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Chapter 7 Equalization
7.1 Equalization : Introduction
7.2 Zero-Forcing Equalizer
7.3 Linear MMSE Equalizer
7.4 Adaptive Transversal Equalizer
7.5 Decision Feedback Equalizer
7.6 Adaptive DFE
7.7 MLSE
7.8 Tomlinson-Harashima Precoding
Appendix : Turbo Equalization
2
7.1 Equalization Techniques -Introduction For transmitting digital signals over an ideal bandlimited
communication channel, each pulse is usually shaped by a raised-cosine transfer function in order to avoid ISI as shown in Fig.7.1, the composite transfer function is expressed by
Hrc(f) = HT(f) HR (f)
However, for non-ideal channels, such as frequency-selective channels and multipath radio channels, intersymbol interferences often occur in the transmittedpulse train ffect.
The received digital signal exhibit distortions , as shown in Fig.7.2.
. The pulse sidelobes do not go through zeros at sampling
instants adjacent to the mainlobe of each pulse.
The distortion can be viewed as positive or negative echoes
occurring both before and after the main lobe. To achieve an
overall raised-cosine transfer function , an equalizing filter is
required at the receiver end to compensate the distortion.
3
Fig. 7.1 Communication system with equalization filter
4
Fig. 7.2Receive pulse exhibiting distortion
5
Equalizers can be classified into two types : linear and
nonlinear . Linear equalizers are usually implemented by
transversal- filter structure. Among nonlinear equalizers,
decision feedback equalizer (DFE) is the most common
because it is fairly easy to implement and usually performs
well. However, on channel with low SNR , DFE suffers from
error propagation when bits are decoded in error ,leading to
poor performance. Both linear equalizer and DFE are
symbol-by-symbol equalizers. They remove ISI from each
symbol and then detect each symbol individually.
The optimal equalization technique is maximum likelihood
sequence estimation (MLSE). Sequence estimators detect
sequence of symbols , so the suppression of ISI is part of the
estimation process .
Unfortunately, the complexity of this technique grows exponentially with the length of delay-spread of the channel.
6
Equalizers may be either preset or adaptive . The
parameters ( or coefficients) of a preset equalizer are
adjusted by making measurements of the channel impulse
response and solving a set of equations for the parameters
using these measurements.
An adaptive equalizer is automatically adjusted by
periodically sending a known training signal ( or
sequence) through the channel and allowing the equalizer
to adjust its own parameters in response to this known
signal,
7
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7.2 Zero-Forcing Equalizer
An ideal zero-ISI equalizer is simply an inverse filter which has a frequency response that is the inverse of the channel’s frequency response. This inverse filter is often approximated by a finite-impulse response (FIR) filter or transversal filter, as shown in Fig.6.19.
Consider a transversal filter with 2N+1 taps , the impulse response of the filter is
h(t) = Σk = -NN ck δ (t- kT ) (6.20)
and the output of the filter z(t) can be expressed as
z(t) = Σk = -NN ck y(t- kT) (6.21)
where T is the time interval between adjacent taps of the
filter .
9
Fig.6.19 Transversal filter structure
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The zero-forcing solution can be applied to the samples of
z(t) taken at time t = kT . These samples are
z( mT) = Σk = -NN ck y(mT- kT ) (6.22)
Since there are 2N +1 equalizer coefficients , we can control only
2N +1 sampled values of z(t) . By selecting the { cn } weights so that the equalizer output is forced to zero at N sample points on either side of the desired pulse .
z( mT ) = Σk = -NN ck y(mT - kT )
1 for m = 0
=
0 for m = ± 1 , ± 2 , ± N (6.23)
11
Denote that
Zeq = [ 0 0 … o 1 0 ... 0 0 ]T
C = [ c-N c-N + 1 … cN ]T (6.24)
and
y(0) y(-T) … y(-2NT)
y(T) y(0) … y((-2N+1)T)
y(2T) y(T) … y((-2N+2)T)
Y = . . .
. . . (6.25)
y(2NT) y((2N-1)T) … y(0)
Then Eq.(5.4) can be expressed as
Zeq = Y C (6.26)
Since the zero-forcing equalizer neglects the effects of noise, it is not always the best
12
Example : A 3-tap equalizer
Given a received distorted set of pulse samples { 0.0, 0.2 , 0.9 , -0.3 , 0.1 }
N=1 ,we have
0 x(0) x(-1) x(-2) c-1
1 = x(1) x(0) x(-1) c0
0 x(2) x(1) x(0) c1
0.9 0.2 0 c-1
= -0.3 0.9 0.2 c0
0.1 -0.3 0.9 c1
Then we obtain
c-1 -0.2140
c0 = 0.9631
c1 0.3448
13
7.3 Linear MMSE Equalizer
Suppose that the desired output from the transversal filter
equalizer is d(t) ,which is a sequence of ±1 –valued pulse of
duration T seconds.
The filter tap-weights are chosen so that the mean-square
error between desired output d(t) and its actual output is
minimized. As shown in Fig.5.7 , the actual output,
including noise, is denoted as z(t). The minimum mean-
square error (MMSE) criterion may be expressed as
ε= E [(z(t)-d(t))2 ] = minimum
14
For a transversal filter input denoted by y(t), which
includes AWGN , the filter output is
z(t) = Σk = -NN ck y(t- kT )
The mean-square error is generally a concave function of
the tap-weights. The optimum weight vector Co can be
obtained by Wiener-Hopf equation ;
Ryy Cop = Ryd or Cop = Ryy-1 Ryd (6.27)
where
Ry y(0) Ryy (T) … Ryy (2N T)
Ryy (T) Ry y(0) … Ryy ((2N-1)T)
Ry y = . (6.28)
.
Ryy (2NT) Ryy ((2N-1) T) … . Ry y(0)
15
Ryd (-N T)
Ryd (-N+1) T)
Ryd = . (6.29)
.
Ryd (NT)
where Ryd (mT) = E [ y (t) d (t+mT ) ]
Ryy (mT) = E [ y (t) y (t+mT ) ]
and the MMSE is given by
εmin = E[ d2(t) – R-1yd Ryy Ryd ] (6.30)
16
Fig.6.20 MMSE equalizer structure
17
7.4 Decision- Feedback Equalizer
The basic limitation of a linear equalizer, such as the transversal filter, is the poor perform on channel having spectral nulls.
A decision-feedback equalizer (DFE), as shown in Fig. 5.7, consists of a feedforward filter W(z) with the received sequence as input ( similar to the linear equalizer) followed by a feedback filter V(z) with the previously detected sequence as input.
Assuming that W(z) has N1 +1 taps and V(z) has N2
taps , we can express the DFE output as
z ( k ) = Σi = -N10 ci y ( k-i) - Σj = 1
N2 cj I k-j (6.32)
where the above equation, { ck } are the tap coefficients of the
filter , I k is an estimate of the k-th information symbol, and
{ I k-1 , I k ,…, I k- N2 } are previously detected symbols.
18
• Qualitative impulse of a discrete channel
19
20
Fig.5.7 Decision feedback equalizer structure
21
A decision –feedback equalizer is a nonlinear equalizer that
uses previous detector decisions to eliminate the ISI on
pulses that are currently being demodulated . The ISI being
removed was caused by the tails of the previous pulses.
The forward filter whitens the noise and produces a
response with post-cursor ISI only. Its task may be viewed
as the elimiation of the precursors.
Since the feedback filter V(z) sits in a feedback loop , it
must be strictly causal or else the system is stable. The
feedback filter of the DFE does not suffer from noise
enhancement because it estimates the channel frequency
response rather than its inverse. For channels with deep
spectral nulls, DFEs generally perform much better than
linear equalizer.
Both zero-forcing criterion and MMSE criterion can be
applied to determine the filter coefficients.
22
Procedure for MMSE-DFE filter design:
(1) Design linear filter HR(f) so that noise is minimized
(2) Design feedback FI R filter so that ISI = zero
In the ideal case (infinite-length feedback filter), all ISI
can be completely eliminated !
In practice, only postcursor ISI from a finite number of
previous decisions can be eliminated .
Precursor ISI can be reduce by linear (precursor) filter and adding delay in the system .
Error propagation :
One decision error in DFE causes a burst of new errors .
The errors only stop after M ( = order of feedback filter) consecutive correct decisions .
23
7.5 Adaptive Transversal Equalizer
Setting the tap-weights of the zero-forcing and MMSE
equalizers involves the solution of a set of simultaneous
equations . In these kind of of equalizer, to solve the
equations, , channel response or the autocorrelation of the
measured data are required. In practice, it may be
difficult or impossible to determine these quantities.
An adaptive equalizer scheme is illustrated in Fig. 5.6 .
The tap-weights of the transversal filter can be adjusted,
iteratively, by the LMS algorithm ,or other adaptive
algorithms.
The LMS adaptive algorithm is given by
ck (n+1) = ck(n) - α y(n) ε(n) (6.31)
24
In adaptive equalization, there are two modes of operation :
training and tracking .
During the training period , the coefficients of the
equalizer are updated at time k based on a known training
sequence that has sent over the channel.
During the tracking mode, the known training sequence is
replaced by the output of the decision device.
25
Fig.5.6 Adaptive Equalizer Scheme
26
7.6 Adaptive DFE Equalizer
In the case of adaptive DFE, the LMS algorithm for filter-
coefficients updating is given by
ck(n+1) = ck(n) +με(n) v (n-k) (6.33)
where ε(n) = I ~ (n) – I ^ (n ) ,
v (n-k) = y(n- k ) for feedforward section, i.e. k = -N1 ,…, 0
I ~ (n-k) for feedback section, i.e., k = 1,2,…, N2
(6.34)
27
7.7 Maximum Likelihood Sequence Estimation
• In symbol-by-symbol equalization, ISI is completely removed ( ZF) or reduced to a large extent (MMSE) . This approach is not the best way to approach the performance of optimum demodulator since the symbol energy that resides in ISI is not used .
• In an AWGN channel with ISI, the energy of a symbol usually spans many symbol intervals. Sequence estimation can be applied in order top use the entire symbol energy .
The best sequence estimation is the maximum likelihood sequence estimation ( MLSE).
Maximum likelihood sequence estimation (MLSE) avoids the problem of noise enhancement because it does not use an equalizing filter, instead it estimates the sequence of transmitted symbols .
28
29
The structure of the MLSE is shown in ,Fig. xx . The received
signal is given by x(t) . The MLSE algorithm chooses the input
sequence { dk } that maximizes the likelihood of x(t).
Let x(t) be denote the composite equivalent lowpass impulse
response consisting of the transmitter pulse shape, channel ,
and the matched filter impulse response.
x(t) = g(t) ★ c(t) ★ gm(-t)
Then the matched filter output is given by
y(t) = d(t) ★ x(t) + ng(t)
where ng(t) = n(t) ★ gm(-t) is an equivalent lowpass noise.
Using a Gram-Schmidt orthogonization procedure , we can
express x(t ) on a time interval [ 0 , LTs ] as
x ( t ) = Σk=1K xk ψk( t )
where {ψn( t ) } form a complete set of orthogonal basis
functions. The number K of functions in this set is a function
of the channel memory, since x(t) on ( 0 ,LTs ) depends on
DL = d0 . d1 …,dL-1 .
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31
With this expansion we have
xk = Σl= 0L-1 dl hkl + νk
where
hkl = ∫0LTs h(t – lTs ) ψk
*( t ) dt
and
νl = ∫0 LTs n(t ) ψl
*( t ) dt
The νk are complex Gaussian random variables with zero
mean and variance E[νl*
νm ]= N0 δ[l-m ] .
Thus x = [ x1 ,…,xK ] has a multivariate Gaussian
distribution .
p( x | d , h( t )
= Πk =1K ( 1/πN0 ) exp{- (1/N0 ) |xk –Σl=0
L dl hkl |2 }
32
Given a received signal x(t) or, equivalently, x , the MLSE decodes this as the symbol sequence d that maximizes the likelihood function p( x | d , h( t ) ( or the log of this function ).
Thus , the MLSE output is given by
d ^ = arg max { log p( x | d , h( t ) }
= arg max { - |xk –Σl dl hkl |2 } = …
= arg max { 2 Re [Σm dm* Σk=1
K xk hkl* ] -
ΣlΣm dl dm * Σk=1
K hklhkm* }
=arg max {2Re[Σm dm* ym ] - ΣmΣl dm dl
* q(m-l) }
where yl = ∫- ∞∞ x(τ) h(τ –lTs ) dτ = Σk=1
K xl hkl*
and q ( l – m ) = ∫- ∞∞ h(τ – lTs ) h
*(τ – mTs ) dτ
= Σk=1K hklhkm
*
q(t) = h (t) * h(-t)* .
33
We see from this equation that the MLSE output depends
only on the sampler output { y[k]} and the channel
parameters q[n-k] = q ( nTs- kTs ) .
c(DL ) = 2 Re[Σm dm* ym ] - ΣmΣl dm dl
* q(m-l)
is also known as correlation metric .
In summary , the MLSE chooses a symbol sequence that
corresponding to the largest correation metric c(DL ) .
Such MLSE employs a front-end matched filter that matches
the symbol waveform g(t )
34
Viterbi Equalizer
The Viterbi algorithm can be used for MLSE to reduce
complexity. However, the complexity of this equalization
technique still grows exponentially with the channel delay
spread.
We assume that the input to the Viterbi equalizer
can be written as
yk = Σl=0Lc di ck-l
+ nk
where n is a Gaussian white noise with variance σn2 .
35
• For a sequence of K received values , the joint probability
density function (PDF) of the vector of received signals u
conditioned on the data vector is given by
p (y|c ;d) = {1/( 1/2 π σn2 )K/2 }
exp {- (1/ 2σn2 ) Σk=1
K| yk -Σl =0Lc dl ck-l|
2 }
The MLSE of c for a given are the values of the vector that
maximize the joint PDF , p (y|c ; d ) .
Since the variables only occur in the exponent , it is suffuicient
to minimize : the term
M =Σk=0K| yk -Σl =0
Lc dl ck-l |2
where is called the distance metric
36
Example : Viterbi Equalizer
Discrete-time impulse response of channel
1
c = ( - 0.5 )
0.3
Tapped delay line model
37
Trellis diagram
38
39
40
41
7.8 Tomlinson-Harashima Precoding
If the channel response is known the transmitter , the
equalizer can be placed at the transmitter end of the
communication system. Thus the noise enhancement that is
generally inherent when the equalizer ( linear or DFE ) is
placed at the receiver is avoided.
In wireline channels , the channel characteristics do not
vary significantly with time. Therefore , it is possible to
place the feedback filter of the DFE at the transmitter and
the feedforward filter at the receiver. This approach has
the advantage that the problem of error propagation due
to the incorrect decision ( in the traditional DFE ) is
completely eliminated.
The linear feedfprward part of the DFE is designed to
compensate for the ISI that results from the small variation
in the channel response.
42
The synthesis of the feedback filter of thhe DFE at the
transmitter side is usually performed after the response of the
channel measured at the receiver by the transmission of a a
channel probe signal and the receiver sends to the transmitter
the coefficients of the feedback filter.
However, this modification may result ina significant increase
in signal dynamic range ,and consequently require larger
transmitter power. This problem can be overcome by precoding
the information symbols prior to transmission as described by
Tomlinson (1971) , Harashima and Miyakawa (1972 ).
Tomlinson, M , “ New Automatic Equalizer employing modulo arithmetics, “
Electronics Letter, vol.7, pp.138-139, March 1971.
Harashima, H. and Miyakawa,H., “Matched-Transmission Technique for Channels
with Intersymbol Interference ,” IEEE Trans. Commun., vol. COM-20, no.4 ,
pp.774-780, 1972.
43
Tomlinson-Harashima Precoding
We consider the precoding technique for a pulse amplitude
modulation (PAM) constellation. The technique can be easily
extended to QAM signal constellation since a square QAM signal
constellation can be viewed as two PAM signal sets on quadrature
carriers.
For simplicity , the feedforward filter in the DFE is assumed to be the
whitening matched filter (WMF) .The channel response ,
characterized by the parameters { ci , ,0 ≦ i ≦ L , is also
assumed perfectly known to the transmitter and receiver.
The information symbols I k are assumed to take the values
{ ± 1 , ± 3,…. , ± (M-1 )} .
44
Fig. 4B. shows a block diagram of the Tomlinson-Harashima
predocing. The ISI due to the postcursor is subtracted from the
symbol to be transmitted and, if the difference falls outside of the
range ( -M , M ), it is reduced to the range by subtracting an integer
multiple of 2 M from this difference.
Hence , the precoder output may be expressed as
ak = I k - Σk =1L c k ak-1 - 2M b k
where {b k } represents the proper integer that brings {ak } to the
desired range . This reduction operation can be performed by a
modulo-2M operation .
By using the z-transform to describe the operation of the
precoding, we have
A(z) = I(z) - [ C(z) -1 ] A(z) + 2M B(z)
where the channel coeffiicient c0 is normalized to unity .
45
Fig. 4B.1Block diagram of Tomlinson-Harashima precoding
46
The transmitted sequence is expressed by
A(z) = { I(z) + 2M B(z) } / C(z)
and the r3eceived signal sequence is expressed by
V(z) = A(z) + W(z)
= { I(z) + 2M B(z) + W(z)
From the above expression , we can see that the received data
sequence at the input of detector is free of ISI and I(z) can be
recoverede from V(z) by use of symbol-by-symbol detector that
decode the symbols modulo -2M .
47
48
Appendix : Turbo Equalization
Turbo decoding principle can be used in equalizer design.
A turbo equalizer iterates between a maximum a posteriori (MAP)
equalizer and a decoder to determine the transmitted symbol. The
MAP equalizer computes the a posteriori probability ( APP ) of the
transmitted symbol given the past channel outputs . The decoder
computes the log- likelihood ratio (LLR) associated with the
transmitted symbol given past channel outputs . The APP and
LLR comprise the soft information exchanged between the
equalizer and decoder in the turbo iteration .
After some iterations the turbo equalizer converges on its
estimate of the the transmitted symbol.
49
• Suppose the transmitter of a digital communication system
employing a binary systematic convolutional encoder followed
by a block interleaver and a modulator, as shown in Fig. 4.xx .
• The channel is is a linear time-dispersive channel that
introduces ISI. In such a case, we may view the channel as an
inner encoder in a serially concatenated code. Hence , we can
apply iterative decoding based on the MAP criterion.
• In Fig.4.xx , the input to the MA equalizer is the sequence {vk }
frrom the WMF. The output of MAP, which computes the LLR of
the coded bits , is denoted as LE( x ^ ) .
The outer decoder receives the extrinsic part of LE( x
^ ) as input. The extrinsic part LeE ( x ^ ) is defined as
LeE ( x ^ ) = LE( x ^ ) - Le
D ( x^’ )
where LeD ( x^’ ) is the extrinsic part of the outer
decoder output after interleaving. LeE ( x ^ ) is
deinterleaved prior to being fed to the outer decoder.
50
51
The outer decoder computes the LLR of the coded bits, denoted by LD ( x^’ ) , and the information bits , denoted as LD ( I ^ ) .
The extrinsic part of LD ( x^’ ) , denoted as LeD ( x^’ ) ,is the
incremental information about the current bit obtained by the decoder after observing all informationfor all the received bits.
The extrinsic information is computed as
LeD ( x ^’ ) = LD( x ^ ) - Le
E ( x^’ )
LeD ( x ^’ ) is interleaved to produce Le
D ( x ^ ) and fed to
the MAP equalizer.
The computation of the LLR iis described ibn the paper by Bauch et al (1997) .
Fig.7.xx illustrates the bit error probability obtained through simulatuion of the five-tap time-invariant channel given Fig. .
The outer decoder used is rate ½ recursive systematic convolutional code with constraint length K =5 . The interleaver is a pseudorandom block interleaver of length N = 4096 bits . Binary PSK was used for modulation .
52
53
• Compared to the Viterbi detector for this channel, the iterative
equalizer hs achieved a performance gain of about 6 dB , aside
from the coding gain due to the convolutional code.
References
1. Bauch,G. ,Khorram, H., and Hagenaur,J., “Iterative Equalization and Decoding in Mobile
Communications Systems, “ in Proc. 2nd Eur. Personal Mobile Comm. Conf., Sep/Oct.
1997, pp. 307-712.
2. Tuchler ,M.,and Singer ,A.C., “ Turbo equalization : An overview “,IEEE Trans. Inform.
Theory Commun.,vol.57, no.2 ,pp,920-952, Feb, 2011.
3. Koetter,R. , Singer , A.C ,and Tuchler ,M.,, “ Turbo Equalization ,IEEE Signal Processing
Magazine, Jan. 2004, pp.67 -80.
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Appendix : Classification of the equalizati structure
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