IEEE SIGNAL PROCESSING LETTERS, VOL. 13, NO. 7, JULY 2006 425

New Multimodulus Blind EqualizationAlgorithm With Relaxation

Shafayat Abrar, Student Member, IEEE, and Syed Ismail Shah, Member, IEEE

Abstract—A new multimodulus algorithm is presented for blindequalization of complex-valued communication channels. Theproposed algorithm is obtained by solving a novel deterministicoptimization criterion that constituted the dispersion minimiza-tion of a priori as well as a posteriori quantities, leading to anupdate equation having a particular zero-memory continuousnonlinearity. Analyses of automatic phase-recovery and inter-ference cancellation capabilities of the proposed algorithm areprovided.

Index Terms—Adaptive equalizers, blind equalizationalgorithms.

I. INTRODUCTION

WE address the problem of blind equalization of linearchannels in digital communication systems that em-

ploy quadrature amplitude modulation (QAM). Assuminga time-invariant channel, the channel and equalizer outputsat the Baud rate are given byand , respectively, where

is the ( -tap FIR) channel impulse response,is the ( -tap FIR) equalizer

vector at time instant , and are the channel inputand additive noise sample, respectively, at time instant ,

, superscripts and denotetranspose and Hermitian transpose, respectively, and subscripts

and denote the real and the imaginary parts of the complexentity, respectively. We are interested in deriving a stochasticgradient blind equalization algorithm based on multimoduluscriterion. Consider first the multimodulus criterion [1]–[3]

(1)

The above cost function yields the following stochastic gradientalgorithm:

(2)

where and. This criterion minimizes the dispersion of the real

Manuscript received October 18, 2005; revised January 9, 2006. Part ofthis work was presented at the IEEE International Conference on Microelec-tronics (ICM’2005) and International Workshop on Frontiers of InformationTechnology (FIT’2005). The associate editor coordinating the review of thismanuscript and approving it for publication was Dr. Philip Schniter.

S. Abrar is with the Department of Electrical Engineering, COMSATSInstitute of Information Technology, Islamabad 44000, Pakistan (e-mail:shafayat@ieee.org).

S. I. Shah is with the Department of Computing and Technology, Iqra Uni-versity, Islamabad 44000, Pakistan (e-mail: ismail@iqraisb.edu.pk).

Digital Object Identifier 10.1109/LSP.2006.871860

and imaginary parts of the a priori output away from statis-tical constants and , respectively. The stochastic gradientalgorithm (2) drops the expectation operator and minimizesthe resulting cost function by performing one iteration persample period. However, after having the filter estimate ,we want to adapt it by considering the following instantaneousoptimization problem:

(3)

where we denote as the a posteriori output of theequalizer. It is obvious that we can minimize this cost functionperfectly, leading to and , ,while leaving largely undetermined. To fix the degree offreedom in , we impose that remains as close aspossible to its prior estimate , while satisfying the constraintsimposed by the new data, leading tosubject to , . Using Lagrange multi-pliers, we may formulate our optimization problem as follows:

(4)

Notice that, for , the above cost function becomes anequivalent form of the cost function used in [4] and [5] to obtaintwo different (normalized) multimodulus algorithms (MMA),where the constraints in (4) were satisfied hardly and softly, re-spectively. Consider a normalized adaptive algorithm

, where is an appropriateblind estimate of the desired signal based on . We obtain

. This shows that thea posteriori output is a convex combination of the a priorioutput and the blind estimate . Hence, will be closerto the than (where the step-size controls theextent to which approaches ). It provides us a heuristicidea that better blind equalization algorithms may be obtained ifboth a posteriori and a priori outputs are forced to come closerto the blind estimate. From a multimodulus perspective, we candevelop a cost function to minimize (some) joint dispersion of

and away from statistical constants, leading to a modifiedform of (4) as follows:

(5)

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426 IEEE SIGNAL PROCESSING LETTERS, VOL. 13, NO. 7, JULY 2006

where , . For tractable derivation, we use and. We differentiate (5) with respect to and set that

to zero:. Defining ,

, , taking transpose (for nota-tional simplicity), and applying the constraints (5), we get thefollowing vector-term expressions:

(6)

(7)

We post-multiply (6) and (7) with and , respectively, andsubtract the resulting expressions to get

. Similarly, we post-multiply (6) and (7)with and , respectively, and add the resulting expressionsto get . At this point, wedeviate a little and modify the minimization procedure of (5) byexploiting the principle of relaxation [6], [7]. This principle in-troduces a step-size so that the constraints are retained as softconstraints unless , leading to

(8)

(9)

Now we post-multiply (8) and (9) with and , re-spectively, and subtract them to get

. Similarly, we post-multiply (8)and (9) with and , respectively, and add them to get

. These results lead tothe following update equation:

(10)

If , then we can assume ,where or , to yield a much simpler expression givenby

. Note that the use of normalizationfactor in the weight adaptation process does not helpmuch in convergence for nonconstant modulus (or multimod-ulus) signals [8]. Based on this result, we suggest to removeit. Second, the four-quadrant symmetry of QAM constellationallows us to write , leading to

(11)

The update (11) represents a new MMA that becomes the sameas (2) for . Next we find the values of and such thatthe mean of the equalizer-deviation is zero. In fact, we requirethat at the equilibrium point of equalizer, whereis the bulk delay. Since , solving for the

Fig. 1. (a) Contour plot of J (12) showing four minima at �R � |R inthe signal-space (p = 2). (b) J (12) minimizes the dispersion of the real andimaginary parts of the equalizer output around two points �R and �R ,respectively.

th tap, we get. Next

, we get

which gives and. Notice that the update rule (11) can

be interpreted as a stochastic approximation of a gradientdescent algorithm for the cost function given by

,leading to

(12)where is constant of integration that ensures , .The stochastic cost function (12) has minimum values at

in the complex plane, as shown in Fig. 1(a). It impliesthat (12) penalizes the dispersion of the real and the imaginarypart of away from and , respectively, as depicted inFig. 1(b).1 Note that many existing algorithms are special casesof the proposed algorithm (11). For example, MMAs reported in[14]–[16] are the same as (11) with . Similarly, MMAs re-ported in [1], [9], and [17] are the same as (11) with . Alsonotice that (11) can be considered as a complex-valued versionof the real-valued algorithm reported in [18, eq. (15)] or a sim-plified version of the block-processing-based MMA reported in[19, eq. (9)-(10)]. In addition, the proposed cost algorithm (11)can be considered as a one-dimensional version of the two-di-mensional algorithm reported in [20] and [21].

II. PHASE RECOVERY CAPABILITY

The inherent phase recovery capability of the MMA (2)has been explored in detail in [22]–[25]. In this section, weinvestigate the phase recovery capability of the proposedMMA (11). Observe that the phase-offset only affects thefirst and the third terms in the cost function (12). Now as-sume QAM alphabet contains four points ,

1There is a misconception about the zero-error contour of MMA. Mistak-enly, MMA was realized of being capable of forcing the equalizer outputs ontostraight-line contours [9, p. 45]]–[11, p. 202]. However, as a matter of fact, theMMA forces the equalizer outputs to lie on four-point contours [12], [13, pp.95-110].

ABRAR AND SHAH: NEW MULTIMODULUS BLIND EQUALIZATION ALGORITHM WITH RELAXATION 427

Fig. 2. Defining � and � in first quadrant for an arbitrary value of p.

, , and ,where , . The functionmaps these four points to the same point as . Theexpectation can be written as

toyield Now assume that the equalizeroutput is subjected to a phase-offset, such that, .For the moment, we restrict to be in a range such that both

and lie in the same quadrant. If ,then this restriction corresponds to

. We defineto describe the effect of

phase-offset on the cost function (12), leading to

since

, . For , is expected to be greater than .However, it is observed that may become smaller than

for some values of . This behavior is illustrated in Fig. 2;observe that it is the middle region (specified with angle ,

, in the first quadrant), where the value of is higherthan . We also observe two lines at which .At the line closer to the -axis, we have , and atthe line closer to the -axis, we have .Considering and together and solving for the lowerline with substitution , we get

(13)

Letting , we take the th root on both sides of (15).Under the limit , we obtain for (15), theLHS ,and the RHS

. Due to considering the lower line,the equation has a unique solu-tion , which implies and

. It is an important result that describesan improvement in phase recovery capability with an incrementin (due to the expansion of the middle region).

III. INTER-SYMBOL INTERFERENCE (ISI) OPTIMIZATION

Let be the exact solution of the blind equalizationproblem, such that subject

to , where is the channel impulse response. Letbe a stochastic approximation estimate based on the partic-

ular (finite) realization of data , such that for

which . The equalization quality in termsof residual inter-symbol interference (ISI) can be expressed asISI ,where is the transmitted data sequence, is the re-ceived sequence, and is the estimation error. As-suming is small and using second-order Taylor expan-sion, the linearization of around gives

.Therefore, the error covariance matrix can be approximatedas . Underperfect signal recovery assumption, , we find

for

i.i.d. property for(14)

which gives , where isan -by- identity matrix. Next we find

fori.i.d. property for

(15)

which gives . Finally, ISI, where

and .Due to QAM four-quadrant symmetry, we have

, leading to ISI. The

evaluation of these expectations can be simplified if we considerthe case of a very high-order square-QAM signal. In this way,the uniformly distributed real and imaginary parts of the signalcan be considered continuous. We use generalized Gaussiandistribution (parameterized in )2 to express these expecta-tions, leading to ISI

. Forcontinuous and uniform distribution, we have , for whichwe can substitute , to yield an upper-bound forISI leading to ISI . It is an importantresult that shows that (even for a very large square-QAM signal)an increment in can yield a reduced residual ISI floor for thegiven filter length.

2For a generalized Gaussian variable X , we have p (x) =(�=(2��(1=�))e and IE[jXj ] = (� �((p + 1)=�)=�(1=�)),where �(a)

:= x e dx. For � � p, the Euler’s reflection formula

gives �(p=�) � �[sin(p�=�)] � �=p.

428 IEEE SIGNAL PROCESSING LETTERS, VOL. 13, NO. 7, JULY 2006

Fig. 3. Residual ISI traces for the existing algorithm (2).

Fig. 4. Residual ISI traces for the proposed algorithm (11).

IV. SIMULATION RESULTS

In [26], we have discussed that the existing MMA (2) ex-hibits poor convergence behavior as goes beyond 4. It is dueto the fact that, for , the error function of (2) (defined as

)becomes insignificant (that is, approaching to zero) for smallvalues of , making the algorithm fail to converge. It is also re-ported in [26] that the error function of the proposed algorithm(11) [defined as ] issignificant and thus offers a successful convergence.

To corroborate the above discussion, we simulated a Baudspaced seven-tap equalizer in a voice-band telephone channel(taken from [27]) with additive white Gaussian noise SNR

dB . With center-spike equalizer initialization, the residualISI is measured for 16-QAM by averaging 200 independent re-alizations of noise and data source. In Fig. 3, the ISI traces ob-tained from (2) are depicted for , 2, 3, and 4. However,for , the algorithm (11) yielded no stable convergence forany value of step-size. On the other hand, as depicted in Fig. 4,the proposed algorithm (11) is yielding stable convergence for

to 6 with consistent lowering in ISI floor (without af-fecting the convergence speed).

ACKNOWLEDGMENT

The authors would like to thank the anonymous reviewers andthe Associate Editor, Dr. P. Schniter, for their feedback, whichhas improved the quality of this manuscript.

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