760 IEEE SIGNAL PROCESSING LETTERS, VOL. 11, NO. 9, SEPTEMBER 2004

Tracking Issues of SomeBlind Equalization Algorithms

Magno T. M. Silva and Maria D. Miranda

Abstract—Due to the growing demand for mobile commu-nications, blind adaptive algorithms have an important role inimproving data transmission efficiency. In this context, the con-vergence and tracking analysis of such algorithms is a problemof interest. Recently, a tracking analysis of the Constant ModulusAlgorithm was presented based on an energy conservation rela-tion. In this letter we extend that analysis to blind quasi-Newtonalgorithms that minimize the Constant Modulus cost function.Under certain conditions, the considered algorithms can reach thesame steady-state mean-square error. Close agreement betweenanalytical and simulation results is shown.

Index Terms—Adaptive filter, blind equalization algorithms,steady-state analysis, tracking analysis.

I. INTRODUCTION

ACOMMUNICATION system model considering afractionally-spaced equalizer is shown in Fig. 1. Under

certain well-known conditions, this model ensures perfectequalization in a noise free environment, e.g., [1], [2]. Thetransmitted signal is assumed i.i.d. and non Gaussian.The channel is modeled by an FIR filter of length whosecoefficients are split as

(1)

where and the represent the impulse re-sponses of the sub-channels. Defining the input channel vector

(2)

the outputs of the sub-channels can be expressed by

(3)

Let an FIR equalizer of length and the input vectorof the form

(4)

Manuscript received December 22, 2003; revised February 11, 2004. Thiswork was supported by FAPESP—São Paulo State Research Council—underGrant 00/12350-6 and Mackpesquisa—Mackenzie Research Council. The as-sociate editor coordinating the review of this manuscript and approving it forpublication was Prof. P. C. Ching.

M. T. M. Silva is with Polytechnic School, University of São Paulo, São Paulo,Brazil (e-mail: magno@lcs.poli.usp.br).

M. D. Miranda is with Electrical Engineering Program, Mackenzie Univer-sity, São Paulo, Brazil (e-mail: mdm@mackenzie.com.br).

Digital Object Identifier 10.1109/LSP.2004.833517

Fig. 1. Communication system model considering a T=L fractionally-spacedequalizer.

in which . The equalizeroutput in a noise free environment can be written as

where is the weight equalizer vector. The blind equal-izer must mitigate the channel effects without the data trainingand recover the signal for some delay and phase

, which is assumed to be without loss of generality.

In this context, the literature contains two popular blindschemes: the Godard [3] and Super-Exponential [4]. As suchschemes are equivalent under certain circumstances [1], weconsider only the Godard cost function, given by

(5)

where .Gradient and quasi-Newton methods were exploited in min-

imizing , leading to several different algorithms containedin the literature, e.g., [3], [5], [6]. The Constant ModulusAlgorithm (CMA) is based on a stochastic gradient approach[3]. The Stochastic Newton-like Algorithm (SNLA) is based onthe quasi-Newton method [6]. The Shalvi-Weinstein Algorithm(SWA) was originally derived in [4] from the Super-Expo-nential cost function and can be interpreted as a stochasticgradient algorithm with an optimal step-size [1]. It can also beinterpreted as a quasi-Newton algorithm if the autocorrelationmatrix, responsible for the whitening of the input sequence [4],is assumed to be an approximation of the Hessian matrix of .These algorithms can be described by the following generalexpression

(6)

where

(7)

and is the stochastic gradient approach,is a symmetric and nonsingular matrix which represents

an estimate for the inverse Hessian, and stands for complex-

1070-9908/04$20.00 © 2004 IEEE

IEEE SIGNAL PROCESSING LETTERS, VOL. 11, NO. 9, SEPTEMBER 2004 761

TABLE IPARAMETERS OF SOME BLIND ALGORITHMS

conjugate. The CMA, SNLA, and SWA consider the instanta-neous estimate of the gradient vector and the three differentHessian approximations shown in Table I. In this table, is theidentity matrix, is the forgetting factor,

resp., for the real (resp., complex) case, andis the SNLA convergence factor. For SNLA and SWA, the

matrix is obtained by using the matrix inversion lemma[7, p. 67].

All adaptive schemes of the form (6) obey the energy conser-vation relation and can be described by a lossless mapping andfeedback loop [7]. Tracking analysis of CMA using that relationwas presented in [2]. In Section II, we extend this analysis byconsidering algorithms of the form (6) with different Hessianapproximations. This is followed by simulation results and con-cluding remarks.

The following notation and definitions are used: stands fortranspose conjugate, stands for the squaredand weighted Euclidean norm of the column vector , with

being the weighting matrix,denotes the autocorrelation matrix, and

the trace of the matrix .

II. TRACKING ANALYSIS

In a nonstationary environment, the variation in the zero-forcing solution is assumed to follow the model [7]

(8)

In this model, is an i.i.d. sequence with positive definiteautocorrelation matrix and is indepen-dent of the initial conditions and offor all [7, Sec. 7.4]. In this case, a small degree of non-stationarity is assumed.

Defining , (6) for the nonstationarycase can be rewritten as

(9)

Let the a priori error be . One measureof filter performance is given by the steady-state mean-square

error (MSE), defined as . The MSEand tracking analysis of algorithms of the form (6) can be basedon the following assumptions:

A1. , and for complex data(circularity condition). This as-

sumption holds for most constellations used in digitalcommunications [4], [7].

A2. Algorithms of the form (6) obey the steady-statecondition given by

[7, Sec. 6.9.3)].A3. and are independent at the steady-

state. This assumption was considered in the MSE andtracking analysis of CMA [2] and according to [8] itis reasonable since it essentially requires the estima-tion error of the equalizer to be insensitive, atsteady-state, to the actual transmitted symbols[8, As.I.1, p. 84].

A4. and are independent at the steady-state. Thisrequires the weighted energy of the input vector to beindependent of the equalizer output. A similar assump-tion is considered in [8, As.I.2, p. 84].

A5. Equation (7) can be rewritten using the approximationsince at the steady-state

.A6. . This

assumption is obtained by using A3, A5 and consid-ering [8, Th. 3 and Th.4].

A7.To obtain this assumption, besides the use of A1,

A3, and A5, the terms which depend onwith , were considered to be small whencompared to [8, Th.3 and Th.4]. It can also be ob-tained by considering with

, in mean-square of (7).Assumptions A6 and A7 show good agreement with simulationsfor nonconstant modulus constellations. Now we can establishthe following theorem.

Theorem 1: Under Assumptions A1–A7, the steady-stateMSE of the algorithms of the form (6) can beapproximated by

(10)Proof: By equating the squared and weighted norms on

both sides of (9), using as a weighting matrix, we ob-tain

(11)

By taking expectations of both sides of (11) using A2 we arriveat

(12)

To simplify the notation, time index was suppressed in the pre-vious equation. Using A4, A6, and A7 in (12) we obtain (10)which completes the proof.

It is relevant to note that the MSE from (10) has twoterms: one for the stationary environment and another thatappears in the nonstationary case. The expressions for the

762 IEEE SIGNAL PROCESSING LETTERS, VOL. 11, NO. 9, SEPTEMBER 2004

TABLE IISTEADY-STATE MSE (�), OPTIMUM ADAPTATION FACTORS (�), AND MINIMUM STEADY-STATE MSE (� ) FOR CMA, SNLA, AND SWA

steady-state MSE for the algorithms of Table I are derived fromTheorem 1 by considering usual approximations forand [7, pp. 320, 382], [9]. In a noise free envi-ronment and for nonconstant modulus constellations, there existoptimal adaptation factors, denoted by , that minimize thesteady-state MSE. These factors are , and , for CMA,SNLA, and SWA, respectively. The corresponding minima,denoted by , can also be evaluated. All these results aresummarized in Table II, where is the equalizer length. TheCMA tracking analysis [2] is considered here for comparison.In SNLA analysis when the following approximationis assumed .

We can observe that SNLA and SWA reach the same MSE atthe steady-state since as shown in Table II. Comparingthe minimum MSE value we obtain the following ratios:

(13)

(14)

Ratio (13) was expected since SWA and SLNA have the samebehavior at the steady-state. Ratio (14) is the same one obtainedin [9], [7] for the RLS and LMS algorithms. This means that, asin the comparison between RLS and LMS, there are situationswhen SWA and SLNA present superior tracking capability com-pared to CMA and vice-versa. In these comparisons, it is usualto consider three different choices of the matrix [9], [7]: 1)

, where the performance of CMA is similar to that ofSWA and SNLA; 2) is a multiple of , where CMA is supe-rior; and 3) is a multiple of , where SWA and SNLA aresuperior.

To close this section we consider the MSE in a stationaryenvironment. Assuming , it follows that

(15)

We may adjust the adaptation factors of CMA and SNLA orSWA to reach the same steady-state MSE. Let be the varianceof the input signal. From (15), with , we obtain

(16)

III. SIMULATION RESULTS

To verify the validity of the tracking analysis for CMA,SNLA, and SWA we assume . We consider the channel

, an FIR filter with tapsas a -fractionally spaced equalizer, , and 6-PAMwith statistics ,and [2]. Fig. 2 shows the measured MSE withvarying adaptation factors considering the theoretical andexperimental results for CMA, SNLA and SWA. The valueof experimental MSE was obtained as the average over 100experiments. The simulation results are in good agreementwith the analysis for all considered algorithms. The theoreticalminimum MSE predicted by the expressions of Table II is

dB which corresponds to optimal adaptationfactors and . Theexperimental values are dB,dB, and dB. These values are close to thosepredicted by the expressions of Table II. The same occurs withthe adaptation factors shown in Fig. 2.

IV. CONCLUSION

The tracking analysis of CMA [2] is extended to blind quasi-Newton algorithms. Under certain conditions, the considered al-gorithms can reach the same steady-state MSE as shown in Sec-tion II. SWA and SNLA present the same attainable minimumMSE. The ratio between the minimum MSE for SWA or SNLAand CMA is the same obtained between the RLS and LMS algo-rithms; this allows a direct extension of the comparison of thesealgorithms to blind context. Through simulations, we verify that

IEEE SIGNAL PROCESSING LETTERS, VOL. 11, NO. 9, SEPTEMBER 2004 763

Fig. 2. Analysis and simulation MSE of (a) CMA, (b) SNLA, and (c) SWA;average over 100 experiments.

theoretical analysis can predict experimental steady-state re-sults. Although the considered assumptions may seem imprac-tical, they agree with experimental results. This analysis is vali-dated in an environment with a small degree of nonstationarity.Due to the nonlinear nature of blind algorithms, their trackinganalysis under a high degree of nonstationarity remains an openproblem.

ACKNOWLEDGMENT

The authors thank Dr. L. A. Baccalá, Dr. V. H. Nascimento,and the anonymous reviewers for their useful suggestions.

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