DC Lecture8

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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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.

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