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CHANNEL ESTIMATION IN DVB-T AND OFDM SYSTEMS
RETHNAKARAN PULIKKOONATTU
ABSTRACT. This document outlines the theoretical framework behind pilot based channel estimation in an OFDM system. Funda-mental concepts of optimal frequency domain estimation of afinite length channel are derived and illustrated. Practical limitationsand implementation complexity of optimal algorithms oftenprompt to design sub-optimal estimation schemes. Some suchcircuitsare discussed. Specifics of DVB-T from a channel estimation perspective is considered while discussing the specifics.
This document is a highly condensed version of the internal technical report on channel estimation prepared by the sameauthors from ST (Genesis Microchip group). Since this document is intended to a wider readership, the IP and implementationdetails of algorithms and design aspects are not discussed.For internal ST employees, the detailed report discussing all aspects ofchannel estimation in IP are available for reference. The scope of this document hence is restricted to the discussion ofgeneralchannel estimation of OFDM systems with specific system under study as DVB-T.
Date: 01 March 2007.Key words and phrases. DVB.ST Microelectronics (Genesis Microchip) India, www.st.com .
1
2 RETHNAKARAN PULIKKOONATTU
CONTENTS
1. Introduction 31.1. Data detection 51.2. Estimating the channel taps (Training) 52. Channel estimation 62.1. OFDM channel estimation: Deterministic noiseless channel 82.2. Deterministic channel with AWGN 93. Fading channel scenario 104. Optimum Channel estimation for a wireless channel 115. Channel estimation implementation 176. Pilot pattern in DVB-T 177. Appendix B: Linear Prediction 198. Simulink implementation 21References 22
CHANNEL ESTIMATION IN DVB-T AND OFDM SYSTEMS 3
1. INTRODUCTION
A block diagram of OFDM system using cylcic prefix (or cyclic suffix) is shown in Figure1 .
-1z
-1z
-1z
h(0) h(1) h(2) h(L − 1)
Cyclic prefix
RemoveDFT
Detection
Data
Estimation
Channel
IDFTADD
Channel
v(i)
Cyclic prefix
H(0)
H(1)
.
.
.H(N − 1)
X(0)X(1)
.
.
.X(N − 1)
y(0)y(1)
.
.
.y(N − 1)
X(0)
X(1)
.
.
.X(N − 1)
Y (0)Y (1)
.
.
.Y (N − 1)
x(0)x(1)
.
.
.x(N − 1)
XN−G+1
.
.
.XN−1X(0)X(1)
.
.
.X(N − 1)
Figure 1. OFDM system model figure
To avoid inter symbol interference1 the cyclic prefix lengthG must be atleast equal to the FIR channel lengthL. Whilea largerG helps in improved detection (in noisy channels), the effective transmission rate and bandwidth are sacrificedwith it. Here for analysis, we nevertheless assumeG ≤ L. ForL > G, channel output vector (receiver input) due ton-thOFDM transmission is given by
0 . . . hL−1 . . . h1 h0 0 . . . 0 00 . . . 0 . . . h2 h1 h0 . . . 0 0...
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
...0 . . . 0 . . . hL−1 hL−2 . . . h1 h0 00 . . . 0 . . . 0 hL−1 hL−2 . . . h1 h0
XN−G+1(m)...
XN−2(m)XN−1(m)X0(m)X1(m)X2(m)X3(m)
...XN−G+1(m)
...XN−2(m)XN−1(m)
+ v(n)
1Cylclic prefix also helps us to realise a simple equalizer structure to reduce ISI damage as illustrated in section xxx. Among the other (added)benefits of cyclic prefix includes data aided frame/time synchronization schemes
4 RETHNAKARAN PULIKKOONATTU
After the removal of cyclic prefix (chop theG columns of matrixC(n) and firstG columns of receive input vector) weget the time domain signal (corruped by AWGN noise) vectory(n) equal to,
h0 . . . 0 . . . 0 hL−1 hL−2 . . . h2 h1
h1 h0 0 . . . 0 0 hL−1 . . . h3 h2
h2 h1 h0 . . . 0 0 0 . . . h4 h3
h3 h2 h1 . . . 0 0 0 . . . h5 h4
......
......
......
......
......
hL−2 hL−3 hL−4 . . . h0 0 0 . . . 0 hL−1
hL−1 hL−2 hL−3 . . . h1 h0 0 . . . 0 0...
......
......
......
......
...0 . . . . . . hL−1 hL−2 hL−3 hL−4 . . . h0 00 . . . 0 0 hL−1 hL−2 hL−3 . . . h1 h0
X0(m)X1(m)X2(m)X3(m)
...XN−G+1(m)
...XN−2(m)XN−1(m)
+
V0(m)V1(m)V2(m)V3(m)
...VN−G+1(m)
...VN−2(m)VN−1(m)
or simply,
y(n) = C(n)x(n) + vN (n)
= C(n)F∗
NX(n) + v(n)(1)
Since the time domain channel matrixC(n) is circulant, it can be decomposed using the well known Fourier diagonaliza-tion to,
(2) C(n) = F∗
N
H1(n) 0 0 0 0
0 H2(n) 0. . . 0
0 0 H3(n). . . 0
.... . .
. . .. . .
...
0 0. . . 0 HN−1(n)
︸ ︷︷ ︸
,Λ(n)
FN
Using Eq.2 on Eq.1, equivalent (and simple) frequency domain representationof the system can be arrived.
(3) Y(n) = Λ(n)X(n) + V(n)
Figure2 and Eq.4 shows the simplified model of OFDM .
X(0)
X(1)
.
.
.X(N − 1)
H(N − 1)
H(1)
H(2)
V (1)
V (2)
V (N − 1)
...
...
...
...
Y (0)
Y (1)
Y (2)
Y (N − 1)
H(0)
...
...
V (0)
Y (0)Y (1)
.
.
.Y (N − 1)
Detection
Data
Channel
Estimation
H(0)
H(1)
.
.
.H(N − 1)
X(0)X(1)
.
.
.X(N − 1)
X(0)
X(1)
X(2)
X(N − 1)
Figure 2. Frequency domain equivalent of OFDM system
CHANNEL ESTIMATION IN DVB-T AND OFDM SYSTEMS 5
1.1. Data detection. Ignoring the subscriptn we could simply write,
(4) Y(n) = Λ(n)X(n) + V(n)
Using results in Appendix (TBD) The LMMSE estimate vectorX of the data vectorX can be obtained.
X = RXXΛ∗ (RV V + ΛRXXΛ∗)−1
Y(5)
= Λ∗ (D + ΛΛ∗)−1
Y(6)
D is again a diagonal matrix defined as,
(7) D ,
1SNR0
0 0 . . . 0
0 1SNR1
0... 0
.... . .
. . ....
...0 0 . . . . . . 1
SNRN−1
Note that, Eq6 is derived from Eq.5 for the AWGN case where,
(8) RV V = Σ =
σ20 0 0 . . . 0
0 σ21 0
... 0...
. . .. . .
......
0 0 . . . . . . σ2N−1
Eq6 is the same as the following (for individual elements of an OFDM symbol),
(9) Xk = H∗
k
(1
SNRk
+ |Hk|2
)−1
Yk, k = 0, 1, 2, . . . , N − 1
1.2. Estimating the channel taps (Training). To recover the transmitted data, the channel taps (Λ) are needed. The mostcomon training method is to allocate some tones to known training data. Channel taps can be estimated using these pilots.
Let usassume that,M (where1 ≤ M ≤ N ) out of theN tones in the OFDM symbolY (n) correspond to a known(training) data (see Figure7 for such an arrangement of pilots, usually called Comb type pilots). Let us denote, theMpilot positions (tones) as{k1, k2, . . . , kM} and the training data containing on these tones as{S1, S2, . . . , SM}. Usingthese we can write the linear relationship,
(10)
S1(n) 0 0 . . . 0
0 S2(n) 0. . . 0
.... . .
. . .. . .
......
. . .. . .
. . . SM (n)
︸ ︷︷ ︸
Sp(n)
1∗
k1
1∗
k2
1∗
k3
...1∗
kM
︸ ︷︷ ︸
,Γ
H0(n)H1(n)H2(n)
...HL−1(n)
+
Vk1(n)Vk2(n)Vk3(n)
...VkM
(n)
=
Yk1(n)Yk2(n)Yk3(n)
...YkM
(n)
Where1ν is a column vector of sizeL×1with all positions except at row numberν are zeros. Theν row contain the value1 . For example12 represents,
(11) 12 =
010...0
1∗
3 =[0 0 1 0 . . . 0
]
(12) Sp(n)Γ(n)H(n) + V (n) = Yp(n)
LS solution to this problem would provide the estimate,
(13) H(n) =(Γ∗S∗
p(n)Sp(n)Γ(n))−1
Γ∗(n)S∗
p(n)Yp(n)
6 RETHNAKARAN PULIKKOONATTU
V (0)
H(N − 1)
H(1)
H(2)
V (1)
V (2)
V (N − 1)
...
...
...
...
S(0)
S(1)
S(2)
Y (0)
Y (1)
Y (2)
Y (N − 1)S(N − 1)
S(0)S(1)
.
.
.S(N − 1)
Y (0)Y (1)
.
.
.Y (N − 1)
H(0)
...
...
Figure 3. Frequency domain equivalent of OFDM system
From a statistical estimation point of view, the channel estimate is not optimum because, the statistical correlations(channel, input/output) are not considered here. Thus the least square algorithm for channel estimation provide only asub optimal esimate. However, this is close to the best one can do, when the statistcs of the channel are unknown at thereceiver.
Using this method, the channel taps2 are estimated on every symbol, assuming that some subcarrier positions (one everysymbol) are allocated for for pilots. In practice channel isnot often estimated on a symbol by symbol scale3, but insteadon every few symbols or so. Similarly, the number of pilot subcarriers on a given (pilot) symbol) decides the closeness ofthe estimate to the actual channel. The higher the numberM of pilot subcarriers, the better the estimate would be. Thepenalty again is wasting the bandwidth, whcih otherwise could have been used for data transmission.
2. CHANNEL ESTIMATION
Most of the practical OFDM based systems have limited numberof pilots in the time-frequency grid. It is from thesepilots, the channel estimate at all time (symbol index) and frequency( sub carrier index) positions have to be computed.Thus the problem in prestine form resembles to a two dimensional estimation problem. The complex channel gain4 variesacross sub carriers (frequency axis) as well as from symbol to symbol (time axis).
It is much easier to visualise multicarrier systems in frequency domain. The task of channel estimation involves thecomputation of the channel gainH(k, n) at certain subcarrierk (i.e., f = k∆f ) for OFDM symbol indexn (i.e., n-thOFDM symbol in time).
While there are non-pilot based schemes studied in the literature [Ratna], in this document, we considered only pilotbased channel estimation methods. Pilot based schemes usesknown5 (to both transmitter and receiver) data sent at knownpositions (fixed subcarriers and fixed symbols). At certain positions on the time frequency grid, the data sent is known.These known data is often called as pilot data and their positions as pilot positions.
Channel estimation techniques greatly works for the given pilot arrangement considered. There are different ways thesepilots cane be arranged on a 2D grid, based on the channel variations in time and frequency. Some of the commonly usedpilot arrangements are shown in Figure??, Figure?? and Figure??.
2Strictly speaking, the frequency domain channel is estimated here. The time domain tap values are obtained by performing a DFT on them. That ish(n) = DFT (H(n)))
3The channel may not vary (drastically) from one symbol to another symbol, due to which the channel need to be estimated only at intervals offew symbols. Performing channel estimation too often demands extensive computational complexity, besides consumingavailable bandwidth for pilots.Design of pilots and their positions are thus based on the channel charactersitcs and also the system complexity.
4Here channel gain refers to the channel attenuation. In general this value is a complex number, which means both amplitude and phase of the inputare distorted by the channel
5known both to transmitter as well as receiver
CHANNEL ESTIMATION IN DVB-T AND OFDM SYSTEMS 7
OFDM symbol index (time)S
ub
carr
ier
ind
ex(F
req
uen
cy)
Figure 4. Block type pilot arrangement for OFDM
OFDM symbol index (time)
Su
bca
rrie
rin
dex
(Fre
qu
ency
)
Figure 5. Rectangular arrangement of pilots in OFDM
.
.
.
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . . . . . . . . . . . . . . . . . . . .
4T T
0 1 2 3 4 5 6 7 8 9 10 11 n + 1n12 13 14 15 16 17 18 19
0
1
2
3
4
5
6
7
8
9
3
12
2045
2046
2047
OFDM symbol index
carr
ier
ind
ex(f
req
uen
cy)
10
11
12
13
14
15
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Figure 6. Diagonal Pilots: Used in DVB-T
8 RETHNAKARAN PULIKKOONATTU
Su
bca
rrie
rin
dex
(Fre
qu
ency
)OFDM symbol index (time)
Figure 7. Comb type pilot arrangement for OFDM
These pilots are usually called scattered pilots (in DVB-T parlance for example) to distinguish them from the continuouspilots ( also known as Comb type pilots).
At certain positions in time and frequency, the modulation symbolsX(k, n) will be replaced by known pilot symbolsS(k, l). At these positions, the channel can be measured. Figure?? shows a rectangular grid with pilot symbols at everythird frequency and every fourth time slot (OFDM symbol). The pilot density is thus1/12, that is,1/12 of the wholecapacity is used for channel estimation. This lowers not only the data rate, but also the available energyEb/N0 . Bothmust be taken into account in the evaluation of the spectral and the power efficiency of such a system.
It is useful, when the power on the pilot symbols are higher than that on data. broadly speaking, this helps in betterdetection and estimation since it can provide higher noise immunity. But they are not good on rectangular pilot arrange-ments like that in Figure?? because that would cause a higher average power for every fourth OFDM symbol, which isnot desirable for reasons of transmitter implementation (burden on power amplifiers and other analog circuits). A diagonallike Figure?? however can have this option availed, as indeed done in DVB-T.
In the build up to channel estimation in practical systems (say DVB-T) we will consider a series of idealistic scenariosto illustrate the concept of OFDM channel estimation.
Let us first consider a deterministic channel in noiseless scenario.
2.1. OFDM channel estimation: Deterministic noiseless channel. In this case the channel in time domain has an FIRstructure withL taps. Stacking up the DFT output (frequency domain representation of channel output) of all theM pilots,we get the vectorYpilot
(14) Ypilot =
Yp1
Yp2
Yp3
...
...YpM−1
YpM
=
Hp1Sp
Hp2Sp
Hp3Sp
...
...HpM−1Sp
HpMSp
Dividing the DFT output vectorYpilot by the known pilot data, we get the frequency domain channel coefficients{Hpi
} at all the pilot positions. Once the channel coefficients at theM pilot positions in the two dimensional grid areobtained, the corresponding values at other points (non-pilot positions on the 2D grid) are obtained using 2D-interpolation.If any prior know-hows on the nature of the channel response,that could be used in the interpolation algorithm. Whensuch informations are unavailable, then various interpolation techiques may be used to find the best fit. Simple holdbased interpolation, polynomial,spline interpolation and sinc interpolation algorithms are some of the commonly usedinterpolation schemes. There is a trade off between complexity and closer match (to ideal channel) have a say in chosinga particular interpolation method.
Let us investigate an example channel response to see how different methods fares with respect to channel estimation.Figure 8 shows the exact channel response in time frequency plane. The channel variation in time (for certain fixed
CHANNEL ESTIMATION IN DVB-T AND OFDM SYSTEMS 9
Figure 8. Ideal Channel response
frequency) is shown in Figure9 and channel variation in frequency (for some fixed time instants) is depicted in Figure10.Figure11 illustrates the comparative performance of various 2D interpolation schemes on the channel estimation.
2.2. Deterministic channel with AWGN. The 2D-interpolation schemes mentioned in the noiseless case may not yieldthe best estimate of the channel, in the presence of (the omniprescent) AWGN in the system (channel model). In suchcases, some optimality criteria is devised to compute the channel estimate. One commonly used optimality crieria is theminimum least square error criteria alias LS esimate. Here again, first a noisy (rough) estimate of the channel at pilotpositions are computed. This is used by the LS estimator to compute the channel gain at all positions in the grid.
As done for the noiseless case, channel estimates at theM pilot positions are estimated first. The DFT output corre-sponding to theseM pilot positions is stacked to form the vectorYpilot given by,
(15) Ypilot =
Yp1
Yp2
Yp3
...
...YpM−1
YpM
=
Hp1Sp + Np1
Hp2Sp + Np1
Hp3Sp + Np1
...
...HpM−1Sp + Np1
HpMSp + Np1
10 RETHNAKARAN PULIKKOONATTU
0
5
10
15
20
01
23
452
2.1
2.2
2.3
OFDM Symbol Index (time)Carrier Index(Frequency)
Cha
nnel
Gai
n
Figure 9. Channel response; Frequency view
On division bySp, the exact channel response at the pilot cannot be obtained,due to the presence of the additivenoise. However, a division is performed and the resultant value is chosen as the initial estimate (noisy nevertheess) atthepilot positions. A conventional 2D interpolation (as done in the noiseless case) may be performed to obtain the channelestimates at all points in the grid, but that is likely to be very crude or atleast sub optimal.A LS estimate result in a 2Dfilter.
3. FADING CHANNEL SCENARIO
Fading is reminiscent of typical wireless communication channel. Such a channel can again be modeled as an FIRfilter, with the tap coefficients non-deterministic (random). It is sometimes possible to model the probability distributionfunction (pdf) of these random coefficients. We will consider a simple fading model without additive noise [4] [5].
Consider the grid of Figure?? for an OFDM system with carrier spacing∆f = 1/T = 1kHz and symbol durationTs = 1250µs [4]. At every third frequency, the channel will be measured once in the time4Ts = 5ms, that is, the unknownsignal (the time-variant channel) is sampled at the sampling frequency of200Hz. For a noise-free channel, we canconclude from the sampling theorem that the signal can be recovered from the samples if the maximum Doppler frequencyvmax fulfills the conditionvmax < 100Hz More generally, for a pilot spacing of4Ts, the conditionvmaxTs < 1/8 mustbe fulfilled. In frequency direction, the sample spacing is3kHz. From the (frequency domain) sampling theorem, weconclude that the delay power spectrum must be inside an interval of the length of333µs. Since the guard interval alreadyhas the length250µs, this condition is automatically fulfilled if we can assume that all the echoes lie within the guardinterval. We can now start the interpolation (according to the sampling theorem) either in time or in frequency directionand then calculate the interpolated values for the other direction.
CHANNEL ESTIMATION IN DVB-T AND OFDM SYSTEMS 11
01
23
45
0
5
102
2.05
2.1
2.15
2.2
2.25
OFDM Symbol Index (time)Carrier Index(Frequency)
Cha
nnel
Gai
n
Figure 10. Channel response; Time view
Simpler interpolations are possible and may be used in practice for a very coherent channel, for example, linear inter-polation or piecewise constant approximation. However, for a really time-variant and frequency-selective channel, thesemethods are not adequate.
For a noisy channel, even the interpolation given by the sampling theorem is not the best choice because the noise isnot taken into account. The optimum linear estimator will bederived in the next subsection.
The density of the grid has to be matched to the incoherency ofthe channel, that is, to the time-frequency fluctuationsdescribed by the scattering function. To illustrate this bya
4. OPTIMUM CHANNEL ESTIMATION FOR A WIRELESS CHANNEL
Practical wireless channels are hostile. It is not possibleto unquley model a wireless channel that represent all practicalcommunication links. Wireless channels are time varying and can be frequency selective. Generally speaking, suchchannels are modeled by a random process with a scattering function of time and frequency.
Optimum estimate in the minimum mean square error (MMSE) sense often leads to nonlinear solution (except in somespecial cases). However a linear MMSE (suboptimal, yet the best MMSE under linearity constraint) can be wroked out. Itis found that such an estimate for a pilot based OFDM channel estimation leads to 2D Wiener filter. A block diagram ofsuch an estimator is shown in Figure12. The estimator for a general pilot arrangement is derived below.
12 RETHNAKARAN PULIKKOONATTU
−3 −2 −1 0 1 2 3
−20
2
0
10
20
30
40
50
60
OFDM Symbol(time)OFDM subcarrier (frequency)
Cha
nnel
Est
imat
es (
Am
plitu
de)
chan at pilot poscubic interplinear interppilothold interp
Figure 11. DVB-T Channel estimation using 2-D interpolation
1 0 0 0 0 . . . . . . 0 00 0 0 0 1 . . . . . . 0 00 0 0 0 0 . . . . . . 0 0...
.... . .
. . .. . . . . . . . .
......
0 0 0 0 0 . . . . . . 0 00 0 0 0 0 . . . 1 0 0...
.... . .
. . .. . . . . . . . .
......
......
. . .. . .
. . . . . . . . ....
...
0 0 0 0 0 . . .. . . 1 0
︸ ︷︷ ︸
Γ
H(0∆f , 0Ts)H(0∆f , Ts)H(0∆f , 2Ts)H(0∆f , 3Ts)
H(0, 4Ts)H(0∆f , 5Ts)
. . .. . ....
H((K − 1)∆f , (N − 3)Ts)H((K − 1)∆f , (N − 2)Ts)H((K − 1)∆f , (N − 1)Ts)
+
V1/S1
V2/S2
V3/S3
.... . ....
VM−1/SM−1
VM/SM
︸ ︷︷ ︸
Vp
=
Y1/S1
Y2/S2
Y3/S3
.... . ....
YM−1/SM−1
YM/SM
︸ ︷︷ ︸
Yp
CHANNEL ESTIMATION IN DVB-T AND OFDM SYSTEMS 13
RHHpilot
(
Rν + RHpilotHpilot
)−1
.
.
.
1
2
KpO
FD
Msu
bca
rrie
rin
dex
(k)
OFDM symbol index (time)
.
.
.
. . .
.
.
.
.
.
.
.
.
.
. . .
. . .
. . .
.
.
.
.
.
.
H
2D Wiener (MMSE) filter
Figure 12. 2D MMSE channel estimator
or simply,
(16) ΓH + Ψp = Hp
HereHp is the noisy channel estimate at the pilot positions, given by,
Hp =
H1
H2
H3
.... . ....
HM−1
HM
=
Y1/S1
Y2/S2
Y3/S3
.... . ....
YM−1/SM−1
YM/SM
(17)
14 RETHNAKARAN PULIKKOONATTU
TheM pilot positions in the channel grid{Hp1 , Hp
2 , . . . . . . HpM} are shown in red colour.
H(0∆f , 0Ts) → Hp1
H(0∆f , Ts)H(0∆f , 2Ts)H(0∆f , 3Ts)
H(0, 4Ts) → Hp2
H(0∆f , 5Ts). . .. . ....
H((K − 1)∆f , (N − 3)Ts)H((K − 1)∆f , (N − 2)Ts) → Hp
M
H((K − 1)∆f , (N − 1)Ts)
(18)
From Eq 16a linear MMSE estimate can be computed as,The correlation matrixRH is given byLet us denote theM pilot positions using a vector pair,
(kp1 , np
1) , (kp2 , np
2) , (kp3 , np
3) , . . . , . . . , (kpM , np
M ), with npj , k
pj ∈ Z and0 ≤ kp
j ≤ K − 1, 0 ≤ npj ≤ N − 1, If we were to
think of the pilot positions in a one dimensional vector form, the pilot positions would be atp1, p2, p3, . . . , . . . , pM , wherepj ∈ Z, 1 ≤ pj ≤ M . It is easy to establish the relationship
(19) pj = 1 +(kp
j − 1)K + kp
j
We will denote the channel estimates at the pilot positions as a single array (vector)Hpilot,
Hpilot =
H(0∆f , 0Ts) → Hp1
H(0∆f , 4Ts) → Hp2
H(0∆f , 8Ts) → Hp3
. . .. . ....
H(∆f , Ts)H(∆f , 5Ts)H(∆f , 9Ts)
. . .. . ....
H((K − 1)∆f , (N − 6)Ts)H((K − 1)∆f , (N − 2)Ts)
(20)
CHANNEL ESTIMATION IN DVB-T AND OFDM SYSTEMS 15
In general for arbitrary pilot positioning, we can write this vector (using the pilot position notationpi) as,
Hpilot =
H(kp1∆f , np
1Ts) → Hp1
H(kp2∆f , np
2Ts) → Hp2
H(kp3∆f , np
3Ts) → Hp3
. . .. . ....
H(kpj ∆f , np
jTs) → Hpj
. . .. . ....
H(kpM−1∆f , np
M−1Ts) → HpM−1
H(kpM∆f , np
MTs) → HpM
(21)
For the DVB-T case, the first few pilots (scattered pilots) inthe frequency-time grid are positioned at(0, 0), (0, 4), (0, 8), . . . , . . . , (3, 1), (3and so on. That is→ Hp1 = H(0∆f , 0Ts), → Hp2 = H(0∆f , 4Ts), → Hp3 = H(0∆f , 8Ts) etc.,
More generally,
(22) RHC∗ =[Colp1 (RH) Colp2 (RH) Colp3 (RH) . . . . . . ColpM
(RH)]
Where, Colpi(RH) is thepi-th column vector of the channel correlation matrixRH .
RThe matrixRHC∗ has dimensionL × M . Mathematically,RHC∗ ∈ CL×M .We find thatRHC∗ is nothing but the cross-correlation matrix formed by correlatingH with Hpilot. That is.,
(23) RHΓ∗ = E [HHpilot]
Similarly,
(24) ΓRHΓ∗ =
Rowp1 (RHΓ∗)Rowp2 (RHΓ∗)Rowp3 (RHΓ∗)
. . ....
. . .RowpM
(RHΓ∗)
In other words,
(25) ΓRHΓ∗ = E[HpilotH
∗
pilot
]
16 RETHNAKARAN PULIKKOONATTU
For our special case (DVB-T pilot positions),
(26) ΓRHΓ∗ =
RH(0∆f , 0Ts) R(0∆f ,−4Ts). . . RH(−3∆f ,−5Ts)
RH(0∆f , 4Ts) R(0∆f , 0Ts). . . RH(−3∆f ,−1Ts)
RH(0∆f , 8Ts) R(0∆f , 4Ts). . . RH(−3∆f , 3Ts)
.... . .
. . .. . .
.... . .
. . .. . .
RH(3∆f , Ts) R(3∆f ,−3Ts). . . RH(0∆f ,−4Ts)
RH(3∆f , 5Ts) R(3∆f , 1Ts). . . RH(0∆f , 0Ts)
.... . .
. . .. . .
.... . .
. . .. . .
RH(kpM∆f , np
MTs) R(kpM∆f , (np
M − 4)Ts). . . RH((kp
M − 3)∆f , (npM − 5)Ts)
(27) HLMMSE = RHΓ∗ (Rν + ΓRHΓ∗)−1
H
In AWGN case,Rν becomes a diagonal matrix with entries of noise to pilot power ratio. In this case,ν = V/S (awgnnoise samples scaled down by the pilot values) Eq 28 becomes,
(28) HLMMSE = RHΓ∗ (D + ΓRHΓ∗)−1
H
where
(29) D = diag(σ21/Es, σ
22/Es, . . . , σ
2M−1/Es)
H = Hp + ν is the vector with noisy estimates at pilot positions.While the optimum channel estimate is mathematically pleasing, the heavy computational burden often force such
techniques from implementation. Moreover, the estimate requires the time-frequency correlation (scattering functions) ofthe channel. While the time correlations and frequency correlations are some times individually known, the 2D scatteringfunctions are difficult to model. If the 2D correlations can be decomposed to two 1D correlations (one in time and one infrequency), then the optimum 2D filtering can be simplified totwo separate 1D estimation filters. To retain optimality, itis mandatory that the decomposition in Figure13hold good. This is very rarely met for arbitrary channel models.
Wt
I
I
I
A
B
N
Np1
Np2
C
0I
N
. . .
. . .
...
....
..
...
.
.
....
...
N
0
.
.
.
...
...
0
0
. . .
0
0
0
0
0 00
0
Wf
Figure 13. 2D to 1D matrix decomposition
CHANNEL ESTIMATION IN DVB-T AND OFDM SYSTEMS 17
Since the optimum 2D filter is prohibitively complex, simplified estimates are tried out. One obvious method is to haveindividual 1D filtering. In this method, first a filtering across time (for a fixed subcarrier pilot) is performed, followedbya frequency domain 1D filtering. The order of filtering (whether time interpolation to be done first or frequency domaininterpolation) is decided by the pilot positioning. In rectangular, symmetric pilot arrangements, the order doesnt alterperformance, but for scattered pilots (diagonal schemes for intance) the order does matter. Figure14illustrates the2× 1Dfiltering concept.
H(., (N − 1)Ts)
RHtH
pt
(
Rν + RH
pt
Hpt
)−1
RHtH
pt
(
Rν + RH
pt
Hpt
)−1
RHtH
pt
(
Rν + RH
pt
Hpt
)−1
RHf H
p
f
(
Rν + RH
pt
Hpt
)−1
RHf H
p
f
(
Rν + RH
pt
Hpt
)−1
RHf H
p
f
(
Rν + RH
pt
Hpt
)−1
.
.
.
.
.
.
...
.
.
.
...
...
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.. . .
Filter Kp
Filter2
Filter1 Filter1
Filter2
Filter N
1
2
Kp
.
.
.
.
.
.
.
.
.
H(., 0Ts)
OF
DM
sub
carr
ier
ind
ex(k
)
OFDM symbol index (time)
.
.
.
.
.
.
.
.
.
H(., Ts)
Figure 14. 2 by 1D Wiener
The advantage of doing individual 1D filtering besides, the computational simplification is that, the correlation matricescan be precomputed, for some channel models. The time autocorrelation function can be derived from the Doppler spec-trum (inverse Fourier transform of Doppler spectrum gives the autocorrelation function in time), whereas the frequencycorrelation can be infered from the delay power spectrum.(Ratna: Some examples to be given)
5. CHANNEL ESTIMATION IMPLEMENTATION
The section from here on wards are not for public use.
6. PILOT PATTERN IN DVB-T
In DVB-T, two kind of pilots are used. Even though, there is nodifference in signal properties (or power), they aredistinguished by the respective names, due to their arrangements on the time-frequency (2D) OFDM grid.
scattered pilots are distributed on the 2D grid in a diagonalfashion. Their time (OFDM symbol indexn) and frequency(sub carrier indexk) form the following simple relationship.
(30) k = kmin + 3(n mod 4) + 12p, n, k, p ∈ Z, kmin ≤ k ≤ kmax, n ∈ [0, 67]
Wherekmin andkmax are pre defined values for the mode of operation. They are different for 2K and 8K modes.kmin
value is 171 and 687 for 2K and 8K modes respectively. The correspondingkmax values are687 and7503 in respectiveorder.(Ratna:Values need to be added/verified). Since one frame composite of68 symbols, the range of valuesn take islimited in the interval[0, 67]
18 RETHNAKARAN PULIKKOONATTU
0 5 10 15 20 25 30 35 40 45
0
5
10
15
20
25
30
35
40
45
Scattered pilots in DVB−T
OFDM Symbol index (n)
OF
DM
car
rier
inde
x (k
)
Figure 15. DVB-T scattered pilots: Scattered pilots shown in magentamarker. Data carriers are shownin green. Only a segment of the complete time frequency grid shown
Mode Scattered Pilot Carrier index2K 0 48 54 87 141 156 192 201 255 279 282 333 432 450 483 525 531 618 636 714 759
765 780 804 873 888 918 939 942 969 984 1050 1101 1107 1110 1137 1140 11461206 1269 1323 1377 1491 1683 1704
8K0 48 54 87 141 156 192 201 201 255 279 279 282 282 333 333 432 432 450 450 483483 525 525 531 531 618 618 636 636 714 714 759 759 765 765 780 780804 804873 873 888 888 918 918 939 939 942 942 969 969 984 984 1050 1050 1101 11011107 1107 1110 1110 1137 1137 1140 1140 1146 1146 1206 1206 1269 1269 13231323 1377 1377 1491 1491 1683 1683 1704 1704 1752 1758 1791 1845 1860 18961905 1959 1983 1986 2037 2136 2154 2187 2229 2235 2322 2340 2418 2463 24692484 2508 2577 2592 2622 2643 2646 2673 2688 2754 2805 2811 2814 2841 28442850 2910 2973 3027 3081 3195 3387 3408 3456 3462 3495 3549 3564 3600 36093663 3687 3690 3741 3840 3858 3891 3933 3939 4026 4044 4122 4167 4173 41884212 4281 4296 4326 4347 4350 4377 4392 4458 4509 4515 4518 4545 4548 45544614 4677 4731 4785 4899 5091 5112 5160 5166 5199 5253 5268 5304 5313 53675391 5394 5445 5544 5562 5595 5637 5643 5730 5748 5826 5871 5877 5892 59165985 6000 6030 6051 6054 6081 6096 6162 6213 6219 6222 6249 6252 6258 63186381 6435 6489 6603 6795 6816
CHANNEL ESTIMATION IN DVB-T AND OFDM SYSTEMS 19
0 10 20 30 40 50 60 70 80 90 1000
10
20
30
40
50
60
70
80
90
100DVB−T Pilot pattern (2K Mode): Scattered plus Continuous
k
n
Figure 16. DVB-T pilots: Scattered pilots shown in magenta and data carriers in green. Continuouspilots shown in blue markers.It may be observed that continuous pilots may coincided with scatteredpilots at the continuous pilot tones. Only a segment of the time frequency grid shown in figure.
7. APPENDIX B: L INEAR PREDICTION
(31) u(n + 1) =
N−1∑
k=0
w(k)u(n − k)
wherew(k) for k = 0, 1, 2, . . . , N − 1 are the coefficients of the prediction filter. The linear predictor may be cast into theWiener filtering problem, where the desired responsed(n) set equal tox(n + 1)
The Wiener filtering setup for (linear) prediction has the Wiener-Hopf system of equations as,
20 RETHNAKARAN PULIKKOONATTU
0
20
40
60
80
100
0
20
40
60
80
100−5
0
5
10
k
DVB−T Pilot pattern (2K Mode): Scattered plus Continuous
n
scal
ar
Figure 17. DVB-T pilots: Scattered pilots shown in magenta and data carriers in green. Continuouspilots shown in blue markers.It may be observed that continuous pilots may coincided with scatteredpilots at the continuous pilot tones. Only a segment of the time frequency grid shown in figure.
−1z −1z−1z
W (z) =
N−1∑
i=0
wkzk d(n)
d(n)
e(n)
d(n)
x(n)
x(n)
w0 w1 w2 wN−1
Figure 18. Wiener filter setup: Prediction
CHANNEL ESTIMATION IN DVB-T AND OFDM SYSTEMS 21
rx(0) r∗x(1) r∗x(2) . . . r∗x(N − 1)rx(1) rx(0) r∗x1 . . . r∗x(N − 2)rx(2) rx(1) rx(0) . . . r∗x(N − 3)
.... . .
. . .. . .
......
. . .. . .
. . ....
rx(N − 1) rx(N − 2) rx(N − 3) . . . rx(0)
︸ ︷︷ ︸
Rxx
w0
w1
w2
...wN−1
︸ ︷︷ ︸
w
=
rx(1)rx(2)rx(3)
...rx(N)
︸ ︷︷ ︸
rdx
Optimum (Wiener/LMMSE) solution is,
(32) w = R−1xx rdx
−1z −1z −1zx(n − 1)x(n) x(n − N + 1)x(n − 2)
e(n) d(n) = x(n + 1)
d(n)
w0(n) w1(n) w2(n) wN−1(n)
Figure 19. LMS for adaptive prediction
The LMS weight vector adaptation equation is given by,
(33) w(n + 1) = w(n) + µE [e(n)x∗(n)]
Individual tap update (say tap weightwi ) can be written similarly,
(34) wi(n + 1) = wi(n) + µE [e(n)x∗(n − i)]
NLMS (Normalised LMS) algorithm has a normalization6 factor in the adaptation. NLMS update equation can thus bewritten as follows.
(35) w(n + 1) = w(n) + µx∗(n)
‖x(n)‖2e(n)
(36) wi(n + 1) = wi(n) + µx∗(n − i)
‖x(n − i)‖2e(n)
To aoid the gradient noise amplification (whenx(n) is high), a slight modified version is used.
(37) w(n + 1) = w(n) + µx∗(n)
ǫ + ‖x(n)‖2e(n)
(38) wi(n + 1) = wi(n) + µx∗(n − i)
ǫ + ‖x(n − i)‖2e(n)
8. SIMULINK IMPLEMENTATION
TBD later
6Here the adaptation coefficientµ is not the same as that used in LMS. Same notation is used to limit the usage of too many symbols
22 RETHNAKARAN PULIKKOONATTU
REFERENCES
[1] ETSI Standard: EN 300 744 V1.5.1 (2004-11), Digital Video Broadcasting (DVB); Framing structure, channel coding and modulation for digitalterrestrial television, download from ETSI.
[2] Goldsmith, Andreas .,Wireless Communications, (Cambridge University Press, 2005).[3] Tse, David and Viswanath, PramodFundamentals of Wireless Communication, (Cambridge University Press, 2005).[4] Henrik Schulze and Christian Luders, Theory and applications of OFDM and CDMA : wideband wireless communications, Chichester, West
Sussex, England Hoboken, NJ : John Wiley, 2005.[5] Khaled Fazel, Stefan Kaiser, Multi-carrier and Spread Spectrum Systems, John Wiley and Sons, 2003, ISBN 0470848995, 9780470848999
DEMOD IP TECHNOLOGYGROUP
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