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Radioelektronika 2005 1
Spread Spectrum Signals in Modern Communications
Jan ŠimšaInstitute of Radio Engineering and Electronics AS
Radioelektronika 2005 2
What signal has spread spread spectrumspectrum?
Any digitally modulated signalAny digitally modulated signal
whose ratio of bandwidth to itswhose ratio of bandwidth to its
data (modulation) symbol ratedata (modulation) symbol rate
is substantially greater than is substantially greater than
one.one.
Spreading of spectrum Spreading of spectrum – – generation of signal using shaping or modulationgeneration of signal using shaping or modulation
Radioelektronika 2005 3
Contents• Motivation Classification of signals and methods of
spreading spectrum Features of SS signals Optimum reception (detection) Synchronization Code division multiplex (multiple access) Conclusions
Radioelektronika 2005 4
Motivation Why is spreading utilized?
Transmission of an additional information? Decrease of error rate in AWGN channel? ?????????? Decrease of error rate in any more disturbing channel? Preserving of unauthorized reception? Advantageous multiple utilization of a channel?
What is the price we pay for these advantageous features? Complexity of a system realization
No (user)
No
Yes
Yes
Yes
Radioelektronika 2005 5
Heavily distorting Heavily distorting communication channelscommunication channels
Time variant channelsMultipath channelsFrequency selective channelsChannels with interferences
Radioelektronika 2005 6
Frequency diversityFrequency selective / nonselective
channelsChannel coherence bandwidth is given by
parameters of the channelNarrowband signals can fade completelyFrequency nonselective selective
f f
H(f), S(f)H(f), S(f)
Radioelektronika 2005 7
Frequency diversityDistortion by frequency selective
channelOne deep fade model – rectangular fade
f
H(j2πf)
1
0 0
f +B/2f -B/2 0
1 1 0 0( ) 1 ( ) ( ) 1 ( , )
0
H j H j where H j for B B
elswhere
Radioelektronika 2005 8
Frequency diversityImpulse response of the rectangular fade
1 1 0
1 sin( ) ( ) 2 cos
2j t Bt
h t H j e d B tBt
Response to one chip of the rectangular shape
1 0 0 0
0 0 0 0
20 0 0
( 1)
0 0
sin( ) 2 cos ( ) sin[ ( ) ]
sin2 sin[ ] cos ( ) cos
sin2 sin[ ] cos
sinsin[ ]
c
c
c
c
i i T c
i T c
t iT
i
t i T
i
By t c B p t iT t d
B
Bc B t p t iT d
B
Bc B t d
B
ct
( )
( 1)
c
c
B t iT
Bt i T
d
Radioelektronika 2005 9
Frequency diversityResponse y1i(t) to positive chip
for B.Tc = 0.01 (solid line )
B.Tc = 0.05 (dotted line)
Radioelektronika 2005 10
Classification of signals and methods of spreading spectrum
Any spread spectrum signal belongs to one of the three categories:
signals without a carriersignals with a single carrier signals with multiple carriers
Radioelektronika 2005 11
Signals without carrierare created by a sequence of very
narrow pulses (< 1ns) of a proper shape (Gaussian, wavelets)
their spectrum bandwidth is by some orders of magnitude wider than the modulation rate. Such signals are often referred to as Ultra Wide-Band (UWB) signals (and related UWB systems). Their bandwidth B≥500 MHz, B/f0 ≥ 0.2
it can reach bandwidth > 5 GHz
Radioelektronika 2005 12
Single carrier signals
Harmonic carrier spread spectrum (SS) signals three basic subgroups
Direct Sequence (DS) – spreading by BPSK, QPSK
keying
Frequency Hopping (FH)
Time Hopping (TH)
- slow (SFH)- fast (FFH)
Radioelektronika 2005 13
Direct Sequence (DS) Spreading
Transmitter Receiver
BP filter
DEM ReplicaSpreading modulation
data
carrier
channel
timing
Spectra - PSD
f0 f0f0 f0
ffff
B ~ 1/T
Radioelektronika 2005 14
Direct Sequence (DS) Spreading
Transmitter Receiver
BP filter
DEM ReplicaSpreading modulation
data
carrier
channel
timing
Spectra - PSD
f0 f0 f0f0 f0
fffff
B ~ 1/T B ~ 1/T = L/Tc c
Radioelektronika 2005 15
Direct Sequence (DS) Spreading
Transmitter Receiver
BP filter
DEM ReplicaSpreading modulation
data
carrier
channel
timing
Spectra - PSD
f0 f0 f0f0 f0
fffff
B ~ 1/T B ~ 1/T = L/Tc c
Spreading (system) gain L = T / T = B / B = R / Rc cc
Radioelektronika 2005 16
Direct Sequence Spreading
Replica of spreading signal
I ntegration interval
Product
Output value of integrator
1
-1
1
-1
1
-1
00
t
t
t
t
Spreading modulation of received signal
Symbol interval = spreading signal period
Radioelektronika 2005 17
Direct Sequence Spreading
Spreading modulation of received signal
Replica of spreading signal
I ntegration interval
Product
Output value of integrator
1
-1
1
-1
1
-1
00
t
t
t
t
Radioelektronika 2005 18
Direct Sequence Spreading
Spreading modulation of received signal
Replica of spreading signal
I ntegration interval
Product
Output value of integrator
1
-1
1
-1
1
-1
0 0
t
t
t
t
1
-1/L
Symbol interval = spreading signal period
Radioelektronika 2005 19
Direct Sequence SpreadingBPSK data modulation b(t) and BPSK
spreading modulation c(t)
0 0( ) ( ) ( ) cos( )s t A c t b t t
Spreading modulation c(t)( ) ( ) ( ) 1( ) 1( )
ci T ci
c t c p t iT where p t t t Data modulation b(t)
( ) ( )k Tk
b t b p t kT where T- symbol interval, Tc- chip intervalThe period of spreading modulation is T=L.Tc
Radioelektronika 2005 20
Optimization of linear receiver
Matched filter (MF) – correlatorIts response
MF impulse response is defined by the signal to which MF is matched - MF[s(t)]
MF response at t=t0=T to finite signal s(t)
whose interval of nonzero values is (0,T),is
0 0( ) ( ) ( ) ( )h t s t t h t s
2
0 0
( ) ( ) ( ) ( )T T
y T s t dt n t s t dt
( ) [ ( ) ( )] ( )y t s n h t d
Radioelektronika 2005 21
Optimization of linear receiver
MF
correlators(t) + n(t)
s(t)
0
T
y(T)
h(t)s(t) + n(t) y(t)
Radioelektronika 2005 22
InterferencesResponse of MF[s(t)] to interfering signal s1
Correlation coef.Orthogonal signals do not cause any
interferenceCorrelation in time domain – frequency
domainParseval’s formula
1 1
0 0
( ) ( ) ( ) ( ) ( )T T
y T s t s t dt n t s t dt
1 2 1 2
1( ) ( ) ( ) ( )
2s t s t dt S j S j d
( ) { ( )} , 1,2i iS j s t i F{
Orthogonal signals have orthogonal spectra
Radioelektronika 2005 23
Code synchronization
Code synchronization is an alignment of the spreading modulation of received signal and the replica at the multiplier producing despreaded signal
It consists of two steps Acquisition – coarse alignment of the
modulation and the replica Tracking – accurate alignment and time-
variations tracking
Radioelektronika 2005 24
DS Code acquisitionwith the aid of a system of correlators - Matched filters Uncertainty region – the interval of prospective alignments. The
position is nonsensitive to shifts by an integer multiple of the spreading signal period (modulo L.Tc )
The goal (penalty) function of code acquisition process optimization
• Mean acquisition time – minimization • Probability of acquisition within given time interval – max
Code acquisition process is defined
• by acquisition detector rule • by search strategy - a sequence of points defining the replica
positions within an uncertainty region Classification according to changes of replica position
Stepping correlator Sliding correlator
Radioelektronika 2005 25
Classification of DS code acquisition detectors
as to the number of channels Single channel detector – for serial search on
uncertainty region Multiple channel detector – for parallel / serio-
parallel search
as to the realization of each channel of the detector
Passive correlator - multiplier/integrator Active correlator - Matched Filter (DSP, SAW)
as to the rule of the final decision making Single dwell detector Multiple dwell detector
Radioelektronika 2005 26
Spreading signal - optimization
Sharp and narrow main lobe of autocorrelationDC component-freeLow- level side lobes of autocorrelationLong linear spanLow croscorrelation between signal and interferencesGenerating constant- envelope signal
Noise-like signals – pseudonoise (PN) signals
M-sequences, Gold codes, Kasami codes, Walsh Hadamard sequences, bent sequences,
Radioelektronika 2005 27
Slow Frequency Hopping
cT T
0 Tc Tc Tc2 3 Tc4 Tc6 Tc8 Tc Tc10 13f1
f
f
f
f
f
f
3
5
7
9
11
13
t
Carrier frequency
Radioelektronika 2005 28
Slow Frequency Hopping
f
Spectrum of subchannels
f f f f1 2 3 7
1i if f B
Radioelektronika 2005 29
Fast Frequency Hopping
0 Tc Tc Tc2 3 Tc4 Tc6 Tc8 Tc Tc10 13f1
f
f
f
f
f
f
3
5
7
9
11
13
t
Carrier frequency
T
cT T
Radioelektronika 2005 30
Time hopping
Packet transmissionFixed length of packetsPseudorandom position of packets
within framePosition control by code sequence
TFTFT
FTF0 2 3 4 t
Radioelektronika 2005 31
Two-path channelSpreading modulation of received signal (fi rst path)
Replica of spreading signal
I ntegration interval
1
-1
1
-1
-0,5
t
t
Symbol interval = spreading signal period
0,5
t
Spreading modulation of the second path signal (Delayed in the channel)
Spreading modulation of the fi rst path signal delayed in the receiver
t
Radioelektronika 2005 32
Multipath channel – Rake receiver
Replica Gen.
Delay T c Delay T c Delay T cDelay T c
Combiner - Weighted summation
• •
• •••
• •
I nput signal
OUT
•• • •
• • ••
Radioelektronika 2005 33
InterferencesOrthogonal signals do not cause interferences
(orthogonality is not generally invariant to mutual shift of signals)
Nonorthogonal (correlated) signals cause interference
Interference is proportional to amplitude of interfering signal
This amplitude can be greater than the amplitude
of useful (target) signal = near-far effect
Efekt nestejných vzdáleností (efekt nestejných amplitud)
Radioelektronika 2005 34
Multicarrier CDMATransmitter of the k-th user
Data inOutput
MC CDMA∑
C k1
C kN
cos 1t
cos Nt
•••
••••
••T
TN
TN
TN = T, = fi+1 – fi =(i+1-i)/2 = T-1
Components are orthogonal on the interval T
Radioelektronika 2005 35
Multicarrier DS CDMA
TN = T/N , = fi+1 – fi =(i+1-i)/2 =Tc-1
Components are orthogonal on the interval Tc
Transmitter of the k-th user
Data inOutput
MC DS CDMA∑
Ck(t)
Ck(t)
cos 1t
cos Nt
•••
•••
T
TN
TN
S/P
Radioelektronika 2005 36
Multitone CDMATransmitter of the k-th user
Data inOutput
MT CDMA∑
cos 1t
cos Nt
•••
T
TN
TN
S/P
ck(t)
TN = N.T, = fi+1 – fi =(i+1-i)/2 =(N• T)-1
Components are orthogonal on the interval TN
Radioelektronika 2005 37
Comparison in frequency domain
Spectra
OFDM
MC CDMA
MC DS CDMA
MT CDMA
f
f
f
f
1/TN1/NT
1/T
=1/Tc=L/TN=L/NT
1/TN
Radioelektronika 2005 38
Shared communication Shared communication channelchannel Multiplex (Multiple Access)
• Frequency division• Time division• Code division
Timing of symbols – identical = synchronous CDMA - nonidentical = symbol asynchronous CDMA
> identical chip timing = chip synchronous CDMA > nonidentical chip timing = chip asynchronous
Radioelektronika 2005 39
Code Division Multiple Access
t
f
B
T
FDM
TDM
CDM
Signal space - dimensionality 2BT = number of mutually orthogonal signals
Radioelektronika 2005 40
Synchronous and Asynchronous CDMA
Signature
Synchronous CDMA (chip synchronous – symbol synchronous)
Asynchronous CDMA
t
t
t
s1(t)
Radioelektronika 2005 41
CDMA signalSignal at detector input
K
i i i i ii 1
y(t) A b (t ) s (t ) n(t)
where bi(t) is data signal of the i-th user, is its delay and si(t) is its spreading modulation, which in CDMA is labeled to as a signaturesignature.Usually, signature has unit energy ( a system of orthogonal signatures is in the same time orthonormal).As ci
2(t)=1, the unit energy signature si(t) is preserved if
i
1( ) ( )i is t c t
T
Radioelektronika 2005 42
Synchronous CDMACDMA is synchronous, iff
1,2, ...i konst for i K
Then K
i i ii 1
y(t) A b s (t) n(t) t (0,T)
Without loss of generality let it be 0i
This signal causes response ym of MF matched to the signature of m-th userT K
m m i i im mi 10
K
m m i i im mi 1i m
y y(t) s (t)dt A b n
A b A b n
MUI
Radioelektronika 2005 43
The second component of the right side of the equation represents Multiuser Interference (MUI / MAI)This component is zero if the signaturesat the detector input are ortogonal.If system designer is not able ensure validity of this condition (it is the usual case as parameters of channel are unknown and time variable)Detector minimizing MUI is not MF any more; it is more complex, usually nonlinear.
To keep the complexity of receiver acceptably low, suboptimum linear detectors are used.
Multiuser interference (MUI)Multiuser interference (MUI)
Radioelektronika 2005 44
Multiuser detectionVector notation
Matrix of amplitudes and correlation matrix
1 2 K
ij
= diag{A ,A ,...,A }
= { }A
R
T1 2 K
T1 2 K
T1 2 K
= [y , y ,..., y ]
= [b ,b ,..., b ]
= [n ,n ,..., n ]
y
b
n
b̂
T K2
i i ii 10
ˆ = { [y(t) - A b s (t)] dt}arg minb
b
Detector estimates vector of dataOptimum detector is described by the equation
Vector y can now be expressed as y ARb n
Radioelektronika 2005 45
It can be expressed using likelihood
where
and
Averaging over random variable bk is not performed . For a priori equiprobable symbols
the above average is
Optimum receiver is nonlinear.
kk (k)b 1,1
b̂ arg max E [P( )]
y b
K1 2P( ) P( b ,b ,...,b )y b y
K1 2 k 1 k 1K1 2(k) b ,b ,...,b ,b ,...,b
E [P( )] E [P( b ,b ,...b )]
y b y
k k1P(b 1) P(b 1) , k 1, 2, ... , K2
K1 k 1 k 1K1 2(k) K 1
b 1,1 b 1,1 b 1,1 b 1,1
1E P( ) ... ... P( b ,b ,...,b )2
yb y
Radioelektronika 2005 46
Using vector notation it isˆ = {2 T T
bb b Ay - b Hb }arg min
where=H ARA
Approximation of the optimum detector - linear detectors
m mb̂ Mysign
Linear detector consists of the bank of filters matched to the signatures of individual users and weighted sum of their outputs is created. Weights in summation are chosen in a way minimizing MUI or noise plus MUI, respectively.
Linear detectors
Radioelektronika 2005 47
Block diagram of linear multiuser detector
MF 1
y(t)
1b̂
2b̂
3b̂
Kb̂
Sync 1
Sync 2
Sync 3
Sync K
M
MF 2
MF 3
MF K
Radioelektronika 2005 48
Conventional detector
M I
This detector is optimum in the case of uncorrelated signatures, i.e. if
k kb̂ sign(y )
If signatures are correlated, MAI is nonzero.
Conventional detector – bank of MFs
Radioelektronika 2005 49
Decorrelating detector Matrix M is the inverse to correlation matrix R
1M R
Then1 1 1 My R RAb R n Ab R n
and k-th component of the detector output isK
k k ik ik i 1A b n
My
MUI component is totally MUI component is totally compensatedcompensated
but noise component has increased.but noise component has increased.
Radioelektronika 2005 50
MMSE detectorIt minimizes men square error of the vector b estimate. We search for the matrix M minimizing this error
0
2
0min E
MM b M y
K*KRarg
where the norm of vector is defined asK2 2 T
ii 1
x trace{ }
x xx
After some manipulation
0
0 0
T0 0
0M
2
0M Mtrace ( )( )
min trace [cov( )]
min E min E
b M y b M y
b M y
b M y
K*K
K*K K*K
R
R R
Radioelektronika 2005 51
We expand T0 0
( )( )E
b M y b M y
T T T T T T0 0 0 0 0 0
( )( ) E[ ] E[ ] E[ ] E[ ]E
b M y b M y bb by M M yb M yy M
After substitution y it can be rearranged to the shape 2 1 2 2 T
0 0 0cov[ ] [ ] ( )[ ]( ) b M y I ARA M M RA R R M M
where 1 2 2 1[ ] M A R A
Minimum MSE is reached forM M
Decision rule of MMSE detector is
2 2 1 2 2 1k k k
k
1b̂ sign {( ) } sign {( ) }A
R A y R A y
MMSEMMSE detectordetector
Radioelektronika 2005 52
Asymptotic cases
AAdaptive methods2 20 R A R
2 2 2 2 R A A MF detectorMF detector
Decorrelating detectorDecorrelating detector
- Trained methods- Blind methods
To avoid periodical repetition of measurements of values of matrices A and R in the case of time variant channeladaptive MMSE methods can be used.
Radioelektronika 2005 53
Spread Spectrum Signals AdvantagesAdvantages
» Interference resistant»Multipath resistant»Selective fades resistant»Sharing communication channel by multiple
users»Unauthorized reception resistant» Interference into other systems reduced
CostsCosts»Code synchronization»More complex system realization
Radioelektronika 2005 54
Spread Spectrum Signals