirculant matrix leading to be a diagonal matrix
=H FCF
( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0
h h L L hh h
⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥
( )( ) ( )
( ) ( )
1 10 1 0 0i
h Lh L h N L
⎢ ⎥⎢ ⎥minus minus⎢ ⎥
equiv= minus minus⎢ ⎥⎢ ⎥
C( ) ( )
( ) ( )
0 1 1 100
0 0 1 1 0
h L h N L
h N L L h N
⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
45
( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦
Synchronization2010722
Signal model (44)Signal model (44)g ( )g ( )
loz In a time-invariant channel the channel matrix H would be a diagonal matrix
k k k kY H X W= +
( )1 1
2
0 0
N Lj kl N
kn l
H h n l e πminus minus
minus
= =
= sumsum
loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI
k k k k kY H X I Wα= + +k k k k k
( ) ( )1 1 1
2 2 N N L
j m k Nj ml Nk mI X h n l e e ππ
minus minus minusminusminus⎛ ⎞
= ⎜ ⎟⎝ ⎠
sum sum sumICI
46
0 0 0m n lm k= = =ne
⎝ ⎠
Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
47 Synchronization2010722
Impact of timing offset Impact of timing offset p gp g
loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively
( )I f s
Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ
( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr
2 fj k Nk k k kY X H e Wπδminus= +
loz For integer part of timing offset k k k kY X H e W+
I sTδ ( ) 0g IN L δminus minus le leFFT
2 Ij k NY X H e Wπδminus= +
( ) ( ) 2 I sFFT
j f TI sh t T H f e π δδ minusminus rarr
48
k k k kY X H e W= +
Synchronization2010722
Impact of timing offset derivationImpact of timing offset derivationp gp g
49 Synchronization2010722
Discussion of symbol boundaryDiscussion of symbol boundaryy yy y
50 Synchronization2010722
Impact of timing offsetImpact of timing offsetp gp g
loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI
2 Ij k NIk k k k
NY X H e ISI W
Nπδδ minusminus
= + +
loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk
e e
p kloz Phase Shift can be resolved by Differential Encoding
Decoding or carrier recovery with pilots
51 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
52 Synchronization2010722
Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )
53 Synchronization2010722
Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )
( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s
i n t i N N T N T nTy y t e
= + + +=
( ) ( )( ) ( ) ( ) 12
sin
sin
g gI f f
I I
i N N N Nj jf N Ni k i k k
f
Y X H e eN
π ε ε π ε
ε ε
πεπε
+ + minus+
minus minus=⎛ ⎞⎜ ⎟
Faded signal attenuated and rotated by CFO
( )( ) ( ) ( ) ( )11 2
sin
sin g gI f I f
i N N N NN j j l kI f
NN
l k π ε ε π ε επ ε ε + + minusminus + + + minus
⎜ ⎟⎝ ⎠
+ + minussum
( )( )( )
( ) ( )2
0
sin
I f I f
I
j j l kI f N Ni l l
l l k I f
X H e el k
NN
π ε ε π ε ε
ε π ε ε
+ + +
= ne minus
+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠
sum
i k
N
W
⎜ ⎟⎝ ⎠
+ Inter-Carrier Interference (ICI)
54 Synchronization2010722
Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )
loz An easier representation of received signal with only p g yfractional CFO is
1N
( ) ( )1
0
0N
k k k l l kl l k
Y H X C H X C l k Wminus
= ne
= + minus +⎡ ⎤⎣ ⎦sum
where the ICI coefficient is given by
( )( )( ) ( )
sin 1exp 1fl k
C l k j l kπ ε
π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )
( )exp 1sin
f
f
C l k j l kNN l k
N
π επ ε
minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠
55 Synchronization2010722
Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )
loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality
loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift
56 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
57 Synchronization2010722
TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )
loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq
58 Synchronization2010722
TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )
loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below
loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and
rN2 can be expressed by
0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+
( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +
59
( )
Synchronization2010722
TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )
loz Without CFO the first half is identical to the second half (in time order)
loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below
( ) 12 in discretesft T f NT
NTφ π π π ε πε
⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟
⎝ ⎠
helliphellip
sNT⎝ ⎠
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)
60
Length=N2
Synchronization2010722
TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )
loz Received signal without CFOg0
0 0
jr a e θ=jθ 2 0
2 2 0
Nj j
N Nr a e a e
θ θ= =
loz Received signal with CFO0 0
0 0
j jr a e eθ φ= Extra phase rotation due to CFO0 0
2 2 20
2 2 0
N N Nj j jj
N Nr a e e a e e
θ φ φθ= =
due to CFO
Phase difference 2 2 0N N
( )2 02 20 2 0 0
Nj j T fN
r r a e a eφ φ πminus Δ= =
which contains the information about CFO
61
0 2 0 0Nr r a e a e
Synchronization2010722
TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )
loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be
loz Note that d is a time index corresponding to the first( ) ( ) 2 1
20
Nd m d m Nm
P d r rminus
+ + +== sum
loz Note that d is a time index corresponding to the first sample in a window of N samples
loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by
( )22 1
20
Nd m NR d rminus
+ += sumloz A timing metric can be defined as
))(()(
)( 2
2
dRdP
dM =
( ) 20 d m Nm + +=sum
62
))(( dR
Synchronization2010722
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
Introduction (13)Introduction (13)( )( )
loz In OFDM systems there are two categories in generaly g gloz Data-Aided methods
loz The drawback of the data-aided scheme is the leakage ofloz The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead
loz Non-Data-Aided methodsloz Non data aided methods or called blind relying on theloz Non-data-aided methods or called blind relying on the
cyclo-stationarity and virtual sub-carriers etcloz The blind estimation requires a large amount ofloz The blind estimation requires a large amount of
computational complexity therefore it may be not be available in short-burst wireless communication
39 Synchronization2010722
Introduction (23)Introduction (23)( )( )
loz Transcevier
40 Synchronization2010722
Introduction (33)Introduction (33)( )( )
loz Synchronization erroryloz Symbol timing offset (TO)
loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)
loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect
41 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg
42 Synchronization2010722
Signal model (14)Signal model (14)g ( )g ( )
loz Time domain sample p1
2
0
1 Nj nk N
n kk
x X eN
πminus
minus= sum
Xk the frequency domain data at the k-th subcarrier0kN =
loz The multipath fading channel
( ) ( ) ( )1L
h t h t δminus
sum
loz Received time domain signal
( ) ( ) ( )0
l ll
h t h tτ τ δ τ τ=
= minussum
loz Received time-domain signal
( ) ( )1
L
n n ly h n l x w nminus
minus= +sum43
( ) ( )0l=sum
Synchronization2010722
Signal model (24)Signal model (24)g ( )g ( )
loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting
12
0
Nj kn N
k nn
Y y e πminus
minus
=
= sum
( )1 1
2
0
N Lj kn N
n l kn l
h n l x e Wπminus minus
minusminus
=
= +sumsum
loz The equation above could be expressed in matrix form
0n l
loz The equation above could be expressed in matrix form
= +Y HX W
44 Synchronization2010722
Signal model (34)Signal model (34)g ( )g ( )
H= = = = +Y Fy FCx FCF X HX W
loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal
y
H=H FCFcirculant matrix leading to be a diagonal matrix
=H FCF
( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0
h h L L hh h
⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥
( )( ) ( )
( ) ( )
1 10 1 0 0i
h Lh L h N L
⎢ ⎥⎢ ⎥minus minus⎢ ⎥
equiv= minus minus⎢ ⎥⎢ ⎥
C( ) ( )
( ) ( )
0 1 1 100
0 0 1 1 0
h L h N L
h N L L h N
⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
45
( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦
Synchronization2010722
Signal model (44)Signal model (44)g ( )g ( )
loz In a time-invariant channel the channel matrix H would be a diagonal matrix
k k k kY H X W= +
( )1 1
2
0 0
N Lj kl N
kn l
H h n l e πminus minus
minus
= =
= sumsum
loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI
k k k k kY H X I Wα= + +k k k k k
( ) ( )1 1 1
2 2 N N L
j m k Nj ml Nk mI X h n l e e ππ
minus minus minusminusminus⎛ ⎞
= ⎜ ⎟⎝ ⎠
sum sum sumICI
46
0 0 0m n lm k= = =ne
⎝ ⎠
Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
47 Synchronization2010722
Impact of timing offset Impact of timing offset p gp g
loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively
( )I f s
Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ
( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr
2 fj k Nk k k kY X H e Wπδminus= +
loz For integer part of timing offset k k k kY X H e W+
I sTδ ( ) 0g IN L δminus minus le leFFT
2 Ij k NY X H e Wπδminus= +
( ) ( ) 2 I sFFT
j f TI sh t T H f e π δδ minusminus rarr
48
k k k kY X H e W= +
Synchronization2010722
Impact of timing offset derivationImpact of timing offset derivationp gp g
49 Synchronization2010722
Discussion of symbol boundaryDiscussion of symbol boundaryy yy y
50 Synchronization2010722
Impact of timing offsetImpact of timing offsetp gp g
loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI
2 Ij k NIk k k k
NY X H e ISI W
Nπδδ minusminus
= + +
loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk
e e
p kloz Phase Shift can be resolved by Differential Encoding
Decoding or carrier recovery with pilots
51 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
52 Synchronization2010722
Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )
53 Synchronization2010722
Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )
( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s
i n t i N N T N T nTy y t e
= + + +=
( ) ( )( ) ( ) ( ) 12
sin
sin
g gI f f
I I
i N N N Nj jf N Ni k i k k
f
Y X H e eN
π ε ε π ε
ε ε
πεπε
+ + minus+
minus minus=⎛ ⎞⎜ ⎟
Faded signal attenuated and rotated by CFO
( )( ) ( ) ( ) ( )11 2
sin
sin g gI f I f
i N N N NN j j l kI f
NN
l k π ε ε π ε επ ε ε + + minusminus + + + minus
⎜ ⎟⎝ ⎠
+ + minussum
( )( )( )
( ) ( )2
0
sin
I f I f
I
j j l kI f N Ni l l
l l k I f
X H e el k
NN
π ε ε π ε ε
ε π ε ε
+ + +
= ne minus
+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠
sum
i k
N
W
⎜ ⎟⎝ ⎠
+ Inter-Carrier Interference (ICI)
54 Synchronization2010722
Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )
loz An easier representation of received signal with only p g yfractional CFO is
1N
( ) ( )1
0
0N
k k k l l kl l k
Y H X C H X C l k Wminus
= ne
= + minus +⎡ ⎤⎣ ⎦sum
where the ICI coefficient is given by
( )( )( ) ( )
sin 1exp 1fl k
C l k j l kπ ε
π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )
( )exp 1sin
f
f
C l k j l kNN l k
N
π επ ε
minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠
55 Synchronization2010722
Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )
loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality
loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift
56 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
57 Synchronization2010722
TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )
loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq
58 Synchronization2010722
TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )
loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below
loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and
rN2 can be expressed by
0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+
( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +
59
( )
Synchronization2010722
TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )
loz Without CFO the first half is identical to the second half (in time order)
loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below
( ) 12 in discretesft T f NT
NTφ π π π ε πε
⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟
⎝ ⎠
helliphellip
sNT⎝ ⎠
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)
60
Length=N2
Synchronization2010722
TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )
loz Received signal without CFOg0
0 0
jr a e θ=jθ 2 0
2 2 0
Nj j
N Nr a e a e
θ θ= =
loz Received signal with CFO0 0
0 0
j jr a e eθ φ= Extra phase rotation due to CFO0 0
2 2 20
2 2 0
N N Nj j jj
N Nr a e e a e e
θ φ φθ= =
due to CFO
Phase difference 2 2 0N N
( )2 02 20 2 0 0
Nj j T fN
r r a e a eφ φ πminus Δ= =
which contains the information about CFO
61
0 2 0 0Nr r a e a e
Synchronization2010722
TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )
loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be
loz Note that d is a time index corresponding to the first( ) ( ) 2 1
20
Nd m d m Nm
P d r rminus
+ + +== sum
loz Note that d is a time index corresponding to the first sample in a window of N samples
loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by
( )22 1
20
Nd m NR d rminus
+ += sumloz A timing metric can be defined as
))(()(
)( 2
2
dRdP
dM =
( ) 20 d m Nm + +=sum
62
))(( dR
Synchronization2010722
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
Introduction (23)Introduction (23)( )( )
loz Transcevier
40 Synchronization2010722
Introduction (33)Introduction (33)( )( )
loz Synchronization erroryloz Symbol timing offset (TO)
loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)
loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect
41 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg
42 Synchronization2010722
Signal model (14)Signal model (14)g ( )g ( )
loz Time domain sample p1
2
0
1 Nj nk N
n kk
x X eN
πminus
minus= sum
Xk the frequency domain data at the k-th subcarrier0kN =
loz The multipath fading channel
( ) ( ) ( )1L
h t h t δminus
sum
loz Received time domain signal
( ) ( ) ( )0
l ll
h t h tτ τ δ τ τ=
= minussum
loz Received time-domain signal
( ) ( )1
L
n n ly h n l x w nminus
minus= +sum43
( ) ( )0l=sum
Synchronization2010722
Signal model (24)Signal model (24)g ( )g ( )
loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting
12
0
Nj kn N
k nn
Y y e πminus
minus
=
= sum
( )1 1
2
0
N Lj kn N
n l kn l
h n l x e Wπminus minus
minusminus
=
= +sumsum
loz The equation above could be expressed in matrix form
0n l
loz The equation above could be expressed in matrix form
= +Y HX W
44 Synchronization2010722
Signal model (34)Signal model (34)g ( )g ( )
H= = = = +Y Fy FCx FCF X HX W
loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal
y
H=H FCFcirculant matrix leading to be a diagonal matrix
=H FCF
( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0
h h L L hh h
⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥
( )( ) ( )
( ) ( )
1 10 1 0 0i
h Lh L h N L
⎢ ⎥⎢ ⎥minus minus⎢ ⎥
equiv= minus minus⎢ ⎥⎢ ⎥
C( ) ( )
( ) ( )
0 1 1 100
0 0 1 1 0
h L h N L
h N L L h N
⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
45
( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦
Synchronization2010722
Signal model (44)Signal model (44)g ( )g ( )
loz In a time-invariant channel the channel matrix H would be a diagonal matrix
k k k kY H X W= +
( )1 1
2
0 0
N Lj kl N
kn l
H h n l e πminus minus
minus
= =
= sumsum
loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI
k k k k kY H X I Wα= + +k k k k k
( ) ( )1 1 1
2 2 N N L
j m k Nj ml Nk mI X h n l e e ππ
minus minus minusminusminus⎛ ⎞
= ⎜ ⎟⎝ ⎠
sum sum sumICI
46
0 0 0m n lm k= = =ne
⎝ ⎠
Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
47 Synchronization2010722
Impact of timing offset Impact of timing offset p gp g
loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively
( )I f s
Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ
( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr
2 fj k Nk k k kY X H e Wπδminus= +
loz For integer part of timing offset k k k kY X H e W+
I sTδ ( ) 0g IN L δminus minus le leFFT
2 Ij k NY X H e Wπδminus= +
( ) ( ) 2 I sFFT
j f TI sh t T H f e π δδ minusminus rarr
48
k k k kY X H e W= +
Synchronization2010722
Impact of timing offset derivationImpact of timing offset derivationp gp g
49 Synchronization2010722
Discussion of symbol boundaryDiscussion of symbol boundaryy yy y
50 Synchronization2010722
Impact of timing offsetImpact of timing offsetp gp g
loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI
2 Ij k NIk k k k
NY X H e ISI W
Nπδδ minusminus
= + +
loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk
e e
p kloz Phase Shift can be resolved by Differential Encoding
Decoding or carrier recovery with pilots
51 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
52 Synchronization2010722
Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )
53 Synchronization2010722
Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )
( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s
i n t i N N T N T nTy y t e
= + + +=
( ) ( )( ) ( ) ( ) 12
sin
sin
g gI f f
I I
i N N N Nj jf N Ni k i k k
f
Y X H e eN
π ε ε π ε
ε ε
πεπε
+ + minus+
minus minus=⎛ ⎞⎜ ⎟
Faded signal attenuated and rotated by CFO
( )( ) ( ) ( ) ( )11 2
sin
sin g gI f I f
i N N N NN j j l kI f
NN
l k π ε ε π ε επ ε ε + + minusminus + + + minus
⎜ ⎟⎝ ⎠
+ + minussum
( )( )( )
( ) ( )2
0
sin
I f I f
I
j j l kI f N Ni l l
l l k I f
X H e el k
NN
π ε ε π ε ε
ε π ε ε
+ + +
= ne minus
+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠
sum
i k
N
W
⎜ ⎟⎝ ⎠
+ Inter-Carrier Interference (ICI)
54 Synchronization2010722
Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )
loz An easier representation of received signal with only p g yfractional CFO is
1N
( ) ( )1
0
0N
k k k l l kl l k
Y H X C H X C l k Wminus
= ne
= + minus +⎡ ⎤⎣ ⎦sum
where the ICI coefficient is given by
( )( )( ) ( )
sin 1exp 1fl k
C l k j l kπ ε
π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )
( )exp 1sin
f
f
C l k j l kNN l k
N
π επ ε
minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠
55 Synchronization2010722
Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )
loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality
loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift
56 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
57 Synchronization2010722
TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )
loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq
58 Synchronization2010722
TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )
loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below
loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and
rN2 can be expressed by
0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+
( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +
59
( )
Synchronization2010722
TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )
loz Without CFO the first half is identical to the second half (in time order)
loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below
( ) 12 in discretesft T f NT
NTφ π π π ε πε
⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟
⎝ ⎠
helliphellip
sNT⎝ ⎠
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)
60
Length=N2
Synchronization2010722
TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )
loz Received signal without CFOg0
0 0
jr a e θ=jθ 2 0
2 2 0
Nj j
N Nr a e a e
θ θ= =
loz Received signal with CFO0 0
0 0
j jr a e eθ φ= Extra phase rotation due to CFO0 0
2 2 20
2 2 0
N N Nj j jj
N Nr a e e a e e
θ φ φθ= =
due to CFO
Phase difference 2 2 0N N
( )2 02 20 2 0 0
Nj j T fN
r r a e a eφ φ πminus Δ= =
which contains the information about CFO
61
0 2 0 0Nr r a e a e
Synchronization2010722
TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )
loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be
loz Note that d is a time index corresponding to the first( ) ( ) 2 1
20
Nd m d m Nm
P d r rminus
+ + +== sum
loz Note that d is a time index corresponding to the first sample in a window of N samples
loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by
( )22 1
20
Nd m NR d rminus
+ += sumloz A timing metric can be defined as
))(()(
)( 2
2
dRdP
dM =
( ) 20 d m Nm + +=sum
62
))(( dR
Synchronization2010722
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
Introduction (33)Introduction (33)( )( )
loz Synchronization erroryloz Symbol timing offset (TO)
loz Due to unknown transmission timeloz Due to unknown transmission timeloz Carrier frequency offset (CFO)
loz Oscillator mismatchloz Oscillator mismatchloz Doppler effect
41 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg
42 Synchronization2010722
Signal model (14)Signal model (14)g ( )g ( )
loz Time domain sample p1
2
0
1 Nj nk N
n kk
x X eN
πminus
minus= sum
Xk the frequency domain data at the k-th subcarrier0kN =
loz The multipath fading channel
( ) ( ) ( )1L
h t h t δminus
sum
loz Received time domain signal
( ) ( ) ( )0
l ll
h t h tτ τ δ τ τ=
= minussum
loz Received time-domain signal
( ) ( )1
L
n n ly h n l x w nminus
minus= +sum43
( ) ( )0l=sum
Synchronization2010722
Signal model (24)Signal model (24)g ( )g ( )
loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting
12
0
Nj kn N
k nn
Y y e πminus
minus
=
= sum
( )1 1
2
0
N Lj kn N
n l kn l
h n l x e Wπminus minus
minusminus
=
= +sumsum
loz The equation above could be expressed in matrix form
0n l
loz The equation above could be expressed in matrix form
= +Y HX W
44 Synchronization2010722
Signal model (34)Signal model (34)g ( )g ( )
H= = = = +Y Fy FCx FCF X HX W
loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal
y
H=H FCFcirculant matrix leading to be a diagonal matrix
=H FCF
( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0
h h L L hh h
⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥
( )( ) ( )
( ) ( )
1 10 1 0 0i
h Lh L h N L
⎢ ⎥⎢ ⎥minus minus⎢ ⎥
equiv= minus minus⎢ ⎥⎢ ⎥
C( ) ( )
( ) ( )
0 1 1 100
0 0 1 1 0
h L h N L
h N L L h N
⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
45
( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦
Synchronization2010722
Signal model (44)Signal model (44)g ( )g ( )
loz In a time-invariant channel the channel matrix H would be a diagonal matrix
k k k kY H X W= +
( )1 1
2
0 0
N Lj kl N
kn l
H h n l e πminus minus
minus
= =
= sumsum
loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI
k k k k kY H X I Wα= + +k k k k k
( ) ( )1 1 1
2 2 N N L
j m k Nj ml Nk mI X h n l e e ππ
minus minus minusminusminus⎛ ⎞
= ⎜ ⎟⎝ ⎠
sum sum sumICI
46
0 0 0m n lm k= = =ne
⎝ ⎠
Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
47 Synchronization2010722
Impact of timing offset Impact of timing offset p gp g
loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively
( )I f s
Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ
( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr
2 fj k Nk k k kY X H e Wπδminus= +
loz For integer part of timing offset k k k kY X H e W+
I sTδ ( ) 0g IN L δminus minus le leFFT
2 Ij k NY X H e Wπδminus= +
( ) ( ) 2 I sFFT
j f TI sh t T H f e π δδ minusminus rarr
48
k k k kY X H e W= +
Synchronization2010722
Impact of timing offset derivationImpact of timing offset derivationp gp g
49 Synchronization2010722
Discussion of symbol boundaryDiscussion of symbol boundaryy yy y
50 Synchronization2010722
Impact of timing offsetImpact of timing offsetp gp g
loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI
2 Ij k NIk k k k
NY X H e ISI W
Nπδδ minusminus
= + +
loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk
e e
p kloz Phase Shift can be resolved by Differential Encoding
Decoding or carrier recovery with pilots
51 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
52 Synchronization2010722
Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )
53 Synchronization2010722
Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )
( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s
i n t i N N T N T nTy y t e
= + + +=
( ) ( )( ) ( ) ( ) 12
sin
sin
g gI f f
I I
i N N N Nj jf N Ni k i k k
f
Y X H e eN
π ε ε π ε
ε ε
πεπε
+ + minus+
minus minus=⎛ ⎞⎜ ⎟
Faded signal attenuated and rotated by CFO
( )( ) ( ) ( ) ( )11 2
sin
sin g gI f I f
i N N N NN j j l kI f
NN
l k π ε ε π ε επ ε ε + + minusminus + + + minus
⎜ ⎟⎝ ⎠
+ + minussum
( )( )( )
( ) ( )2
0
sin
I f I f
I
j j l kI f N Ni l l
l l k I f
X H e el k
NN
π ε ε π ε ε
ε π ε ε
+ + +
= ne minus
+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠
sum
i k
N
W
⎜ ⎟⎝ ⎠
+ Inter-Carrier Interference (ICI)
54 Synchronization2010722
Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )
loz An easier representation of received signal with only p g yfractional CFO is
1N
( ) ( )1
0
0N
k k k l l kl l k
Y H X C H X C l k Wminus
= ne
= + minus +⎡ ⎤⎣ ⎦sum
where the ICI coefficient is given by
( )( )( ) ( )
sin 1exp 1fl k
C l k j l kπ ε
π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )
( )exp 1sin
f
f
C l k j l kNN l k
N
π επ ε
minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠
55 Synchronization2010722
Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )
loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality
loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift
56 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
57 Synchronization2010722
TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )
loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq
58 Synchronization2010722
TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )
loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below
loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and
rN2 can be expressed by
0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+
( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +
59
( )
Synchronization2010722
TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )
loz Without CFO the first half is identical to the second half (in time order)
loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below
( ) 12 in discretesft T f NT
NTφ π π π ε πε
⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟
⎝ ⎠
helliphellip
sNT⎝ ⎠
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)
60
Length=N2
Synchronization2010722
TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )
loz Received signal without CFOg0
0 0
jr a e θ=jθ 2 0
2 2 0
Nj j
N Nr a e a e
θ θ= =
loz Received signal with CFO0 0
0 0
j jr a e eθ φ= Extra phase rotation due to CFO0 0
2 2 20
2 2 0
N N Nj j jj
N Nr a e e a e e
θ φ φθ= =
due to CFO
Phase difference 2 2 0N N
( )2 02 20 2 0 0
Nj j T fN
r r a e a eφ φ πminus Δ= =
which contains the information about CFO
61
0 2 0 0Nr r a e a e
Synchronization2010722
TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )
loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be
loz Note that d is a time index corresponding to the first( ) ( ) 2 1
20
Nd m d m Nm
P d r rminus
+ + +== sum
loz Note that d is a time index corresponding to the first sample in a window of N samples
loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by
( )22 1
20
Nd m NR d rminus
+ += sumloz A timing metric can be defined as
))(()(
)( 2
2
dRdP
dM =
( ) 20 d m Nm + +=sum
62
))(( dR
Synchronization2010722
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Timing offset estimation algorithmloz CFO estimation algorithmg
42 Synchronization2010722
Signal model (14)Signal model (14)g ( )g ( )
loz Time domain sample p1
2
0
1 Nj nk N
n kk
x X eN
πminus
minus= sum
Xk the frequency domain data at the k-th subcarrier0kN =
loz The multipath fading channel
( ) ( ) ( )1L
h t h t δminus
sum
loz Received time domain signal
( ) ( ) ( )0
l ll
h t h tτ τ δ τ τ=
= minussum
loz Received time-domain signal
( ) ( )1
L
n n ly h n l x w nminus
minus= +sum43
( ) ( )0l=sum
Synchronization2010722
Signal model (24)Signal model (24)g ( )g ( )
loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting
12
0
Nj kn N
k nn
Y y e πminus
minus
=
= sum
( )1 1
2
0
N Lj kn N
n l kn l
h n l x e Wπminus minus
minusminus
=
= +sumsum
loz The equation above could be expressed in matrix form
0n l
loz The equation above could be expressed in matrix form
= +Y HX W
44 Synchronization2010722
Signal model (34)Signal model (34)g ( )g ( )
H= = = = +Y Fy FCx FCF X HX W
loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal
y
H=H FCFcirculant matrix leading to be a diagonal matrix
=H FCF
( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0
h h L L hh h
⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥
( )( ) ( )
( ) ( )
1 10 1 0 0i
h Lh L h N L
⎢ ⎥⎢ ⎥minus minus⎢ ⎥
equiv= minus minus⎢ ⎥⎢ ⎥
C( ) ( )
( ) ( )
0 1 1 100
0 0 1 1 0
h L h N L
h N L L h N
⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
45
( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦
Synchronization2010722
Signal model (44)Signal model (44)g ( )g ( )
loz In a time-invariant channel the channel matrix H would be a diagonal matrix
k k k kY H X W= +
( )1 1
2
0 0
N Lj kl N
kn l
H h n l e πminus minus
minus
= =
= sumsum
loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI
k k k k kY H X I Wα= + +k k k k k
( ) ( )1 1 1
2 2 N N L
j m k Nj ml Nk mI X h n l e e ππ
minus minus minusminusminus⎛ ⎞
= ⎜ ⎟⎝ ⎠
sum sum sumICI
46
0 0 0m n lm k= = =ne
⎝ ⎠
Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
47 Synchronization2010722
Impact of timing offset Impact of timing offset p gp g
loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively
( )I f s
Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ
( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr
2 fj k Nk k k kY X H e Wπδminus= +
loz For integer part of timing offset k k k kY X H e W+
I sTδ ( ) 0g IN L δminus minus le leFFT
2 Ij k NY X H e Wπδminus= +
( ) ( ) 2 I sFFT
j f TI sh t T H f e π δδ minusminus rarr
48
k k k kY X H e W= +
Synchronization2010722
Impact of timing offset derivationImpact of timing offset derivationp gp g
49 Synchronization2010722
Discussion of symbol boundaryDiscussion of symbol boundaryy yy y
50 Synchronization2010722
Impact of timing offsetImpact of timing offsetp gp g
loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI
2 Ij k NIk k k k
NY X H e ISI W
Nπδδ minusminus
= + +
loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk
e e
p kloz Phase Shift can be resolved by Differential Encoding
Decoding or carrier recovery with pilots
51 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
52 Synchronization2010722
Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )
53 Synchronization2010722
Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )
( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s
i n t i N N T N T nTy y t e
= + + +=
( ) ( )( ) ( ) ( ) 12
sin
sin
g gI f f
I I
i N N N Nj jf N Ni k i k k
f
Y X H e eN
π ε ε π ε
ε ε
πεπε
+ + minus+
minus minus=⎛ ⎞⎜ ⎟
Faded signal attenuated and rotated by CFO
( )( ) ( ) ( ) ( )11 2
sin
sin g gI f I f
i N N N NN j j l kI f
NN
l k π ε ε π ε επ ε ε + + minusminus + + + minus
⎜ ⎟⎝ ⎠
+ + minussum
( )( )( )
( ) ( )2
0
sin
I f I f
I
j j l kI f N Ni l l
l l k I f
X H e el k
NN
π ε ε π ε ε
ε π ε ε
+ + +
= ne minus
+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠
sum
i k
N
W
⎜ ⎟⎝ ⎠
+ Inter-Carrier Interference (ICI)
54 Synchronization2010722
Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )
loz An easier representation of received signal with only p g yfractional CFO is
1N
( ) ( )1
0
0N
k k k l l kl l k
Y H X C H X C l k Wminus
= ne
= + minus +⎡ ⎤⎣ ⎦sum
where the ICI coefficient is given by
( )( )( ) ( )
sin 1exp 1fl k
C l k j l kπ ε
π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )
( )exp 1sin
f
f
C l k j l kNN l k
N
π επ ε
minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠
55 Synchronization2010722
Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )
loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality
loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift
56 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
57 Synchronization2010722
TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )
loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq
58 Synchronization2010722
TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )
loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below
loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and
rN2 can be expressed by
0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+
( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +
59
( )
Synchronization2010722
TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )
loz Without CFO the first half is identical to the second half (in time order)
loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below
( ) 12 in discretesft T f NT
NTφ π π π ε πε
⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟
⎝ ⎠
helliphellip
sNT⎝ ⎠
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)
60
Length=N2
Synchronization2010722
TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )
loz Received signal without CFOg0
0 0
jr a e θ=jθ 2 0
2 2 0
Nj j
N Nr a e a e
θ θ= =
loz Received signal with CFO0 0
0 0
j jr a e eθ φ= Extra phase rotation due to CFO0 0
2 2 20
2 2 0
N N Nj j jj
N Nr a e e a e e
θ φ φθ= =
due to CFO
Phase difference 2 2 0N N
( )2 02 20 2 0 0
Nj j T fN
r r a e a eφ φ πminus Δ= =
which contains the information about CFO
61
0 2 0 0Nr r a e a e
Synchronization2010722
TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )
loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be
loz Note that d is a time index corresponding to the first( ) ( ) 2 1
20
Nd m d m Nm
P d r rminus
+ + +== sum
loz Note that d is a time index corresponding to the first sample in a window of N samples
loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by
( )22 1
20
Nd m NR d rminus
+ += sumloz A timing metric can be defined as
))(()(
)( 2
2
dRdP
dM =
( ) 20 d m Nm + +=sum
62
))(( dR
Synchronization2010722
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
Signal model (14)Signal model (14)g ( )g ( )
loz Time domain sample p1
2
0
1 Nj nk N
n kk
x X eN
πminus
minus= sum
Xk the frequency domain data at the k-th subcarrier0kN =
loz The multipath fading channel
( ) ( ) ( )1L
h t h t δminus
sum
loz Received time domain signal
( ) ( ) ( )0
l ll
h t h tτ τ δ τ τ=
= minussum
loz Received time-domain signal
( ) ( )1
L
n n ly h n l x w nminus
minus= +sum43
( ) ( )0l=sum
Synchronization2010722
Signal model (24)Signal model (24)g ( )g ( )
loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting
12
0
Nj kn N
k nn
Y y e πminus
minus
=
= sum
( )1 1
2
0
N Lj kn N
n l kn l
h n l x e Wπminus minus
minusminus
=
= +sumsum
loz The equation above could be expressed in matrix form
0n l
loz The equation above could be expressed in matrix form
= +Y HX W
44 Synchronization2010722
Signal model (34)Signal model (34)g ( )g ( )
H= = = = +Y Fy FCx FCF X HX W
loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal
y
H=H FCFcirculant matrix leading to be a diagonal matrix
=H FCF
( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0
h h L L hh h
⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥
( )( ) ( )
( ) ( )
1 10 1 0 0i
h Lh L h N L
⎢ ⎥⎢ ⎥minus minus⎢ ⎥
equiv= minus minus⎢ ⎥⎢ ⎥
C( ) ( )
( ) ( )
0 1 1 100
0 0 1 1 0
h L h N L
h N L L h N
⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
45
( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦
Synchronization2010722
Signal model (44)Signal model (44)g ( )g ( )
loz In a time-invariant channel the channel matrix H would be a diagonal matrix
k k k kY H X W= +
( )1 1
2
0 0
N Lj kl N
kn l
H h n l e πminus minus
minus
= =
= sumsum
loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI
k k k k kY H X I Wα= + +k k k k k
( ) ( )1 1 1
2 2 N N L
j m k Nj ml Nk mI X h n l e e ππ
minus minus minusminusminus⎛ ⎞
= ⎜ ⎟⎝ ⎠
sum sum sumICI
46
0 0 0m n lm k= = =ne
⎝ ⎠
Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
47 Synchronization2010722
Impact of timing offset Impact of timing offset p gp g
loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively
( )I f s
Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ
( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr
2 fj k Nk k k kY X H e Wπδminus= +
loz For integer part of timing offset k k k kY X H e W+
I sTδ ( ) 0g IN L δminus minus le leFFT
2 Ij k NY X H e Wπδminus= +
( ) ( ) 2 I sFFT
j f TI sh t T H f e π δδ minusminus rarr
48
k k k kY X H e W= +
Synchronization2010722
Impact of timing offset derivationImpact of timing offset derivationp gp g
49 Synchronization2010722
Discussion of symbol boundaryDiscussion of symbol boundaryy yy y
50 Synchronization2010722
Impact of timing offsetImpact of timing offsetp gp g
loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI
2 Ij k NIk k k k
NY X H e ISI W
Nπδδ minusminus
= + +
loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk
e e
p kloz Phase Shift can be resolved by Differential Encoding
Decoding or carrier recovery with pilots
51 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
52 Synchronization2010722
Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )
53 Synchronization2010722
Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )
( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s
i n t i N N T N T nTy y t e
= + + +=
( ) ( )( ) ( ) ( ) 12
sin
sin
g gI f f
I I
i N N N Nj jf N Ni k i k k
f
Y X H e eN
π ε ε π ε
ε ε
πεπε
+ + minus+
minus minus=⎛ ⎞⎜ ⎟
Faded signal attenuated and rotated by CFO
( )( ) ( ) ( ) ( )11 2
sin
sin g gI f I f
i N N N NN j j l kI f
NN
l k π ε ε π ε επ ε ε + + minusminus + + + minus
⎜ ⎟⎝ ⎠
+ + minussum
( )( )( )
( ) ( )2
0
sin
I f I f
I
j j l kI f N Ni l l
l l k I f
X H e el k
NN
π ε ε π ε ε
ε π ε ε
+ + +
= ne minus
+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠
sum
i k
N
W
⎜ ⎟⎝ ⎠
+ Inter-Carrier Interference (ICI)
54 Synchronization2010722
Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )
loz An easier representation of received signal with only p g yfractional CFO is
1N
( ) ( )1
0
0N
k k k l l kl l k
Y H X C H X C l k Wminus
= ne
= + minus +⎡ ⎤⎣ ⎦sum
where the ICI coefficient is given by
( )( )( ) ( )
sin 1exp 1fl k
C l k j l kπ ε
π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )
( )exp 1sin
f
f
C l k j l kNN l k
N
π επ ε
minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠
55 Synchronization2010722
Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )
loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality
loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift
56 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
57 Synchronization2010722
TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )
loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq
58 Synchronization2010722
TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )
loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below
loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and
rN2 can be expressed by
0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+
( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +
59
( )
Synchronization2010722
TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )
loz Without CFO the first half is identical to the second half (in time order)
loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below
( ) 12 in discretesft T f NT
NTφ π π π ε πε
⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟
⎝ ⎠
helliphellip
sNT⎝ ⎠
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)
60
Length=N2
Synchronization2010722
TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )
loz Received signal without CFOg0
0 0
jr a e θ=jθ 2 0
2 2 0
Nj j
N Nr a e a e
θ θ= =
loz Received signal with CFO0 0
0 0
j jr a e eθ φ= Extra phase rotation due to CFO0 0
2 2 20
2 2 0
N N Nj j jj
N Nr a e e a e e
θ φ φθ= =
due to CFO
Phase difference 2 2 0N N
( )2 02 20 2 0 0
Nj j T fN
r r a e a eφ φ πminus Δ= =
which contains the information about CFO
61
0 2 0 0Nr r a e a e
Synchronization2010722
TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )
loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be
loz Note that d is a time index corresponding to the first( ) ( ) 2 1
20
Nd m d m Nm
P d r rminus
+ + +== sum
loz Note that d is a time index corresponding to the first sample in a window of N samples
loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by
( )22 1
20
Nd m NR d rminus
+ += sumloz A timing metric can be defined as
))(()(
)( 2
2
dRdP
dM =
( ) 20 d m Nm + +=sum
62
))(( dR
Synchronization2010722
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
Signal model (24)Signal model (24)g ( )g ( )
loz At receiver the guard interval is removed and DFT gdemodulation is performed resulting
12
0
Nj kn N
k nn
Y y e πminus
minus
=
= sum
( )1 1
2
0
N Lj kn N
n l kn l
h n l x e Wπminus minus
minusminus
=
= +sumsum
loz The equation above could be expressed in matrix form
0n l
loz The equation above could be expressed in matrix form
= +Y HX W
44 Synchronization2010722
Signal model (34)Signal model (34)g ( )g ( )
H= = = = +Y Fy FCx FCF X HX W
loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal
y
H=H FCFcirculant matrix leading to be a diagonal matrix
=H FCF
( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0
h h L L hh h
⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥
( )( ) ( )
( ) ( )
1 10 1 0 0i
h Lh L h N L
⎢ ⎥⎢ ⎥minus minus⎢ ⎥
equiv= minus minus⎢ ⎥⎢ ⎥
C( ) ( )
( ) ( )
0 1 1 100
0 0 1 1 0
h L h N L
h N L L h N
⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
45
( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦
Synchronization2010722
Signal model (44)Signal model (44)g ( )g ( )
loz In a time-invariant channel the channel matrix H would be a diagonal matrix
k k k kY H X W= +
( )1 1
2
0 0
N Lj kl N
kn l
H h n l e πminus minus
minus
= =
= sumsum
loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI
k k k k kY H X I Wα= + +k k k k k
( ) ( )1 1 1
2 2 N N L
j m k Nj ml Nk mI X h n l e e ππ
minus minus minusminusminus⎛ ⎞
= ⎜ ⎟⎝ ⎠
sum sum sumICI
46
0 0 0m n lm k= = =ne
⎝ ⎠
Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
47 Synchronization2010722
Impact of timing offset Impact of timing offset p gp g
loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively
( )I f s
Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ
( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr
2 fj k Nk k k kY X H e Wπδminus= +
loz For integer part of timing offset k k k kY X H e W+
I sTδ ( ) 0g IN L δminus minus le leFFT
2 Ij k NY X H e Wπδminus= +
( ) ( ) 2 I sFFT
j f TI sh t T H f e π δδ minusminus rarr
48
k k k kY X H e W= +
Synchronization2010722
Impact of timing offset derivationImpact of timing offset derivationp gp g
49 Synchronization2010722
Discussion of symbol boundaryDiscussion of symbol boundaryy yy y
50 Synchronization2010722
Impact of timing offsetImpact of timing offsetp gp g
loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI
2 Ij k NIk k k k
NY X H e ISI W
Nπδδ minusminus
= + +
loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk
e e
p kloz Phase Shift can be resolved by Differential Encoding
Decoding or carrier recovery with pilots
51 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
52 Synchronization2010722
Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )
53 Synchronization2010722
Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )
( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s
i n t i N N T N T nTy y t e
= + + +=
( ) ( )( ) ( ) ( ) 12
sin
sin
g gI f f
I I
i N N N Nj jf N Ni k i k k
f
Y X H e eN
π ε ε π ε
ε ε
πεπε
+ + minus+
minus minus=⎛ ⎞⎜ ⎟
Faded signal attenuated and rotated by CFO
( )( ) ( ) ( ) ( )11 2
sin
sin g gI f I f
i N N N NN j j l kI f
NN
l k π ε ε π ε επ ε ε + + minusminus + + + minus
⎜ ⎟⎝ ⎠
+ + minussum
( )( )( )
( ) ( )2
0
sin
I f I f
I
j j l kI f N Ni l l
l l k I f
X H e el k
NN
π ε ε π ε ε
ε π ε ε
+ + +
= ne minus
+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠
sum
i k
N
W
⎜ ⎟⎝ ⎠
+ Inter-Carrier Interference (ICI)
54 Synchronization2010722
Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )
loz An easier representation of received signal with only p g yfractional CFO is
1N
( ) ( )1
0
0N
k k k l l kl l k
Y H X C H X C l k Wminus
= ne
= + minus +⎡ ⎤⎣ ⎦sum
where the ICI coefficient is given by
( )( )( ) ( )
sin 1exp 1fl k
C l k j l kπ ε
π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )
( )exp 1sin
f
f
C l k j l kNN l k
N
π επ ε
minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠
55 Synchronization2010722
Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )
loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality
loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift
56 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
57 Synchronization2010722
TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )
loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq
58 Synchronization2010722
TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )
loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below
loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and
rN2 can be expressed by
0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+
( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +
59
( )
Synchronization2010722
TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )
loz Without CFO the first half is identical to the second half (in time order)
loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below
( ) 12 in discretesft T f NT
NTφ π π π ε πε
⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟
⎝ ⎠
helliphellip
sNT⎝ ⎠
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)
60
Length=N2
Synchronization2010722
TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )
loz Received signal without CFOg0
0 0
jr a e θ=jθ 2 0
2 2 0
Nj j
N Nr a e a e
θ θ= =
loz Received signal with CFO0 0
0 0
j jr a e eθ φ= Extra phase rotation due to CFO0 0
2 2 20
2 2 0
N N Nj j jj
N Nr a e e a e e
θ φ φθ= =
due to CFO
Phase difference 2 2 0N N
( )2 02 20 2 0 0
Nj j T fN
r r a e a eφ φ πminus Δ= =
which contains the information about CFO
61
0 2 0 0Nr r a e a e
Synchronization2010722
TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )
loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be
loz Note that d is a time index corresponding to the first( ) ( ) 2 1
20
Nd m d m Nm
P d r rminus
+ + +== sum
loz Note that d is a time index corresponding to the first sample in a window of N samples
loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by
( )22 1
20
Nd m NR d rminus
+ += sumloz A timing metric can be defined as
))(()(
)( 2
2
dRdP
dM =
( ) 20 d m Nm + +=sum
62
))(( dR
Synchronization2010722
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
Signal model (34)Signal model (34)g ( )g ( )
H= = = = +Y Fy FCx FCF X HX W
loz In a time-invariant channel the matrix C would be a circulant matrix leading to be a diagonal
y
H=H FCFcirculant matrix leading to be a diagonal matrix
=H FCF
( ) ( ) ( )( ) ( )00 0 0 1 1 1101 10 0
h h L L hh h
⎡ minus + minus minus ⎤⎢ ⎥⎢ ⎥
( )( ) ( )
( ) ( )
1 10 1 0 0i
h Lh L h N L
⎢ ⎥⎢ ⎥minus minus⎢ ⎥
equiv= minus minus⎢ ⎥⎢ ⎥
C( ) ( )
( ) ( )
0 1 1 100
0 0 1 1 0
h L h N L
h N L L h N
⎢ ⎥minus minus +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
45
( ) ( )0 0 1 10h N L L h Nminus minus minus⎣ ⎦
Synchronization2010722
Signal model (44)Signal model (44)g ( )g ( )
loz In a time-invariant channel the channel matrix H would be a diagonal matrix
k k k kY H X W= +
( )1 1
2
0 0
N Lj kl N
kn l
H h n l e πminus minus
minus
= =
= sumsum
loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI
k k k k kY H X I Wα= + +k k k k k
( ) ( )1 1 1
2 2 N N L
j m k Nj ml Nk mI X h n l e e ππ
minus minus minusminusminus⎛ ⎞
= ⎜ ⎟⎝ ⎠
sum sum sumICI
46
0 0 0m n lm k= = =ne
⎝ ⎠
Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
47 Synchronization2010722
Impact of timing offset Impact of timing offset p gp g
loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively
( )I f s
Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ
( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr
2 fj k Nk k k kY X H e Wπδminus= +
loz For integer part of timing offset k k k kY X H e W+
I sTδ ( ) 0g IN L δminus minus le leFFT
2 Ij k NY X H e Wπδminus= +
( ) ( ) 2 I sFFT
j f TI sh t T H f e π δδ minusminus rarr
48
k k k kY X H e W= +
Synchronization2010722
Impact of timing offset derivationImpact of timing offset derivationp gp g
49 Synchronization2010722
Discussion of symbol boundaryDiscussion of symbol boundaryy yy y
50 Synchronization2010722
Impact of timing offsetImpact of timing offsetp gp g
loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI
2 Ij k NIk k k k
NY X H e ISI W
Nπδδ minusminus
= + +
loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk
e e
p kloz Phase Shift can be resolved by Differential Encoding
Decoding or carrier recovery with pilots
51 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
52 Synchronization2010722
Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )
53 Synchronization2010722
Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )
( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s
i n t i N N T N T nTy y t e
= + + +=
( ) ( )( ) ( ) ( ) 12
sin
sin
g gI f f
I I
i N N N Nj jf N Ni k i k k
f
Y X H e eN
π ε ε π ε
ε ε
πεπε
+ + minus+
minus minus=⎛ ⎞⎜ ⎟
Faded signal attenuated and rotated by CFO
( )( ) ( ) ( ) ( )11 2
sin
sin g gI f I f
i N N N NN j j l kI f
NN
l k π ε ε π ε επ ε ε + + minusminus + + + minus
⎜ ⎟⎝ ⎠
+ + minussum
( )( )( )
( ) ( )2
0
sin
I f I f
I
j j l kI f N Ni l l
l l k I f
X H e el k
NN
π ε ε π ε ε
ε π ε ε
+ + +
= ne minus
+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠
sum
i k
N
W
⎜ ⎟⎝ ⎠
+ Inter-Carrier Interference (ICI)
54 Synchronization2010722
Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )
loz An easier representation of received signal with only p g yfractional CFO is
1N
( ) ( )1
0
0N
k k k l l kl l k
Y H X C H X C l k Wminus
= ne
= + minus +⎡ ⎤⎣ ⎦sum
where the ICI coefficient is given by
( )( )( ) ( )
sin 1exp 1fl k
C l k j l kπ ε
π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )
( )exp 1sin
f
f
C l k j l kNN l k
N
π επ ε
minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠
55 Synchronization2010722
Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )
loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality
loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift
56 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
57 Synchronization2010722
TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )
loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq
58 Synchronization2010722
TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )
loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below
loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and
rN2 can be expressed by
0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+
( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +
59
( )
Synchronization2010722
TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )
loz Without CFO the first half is identical to the second half (in time order)
loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below
( ) 12 in discretesft T f NT
NTφ π π π ε πε
⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟
⎝ ⎠
helliphellip
sNT⎝ ⎠
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)
60
Length=N2
Synchronization2010722
TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )
loz Received signal without CFOg0
0 0
jr a e θ=jθ 2 0
2 2 0
Nj j
N Nr a e a e
θ θ= =
loz Received signal with CFO0 0
0 0
j jr a e eθ φ= Extra phase rotation due to CFO0 0
2 2 20
2 2 0
N N Nj j jj
N Nr a e e a e e
θ φ φθ= =
due to CFO
Phase difference 2 2 0N N
( )2 02 20 2 0 0
Nj j T fN
r r a e a eφ φ πminus Δ= =
which contains the information about CFO
61
0 2 0 0Nr r a e a e
Synchronization2010722
TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )
loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be
loz Note that d is a time index corresponding to the first( ) ( ) 2 1
20
Nd m d m Nm
P d r rminus
+ + +== sum
loz Note that d is a time index corresponding to the first sample in a window of N samples
loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by
( )22 1
20
Nd m NR d rminus
+ += sumloz A timing metric can be defined as
))(()(
)( 2
2
dRdP
dM =
( ) 20 d m Nm + +=sum
62
))(( dR
Synchronization2010722
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
Signal model (44)Signal model (44)g ( )g ( )
loz In a time-invariant channel the channel matrix H would be a diagonal matrix
k k k kY H X W= +
( )1 1
2
0 0
N Lj kl N
kn l
H h n l e πminus minus
minus
= =
= sumsum
loz In a fast fading channel ie the channel coherent time lt OFDM symbol period the Doppler effect must be take into consideration H h h l i H ld b di l iHence the channel matrix H would not be a diagonal matrix resulting ICI
k k k k kY H X I Wα= + +k k k k k
( ) ( )1 1 1
2 2 N N L
j m k Nj ml Nk mI X h n l e e ππ
minus minus minusminusminus⎛ ⎞
= ⎜ ⎟⎝ ⎠
sum sum sumICI
46
0 0 0m n lm k= = =ne
⎝ ⎠
Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
47 Synchronization2010722
Impact of timing offset Impact of timing offset p gp g
loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively
( )I f s
Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ
( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr
2 fj k Nk k k kY X H e Wπδminus= +
loz For integer part of timing offset k k k kY X H e W+
I sTδ ( ) 0g IN L δminus minus le leFFT
2 Ij k NY X H e Wπδminus= +
( ) ( ) 2 I sFFT
j f TI sh t T H f e π δδ minusminus rarr
48
k k k kY X H e W= +
Synchronization2010722
Impact of timing offset derivationImpact of timing offset derivationp gp g
49 Synchronization2010722
Discussion of symbol boundaryDiscussion of symbol boundaryy yy y
50 Synchronization2010722
Impact of timing offsetImpact of timing offsetp gp g
loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI
2 Ij k NIk k k k
NY X H e ISI W
Nπδδ minusminus
= + +
loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk
e e
p kloz Phase Shift can be resolved by Differential Encoding
Decoding or carrier recovery with pilots
51 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
52 Synchronization2010722
Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )
53 Synchronization2010722
Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )
( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s
i n t i N N T N T nTy y t e
= + + +=
( ) ( )( ) ( ) ( ) 12
sin
sin
g gI f f
I I
i N N N Nj jf N Ni k i k k
f
Y X H e eN
π ε ε π ε
ε ε
πεπε
+ + minus+
minus minus=⎛ ⎞⎜ ⎟
Faded signal attenuated and rotated by CFO
( )( ) ( ) ( ) ( )11 2
sin
sin g gI f I f
i N N N NN j j l kI f
NN
l k π ε ε π ε επ ε ε + + minusminus + + + minus
⎜ ⎟⎝ ⎠
+ + minussum
( )( )( )
( ) ( )2
0
sin
I f I f
I
j j l kI f N Ni l l
l l k I f
X H e el k
NN
π ε ε π ε ε
ε π ε ε
+ + +
= ne minus
+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠
sum
i k
N
W
⎜ ⎟⎝ ⎠
+ Inter-Carrier Interference (ICI)
54 Synchronization2010722
Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )
loz An easier representation of received signal with only p g yfractional CFO is
1N
( ) ( )1
0
0N
k k k l l kl l k
Y H X C H X C l k Wminus
= ne
= + minus +⎡ ⎤⎣ ⎦sum
where the ICI coefficient is given by
( )( )( ) ( )
sin 1exp 1fl k
C l k j l kπ ε
π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )
( )exp 1sin
f
f
C l k j l kNN l k
N
π επ ε
minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠
55 Synchronization2010722
Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )
loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality
loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift
56 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
57 Synchronization2010722
TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )
loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq
58 Synchronization2010722
TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )
loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below
loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and
rN2 can be expressed by
0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+
( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +
59
( )
Synchronization2010722
TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )
loz Without CFO the first half is identical to the second half (in time order)
loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below
( ) 12 in discretesft T f NT
NTφ π π π ε πε
⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟
⎝ ⎠
helliphellip
sNT⎝ ⎠
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)
60
Length=N2
Synchronization2010722
TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )
loz Received signal without CFOg0
0 0
jr a e θ=jθ 2 0
2 2 0
Nj j
N Nr a e a e
θ θ= =
loz Received signal with CFO0 0
0 0
j jr a e eθ φ= Extra phase rotation due to CFO0 0
2 2 20
2 2 0
N N Nj j jj
N Nr a e e a e e
θ φ φθ= =
due to CFO
Phase difference 2 2 0N N
( )2 02 20 2 0 0
Nj j T fN
r r a e a eφ φ πminus Δ= =
which contains the information about CFO
61
0 2 0 0Nr r a e a e
Synchronization2010722
TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )
loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be
loz Note that d is a time index corresponding to the first( ) ( ) 2 1
20
Nd m d m Nm
P d r rminus
+ + +== sum
loz Note that d is a time index corresponding to the first sample in a window of N samples
loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by
( )22 1
20
Nd m NR d rminus
+ += sumloz A timing metric can be defined as
))(()(
)( 2
2
dRdP
dM =
( ) 20 d m Nm + +=sum
62
))(( dR
Synchronization2010722
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
47 Synchronization2010722
Impact of timing offset Impact of timing offset p gp g
loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively
( )I f s
Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ
( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr
2 fj k Nk k k kY X H e Wπδminus= +
loz For integer part of timing offset k k k kY X H e W+
I sTδ ( ) 0g IN L δminus minus le leFFT
2 Ij k NY X H e Wπδminus= +
( ) ( ) 2 I sFFT
j f TI sh t T H f e π δδ minusminus rarr
48
k k k kY X H e W= +
Synchronization2010722
Impact of timing offset derivationImpact of timing offset derivationp gp g
49 Synchronization2010722
Discussion of symbol boundaryDiscussion of symbol boundaryy yy y
50 Synchronization2010722
Impact of timing offsetImpact of timing offsetp gp g
loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI
2 Ij k NIk k k k
NY X H e ISI W
Nπδδ minusminus
= + +
loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk
e e
p kloz Phase Shift can be resolved by Differential Encoding
Decoding or carrier recovery with pilots
51 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
52 Synchronization2010722
Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )
53 Synchronization2010722
Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )
( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s
i n t i N N T N T nTy y t e
= + + +=
( ) ( )( ) ( ) ( ) 12
sin
sin
g gI f f
I I
i N N N Nj jf N Ni k i k k
f
Y X H e eN
π ε ε π ε
ε ε
πεπε
+ + minus+
minus minus=⎛ ⎞⎜ ⎟
Faded signal attenuated and rotated by CFO
( )( ) ( ) ( ) ( )11 2
sin
sin g gI f I f
i N N N NN j j l kI f
NN
l k π ε ε π ε επ ε ε + + minusminus + + + minus
⎜ ⎟⎝ ⎠
+ + minussum
( )( )( )
( ) ( )2
0
sin
I f I f
I
j j l kI f N Ni l l
l l k I f
X H e el k
NN
π ε ε π ε ε
ε π ε ε
+ + +
= ne minus
+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠
sum
i k
N
W
⎜ ⎟⎝ ⎠
+ Inter-Carrier Interference (ICI)
54 Synchronization2010722
Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )
loz An easier representation of received signal with only p g yfractional CFO is
1N
( ) ( )1
0
0N
k k k l l kl l k
Y H X C H X C l k Wminus
= ne
= + minus +⎡ ⎤⎣ ⎦sum
where the ICI coefficient is given by
( )( )( ) ( )
sin 1exp 1fl k
C l k j l kπ ε
π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )
( )exp 1sin
f
f
C l k j l kNN l k
N
π επ ε
minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠
55 Synchronization2010722
Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )
loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality
loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift
56 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
57 Synchronization2010722
TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )
loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq
58 Synchronization2010722
TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )
loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below
loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and
rN2 can be expressed by
0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+
( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +
59
( )
Synchronization2010722
TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )
loz Without CFO the first half is identical to the second half (in time order)
loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below
( ) 12 in discretesft T f NT
NTφ π π π ε πε
⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟
⎝ ⎠
helliphellip
sNT⎝ ⎠
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)
60
Length=N2
Synchronization2010722
TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )
loz Received signal without CFOg0
0 0
jr a e θ=jθ 2 0
2 2 0
Nj j
N Nr a e a e
θ θ= =
loz Received signal with CFO0 0
0 0
j jr a e eθ φ= Extra phase rotation due to CFO0 0
2 2 20
2 2 0
N N Nj j jj
N Nr a e e a e e
θ φ φθ= =
due to CFO
Phase difference 2 2 0N N
( )2 02 20 2 0 0
Nj j T fN
r r a e a eφ φ πminus Δ= =
which contains the information about CFO
61
0 2 0 0Nr r a e a e
Synchronization2010722
TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )
loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be
loz Note that d is a time index corresponding to the first( ) ( ) 2 1
20
Nd m d m Nm
P d r rminus
+ + +== sum
loz Note that d is a time index corresponding to the first sample in a window of N samples
loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by
( )22 1
20
Nd m NR d rminus
+ += sumloz A timing metric can be defined as
))(()(
)( 2
2
dRdP
dM =
( ) 20 d m Nm + +=sum
62
))(( dR
Synchronization2010722
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
Impact of timing offset Impact of timing offset p gp g
loz Timing offset ( )I f sTδ δ+gwhere and are the integer part and fractional part of timing offset respectively
( )I f s
Iδ fδof timing offset respectivelyloz For fractional part of timing offset f sTδ
( ) 2FFT j f Tδ( ) ( ) 2 f sj f Tf sh t T H f e π δδ minusminus rarr
2 fj k Nk k k kY X H e Wπδminus= +
loz For integer part of timing offset k k k kY X H e W+
I sTδ ( ) 0g IN L δminus minus le leFFT
2 Ij k NY X H e Wπδminus= +
( ) ( ) 2 I sFFT
j f TI sh t T H f e π δδ minusminus rarr
48
k k k kY X H e W= +
Synchronization2010722
Impact of timing offset derivationImpact of timing offset derivationp gp g
49 Synchronization2010722
Discussion of symbol boundaryDiscussion of symbol boundaryy yy y
50 Synchronization2010722
Impact of timing offsetImpact of timing offsetp gp g
loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI
2 Ij k NIk k k k
NY X H e ISI W
Nπδδ minusminus
= + +
loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk
e e
p kloz Phase Shift can be resolved by Differential Encoding
Decoding or carrier recovery with pilots
51 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
52 Synchronization2010722
Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )
53 Synchronization2010722
Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )
( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s
i n t i N N T N T nTy y t e
= + + +=
( ) ( )( ) ( ) ( ) 12
sin
sin
g gI f f
I I
i N N N Nj jf N Ni k i k k
f
Y X H e eN
π ε ε π ε
ε ε
πεπε
+ + minus+
minus minus=⎛ ⎞⎜ ⎟
Faded signal attenuated and rotated by CFO
( )( ) ( ) ( ) ( )11 2
sin
sin g gI f I f
i N N N NN j j l kI f
NN
l k π ε ε π ε επ ε ε + + minusminus + + + minus
⎜ ⎟⎝ ⎠
+ + minussum
( )( )( )
( ) ( )2
0
sin
I f I f
I
j j l kI f N Ni l l
l l k I f
X H e el k
NN
π ε ε π ε ε
ε π ε ε
+ + +
= ne minus
+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠
sum
i k
N
W
⎜ ⎟⎝ ⎠
+ Inter-Carrier Interference (ICI)
54 Synchronization2010722
Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )
loz An easier representation of received signal with only p g yfractional CFO is
1N
( ) ( )1
0
0N
k k k l l kl l k
Y H X C H X C l k Wminus
= ne
= + minus +⎡ ⎤⎣ ⎦sum
where the ICI coefficient is given by
( )( )( ) ( )
sin 1exp 1fl k
C l k j l kπ ε
π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )
( )exp 1sin
f
f
C l k j l kNN l k
N
π επ ε
minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠
55 Synchronization2010722
Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )
loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality
loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift
56 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
57 Synchronization2010722
TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )
loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq
58 Synchronization2010722
TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )
loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below
loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and
rN2 can be expressed by
0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+
( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +
59
( )
Synchronization2010722
TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )
loz Without CFO the first half is identical to the second half (in time order)
loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below
( ) 12 in discretesft T f NT
NTφ π π π ε πε
⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟
⎝ ⎠
helliphellip
sNT⎝ ⎠
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)
60
Length=N2
Synchronization2010722
TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )
loz Received signal without CFOg0
0 0
jr a e θ=jθ 2 0
2 2 0
Nj j
N Nr a e a e
θ θ= =
loz Received signal with CFO0 0
0 0
j jr a e eθ φ= Extra phase rotation due to CFO0 0
2 2 20
2 2 0
N N Nj j jj
N Nr a e e a e e
θ φ φθ= =
due to CFO
Phase difference 2 2 0N N
( )2 02 20 2 0 0
Nj j T fN
r r a e a eφ φ πminus Δ= =
which contains the information about CFO
61
0 2 0 0Nr r a e a e
Synchronization2010722
TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )
loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be
loz Note that d is a time index corresponding to the first( ) ( ) 2 1
20
Nd m d m Nm
P d r rminus
+ + +== sum
loz Note that d is a time index corresponding to the first sample in a window of N samples
loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by
( )22 1
20
Nd m NR d rminus
+ += sumloz A timing metric can be defined as
))(()(
)( 2
2
dRdP
dM =
( ) 20 d m Nm + +=sum
62
))(( dR
Synchronization2010722
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
Impact of timing offset derivationImpact of timing offset derivationp gp g
49 Synchronization2010722
Discussion of symbol boundaryDiscussion of symbol boundaryy yy y
50 Synchronization2010722
Impact of timing offsetImpact of timing offsetp gp g
loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI
2 Ij k NIk k k k
NY X H e ISI W
Nπδδ minusminus
= + +
loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk
e e
p kloz Phase Shift can be resolved by Differential Encoding
Decoding or carrier recovery with pilots
51 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
52 Synchronization2010722
Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )
53 Synchronization2010722
Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )
( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s
i n t i N N T N T nTy y t e
= + + +=
( ) ( )( ) ( ) ( ) 12
sin
sin
g gI f f
I I
i N N N Nj jf N Ni k i k k
f
Y X H e eN
π ε ε π ε
ε ε
πεπε
+ + minus+
minus minus=⎛ ⎞⎜ ⎟
Faded signal attenuated and rotated by CFO
( )( ) ( ) ( ) ( )11 2
sin
sin g gI f I f
i N N N NN j j l kI f
NN
l k π ε ε π ε επ ε ε + + minusminus + + + minus
⎜ ⎟⎝ ⎠
+ + minussum
( )( )( )
( ) ( )2
0
sin
I f I f
I
j j l kI f N Ni l l
l l k I f
X H e el k
NN
π ε ε π ε ε
ε π ε ε
+ + +
= ne minus
+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠
sum
i k
N
W
⎜ ⎟⎝ ⎠
+ Inter-Carrier Interference (ICI)
54 Synchronization2010722
Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )
loz An easier representation of received signal with only p g yfractional CFO is
1N
( ) ( )1
0
0N
k k k l l kl l k
Y H X C H X C l k Wminus
= ne
= + minus +⎡ ⎤⎣ ⎦sum
where the ICI coefficient is given by
( )( )( ) ( )
sin 1exp 1fl k
C l k j l kπ ε
π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )
( )exp 1sin
f
f
C l k j l kNN l k
N
π επ ε
minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠
55 Synchronization2010722
Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )
loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality
loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift
56 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
57 Synchronization2010722
TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )
loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq
58 Synchronization2010722
TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )
loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below
loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and
rN2 can be expressed by
0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+
( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +
59
( )
Synchronization2010722
TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )
loz Without CFO the first half is identical to the second half (in time order)
loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below
( ) 12 in discretesft T f NT
NTφ π π π ε πε
⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟
⎝ ⎠
helliphellip
sNT⎝ ⎠
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)
60
Length=N2
Synchronization2010722
TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )
loz Received signal without CFOg0
0 0
jr a e θ=jθ 2 0
2 2 0
Nj j
N Nr a e a e
θ θ= =
loz Received signal with CFO0 0
0 0
j jr a e eθ φ= Extra phase rotation due to CFO0 0
2 2 20
2 2 0
N N Nj j jj
N Nr a e e a e e
θ φ φθ= =
due to CFO
Phase difference 2 2 0N N
( )2 02 20 2 0 0
Nj j T fN
r r a e a eφ φ πminus Δ= =
which contains the information about CFO
61
0 2 0 0Nr r a e a e
Synchronization2010722
TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )
loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be
loz Note that d is a time index corresponding to the first( ) ( ) 2 1
20
Nd m d m Nm
P d r rminus
+ + +== sum
loz Note that d is a time index corresponding to the first sample in a window of N samples
loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by
( )22 1
20
Nd m NR d rminus
+ += sumloz A timing metric can be defined as
))(()(
)( 2
2
dRdP
dM =
( ) 20 d m Nm + +=sum
62
))(( dR
Synchronization2010722
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
Discussion of symbol boundaryDiscussion of symbol boundaryy yy y
50 Synchronization2010722
Impact of timing offsetImpact of timing offsetp gp g
loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI
2 Ij k NIk k k k
NY X H e ISI W
Nπδδ minusminus
= + +
loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk
e e
p kloz Phase Shift can be resolved by Differential Encoding
Decoding or carrier recovery with pilots
51 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
52 Synchronization2010722
Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )
53 Synchronization2010722
Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )
( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s
i n t i N N T N T nTy y t e
= + + +=
( ) ( )( ) ( ) ( ) 12
sin
sin
g gI f f
I I
i N N N Nj jf N Ni k i k k
f
Y X H e eN
π ε ε π ε
ε ε
πεπε
+ + minus+
minus minus=⎛ ⎞⎜ ⎟
Faded signal attenuated and rotated by CFO
( )( ) ( ) ( ) ( )11 2
sin
sin g gI f I f
i N N N NN j j l kI f
NN
l k π ε ε π ε επ ε ε + + minusminus + + + minus
⎜ ⎟⎝ ⎠
+ + minussum
( )( )( )
( ) ( )2
0
sin
I f I f
I
j j l kI f N Ni l l
l l k I f
X H e el k
NN
π ε ε π ε ε
ε π ε ε
+ + +
= ne minus
+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠
sum
i k
N
W
⎜ ⎟⎝ ⎠
+ Inter-Carrier Interference (ICI)
54 Synchronization2010722
Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )
loz An easier representation of received signal with only p g yfractional CFO is
1N
( ) ( )1
0
0N
k k k l l kl l k
Y H X C H X C l k Wminus
= ne
= + minus +⎡ ⎤⎣ ⎦sum
where the ICI coefficient is given by
( )( )( ) ( )
sin 1exp 1fl k
C l k j l kπ ε
π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )
( )exp 1sin
f
f
C l k j l kNN l k
N
π επ ε
minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠
55 Synchronization2010722
Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )
loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality
loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift
56 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
57 Synchronization2010722
TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )
loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq
58 Synchronization2010722
TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )
loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below
loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and
rN2 can be expressed by
0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+
( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +
59
( )
Synchronization2010722
TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )
loz Without CFO the first half is identical to the second half (in time order)
loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below
( ) 12 in discretesft T f NT
NTφ π π π ε πε
⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟
⎝ ⎠
helliphellip
sNT⎝ ⎠
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)
60
Length=N2
Synchronization2010722
TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )
loz Received signal without CFOg0
0 0
jr a e θ=jθ 2 0
2 2 0
Nj j
N Nr a e a e
θ θ= =
loz Received signal with CFO0 0
0 0
j jr a e eθ φ= Extra phase rotation due to CFO0 0
2 2 20
2 2 0
N N Nj j jj
N Nr a e e a e e
θ φ φθ= =
due to CFO
Phase difference 2 2 0N N
( )2 02 20 2 0 0
Nj j T fN
r r a e a eφ φ πminus Δ= =
which contains the information about CFO
61
0 2 0 0Nr r a e a e
Synchronization2010722
TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )
loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be
loz Note that d is a time index corresponding to the first( ) ( ) 2 1
20
Nd m d m Nm
P d r rminus
+ + +== sum
loz Note that d is a time index corresponding to the first sample in a window of N samples
loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by
( )22 1
20
Nd m NR d rminus
+ += sumloz A timing metric can be defined as
))(()(
)( 2
2
dRdP
dM =
( ) 20 d m Nm + +=sum
62
))(( dR
Synchronization2010722
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
Impact of timing offsetImpact of timing offsetp gp g
loz Once upon the boundary of DFT window is not located in the ISI-free region it will induce some extra ISI
2 Ij k NIk k k k
NY X H e ISI W
Nπδδ minusminus
= + +
loz The phase shift caused by integer part and fractional part of timing offset and 2 Ij k Ne πδminus 2 fj k Ne πδminusgloz Both are depend on frequency index k loz Not Differentiable from the phase of Hk
e e
p kloz Phase Shift can be resolved by Differential Encoding
Decoding or carrier recovery with pilots
51 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
52 Synchronization2010722
Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )
53 Synchronization2010722
Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )
( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s
i n t i N N T N T nTy y t e
= + + +=
( ) ( )( ) ( ) ( ) 12
sin
sin
g gI f f
I I
i N N N Nj jf N Ni k i k k
f
Y X H e eN
π ε ε π ε
ε ε
πεπε
+ + minus+
minus minus=⎛ ⎞⎜ ⎟
Faded signal attenuated and rotated by CFO
( )( ) ( ) ( ) ( )11 2
sin
sin g gI f I f
i N N N NN j j l kI f
NN
l k π ε ε π ε επ ε ε + + minusminus + + + minus
⎜ ⎟⎝ ⎠
+ + minussum
( )( )( )
( ) ( )2
0
sin
I f I f
I
j j l kI f N Ni l l
l l k I f
X H e el k
NN
π ε ε π ε ε
ε π ε ε
+ + +
= ne minus
+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠
sum
i k
N
W
⎜ ⎟⎝ ⎠
+ Inter-Carrier Interference (ICI)
54 Synchronization2010722
Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )
loz An easier representation of received signal with only p g yfractional CFO is
1N
( ) ( )1
0
0N
k k k l l kl l k
Y H X C H X C l k Wminus
= ne
= + minus +⎡ ⎤⎣ ⎦sum
where the ICI coefficient is given by
( )( )( ) ( )
sin 1exp 1fl k
C l k j l kπ ε
π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )
( )exp 1sin
f
f
C l k j l kNN l k
N
π επ ε
minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠
55 Synchronization2010722
Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )
loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality
loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift
56 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
57 Synchronization2010722
TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )
loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq
58 Synchronization2010722
TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )
loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below
loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and
rN2 can be expressed by
0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+
( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +
59
( )
Synchronization2010722
TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )
loz Without CFO the first half is identical to the second half (in time order)
loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below
( ) 12 in discretesft T f NT
NTφ π π π ε πε
⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟
⎝ ⎠
helliphellip
sNT⎝ ⎠
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)
60
Length=N2
Synchronization2010722
TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )
loz Received signal without CFOg0
0 0
jr a e θ=jθ 2 0
2 2 0
Nj j
N Nr a e a e
θ θ= =
loz Received signal with CFO0 0
0 0
j jr a e eθ φ= Extra phase rotation due to CFO0 0
2 2 20
2 2 0
N N Nj j jj
N Nr a e e a e e
θ φ φθ= =
due to CFO
Phase difference 2 2 0N N
( )2 02 20 2 0 0
Nj j T fN
r r a e a eφ φ πminus Δ= =
which contains the information about CFO
61
0 2 0 0Nr r a e a e
Synchronization2010722
TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )
loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be
loz Note that d is a time index corresponding to the first( ) ( ) 2 1
20
Nd m d m Nm
P d r rminus
+ + +== sum
loz Note that d is a time index corresponding to the first sample in a window of N samples
loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by
( )22 1
20
Nd m NR d rminus
+ += sumloz A timing metric can be defined as
))(()(
)( 2
2
dRdP
dM =
( ) 20 d m Nm + +=sum
62
))(( dR
Synchronization2010722
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
52 Synchronization2010722
Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )
53 Synchronization2010722
Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )
( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s
i n t i N N T N T nTy y t e
= + + +=
( ) ( )( ) ( ) ( ) 12
sin
sin
g gI f f
I I
i N N N Nj jf N Ni k i k k
f
Y X H e eN
π ε ε π ε
ε ε
πεπε
+ + minus+
minus minus=⎛ ⎞⎜ ⎟
Faded signal attenuated and rotated by CFO
( )( ) ( ) ( ) ( )11 2
sin
sin g gI f I f
i N N N NN j j l kI f
NN
l k π ε ε π ε επ ε ε + + minusminus + + + minus
⎜ ⎟⎝ ⎠
+ + minussum
( )( )( )
( ) ( )2
0
sin
I f I f
I
j j l kI f N Ni l l
l l k I f
X H e el k
NN
π ε ε π ε ε
ε π ε ε
+ + +
= ne minus
+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠
sum
i k
N
W
⎜ ⎟⎝ ⎠
+ Inter-Carrier Interference (ICI)
54 Synchronization2010722
Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )
loz An easier representation of received signal with only p g yfractional CFO is
1N
( ) ( )1
0
0N
k k k l l kl l k
Y H X C H X C l k Wminus
= ne
= + minus +⎡ ⎤⎣ ⎦sum
where the ICI coefficient is given by
( )( )( ) ( )
sin 1exp 1fl k
C l k j l kπ ε
π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )
( )exp 1sin
f
f
C l k j l kNN l k
N
π επ ε
minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠
55 Synchronization2010722
Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )
loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality
loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift
56 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
57 Synchronization2010722
TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )
loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq
58 Synchronization2010722
TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )
loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below
loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and
rN2 can be expressed by
0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+
( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +
59
( )
Synchronization2010722
TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )
loz Without CFO the first half is identical to the second half (in time order)
loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below
( ) 12 in discretesft T f NT
NTφ π π π ε πε
⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟
⎝ ⎠
helliphellip
sNT⎝ ⎠
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)
60
Length=N2
Synchronization2010722
TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )
loz Received signal without CFOg0
0 0
jr a e θ=jθ 2 0
2 2 0
Nj j
N Nr a e a e
θ θ= =
loz Received signal with CFO0 0
0 0
j jr a e eθ φ= Extra phase rotation due to CFO0 0
2 2 20
2 2 0
N N Nj j jj
N Nr a e e a e e
θ φ φθ= =
due to CFO
Phase difference 2 2 0N N
( )2 02 20 2 0 0
Nj j T fN
r r a e a eφ φ πminus Δ= =
which contains the information about CFO
61
0 2 0 0Nr r a e a e
Synchronization2010722
TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )
loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be
loz Note that d is a time index corresponding to the first( ) ( ) 2 1
20
Nd m d m Nm
P d r rminus
+ + +== sum
loz Note that d is a time index corresponding to the first sample in a window of N samples
loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by
( )22 1
20
Nd m NR d rminus
+ += sumloz A timing metric can be defined as
))(()(
)( 2
2
dRdP
dM =
( ) 20 d m Nm + +=sum
62
))(( dR
Synchronization2010722
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
Impact of carrier frequency offset (14)Impact of carrier frequency offset (14)p q y ( )p q y ( )
53 Synchronization2010722
Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )
( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s
i n t i N N T N T nTy y t e
= + + +=
( ) ( )( ) ( ) ( ) 12
sin
sin
g gI f f
I I
i N N N Nj jf N Ni k i k k
f
Y X H e eN
π ε ε π ε
ε ε
πεπε
+ + minus+
minus minus=⎛ ⎞⎜ ⎟
Faded signal attenuated and rotated by CFO
( )( ) ( ) ( ) ( )11 2
sin
sin g gI f I f
i N N N NN j j l kI f
NN
l k π ε ε π ε επ ε ε + + minusminus + + + minus
⎜ ⎟⎝ ⎠
+ + minussum
( )( )( )
( ) ( )2
0
sin
I f I f
I
j j l kI f N Ni l l
l l k I f
X H e el k
NN
π ε ε π ε ε
ε π ε ε
+ + +
= ne minus
+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠
sum
i k
N
W
⎜ ⎟⎝ ⎠
+ Inter-Carrier Interference (ICI)
54 Synchronization2010722
Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )
loz An easier representation of received signal with only p g yfractional CFO is
1N
( ) ( )1
0
0N
k k k l l kl l k
Y H X C H X C l k Wminus
= ne
= + minus +⎡ ⎤⎣ ⎦sum
where the ICI coefficient is given by
( )( )( ) ( )
sin 1exp 1fl k
C l k j l kπ ε
π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )
( )exp 1sin
f
f
C l k j l kNN l k
N
π επ ε
minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠
55 Synchronization2010722
Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )
loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality
loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift
56 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
57 Synchronization2010722
TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )
loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq
58 Synchronization2010722
TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )
loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below
loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and
rN2 can be expressed by
0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+
( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +
59
( )
Synchronization2010722
TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )
loz Without CFO the first half is identical to the second half (in time order)
loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below
( ) 12 in discretesft T f NT
NTφ π π π ε πε
⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟
⎝ ⎠
helliphellip
sNT⎝ ⎠
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)
60
Length=N2
Synchronization2010722
TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )
loz Received signal without CFOg0
0 0
jr a e θ=jθ 2 0
2 2 0
Nj j
N Nr a e a e
θ θ= =
loz Received signal with CFO0 0
0 0
j jr a e eθ φ= Extra phase rotation due to CFO0 0
2 2 20
2 2 0
N N Nj j jj
N Nr a e e a e e
θ φ φθ= =
due to CFO
Phase difference 2 2 0N N
( )2 02 20 2 0 0
Nj j T fN
r r a e a eφ φ πminus Δ= =
which contains the information about CFO
61
0 2 0 0Nr r a e a e
Synchronization2010722
TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )
loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be
loz Note that d is a time index corresponding to the first( ) ( ) 2 1
20
Nd m d m Nm
P d r rminus
+ + +== sum
loz Note that d is a time index corresponding to the first sample in a window of N samples
loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by
( )22 1
20
Nd m NR d rminus
+ += sumloz A timing metric can be defined as
))(()(
)( 2
2
dRdP
dM =
( ) 20 d m Nm + +=sum
62
))(( dR
Synchronization2010722
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
Impact of carrier frequency offset (24)Impact of carrier frequency offset (24)p q y ( )p q y ( )
( ) 2 |j fty y t e πΔ= ( ) ( ) |g s g s s
i n t i N N T N T nTy y t e
= + + +=
( ) ( )( ) ( ) ( ) 12
sin
sin
g gI f f
I I
i N N N Nj jf N Ni k i k k
f
Y X H e eN
π ε ε π ε
ε ε
πεπε
+ + minus+
minus minus=⎛ ⎞⎜ ⎟
Faded signal attenuated and rotated by CFO
( )( ) ( ) ( ) ( )11 2
sin
sin g gI f I f
i N N N NN j j l kI f
NN
l k π ε ε π ε επ ε ε + + minusminus + + + minus
⎜ ⎟⎝ ⎠
+ + minussum
( )( )( )
( ) ( )2
0
sin
I f I f
I
j j l kI f N Ni l l
l l k I f
X H e el k
NN
π ε ε π ε ε
ε π ε ε
+ + +
= ne minus
+⎛ ⎞+ + minus⎜ ⎟⎜ ⎟⎝ ⎠
sum
i k
N
W
⎜ ⎟⎝ ⎠
+ Inter-Carrier Interference (ICI)
54 Synchronization2010722
Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )
loz An easier representation of received signal with only p g yfractional CFO is
1N
( ) ( )1
0
0N
k k k l l kl l k
Y H X C H X C l k Wminus
= ne
= + minus +⎡ ⎤⎣ ⎦sum
where the ICI coefficient is given by
( )( )( ) ( )
sin 1exp 1fl k
C l k j l kπ ε
π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )
( )exp 1sin
f
f
C l k j l kNN l k
N
π επ ε
minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠
55 Synchronization2010722
Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )
loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality
loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift
56 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
57 Synchronization2010722
TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )
loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq
58 Synchronization2010722
TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )
loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below
loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and
rN2 can be expressed by
0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+
( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +
59
( )
Synchronization2010722
TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )
loz Without CFO the first half is identical to the second half (in time order)
loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below
( ) 12 in discretesft T f NT
NTφ π π π ε πε
⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟
⎝ ⎠
helliphellip
sNT⎝ ⎠
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)
60
Length=N2
Synchronization2010722
TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )
loz Received signal without CFOg0
0 0
jr a e θ=jθ 2 0
2 2 0
Nj j
N Nr a e a e
θ θ= =
loz Received signal with CFO0 0
0 0
j jr a e eθ φ= Extra phase rotation due to CFO0 0
2 2 20
2 2 0
N N Nj j jj
N Nr a e e a e e
θ φ φθ= =
due to CFO
Phase difference 2 2 0N N
( )2 02 20 2 0 0
Nj j T fN
r r a e a eφ φ πminus Δ= =
which contains the information about CFO
61
0 2 0 0Nr r a e a e
Synchronization2010722
TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )
loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be
loz Note that d is a time index corresponding to the first( ) ( ) 2 1
20
Nd m d m Nm
P d r rminus
+ + +== sum
loz Note that d is a time index corresponding to the first sample in a window of N samples
loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by
( )22 1
20
Nd m NR d rminus
+ += sumloz A timing metric can be defined as
))(()(
)( 2
2
dRdP
dM =
( ) 20 d m Nm + +=sum
62
))(( dR
Synchronization2010722
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
Impact of carrier frequency offset (34)Impact of carrier frequency offset (34)p q y ( )p q y ( )
loz An easier representation of received signal with only p g yfractional CFO is
1N
( ) ( )1
0
0N
k k k l l kl l k
Y H X C H X C l k Wminus
= ne
= + minus +⎡ ⎤⎣ ⎦sum
where the ICI coefficient is given by
( )( )( ) ( )
sin 1exp 1fl k
C l k j l kπ ε
π ε+ minus ⎛ ⎞⎛ ⎞minus = sdot minus + minus⎜ ⎟⎜ ⎟( )( )
( )exp 1sin
f
f
C l k j l kNN l k
N
π επ ε
minus = sdot minus + minus⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠+ minus⎜ ⎟⎝ ⎠
55 Synchronization2010722
Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )
loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality
loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift
56 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
57 Synchronization2010722
TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )
loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq
58 Synchronization2010722
TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )
loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below
loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and
rN2 can be expressed by
0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+
( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +
59
( )
Synchronization2010722
TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )
loz Without CFO the first half is identical to the second half (in time order)
loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below
( ) 12 in discretesft T f NT
NTφ π π π ε πε
⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟
⎝ ⎠
helliphellip
sNT⎝ ⎠
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)
60
Length=N2
Synchronization2010722
TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )
loz Received signal without CFOg0
0 0
jr a e θ=jθ 2 0
2 2 0
Nj j
N Nr a e a e
θ θ= =
loz Received signal with CFO0 0
0 0
j jr a e eθ φ= Extra phase rotation due to CFO0 0
2 2 20
2 2 0
N N Nj j jj
N Nr a e e a e e
θ φ φθ= =
due to CFO
Phase difference 2 2 0N N
( )2 02 20 2 0 0
Nj j T fN
r r a e a eφ φ πminus Δ= =
which contains the information about CFO
61
0 2 0 0Nr r a e a e
Synchronization2010722
TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )
loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be
loz Note that d is a time index corresponding to the first( ) ( ) 2 1
20
Nd m d m Nm
P d r rminus
+ + +== sum
loz Note that d is a time index corresponding to the first sample in a window of N samples
loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by
( )22 1
20
Nd m NR d rminus
+ += sumloz A timing metric can be defined as
))(()(
)( 2
2
dRdP
dM =
( ) 20 d m Nm + +=sum
62
))(( dR
Synchronization2010722
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
Impact of carrier frequency offset (44)Impact of carrier frequency offset (44)p q y ( )p q y ( )
loz Fractional CFOloz Phase shift in time domainloz Induce the magnitude attenuation and ICIgloz Loss of orthogonality
loz Integer CFOloz Integer CFO loz Phase shift in time domainloz No effect on the orthogonalityloz No effect on the orthogonalityloz Index shift
56 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
57 Synchronization2010722
TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )
loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq
58 Synchronization2010722
TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )
loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below
loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and
rN2 can be expressed by
0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+
( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +
59
( )
Synchronization2010722
TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )
loz Without CFO the first half is identical to the second half (in time order)
loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below
( ) 12 in discretesft T f NT
NTφ π π π ε πε
⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟
⎝ ⎠
helliphellip
sNT⎝ ⎠
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)
60
Length=N2
Synchronization2010722
TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )
loz Received signal without CFOg0
0 0
jr a e θ=jθ 2 0
2 2 0
Nj j
N Nr a e a e
θ θ= =
loz Received signal with CFO0 0
0 0
j jr a e eθ φ= Extra phase rotation due to CFO0 0
2 2 20
2 2 0
N N Nj j jj
N Nr a e e a e e
θ φ φθ= =
due to CFO
Phase difference 2 2 0N N
( )2 02 20 2 0 0
Nj j T fN
r r a e a eφ φ πminus Δ= =
which contains the information about CFO
61
0 2 0 0Nr r a e a e
Synchronization2010722
TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )
loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be
loz Note that d is a time index corresponding to the first( ) ( ) 2 1
20
Nd m d m Nm
P d r rminus
+ + +== sum
loz Note that d is a time index corresponding to the first sample in a window of N samples
loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by
( )22 1
20
Nd m NR d rminus
+ += sumloz A timing metric can be defined as
))(()(
)( 2
2
dRdP
dM =
( ) 20 d m Nm + +=sum
62
))(( dR
Synchronization2010722
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
57 Synchronization2010722
TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )
loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq
58 Synchronization2010722
TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )
loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below
loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and
rN2 can be expressed by
0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+
( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +
59
( )
Synchronization2010722
TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )
loz Without CFO the first half is identical to the second half (in time order)
loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below
( ) 12 in discretesft T f NT
NTφ π π π ε πε
⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟
⎝ ⎠
helliphellip
sNT⎝ ⎠
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)
60
Length=N2
Synchronization2010722
TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )
loz Received signal without CFOg0
0 0
jr a e θ=jθ 2 0
2 2 0
Nj j
N Nr a e a e
θ θ= =
loz Received signal with CFO0 0
0 0
j jr a e eθ φ= Extra phase rotation due to CFO0 0
2 2 20
2 2 0
N N Nj j jj
N Nr a e e a e e
θ φ φθ= =
due to CFO
Phase difference 2 2 0N N
( )2 02 20 2 0 0
Nj j T fN
r r a e a eφ φ πminus Δ= =
which contains the information about CFO
61
0 2 0 0Nr r a e a e
Synchronization2010722
TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )
loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be
loz Note that d is a time index corresponding to the first( ) ( ) 2 1
20
Nd m d m Nm
P d r rminus
+ + +== sum
loz Note that d is a time index corresponding to the first sample in a window of N samples
loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by
( )22 1
20
Nd m NR d rminus
+ += sumloz A timing metric can be defined as
))(()(
)( 2
2
dRdP
dM =
( ) 20 d m Nm + +=sum
62
))(( dR
Synchronization2010722
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
TO estimation algorithm (110)TO estimation algorithm (110)g ( )g ( )
loz Schmidlrsquos Method [1] First of all we could design a [ ] gtraining symbol which contains a PN sequence on the odd frequenciesq
58 Synchronization2010722
TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )
loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below
loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and
rN2 can be expressed by
0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+
( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +
59
( )
Synchronization2010722
TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )
loz Without CFO the first half is identical to the second half (in time order)
loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below
( ) 12 in discretesft T f NT
NTφ π π π ε πε
⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟
⎝ ⎠
helliphellip
sNT⎝ ⎠
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)
60
Length=N2
Synchronization2010722
TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )
loz Received signal without CFOg0
0 0
jr a e θ=jθ 2 0
2 2 0
Nj j
N Nr a e a e
θ θ= =
loz Received signal with CFO0 0
0 0
j jr a e eθ φ= Extra phase rotation due to CFO0 0
2 2 20
2 2 0
N N Nj j jj
N Nr a e e a e e
θ φ φθ= =
due to CFO
Phase difference 2 2 0N N
( )2 02 20 2 0 0
Nj j T fN
r r a e a eφ φ πminus Δ= =
which contains the information about CFO
61
0 2 0 0Nr r a e a e
Synchronization2010722
TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )
loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be
loz Note that d is a time index corresponding to the first( ) ( ) 2 1
20
Nd m d m Nm
P d r rminus
+ + +== sum
loz Note that d is a time index corresponding to the first sample in a window of N samples
loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by
( )22 1
20
Nd m NR d rminus
+ += sumloz A timing metric can be defined as
))(()(
)( 2
2
dRdP
dM =
( ) 20 d m Nm + +=sum
62
))(( dR
Synchronization2010722
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
TO estimation algorithm (210)TO estimation algorithm (210)g ( )g ( )
loz Due to the property of IDFT the resulting time domain training symbol would have a repetition form as shown below
loz After sampling the complex samples are denoted as rmloz Let the multipath channel L = [h0 h1] the received sample r0 and
rN2 can be expressed by
0 0 0 1 1r x h x h= +0 0 0 1 1r x h x hminus+
( ) 2 2 0 1 0 0 1 12 1N N Nr x h x h x h x hminusminus= + = +
59
( )
Synchronization2010722
TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )
loz Without CFO the first half is identical to the second half (in time order)
loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below
( ) 12 in discretesft T f NT
NTφ π π π ε πε
⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟
⎝ ⎠
helliphellip
sNT⎝ ⎠
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)
60
Length=N2
Synchronization2010722
TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )
loz Received signal without CFOg0
0 0
jr a e θ=jθ 2 0
2 2 0
Nj j
N Nr a e a e
θ θ= =
loz Received signal with CFO0 0
0 0
j jr a e eθ φ= Extra phase rotation due to CFO0 0
2 2 20
2 2 0
N N Nj j jj
N Nr a e e a e e
θ φ φθ= =
due to CFO
Phase difference 2 2 0N N
( )2 02 20 2 0 0
Nj j T fN
r r a e a eφ φ πminus Δ= =
which contains the information about CFO
61
0 2 0 0Nr r a e a e
Synchronization2010722
TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )
loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be
loz Note that d is a time index corresponding to the first( ) ( ) 2 1
20
Nd m d m Nm
P d r rminus
+ + +== sum
loz Note that d is a time index corresponding to the first sample in a window of N samples
loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by
( )22 1
20
Nd m NR d rminus
+ += sumloz A timing metric can be defined as
))(()(
)( 2
2
dRdP
dM =
( ) 20 d m Nm + +=sum
62
))(( dR
Synchronization2010722
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
TO estimation algorithm (310)TO estimation algorithm (310)g ( )g ( )
loz Without CFO the first half is identical to the second half (in time order)
loz With CFO if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T2 seconds later) there will be an extra phase difference caused by the CFO as shown belowthe CFO as shown below
( ) 12 in discretesft T f NT
NTφ π π π ε πε
⎛ ⎞= Δ = Δ ⎯⎯⎯⎯rarr =⎜ ⎟
⎝ ⎠
helliphellip
sNT⎝ ⎠
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r(N2)+1) helliphellip r(N-2) r(N-1)
60
Length=N2
Synchronization2010722
TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )
loz Received signal without CFOg0
0 0
jr a e θ=jθ 2 0
2 2 0
Nj j
N Nr a e a e
θ θ= =
loz Received signal with CFO0 0
0 0
j jr a e eθ φ= Extra phase rotation due to CFO0 0
2 2 20
2 2 0
N N Nj j jj
N Nr a e e a e e
θ φ φθ= =
due to CFO
Phase difference 2 2 0N N
( )2 02 20 2 0 0
Nj j T fN
r r a e a eφ φ πminus Δ= =
which contains the information about CFO
61
0 2 0 0Nr r a e a e
Synchronization2010722
TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )
loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be
loz Note that d is a time index corresponding to the first( ) ( ) 2 1
20
Nd m d m Nm
P d r rminus
+ + +== sum
loz Note that d is a time index corresponding to the first sample in a window of N samples
loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by
( )22 1
20
Nd m NR d rminus
+ += sumloz A timing metric can be defined as
))(()(
)( 2
2
dRdP
dM =
( ) 20 d m Nm + +=sum
62
))(( dR
Synchronization2010722
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
TO estimation algorithm (410)TO estimation algorithm (410)g ( )g ( )
loz Received signal without CFOg0
0 0
jr a e θ=jθ 2 0
2 2 0
Nj j
N Nr a e a e
θ θ= =
loz Received signal with CFO0 0
0 0
j jr a e eθ φ= Extra phase rotation due to CFO0 0
2 2 20
2 2 0
N N Nj j jj
N Nr a e e a e e
θ φ φθ= =
due to CFO
Phase difference 2 2 0N N
( )2 02 20 2 0 0
Nj j T fN
r r a e a eφ φ πminus Δ= =
which contains the information about CFO
61
0 2 0 0Nr r a e a e
Synchronization2010722
TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )
loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be
loz Note that d is a time index corresponding to the first( ) ( ) 2 1
20
Nd m d m Nm
P d r rminus
+ + +== sum
loz Note that d is a time index corresponding to the first sample in a window of N samples
loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by
( )22 1
20
Nd m NR d rminus
+ += sumloz A timing metric can be defined as
))(()(
)( 2
2
dRdP
dM =
( ) 20 d m Nm + +=sum
62
))(( dR
Synchronization2010722
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
TO estimation algorithm (510)TO estimation algorithm (510)g ( )g ( )
loz Let there be N2 complex samples in one-half of the p ptraining symbol (excluding the cyclic prefix) and let the sum of the pairs of products be
loz Note that d is a time index corresponding to the first( ) ( ) 2 1
20
Nd m d m Nm
P d r rminus
+ + +== sum
loz Note that d is a time index corresponding to the first sample in a window of N samples
loz The received energy for the second half symbol is definedloz The received energy for the second half-symbol is defined by
( )22 1
20
Nd m NR d rminus
+ += sumloz A timing metric can be defined as
))(()(
)( 2
2
dRdP
dM =
( ) 20 d m Nm + +=sum
62
))(( dR
Synchronization2010722
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
TO estimation algorithm (610)TO estimation algorithm (610)g ( )g ( )
63 Synchronization2010722
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
TO estimation algorithm (710)TO estimation algorithm (710)g ( )g ( )
loz If drsquo is the correct symbol timing offset
( )2
2 2
0 1 12
j T f j T f j T fN
P d a e a e a eπ π πΔ Δ Δ
minus= + + +
22
2 2
0 1 12
j T fN
e a a aπ Δ
minus
⎧ ⎫⎪ ⎪= + + +⎨ ⎬⎪ ⎪⎩ ⎭
loz If drsquo falls behind the correct symbol timing offset 1 sample
2⎪ ⎪⎩ ⎭
64 Synchronization2010722
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
TO estimation algorithm (810)TO estimation algorithm (810)g ( )g ( )
65 Synchronization2010722
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
TO estimation algorithm (910)TO estimation algorithm (910)g ( )g ( )
loz Drawback Plateau effect
Since CP is the copy of the last few samples these twolast few samples these two observation windows will the same correlation(without noise)noise)
((N2) ((N2) ((N2)
helliphellip
r(0) r(1) helliphellipr((N2)-
2)r((N2)-
1) r(N2) r((N2)+1) helliphellip r(N-2) r(N-1)r(N-1)
Sliding window (length = N)
66 Synchronization2010722
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
TO estimation algorithm (1010)TO estimation algorithm (1010)g ( )g ( )
loz Advantage Low computational complexity The product g p p y pP(d) can be implemented with the iterative formula
( ) ( ) ( ) ( ) 1P d P d r r r r+ = + minus
扣掉相加
( ) ( ) ( ) ( ) 2 21 d N d N d d NP d P d r r r r+ + ++ = +
loz The method also called delay correlator
67 Synchronization2010722
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
IEEE 80211aIEEE 80211a
68 Synchronization2010722
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
Other TO estimation algorithms (14)Other TO estimation algorithms (14)g ( )g ( )
loz Minnrsquos Method [2][ ]B B -B -B
( )( )
( )
22
2Minn
P dM d =
1 4 1N minus
( )( )( )2
2
MinnR d
( ) ( ) ( ) ( )
1 4 1
2 2 2 40 0
N
d N m k d N m k Nm k
P d r r+ + + + += =
= sum sum
( ) ( ) ( )2
21 4 1
2 40 0
N
d N m k Nk
R d rminus
+ + += sum sum69
0 0m k= =
Synchronization2010722
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
Other TO estimation algorithms (24)Other TO estimation algorithms (24)g ( )g ( )
loz Parkrsquos Method [3][ ]
where D is the symmetric version of CC D C D
where D is the symmetric version of C
( )( ) 2
3k
P dM d =( )
( )( )23
ParkM dR d
2N
( ) 2
30
N
d k d kk
P d r rminus +=
= sdotsum
( )3
22
0
N
d kk
R d r += sum70
0k=
Synchronization2010722
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
Other TO estimation algorithms (34)Other TO estimation algorithms (34)g ( )g ( )
loz Renrsquos Method [4][ ]
AS AS
where S is the PN sequence weighted of the original bl 2preamble
( )( )( )( )
24
Re 24
n
P dM d
R d=
( )( )4R d
( ) 2 1
4 2 2
N
k k N d k d k NP d s s r rminus
+ + + += sum0k=sum
( )21
41 N
d kR d rminus
= sum71
( )402 d k
kR d r +
=sum
Synchronization2010722
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
Other TO estimation algorithms (44)Other TO estimation algorithms (44)g ( )g ( )
72 Synchronization2010722
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
OutlineOutline
loz Introductionloz Signal modelloz Impact of timing offsetloz Impact of timing offsetloz Impact of carrier frequency offsetloz Symbol timing estimation algorithmloz CFO estimation algorithmg
73 Synchronization2010722
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
Fractional CFO estimation Fractional CFO estimation
loz Fractional CFO estimation can be accomplished when pthe symbol boundary is detected
11 Rminus⎛ ⎞sum
h L i th di t b t t id ti l bl k R i
0
12 d m d m L
ms
f r rLTπ + + +
=
⎛ ⎞= ang⎜ ⎟⎝ ⎠sum
where L is the distance between two identical block R is the block size
loz For exampleNN cp
L = N R = NL = N R = Ncp74 Synchronization2010722
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
Estimation rangeEstimation rangegg
loz The acquisition range of is ˆ fε ( ) ( )1 2 1 2s sLT LTminus⎡ ⎤⎣ ⎦q g which depends on the repetition interval
f ( ) ( )s s⎣ ⎦
loz For example802 16 2005 (L N4) [ ]0 5 0 5f floz 80216e-2005 (L=N4)
loz DVB-T (L=N) 802 11 (L N4)
[ ]05 05s sf fminus
[ ]2 2s sf fminus[ ]0 5 0 5f floz 80211a (L=N4)
(L=N) [ ]2 2s sf fminus[ ]05 05s sf fminus
loz Note that is the subcarrier spacing1sf NT=
75
sNT
Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80211aloz Oscillator deviation 20ppmloz Highest carrier frequency 6 GHz
plusmng q y
loz Maximum CFO 40 ppm x 6 GHz = 240 KHzloz 240 KHz subcarrier spacingplusmn 1lt plusmnloz 240 KHz subcarrier spacingplusmn 1lt plusmn
76 Synchronization2010722
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
Maximum CFOMaximum CFO
loz Maximum CFO in IEEE 80216e WiMAX OFDM modeloz Oscillator deviation 8ppmloz Highest carrier frequency 1068 GHz
plusmng q y
loz Maximum CFO 16 ppm x 1068 GHz = 171 KHzloz 171 KHz subcarrier spacingplusmn 11asymp plusmnloz 171 KHz subcarrier spacingloz With estimation for
subcarrier spacing is required
plusmn 11plusmn2 2s f sf fεminus le le 4 8 and 12plusmn plusmn plusmn
p g qloz CFO = 17 fs = (0 + 17) fs CFO = 27 fs = (4 ndash 13) fs
77 Synchronization2010722
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
Integer CFO estimation algorithm (12)Integer CFO estimation algorithm (12)g g ( )g g ( )
loz Time domain correlationloz Match filters with
coefficient of conjugated preamble waveform modulated by different integer CFOCFO
loz Example CFO is 42 ffs
78 Synchronization2010722
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
Integer CFO estimation algorithm (22)Integer CFO estimation algorithm (22)g g ( )g g ( )
loz Frequency domain autocorrelationq yloz In DVB-T take advantage of continual pilot subcarriersloz Assume similar channel frequency response (CFR) in two q y p ( )
consecutive symbols
( )1
0 1 2J
g Z Z gminus
Φ = = plusmn plusmnsum( ) 10
0 1 2j ji g i g
jg Z Z gα α+ minus +
=
Φ = = plusmn plusmnsum hellip
( )ˆ arg maxg g= Φ( )g
79 Synchronization2010722
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
IEEE 80211aIEEE 80211a
80 Synchronization2010722
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
IEEE 80211aIEEE 80211a
2010722 Synchronization81
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
Short OFDM training symbol Short OFDM training symbol g yg y
loz A short OFDM training symbol consists of 12 g ysubcarriers which are modulated by the elements of the sequence S given byq g yS-2626= 0 0 1+j 0 0 0 -1-j 0 0 0 1+j
0 0 0 1 j 0 0 0 1 j 0 0 0 1+j 013 6times0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 0 0 0 0 -1-j 0 0 0 -1-j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0 0 1+j 0 0
2010722 Synchronization82
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
Long OFDM training symbolLong OFDM training symbolg g yg g y
loz A long OFDM training symbol consists of 53 g g ysubcarriers (including a zero value at DC) which are modulated by the elements of the sequence L given byy q g yL-2626=1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1
-1 -1 1 1 -1 1 -1 1 1 1 1 0 1 -1 -11 1 1 1 1 1 1 1 1 1 1 0 1 1 11 1 -1 1 -1 1 -1 -1 -1 -1 -1 -111 1 1 1 1 1 1 1 1 1 11 -1 -1 1 -1 1 -1 1 1 1 1
2010722 Synchronization83
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
Pilot subcarriersPilot subcarriers
loz In each OFDM symbol four of the subcarriers are y dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase g q y pnoise
loz These pilot signals shall be put in subcarriers ndash21 ndash7 7 and 21and 21
loz The pilots shall be BPSK modulated by a pseudo binary sequence to prevent the generation of spectral lines
2010722 Synchronization84
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
Pilot subcarriersPilot subcarriers
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
2010722 Synchronization85
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
ReferenceReference
[1] T M Schmidl and D C Cox ldquoRobust Frequency and Timing Synchronization for OFDM rdquo IEEE Transaction on Communications vol 45 no 12 Dec 1997OFDM IEEE Transaction on Communications vol 45 no 12 Dec 1997
[2] H Minn V K Bhargava and K B Letaief ldquoA Robust Timing and Frequency Synchronization for OFDM Systemsrdquo IEEE Transaction on Wireless Communications vol 2 no 4 Jul 2003
[3] B Park H Cheon and C Kang ldquoA Novel Timing Estimation Method for OFDM Systemsrdquo IEEE Transaction on Communications Letters vol 7 no 5 May 2003
[4] G Ren Y Chang and H Zhang ldquoSynchronization Method Based on a New Constant [ ] g g yEnvelop Preamble for OFDM Systemsrdquo IEEE Transaction on Braodcasting vol 51 no 1 Mar 2005
[5] H Minn V K Bhargava and K B Letaief ldquoOn Timing Offset Estimation for OFDM Systemsrdquo IEEE Transaction on communications Letters vol 4 no 7 Jul 2000
[6] J-J van de Beek M Sandell and P O Borjesson ldquoML Estimation of Time and Frequency Offset in OFDM Systemsrdquo IEEE Transactions on Signal Processing vol 45 7 1800 1805 J l 199745 no 7 pp 1800-1805 Jul 1997
[7] J Lee H Lou and D Toumpakaris ldquoMaximum Likelihood Estimation of Time and Frequency Offset for OFDM Systemsrdquo Electronics Letters vol 40 no 22 pp 1428-1429 Oct 20041429 Oct 2004
86 Synchronization2010722
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
Channel EstimationChannel EstimationWireless Information Transmission System LabWireless Information Transmission System LabInstitute of Communications EngineeringInstitute of Communications Engineering
Channel EstimationChannel Estimation
g gg gNational Sun National Sun YatYat--sensen UniversityUniversity
2010072220100722王森弘
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
882010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
Small scale fadingSmall scale fadinggg
loz Multi-path channelp
892010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
Channel impulse responseChannel impulse responsep pp p
In the absence of noise the received signal can be expressed asloz g p
( ) ( ) ( ) ( ) ( )y t x t h t x h t dτ τ τinfin
= otimes = minusintwhere
( ) is the channel impulse resonseh t
minusinfin
( ) is the channel impulse resonse ( ) is the transmitted signal
( ) i
h tx ty t s the received signal ( ) iy t
1
s the received signalAfter sampling the discrete received signal is given by
L 1
0
[ ] [ ] [ ]L
k
y n x k h n kminus
=
= minussum
902010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The wireless stationary channel impulse response is given by where L is the total number of resolvable paths
( ) ( ) ( )[ 0 1 1 ]Th h h L= minush
( )h lloz We assume that each tap of the channel impulse responses are independently distributed complex Gaussian
d i bl ith d i
( )h l0 1l Lle le minus
2random variables with zero-mean and variance ( )2h lσ
0 5
1
gnitu
de
1 2 3 4 5 6 7 80
05
the
mag
91
the lth tap of the channel
2010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
The multiThe multi--path channel effectpath channel effectpp
loz The multi-path channel effectp
922010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
The transmitted OFDM signal The transmitted OFDM signal
loz After the inverse discrete Fourier transform (IDFT) operation the
gg
ith transmitted OFDM symbol in time domain can be expressed by
H
Hi i=x F X
where and are an vector and an matrix standing for modulated symbols and an IDFT matrix respectively
1N timesiX HF N Ntimes
932010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix G is constructed as follows
( )0 0 0h⎡ ⎤⎢ ⎥( )( ) ( )
01 0
hh L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎢ ⎥G( ) ( )
( ) ( )0 1 00
h L h= ⎢ ⎥
minus⎢ ⎥⎢ ⎥⎢ ⎥
G
( ) ( )0
0 0 1 0h L h⎢ ⎥
minus⎢ ⎥⎣ ⎦
942010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz The matrix is constructed as follows tailG
( ) ( )0 0 1 1h L h⎡ ⎤minus⎢ ⎥
hellip
( )il
01h L
⎢ ⎥⎢ ⎥⎢ ⎥minus
= ⎢ ⎥G tail 0⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
G
0 0⎢ ⎥⎢ ⎥⎣ ⎦
952010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
The matrix form of channelThe matrix form of channel
loz Furthermore a circular convolution matrix can be circularGobtained
circular
circ lar tail=G G + G
( ) ( ) ( )( ) ( )
circular tail
0 0 0 1 11 0 0
h h L hh h
⎡ ⎤minus⎢ ⎥⎢ ⎥
G G G
( ) ( )( )
( ) ( )
1 0 01
1 0 0
h hh L
h L h
⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥( ) ( )
( ) ( )1 0 0
0 1 0h L h
h L h= minus⎢ ⎥⎢ ⎥minus⎢ ⎥⎢ ⎥
( ) ( )0
0 0 1 0h L h
⎢ ⎥⎢ ⎥⎢ ⎥minus⎣ ⎦
962010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
The received OFDM signalThe received OFDM signalgg
loz Assuming the synchronization is perfect the ith received OFDM symbol can be expressed as
tail 1i i i i= + +r Gx G x wtail 1
circular tail tail 1 i i i i
i i i i
minus
minus= minus + +
= otimes + +
G G wG x G x G x wx h z w
where denotes the circular convolution denotes an white
i N i i= otimes + +x h z w
Notimes iwGaussian noise vector in the time domain with zero mean and variance and stands for interference including ISI and ICI in the ith OFDM symbol caused by multi path channel
2wσ iz
in the ith OFDM symbol caused by multi-path channel
tail tail 1i i iminus= minus +z G x G x
972010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbolyy
loz Assuming the synchronization is perfect and CP is adopted the received ith OFDM symbol can be expressed as
+ +r Gx G x wtail
circular i i i i
i i
= + += +
r Gx G x wG x w
where denotes the circular convolution We note that there is i N i= otimes +x h w
Notimesno ICI and ISI in each OFDM symbol
982010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Assuming the synchronization is perfect and CP is adopted the
yy
received ith OFDM symbol in the frequency domain can be expressed as
=R Fr
( )i i
i N i
=
= otimes +
R FrF x h w
where Wi is the AWGN in the frequency domain and H is a di l i d i h h l i f
N Ntimesi i= +HX W
diagonal matrix denoting the channel response in frequency domainThe above result can be verified byloz The above result can be verified byloz DSP theorem
Li l bloz Linear algebra
992010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
The received CPThe received CP--OFDM symbolOFDM symbol
loz Any circular matrix can be diagonalized by DFT matrix
yy
( )circular
0 0 0
H
H
FG F H=
⎡ ⎤( )( )
0 0 00 1
0
HH
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥
( )0
0 0 1H N
⎢ ⎥⎢ ⎥
minus⎢ ⎥⎣ ⎦
loz The received signal can be expressed as
( ) ( )i i l i li i i i i= = + = +R Fr F G x w F G FX w( ) ( )i circular circular
circular
i i i i i
i i i i
+ +
= + = +
R Fr F G x w F G FX wFG FX Fw HX W
1002010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
The frequency response of the channelThe frequency response of the channelq y pq y p
0
1
art
-2
-1real
pa
0 10 20 30 40 50 60 70-2
sub-carrier index
2
0
1
imag
par
t
0 10 20 30 40 50 60 70-1
sub-carrier index
1012010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
The received OFDM signal in frequency domainThe received OFDM signal in frequency domaing q yg q y
1
2
art
-2
-1
0re
al p
a
0 10 20 30 40 50 60 70-2
sub-carrier index
2
-1
0
1
imag
par
t
0 10 20 30 40 50 60 70-2
sub-carrier index
1022010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
ReferenceReference
[1] Z Wang and G B Giannakis ldquoWireless multicarrier communications where Fourier meets Shannonrdquo IEEE Signal Process Magazine pp 29ndash48 May 2000
1032010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz System modelyloz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1042010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
System modelSystem modelyy
loz Generally the pilot symbols are multiplexed into an OFDM symbol in frequency domain
| 0 1 1P k qT q Qτ⎧ isin + = | 01 1 otherwise
kk
k
P k qT q QX
Sτ⎧ isin + = minus
= ⎨⎩
loz In addition the power allocation of data and pilot symbols are given by
β⎧
( )
2
2
| 01 1
1
k
k
P k qT q QQ
X
βρ τ
β
⎧ = isin + = minus⎪⎪= ⎨ ( )2 1
otherwisek
kSN Qβ ρ⎨ minus⎪ =⎪ minus⎩
1052010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
System modelSystem modelyy
loz After the inverse discrete Fourier transform (IDFT) the ith( )transmitted OFDM symbol in discrete-time domain excluding the cyclic prefix could be expressed as
where denotes the Hermitian transpose and F stands for
Hi i=x F X
( )Hi p the normalized discrete Fourier transform (DFT) matrix
( )N Ntimes
1062010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
System modelSystem modelyy
loz If system is perfectly synchronized and the CP is added and removed appropriately there is no ISI and inter-carrier interference (ICI) As a result the ith received OFDM symbol ft DFT b dafter DFT can be expressed as
i i i= +R ΛX W
where is a diagonal channel matrix with the kth element standing for the channel frequency response of the kth sub carrier
i i i
Λ N Ntimesstanding for the channel frequency response of the kth sub-carrier and W is a complex white Gaussian noise vector with covariance matrix 2
W W Nσ=C IW W NσC I
1072010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
The LS channel estimator The LS channel estimator
loz Define where denotes the received pilot signaliRM
iequivsumR R
loz The channel estimator based on the LS method is given by
1i
i=sum
11
LS1
M M
minusminus= = +
Γ RH H Γ W
where denotes a diagonal matrix whose diagonal
M M
Γ Q Qtimes g gelements are given by
01 1q q kP k qT q QτΓ = isin + = minus
1082010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz The mean square error (MSE) of the LS channel estimator can be expressed as
22 W Qσ2LS
W QM
σρ
= sdot
1092010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
The LS channel estimatorThe LS channel estimator
loz It is worthy of note that the performance of the LS channel i i d i d l b h l f il bestimator is determined not only by the total power of pilot sub-
carriers but also by the number of pilot sub-carriers
loz As a result if the channel order is known at the transmitter and the interpolation doesnrsquot causes extra performance loss the p pminimum MSE of the LS channel estimation can be achieved by superimposing Q = L equal-power and equal-space pilots in frequency domainfrequency domain
1102010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
loz The modified LS (MLS) channel estimator is given by
MLS LSH
L= sdotH FΔ F H
where is a diagonal matrix The entries of are LΔ tδ LΔ
1 01 1t L= minus⎧1 01 10 1t
t Lt L N
δ⎧
= ⎨ = minus⎩
loz The MLS channel estimator can be considered as a low-pass filter which is also termed as DFT-based scheme
1112010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
The modified LS (MLS) channel estimator The modified LS (MLS) channel estimator ( )( )
012
012
012
012
FHL
L+1FL-1 L-1
LSH MLSH
N-1 N-1 N-1
1122010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
The MLS channel estimatorThe MLS channel estimator
loz The MSE of the MLS channel estimator can be easily yobtained by definition
( )( )2 1 t EH⎧ ⎫⎡ ⎤
⎨ ⎬H H H H( )( )2MLS MLSMLS tr E
1NL
σ ⎧ ⎫⎡ ⎤= minus minus⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎛ ⎞
H H H H
1=SINR
LN
⎛ ⎞sdot⎜ ⎟⎝ ⎠
1132010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
ReferencesReferences
[1] M Morelli and U Mengali ldquoA comparison of pilot-aided channel estimation methods for OFDM systemsrdquo IEEE Trans Signal Process vol 49 no 12 pp 3065-3073 Dec 2001
[2] S Ohno and G B Giannakis ldquoOptimal training and redundant precoding for block transmissions with application to wireless OFDM rdquo IEEE Trans Commun vol 50 pp 2113 2123 DecOFDM IEEE Trans Commun vol 50 pp 2113-2123 Dec 2002
[3] S Ohno and G B Giannakis ldquoCapacity maximizing MMSE-[3] S Ohno and G B Giannakis Capacity maximizing MMSEoptimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channelsrdquo IEEE Trans Inform Theory vol 50 pp 2138-2145 Sep 2004
1142010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1152010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
IntroductionIntroduction
loz The channel estimation can be performed by either p yinserting pilot tones into all of the subcarriers of OFDM symbols with a specific period (block type) or inserting y p p ( yp ) gpilot tones into each OFDM symbol (comb type)
1162010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
System architectureSystem architectureyy
1172010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
System architectureSystem architectureyy
1 Input to Time Domain
( ) ( ) 1210 minus=
=Nn
kXIDFTnx
( ) ( )( )⎩
⎨⎧
minus=minus+minusminus=+
=110
11NnnxNNnnNx
nx ggf ( ) ( ) ( )nwnhnxy ff +otimes=
32 Guard Interval Channel
( )⎩ 0 Nnnx
( ) ( ) 110 Nnnyny( ) ( ) = nyDFTkY
54 Guard Removal Output to Frequency Domain
( ) ( ) 110 minus== Nnnyny f 1210 minus= Nk
76 Output Channel EstimationAWGNChannel Estimated Ch nn l
( ) ( ) ( ) ( )01 1
Y k X k H k W kk N
= +
= minus( ) ( )
( ) 110 minus== NkkH
kYkXe
e
Channel
1182010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
Pilot arrangementPilot arrangementgg
loz Comb Typeyploz Some sub-carriers are
reserved for pilots for each symbol
loz Block Typeloz All sub-carriers reserved
for pilots with a specific period
1192010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
DecisionDecision--feedback channel estimationfeedback channel estimation
loz When the channel is slow fading the channel estimation i id th bl k b d t d i th d i i f db kinside the block can be updated using the decision feedback equalizer at each sub-carrier
Decision Feedback EqualizerDecision Feedback Equalizer
110)(
)()( minus== NkkH
kYkX e )(kHe
( ) signal demapper signal mapper ( )e pureX k X krarr rarr rarr
( )( ) 01 1( )e
Y kH k k NX k
= = minus
loz For fast fading the comb-type estimation performs much better
( )pureX k
better
1202010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
Linear interpolationLinear interpolation
loz Linear Interpolation
pp
p
( ) ( )e eH k H mL l= +
( ) ( )( ) ( )1p p plH m H m H mL
= + minus +
0 l Lle lt
1212010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
Second order interpolationSecond order interpolationpp
loz Second Order Interpolationp( ) ( )
( 1) ( ) ( 1)e eH k H mL l
c H m c H m c H m= minus
+ + +1 0 1( 1) ( ) ( 1)p p pc H m c H m c H mminus= minus + + +
( 1)α αwhere 1( 1)
2( 1)( 1)
c
c
α α
α α
minus=
=minus minus +0
1
( 1)( 1)( 1)
2
c
c
α α
α αminus
=minus minus +
+=
2lN
α=
122
N
2010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
BlockBlock--type pilot insertiontype pilot insertionyp pyp p
Block type
1232010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
CombComb--type pilot insertiontype pilot insertionyp pyp p
d0 DC d24d29 d30 d42 d43 d47d18 d23P-7 P7 P21d17d5P-21d4
-26 -21 -7 7 21 260
1242010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
OutlineOutline
loz Effect of multi-path channelloz Frequency domain channel estimation
loz LS channel estimationloz MLS channel estimationloz MMSE channel estimation
loz Pilot arrangementloz Block typeloz Comb type
loz Conclusions
1252010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
ConclusionsConclusions
loz OFDM systemloz Block type
loz Direct or Decision Feedbackloz Comb type
loz LS or LMS estimation at pilot frequenciesp qloz Interpolation techniques
loz Linearloz Second Orderloz DFT-based
1262010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation
ReferencesReferences
[1] Steven M Kay ldquoFundamentals of Statistical Signal Processing E Th rdquo P i H ll 1993Estimation Theoryrdquo Prentice Hall 1993
[2] Erwin Kreyszig ldquoAdvanced Engineering MathematicsrdquoJohnwiley amp Sons 1983wiley amp Sons 1983
[3] Yuping Zhao Aiping Huang ldquoA novel channel estimation method for OFDM mobile communication systems based on pilot y psignals and transform-domain processing rdquo IEEE VTC Vol 3 May 1997
[4] Si C l i M t f E A j P i d Ah d B h i[4] Sinem Coleri Mustafa Ergen Anuj Puri and Ahmad Bahai ldquoChannel Estimation Techniques Based on Pilot Arrangement in OFDM Systemsrdquo IEEE Transactions on Broadcasting Vol 48 y g No 3 September 2002
1272010722 Channel Estimation