Equalization Tutorial

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) . . . rLf−1(k)

¤T=

£r0(k) r0(k − 1) . . . r0(k − Lf + 1)

¤T

fT (k) =£f0(k) f1(k) . . . f(Lf−1)(k)

¤

y(k) =

Lf−1Xi=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 = E©e2(k)

ª= E

©d2(k)− 2d(k)y(k) + y2(k)ª

= E©d2(k)

ª− 2E ©d(k)fT (k)r(k)ª+E ©fT (k)r(k)rT (k)f(k)ª

JMSE = E©d2(k)

ª− 2fT E {d(k)r(k)}| {z }p

+fT E©r(k)rT (k)

ª| {z }R

f

= E©d2(k)

ª− 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=

·∂JMSE

∂f0

∂JMSE

∂f1. . .

∂JMSE

∂fLf−1

¸= −2p+ 2Rf

0 = 2Rfo − 2p→ fo = R−1p





Finally, the MMSE is expressed as follows:

Questions:Why is the MSE cost function so popular?

Is the calculation of ‘fo’ practical?

ξmin = E©d2(k)

ª− 2fTo p+ fTo Rfo= E

©d2(k)

ª− 2 £R−1p¤T p+ £R−1p¤T R £R−1p¤= E

©d2(k)

ª− 2pTR−1p+ pTR−1p= E

©d2(k)

ª− pTR−1p





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

f(k + 1) = f(k) + µ · ¡−∇fJMSE¢

Method of Steepest Descent




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

Where λmax 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

0 < µ <1

λmax

Method of Steepest Descent




The least-mean-squares (LMS) algorithm simplifies the gradient calculation by using instantaneous quantities instead of expected quantities [02]

Let us define the following estimates of ‘p’ and ‘R’:

Substituting these estimates, the gradient becomes:

R̂ = r(k)rT (k)

p̂ = d(k)r(k)

∇fJLMS = −2p̂+ 2R̂f(k)= −2 (d(k)r(k)) + 2 ¡r(k)rT (k)¢ f(k)= −2r(k) ¡d(k)− rT (k)f(k)¢| {z }

e(k)

= −2e(k)r(k)

Least-Mean-Squares Algorithm (LMS)




The LMS equalizer tap adjustment is as follows:

The LMS algorithm has two modes of operation: a training mode and a tracking or decision-directed mode

In the following example uses Proakis channel B [04], a stepsize of 5x10-3, and a 2-tap LMS equalizer

f(k + 1) = f(k) + µ · ¡−∇fJLMS¢

= f(k) + µ · e(k)r(k)

0.404 0.404

0.815





f0

f1MSE Surface

−0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8





0 1000 2000 3000 4000 5000−12

−10

−8

−6

−4

−2

0

2

4

Smoothed squared−error history

iteration number

dB

MSE bound





Questions:What is the relationship between steady-state MSE, the

time-to-convergence and the stepsize?





The generalized Sato algorithm is the first of three blind algorithms that we will be discussing

Blind algorithms achieve channel equalization without the transmission of a training sequence

The generalized Sato equalizer tap update for complex signals is defined as [05][06]:

Where ‘csgn(·)’ is the complex sign operator, ‘γ’ is a constant of the source signal, and ‘*’ is the complex conjugate operator

f (k + 1) = f(k) + µ · (csgn(y(k))γ − y(k))| {z }−∇fJGSA=eGSA(k)

r∗(k)

Generalized Sato Algorithm (GSA)




The CMA is a carrier-phase independent blind algorithm that is based on the signal modulus

The CMA equalizer tap update is defined as [07][08][09]:

As illustrated in the figure to follow, the CMA requires phase-recovery after convergence in order to rotate the constellation

f(k + 1) = f(k) + µ · y(k)(γ2 − |y(k)|2)| {z }−∇fJCMA=eCMA(k)

r∗(k)

Constant Modulus Algorithm (CMA)




−2 0 2−2

−1

0

1

2

Re{s(n)}

Im{s

(n)}

Sent Signal Constellation

−2 0 2−2

−1

0

1

2

Re{s(n)}

Im{s

(n)}

Received Signal Constellation

−2 0 2−2

−1

0

1

2

Re{s(n)}

Im{s

(n)}

Equalized Output (CMA)

−2 0 2−2

−1

0

1

2

Re{s(n)}

Im{s

(n)}

Equalized Output with Carrier Recovery

Constant Modulus Algorithm (CMA)




The MMA minimizes dispersion of the equalizer output around separate straight contours

The MMA equalizer tap update is defined as [10]:

Where ‘R’ and ‘I’ correspond to the real and imaginary components, respectively

fR(k + 1) = fR(k) + µ · yR(k)(γ2 − y2R(k))| {z }−∇fJMMA

R =eMMAR (k)

r∗(k)

fI(k + 1) = fI(k) + µ · yI(k)(γ2 − y2I (k)})| {z }−∇fJMMA

I =eMMAI (k)

r∗(k)

f (k + 1) = fR(k + 1) + j · fI(k + 1)

Multimodulus Algorithm (MMA)




0 0.5 1 1.5 2

x 104

−30

−20

−10

0

10MSE Curves for Blind Algorithms

samples

MS

E (

dB)

GSA CMAMMA

Stepsize: µ=10−3

SNR: 30dBSignal: 16−QAMChannel: SPIB #2Lf: 16

Simulation of Blind Algorithms




Questions:What are the advantages of blind equalization?

Drawbacks?

Blind Equalization




For more information on adaptive equalization in general, check out the following tutorials:

Adaptive Equalization [11]Equalization in High-Speed Communication Systems [12]

For more information on blind equalization, check out the following tutorials:

Blind Equalization for Broadband Access [13]A comparative performance study of several blind equalization algorithms [06]

Equalization Tutorials




[01] J.R. Treichler, M.G. Larimore and J.C. Harp, “Practical Blind Demodulators for High-order QAM signals", Proceedings of the IEEE special issue on Blind System Identification and Estimation, vol. 86, pp. 1907-1926, Oct. 1998

[02] B. Widrow and S.D. Sterns, Adaptive Signal Processing, Prentice Hall, New York, 1985.

[03] P.S.R. Diniz, Adaptive Filtering, Kluwar Academic Publishers, Norwell, Massachusetts, 2002.

[04] J.G. Proakis, Digital Communications, McGraw Hill, New York, 2001

[05] Y. Sato, “A method of self-recovering equalization for multilevel amplitude-modulation systems", IEEE Trans. on Communications, Vol. 23, June 1975, pp. 679-682.

[06] J.J. Shynk, R.P. Gooch, G. Krishnamurthy, and C.K. Chan, "A comparative performance study of several blind equalization algorithms", SPIE, Vol. 1565, pp. 102-117, 1991

References




[07] D.N. Godard, “Self-recovering equalization and carrier tracking in two-dimensional data communication systems”, IEEE trans. on comm., Vol 28, No. 11, November 1980

[08] J. R. Treichler and B. G. Agee, "A new approach to multipath correction of constant modulus signals", IEEE Trans. on Acoust., Speech, Signal Processing, Vol. ASSP-31, No. 2, April 1983, pp. 459-472.

[09] R. Johnson, Jr., P. Schniter, T.J. Endres, J.D. Behm, D.R. Brown, and R.A. Casas, ”Blind equalization using the constant modulus criterion: a review”, Proceedings of the IEEE, Vol. 86, No. 10, Oct. 1998, pp. 1927-1950.

[10] J. Yang, J.J. Werner and G.A. Dumont, “The Mulitimodulus Blind Equalization and Its Generalized Algorithms", IEEE Journal on selected areas in communication, Vol 20, No. 5, June 2002, pp. 997-1015.

[11] S.U.H. Qureshi, "Adaptive equalization", Proceedings of the IEEE, Vol. 73, No. 9, September 1985, pp. 1349-1387.

[12] J. Liu and X. Lin, "Equalization in High-Speed Communication Systems", IEEE Circuits and Systems Magazine, 2004, pp. 4-17.

References




[13] J.-J. Werner, J. Yang, D. Harman, and G.A. Dumont, "Blind Equalization for Broadband Access", IEEE Communications Magazine, 1999, pp. 87-93.

[14] Signal Processing Information Base. http://spib.rice.edu/spib/directory.html

[15] P. Schniter, Adaptive Linear Identifier (ALI) Laboratory, http://www.ece.osu.edu/~schniter/research.html.

References

Equalization Tutorial

Documents

Transcript of Equalization Tutorial