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8/12/2019 DC Lecture8
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Lecture # 8
Equalization
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To compensate for channel induced ISI we use a process known as
Equalization: a technique of correcting the frequency response of
the channel The filter used to perform such a process is called an equalizer
Since HR(f) is matched to HT(f), we usually worry about HC(f)
The goal is to pick the frequency response HEQ(f) of the equalizer
such that
where
and the phase characteristics
)(
)(1)(1)()(
fj
C
EQEQcce
fHfHfHfH
|)(|
1|)(|
fHfH
C
EQ )()( ff CEQ
3 4 Equalization
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Equalization Process Apply a filter that results in an equalized impulse response having
zero ISI and channel distortion.
This means that convolution of the channel impulse response andthe equalizer impulse response must equal 1 at the center tap andhave nulls at the other sample points within the filter span.
It can be difficult to determine the inverse of the channel response if
the channel response is zero at any frequency, then the inverse is
not defined at that frequency.
The receiver generally does not know what the channel response is.
Channel changes in real time so equalization must be adaptive
The equalizer can have an infinite impulse response even if the
channel has a finite impulse response
The impulse response of the equalizer must usually be truncated
Problems with Equalization
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Equalization Techniques or Structures Three Basic Equalization Structures
Linear Transversal Filter Simple implementation using Tap Delay Line or FIR filters
FIR filter has guaranteed stability (although adaptive algorithm
which determines coefficients may still be unstable)
Decision Feedback Equalizer
Extra step in subtracting estimated residual error from signal
Maximal Likelihood Sequence Estimator (Viterbi)
Optimal performance
High complexity and implementation problem (not heavily used)
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Linear Transversal Equalizer This is simply a linear filter with adjustable parameters
The parameters are adjusted on the basis of the measurement ofthe channel characteristics
A common choice for implementation is the transversal fi l ter(TapDelay Line) or the FIR filter with adjustable tap coefficient
Total number of taps = 2N+1
Total delay = 2Nt
Fig. 3.26
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N is chosen sufficiently large so that equalizer spans length of the ISI.
Normally the ISI is assumed to be limited to a finite number of samples
The output ykof the Tap Delay Line equalizer in response to the inputsequence {xk} is
where cnis the weight of the nthtap
Ideally, we would like the equalizer to eliminate ISI resulting in
But this cannot be achieved in practice.
N
Nnnknk NNkxcy 2,......2,
0,0
0,1
k
kyk
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However, the tap gains can be chosen such that
There are two types of such equalizer (i.e., linear equalizers)
Preset Equalizer:
Transmits a training sequence that is compared at the receiver
with a locally generated sequence Requires an initial training sequence
Differences between sequences are used to update the
coefficient cnAdaptive Equalizer:
Equalizer adjust itself periodically during transmission of data
The tap weights constitute the adaptive filter coefficient
Nk
kyk
,......2,1,0
0,1
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The two techniques can be combined into a robust equalizer. In this
case, there are two modes of operation:
Training Mode
For the training mode, a known sequence is transmitted and a
synchronized version is generated at the receiver
Decision -directed mode
When training mode is complete, the adaptive algorithm is
switched on The tap weights are then adjusted with info from training mode
The impulse response of the transversal filter is
N
Nn
fnj
neq
N
Nn
neq
ecfH
ntcth
t
t
2)(
)()(
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Ifx(t) is the signal pulse corresponding to
then the equalized output signal is
Nyquist zero ISI condition implies that
)()()()( fHfHfHfX RCT
N
Nnn ntxcty )()(
t
Nk
knkTxckTyyN
Nn
nk,....,2,1,0
0,1)()( t
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Since there are 2N+1 equalizer coefficients, we may express inmatrix form as:
y=Xc
where:
X = (2N+1) x(2N+1) matrix with elementsx(kT - nt)
c = (2N+1) column coefficient vector
y = (2N+1) column vector
Since this design forces the ISI to be zero at sampling instants t =kT, the equalizer is called zero-for cin g equalizer (ZFE)
Thus we obtain a set of (2N+1) linear equations for the ZFE
In Figure 3.26 tis chosen as high as T
t= T Symbol-spaced equalizer; t< T Fractional-spacedequalizer
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Zero-Forcing Solution
For N=1
))1(0())0(0())1(0()0(,0 101 xcxcxcyk
))1(1())0(1())1(1()1(,1101 cxcxcyk
))1(1())0(1())1(1()1(,1 101 xcxcxcyk
1
0
1
)0()1()2(
)1()0()1(
)2()1()0(
)1(
)0(
)1(
c
c
c
xxx
xxx
xxx
y
y
y
1)12()12()12( NNN
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For N=2
Generalizing results:
2
1
1
2
0
)0()1()2()3()4(
)1()0()1()2()3(
)2()1()0()1()2(
)3()2()1()0()1(
)4()3()2()1()0(
)2(
)1(
)0(
)1(
)2(
c
c
c
c
c
xxxxx
xxxxx
xxxxx
xxxxx
xxxxx
y
y
y
y
y
1
0
1
0
Nforwhere z
zXc 1
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Minimum MSE Solution
A more robust equalizer can be obtained if {cn} tap weights are
chosen to minimize the mean square error(MSE) of all ISI terms plusnoise power at the output of equalizer
MSE is defined as:
the expected value of the squared difference between
the desired data symbol and estimated data symbol
Whereas xc )()( nzne
]|)([| 2neEMSE
])()([ 2 cx2xccx TTT nznzE
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cxcxxc TTT2 ])([2][)]([ nzEEnzE
cRccRx
T
xx zz 22
0
c
MSE
022 xxx RcR z
xxxRRc z1
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Deterministic Case:
Example 3.6: A Minimum 7-Tap Equalizer
Consider that the tap weights of an equalizing transversal filter are
to be determined by transmitting a single impulse as a trainingsignal. Let the equalizer circuit be made up of 7 taps. Given areceived distorted set of pulse samples{x(k)}, with values 0.0108, -0.0558, 0.1617,1.0000, -0.1749, 0.0227, 0.0110, use a minimumMSE solution to find the weights {cn} that will minimize the ISI. Withthese weights, calculate the resulting values of the equalized pulse
samples at the following times:
What is the largest magnitude sample contributing to ISI, and what isthe sum of all the ISI magnitudes?
T
x xR )(nzz
xxR T
xx
}6.....,,2,1,0{ k
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Solution: For a 7-tap filter (N=3)
Dimensions for matrix x will be 4N+1 by 2N+1 = 13x7
0108.0000000
0558.00108.000000
1617.00558.00108.000000000.11617.00558.00108.0000
1749.00000.11617.00558.00108.000
0227.01749.00000.11617.00558.00108.00
0110.00227.01749.00000.11617.00558.00108.0
00110.00227.01749.00000.11617.00558.0
000110.00227.01749.00000.11617.0
0000110.00227.01749.00000.1
00000110.00227.01749.0
000000110.00227.0
0000000110.0
x
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Using matrix x, form autocorrelation matrix Rxxand cross
correlation matrix Rzx. Solution for tap weights is:
Using these weights, the 13 equalized samples {y(k)} at times
:
The largest magnitude sample contributing to ISI : 0.0095 The sum of all the ISI magnitudes : 0.0195
}{ ,3,2,1,0,1,2,3 ccccccc
0269.0,0670.0,1318.0,9495.0,1659.0,0108.0,0116.0
}6.....,,2,1,0{ k
0003.0,0022.0,0095.0,0015.0,0007.0
,0003.0,0000.1,0000.0,0000.0,0007.0,0041.0,0001.0,0001.0
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Steepest Descent Algorithm
Difficult to find the inverse of a large matrix.
Use gradient based iterative techniques Cost function
Start with an initial estimate of c0and update it bymoving in the opposite direction of the gradient of J.
Keep on updating the old estimate till convergence isreached.
]|)([| 2neEMSEJ
cRccR xT
xx zz 22
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Steepest Descent Algorithm
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Steepest Descent Algorithm
Consider the coefficients:
The steepest descent algorithm is given by:
Where
If we use instantaneous estimate of we have:
Which is called the LMS algorithm. As in the previous case zis the
desired signal andx is the received signal.
],........,,......,[ 0 NN ccc
iii 2
11cc
)(2 zxi
xx
iRcR
i
))((2 TT xcxx nzii
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clear all;close all
L=3; % Signal Duration in seconds
fs=8000; % Sampling frequencyN=21; % Number of filter taps
%Training Signal
z=rand(1,L*fs);
% Impulse response of the channel
h=[1,0.7,0.2,-0.5,-0.8,-0.4,0,0.25,0.1,0.05,0,0];
x_r=filter(h,1,z);
% Intialization
% Delay line
x=zeros(1,N);
% Filter Coefficients
c=zeros(1,N);
% Step Size
mu=0.001;
Matlab Example :
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% LMS Algorithm
for n=1:L*fs
x=[x(2:N) x_r(n)];
e(n)=z(n)-c*x;
c=c+mu*e(n)*x;
end
figure(1)
H=fftshift(fft(h,fs));H=abs(H);H=H/max(H);
plot(0:fs/2-1,H(fs/2+1:fs));hold on
Cf=fftshift(fft(c,fs));Cf=abs(Cf);Cf=Cf/max(Cf);
plot(0:fs/2-1,Cf(fs/2+1:fs),'r');grid;xlabel('Frequency (Hz)');
gtext('|H(f)|')gtext('|C(f)|')
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0 500 1000 1500 2000 2500 3000 3500 40000.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Frequency (Hz)
|H(f)|
|C(f)|
Results:
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Training Mode vs. Decision Directed mode
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Fractionally Spaced Equalizer
The spectrum property of the baud-rate and fractionally spaced equalizer.
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Decision Feedback Equalizer A decis ion -feedback equalizer(DFE)is a nonlinear equalizer that
employs previous decisions to eliminate the ISI caused by
previously detected symbol It consists of a feedforward section a feedback section and a
detector connected together as shown
The filters are usually fractionally spaced FIR with adjustable tapcoefficients
The detector is a symbol-by-symbol detector
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DFE is based on the principle that once you have determined the
value of the current transmitted symbol, you can exactly remove the
ISI contribution of that symbol to future received symbols
The nonlinear feature is due to the decision device, which attempts
to determine which symbol of a set of discrete levels was actually
transmitted.
Once the current symbol has been decided, the filter structure can
calculate the ISI effect it would tend to have on subsequent received
symbols and compensate the input to the decision device for the
next samples.
This postcursor ISI removal is accomplished by the use of a
feedback filter structure.
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Adaptive Equalization for Digital CellularTelephony The direct sequence spreading employed by CDMA (IS-95) obviates
the need for a traditional equalizer.
The TDMA systems (for example, GSM and IS-54), on the other
hand, make great use of equalization to contend with the effects of:
multipath-induced fading,
ISI due to channel spreading,
additive received noise,
channel-induced spectral distortion, etc
Of the nonlinear equalizers, the DFE is currently the most practical
system to implement in a consumer system. Other designs that outperform the DFE in terms of convergence or
noise performance, but these generally come at the expense of
greatly increased system complexity.