Adaptive Equalization Tutorial: Trained and Blind Algorithms

RESEARCH CENTRE FOR INTEGRATED MICROSYSTEMS UNIVERSITY OF WINDSOR

Kevin Banovic

October 14, 2005

Department of Electrical and Computer Engineering,

University of Windsor, Windsor, Ontario, Canada N9B 3P4

ADAPTIVE EQUALIZATION:A TUTORIAL

KEVIN BANOVIC Slide 2


EQUALIZATION TUTORIAL

Adaptive equalizers compensate for signal distortion attributed to intersymbol interference (ISI), which iscaused by multipath within time-dispersive channels.

Typically employed in high-speed communication systems, which do not use differential modulation schemes or frequency division multiplexing

The equalizer is the most expensive component of a data demodulator and can consume over 80% of the total computations needed to demodulate a given signal [01]

Adaptive Equalization




Channel

EqualizerAdjustment

FIREqualizer

DecisionDevice

ErrorComputation

s k( )y k( )

e k( )

r k( )s k( )

TrainingSequence

SymbolStatistics

Blind Mode

Decision-DirectedModeTraining Mode





The following quantities are defined for a linear equalizer with a real input signal: Equalizer tap coefficient vector:

Equalizer input samples in the tapped delay line:

Equalizer output: (Lf = equalizer length)

r(k) =r0(k) r1(k) . . . rLf1(k)

T=

r0(k) r0(k 1) . . . r0(k Lf + 1)

T

fT (k) =f0(k) f1(k) . . . f(Lf1)(k)

y(k) =

Lf1Xi=0

fi(k) r0(k i) = fT (k)r(k)





Error signal:

where d(k) is the desired signal

e(k) = d(k) y(k)= d(k) fT (k)r(k)





The mean-squared-error cost function is defined as [02]:

When the filter coefficients are fixed, the cost function can be rewritten as follows:

Where p is the cross-correlation vector and R is the input signal correlation matrix

JMSE = Ee2(k)

= E

d2(k) 2d(k)y(k) + y2(k)

= E

d2(k)

2E

d(k)fT (k)r(k)

+E

fT (k)r(k)rT (k)f(k)

JMSE = Ed2(k)

2fT E {d(k)r(k)}| {z }

p

+fT Er(k)rT (k)

| {z }R

f

= Ed2(k)

2fTp+ fTRf

Minimum Mean-Squared-Error (MMSE) Equalization




The gradient of the MSE cost function with respect to the equalizer tap weights is defined as follows:

The optimal equalizer taps fo required to obtain the MMSE can be determined by replacing f with fo and setting the gradient above to zero:

fJMSE =JMSE

f =JMSEf0

JMSEf1

. . .JMSEfLf1

= 2p+ 2Rf

0 = 2Rfo 2p fo = R1p





Finally, the MMSE is expressed as follows:

Questions:Why is the MSE cost function so popular?

Is the calculation of fo practical?

min = Ed2(k)

2fTo p+ fTo Rfo

= Ed2(k)

2

R1p

Tp+

R1p

TRR1p

= E

d2(k)

2pTR1p+ pTR1p

= Ed2(k)

pTR1p





In practical situations, an analytic description of the cost surface is not available

However, points can be estimated by time-averaging and search algorithms are used to descend the surface

The method of steepest descent is a gradient search algorithm that adjusts the equalizer tap weights in direction of the negative gradient as follows [02][03]:

Where is constant stepsize that controls the speed and accuracy of the equalizer tap adaptation.

f(k + 1) = f(k) + fJMSE

Method of Steepest Descent




For convergence, is chosen as follows [02][03]:

Where max is the maximum eigenvalue of R At the minimum, this method requires a noisy estimate of

the gradient during each iteration, which hinders its application in real applications

However, it serves as the basis for an entire class of practical algorithms, including the algorithms to follow

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Adaptive Equalization Tutorial: Trained and Blind Algorithms