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IEEE SIGNAL PROCESSING LETTERS, VOL. 17, NO. 1, JANUARY 2010 1

An Adaptive Constant Modulus Blind EqualizationAlgorithm and Its Stochastic Stability Analysis

Shafayat Abrar, Graduate Student Member, IEEE, and Asoke K. Nandi, Senior Member, IEEE

Abstract—A constant modulus algorithm is presented for blindequalization of complex-valued communication channels. The pro-posed algorithm is obtained by solving a novel deterministic opti-mization criterion which comprises the minimization of a priori aswell as a posteriori dispersion error, leading to an update equationhaving a particular zero-memory continuous Bussgang-type non-linearity. We also derive a stochastic bound for the range of step-sizes for a generic Bussgang-type constant modulus algorithm. Thetheoretical result is validated through computer simulations.

Index Terms—Adaptive equalizer, blind equalization, constantmodulus algorithm, stochastic stability.

I. INTRODUCTION

E QUALIZING a propagating channel without trainingsequence is known as blind equalization. Among

the existing stochastic-gradient based adaptive algorithmsfor blind equalization, the constant modulus algorithm(CMA) [1], [2] is the most famous and widely studiedalgorithm. Consider a baseband, linear, time-invariant,single-input-single-output discrete-time channel and an adap-tive blind transversal equalizer. Assume that the transmittedsequence is independently and identically-distributed,and takes values of quadrature amplitude modulation (QAM)symbols with equal probability. The received signal is ex-pressed as ,whereis the vector of the symbol-rate impulse response of themoving-average channel, is the channel length andis additive white Gaussian noise. The output of the blindequalizer is , where

is the vector of the im-pulse response of the equalizer, is the number of taps ofthe equalizer and is the vector

Manuscript received May 29, 2009; revised August 11, 2009. The work of S.Abrar was supported by the ORSAS (U.K.), the University of Liverpool and bythe COMSATS Institute of Information Technology. This work was presented inpart at the European Signal Processing Conference, 2008, ZLausanne, Switzer-land. The associate editor coordinating the review of this manuscript and ap-proving it for publication was Prof. Behrouz Farhang-Boroujeny.

S. Abrar is with the Department of Electrical Engineering and Electronics,The University of Liverpool Liverpool L69 3GJ, U.K., on leave from the COM-SATS Institute of Information Technology (CIIT), Islamabad, Pakistan (e-mail:shafayat@liverpool.ac.uk).

A. K. Nandi is with the Department of Electrical Engineering and Elec-tronics, The University of Liverpool (UoL), Liverpool L69 3GJ, U.K. (e-mail:aknandi@liverpool.ac.uk).

Digital Object Identifier 10.1109/LSP.2009.2031765

of channel observations. A CMA equalizer minimizes thefollowing cost function [1]:

(1)

where is defined as the th-order a priori dispersion error.The criterion (1) minimizes the dispersion of the modulus ofa priori output away from a statistical constant . The costyields the following stochastic-gradient adaptive algorithm:

(2)

where is dispersion constant. For ,note that the factor in CMA , which contributesonly in the magnitude of adaptation, is usually enormous forhigher-order QAM signals and leads to a high steady-state fluc-tuation even if the equalizer converges successfully [3].

II. PROPOSED ALGORITHM

Let be a generic weight-up-date where is a memory-less non-linear blind estimate of the prediction error, and the nonlin-earity be selected such that, upon convergence, equal-izer restores the actual signal power [3]. Let bethe a posteriori output of the equalizer, it is easy to show that

, where . This showsthat the a posteriori output is a linear combination of the apriori output and the blind estimate ; hence, will becloser to the than , where controls the ex-tent to which approaches . Let bethe th-order a posteriori dispersion error. Consider CMA(2,2)equalizer where we have . Exemplary, ifwe assume , thenand will be positiveand less than , provided .Clearly, by considering a posteriori quantities it should be pos-sible to enhance the quality of equalization.

In the past, exploiting a posteriori quantities, determin-istic cost functions have been proposed to obtain vari-ants of normalized CMA for constant modulus [4] andnon-constant modulus signals [5]. In this work, we insteadsuggest to formulate a deterministic cost function consti-tuting both a priori and a posteriori quantities. For thispurpose, we propose a th-order joint dispersion error,

, where and are positiveintegers. Notice that the can be expressed in terms of

and as ; it

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indicates that is a sort of weighted sum of and .After having the equalizer estimate , we aim to minimizean instantaneous deterministic cost: ; it isobvious that this cost can perfectly be minimized while leaving

largely undetermined. To fix the degree of freedom in, we can impose that remains as close as possible to

its prior estimate , while satisfying the constraints imposedby the new data, i.e., . Using Lagrange multipliers,we formulate the following problem:

(3)

For a tractable derivation, we suggest to use and, differentiating (3) with respect to and setting the

result equal to zero,

(4)

Transposing (4) and post-multiplying it with leads to

(5)

Solving (5) yields the optimum Lagrange multiplier, ,

(6)

and the update reads . At each, the hard constraint in (3) enforces . There-

fore the optimum Lagrange multiplier in (6) is

(7)

At this stage, motivated by the work in [5], we introduce a re-laxation factor, , in (7) to control the degree of constraint sat-isfaction. It implies that the constraint on is now retained asa soft constraint. By introducing , we have a relaxed Lagrangemultiplier, and the corresponding update equation reads

(8)

Taking Hermitian transpose and post-multiplying (8) withwe obtain , which leads to

(9)

The computational complexity of the algorithm (9) is littlehigher than that of . This complexity can be reducedby observing that, for , the denominator can be approxi-mated as . Also, by removing the normalizationfactor, the following simplified variant is obtained

(10)

where is step-size. We denote (10) as th-ordersoft constraint satisfaction constant modulus algorithm,SCS-CMA . The dispersion constant in (10) is evaluated togive . The major difference between

and SCS-CMA is that, the later one lacks thefactor . Obviously, this factor has no effect on thedirection of adaptation; removing it, may have an advantageouseffect of reducing the magnitude of adaptation. Also, the twoalgorithms are equivalent only for . For higher value of, the SCS-CMA requires to use a smaller step-size for its

stability. In a noise-free scenario, however, the SCS-CMAcan be ensured to adapt stochastically stable if ,where

(11)

and is the autocorrelation matrix. The derivationof expression (11) is described in Appendix I.

III. SIMULATION EXAMPLE

We present simulation performance of andSCS-CMA for various values of . We use a complex-valuedseven-tap equalizer and initialize it so that the center tap isset to one and other taps are set to zero. The propagationchannel is a (short) voice-band seven-tap telephone channeland is taken from [6]. The signal to noise ratio (SNR) istaken as 30 dB at the input of the equalizer. The residualinter-symbol interference (ISI) [7] is measured for an 8-QAMsignal and compared. Signal alphabets belong to the set

. Each ISI trace is theensemble average of 400 independent runs with random initial-ization of noise and data source. Fig. 1 depicts the residual ISIperformances of and SCS-CMA and also showsthe values of step-sizes used in the simulations. Note that theperformance gets better in terms of steady-state residual ISIwhen larger is used in both cases. The yieldedstable performance for and it failed to give anystable convergence for . Also, for , the somewhatdelayed convergence is due to the fact that CMA(5,2) cannot beforced any further to yield faster but stable convergence. Theconvergence behavior of SCS-CMA is even more attractivein the sense that it provided a smooth tradeoff between thecomplexity and performance; we can go up to in thisspecific experiment.

Next we validate the upper-bound (11) for and . Inaddition to the channel we used in Fig. 1, we also considerthe first thirty odd-indexed coefficients of a (long) microwavechannel and it is taken from SPIB database [8]. Inall cases, the simulations were performed with 5000 iterations,

runs, and no noise. In Fig. 2, we plot the proba-bilities of divergence for four different equalizer lengths,against the normalized step-size, . Theis estimated as , where indicates thenumber of times equalizer diverged. In our simulations, we labela given run of the algorithm as “diverging” if overflows (wecheck for in MATLAB). Equalizers were initialized closeto the zero-forcing solution and step-sizes were varied in therange . It can be seen that theapproximate bound does guarantee a stable performance when

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ABRAR AND NANDI: ADAPTIVE CONSTANT MODULUS BLIND EQUALIZATION ALGORITHM 3

Fig. 1. Residual ISI traces: (top) ������ �� and (bottom) SCS-CMA���.

, equalizer and channel are long enough, and issmall.

IV. CONCLUSION

By solving a novel deterministic optimization criterion andsatisfying the constraint in a soft manner, we have obtained anew constant modulus algorithm for blind equalization of QAMsignals. We have derived an approximate bound for the range ofstep-sizes for which a generic complex-valued constant mod-ulus algorithm would remain stable if initialized close to a zero-forcing solution. We have validated our theoretical result forshort as well as long channels and equalizers.

APPENDIX ISTOCHASTIC BOUND ON STEP-SIZE

Let be a generic Bussgang-type[3] weight-update. Subtracting the zero-forcing solutionfrom both sides, we obtain , where

. For an algorithm in CM family, a genericerror-function can be expressed as [9],where is a real function about . With

being small enough, a first-order com-plex-valued Taylor series expansion of can be written as [10]

(12)

Fig. 2. SCS-CMA��� � � versus � for � � ��.

Using second-order odd-symmetry property of QAM signal([10], Lemma 1), we can show that

(13)

Exploiting (12)–(13), we get

(14)

Taking the expected value of this expression, we can find a re-cursion for . By virtue of the Bussgang property, we have

(refer to [3] for proof); this leads to

(15)

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4 IEEE SIGNAL PROCESSING LETTERS, VOL. 17, NO. 1, JANUARY 2010

where we have assumed that the vectors and are inde-pendent of each other. The stability of (15) requires that

(16)

Denoting , we express (15) as, where is the

identity matrix. Following the classical reasoning in leastmean squares adaptive filtering, an estimation of the timeconstant can be obtained as and thealgorithm converges in the mean if the step-size is selectedsuch that , where andare respectively the minimum and maximum eigenvalues of

.Multiplying (14) by its conjugate transpose and taking the

expected value, we obtain a recursion for the autocorrelation:

(17)

where stands for the trace of the bracketed matrix and. The evaluation of (17) assumes that the channel

is long enough for the fourth-order moments of to be wellapproximated by those of a Gaussian vector [11]. Now usingthe Fisher’s diagonalizing theorem [11], we can transform (17)to a diagonalized-matrix difference equation

(18)

where, by using orthogonal transformation , we diagonalizeand , such that, and . Defining

andand equating the diagonal elements of the

matrix on the left side of (18) with the corresponding diagonalelements of the matrix sum on the right side of this equalityyields the vector difference equation

(19)

The convergence of depends on the matrix . It will con-verge if and only if the eigenvalues of are all within the unitcircle. Following the steps provided in [11] and ensuring eigen-values lie within the unit circle, the range of step-size that guar-antees stability of (19) is thus obtained as

(20)

The mean-square stochastic stability bound (20) generalizesthe work in [12] in two aspects; firstly, we considered com-plex-valued quantities and (due to which) our result differs fromthe real-valued case in [12], and secondly, we presented the re-sult for an arbitrary (constant modulus) Bussgang error-func-tion. Also, comparing our result (20) with the bound evaluatedin [13] for real-valued CMA(2,2), it is noticed that our evalu-ation procedure as well as the result (20) are noticeably muchsimpler and more meaningful. Moreover, the intermediate re-sult (the time constant ) can be seen to be in agreement with,and a further generalization of, the result reported in [14] forcomplex-valued CMA(2,2). In SCS-CMA , where we have

, the requirement for sta-bility

(21)

is always true for QAM signals due to their sub-Gaussian nature.Substituting the values of and , we get (11).

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[1] D. N. Godard, “Self-recovering equalization and carrier trackingin two-dimensional data communications systems,” IEEE Trans.Commun., vol. COM-28, pp. 1867–1875, 1980.

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[8] Sig. Process. Information Base [Online]. Available: http://spib.rice.edu/spib/microwave.html

[9] B. Baykal, O. Tanrikulu, A. G. Constantinides, and J. A. Chambers,“A new family of blind adaptive equalization algorithms,” IEEE SignalProcess. Lett., vol. 3, no. 4, pp. 109–110, 1999.

[10] B. Lin, R. He, and B. Wang, “The excess mean-square error analysesfor Bussgang algorithm,” IEEE Signal Process. Lett., vol. 15, pp.793–796, 2008.

[11] B. Fisher and N. J. Bershad, “The complex LMS adaptive algorithm-transient weight mean and covariance with applications to the ALE,”IEEE Trans. Commun., vol. ASSP-31, no. 1, pp. 34–44, Feb. 1983.

[12] V. H. Nascimento and M. T. M. Silva, “Stochastic stability analysis forthe constant-modulus algorithm,” IEEE Trans. Signal Process., vol. 56,no. 10, pp. 4984–4989, 2008.

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