IEEE SIGNAL PROCESSING LETTERS, VOL. 13, NO. 12, DECEMBER 2006 745

Blind Equalization of Cross-QAM SignalsShafayat Abrar, Student Member, IEEE, and Ijaz Mansoor Qureshi

Abstract—This letter presents two new blind equalization al-gorithms that are specifically designed for cross-QAM signals.Proposed algorithms minimize the dispersion of the equalizeroutput with respect to single or multiple cross-shaped zero errorcontour(s). Simulations and analysis demonstrate the good perfor-mance of the proposed algorithms.

Index Terms—Adaptive equalizers, blind equalization.

I. INTRODUCTION

WE ADDRESS the problem of blind equalization of linearchannels in digital communication systems that employ

(odd-bit) cross-quadrature amplitude modulation (QAM). As-suming a time-invariant channel, the channel and the equalizeroutputs at time instant are given byand , respectively, where

and are the channel and equalizer im-pulse response vectors, respectively, and are the channelinput and additive noise samples, respectively, and subscriptsand denote the in-phase and the quadrature components of thecomplex entity, respectively.

There exist many multimodulus blind equalization algorithms(MMA) [1]–[3] that separately minimize the dispersion of thein-phase and the quadrature levels of received QAM signals.For square-QAM signals, where the in-phase and the quadraturelevels are independent of each other, these MMA work reason-ably well. However, for cross-QAM signals, where the in-phaseand the quadrature levels are not independent of each other, theperformances of different MMA are not very impressive. Notethat, in a (distortion-free) cross-QAM, if the in-phase level be-longs to the set

, then the quadrature level must belong to the setand vice versa (where de-

notes the number of different symbols on QAM constellation).To develop a blind equalization algorithm for cross-QAM, the

authors in [4] presented an idea to separately minimize the dis-persion of the disjoint sets and . This idea has also been thesubject of a U.S. patent [5]. However, the authors did not pro-vide any general expression for the evaluation of dispersion con-stants and the threshold that was required to determine whichdispersion constant to be used. In another attempt, the authorof [6] proposed to use a separate dispersion constant for eachsymbol such that the resulting zero-error contour was -point

Manuscript received March 2, 2006; revised May 17, 2006. The associate ed-itor coordinating the review of this manuscript and approving it for publicationwas Dr. Philip Schniter.

S. Abrar is with the Electrical Engineering Department, COMSATS Instituteof Information Technology, Islamabad 44000, Pakistan (e-mail: shafayat@ieee.org; sabrar@comsats.edu.pk).

I. M. Qureshi is with the Electrical Engineering Department, Mohammad AliJinnah University, Islamabad 44000, Pakistan.

Digital Object Identifier 10.1109/LSP.2006.879828

Fig. 1. Illustration of the cross.

cross-shaped. However, the convergence capability of that algo-rithm was found to be poor for high-order QAM signals.

II. PROPOSED ALGORITHMS

Recently, three separate groups of researchers indepen-dently presented a novel idea for the blind equalization ofsquare-QAM signals. They proposed to minimize the disper-sion of the in-phase or the quadrature level, only one at a time,depending on their relative absolute amplitudes [7]–[9, Chap.15]. This scheme resulted in the minimization of the dispersionw.r.t. a square-shaped zero-error contour, and for this reason,it was termed as the constant square algorithm (CQA) [7]. Theidea was realized by minimizing the following cost function:1

CQA (1)

Notice that is an expression ofsquare with radius . If we replace this expression with the ex-pression of cross, then the resulting cost function will be able toyield a cross-shaped zero-error contour over the signal-space.As a result, the equalizer will be able to minimize the disper-sion w.r.t. a cross-shaped contour without determining the set

or to which symbols belong. Consider the expression ofthe cross on -plane as follows:

where , , and(refer to Fig. 1). Using the expression of cross, we

can develop the following criterion for the blind equalization ofcross-QAM signal:

(2)

where is given by

(3)

The zero-error cross-shaped contour, exhibited by (2) is de-picted in Fig. 2(a) for . To obtain a stochastic gradient

1Readers are referred to [10] for a detail analysis of (1).

1070-9908/$20.00 © 2006 IEEE

746 IEEE SIGNAL PROCESSING LETTERS, VOL. 13, NO. 12, DECEMBER 2006

adaptive algorithm, we drop the expectation in (2) and minimizeit w.r.t to obtain the following:

(4)

where , and and are obtained as follows:

The algorithm (4) is named constant cross algorithm (CXA).The optimum values of and in (4) are described inAppendices A and B, respectively. Note that CXA minimizesthe dispersion of the equalized symbols w.r.t. a cross-shapedcontour. However, due to the presence of single contour, CXAresults in a significant misadjustment in steady state for largeQAM constellations, even if the equalizer has converged suc-cessfully with correct orientation.

The problem of misadjustment in Bussgang-type algorithmsdue to the single contour can be minimized by introducing mul-tiple contours. This idea has recently been discussed in the sce-nario of a multimodulus algorithm [11]. According to this idea,increased number of contours can be obtained by the joint useof dispersion constant and the sliced symbol (where isthe decision made on ). The idea was realized by replacingthe dispersion constant with , where is some suit-able positive-valued even-symmetrical function. As a result, thevalues of the resulting contours become proportional to the mag-nitude of . So the dispersion in the small and large values of

are, respectively, minimized w.r.t. small- and large-valuedcontours.

To introduce multiple contours in CXA, we usein the CXA cost function (2) as follows:2

(5)

where is given by

(6)

The adaptive algorithm for (6) is given by

(7)

The algorithm (7) is named sliced constant cross algorithm(SCXA). The mesh and the contour plots of (5) are depicted in

2If J = (1=pq) [jjf(y)j � R j ] is a cost function for some Buss-gang-type blind equalization algorithm, then its (heuristic) sliced version can beobtained as J = (1=pq) [jjf (y)j �jf(a)j R j ], where r � s = p�r.If p is an even number, then r and s can be selected as r = s = p=2 toease the computational cost. Moreover, based on an idea which is recently pre-sented in a U.S. patent [12], it is deduced that the slicing can also be obtainedas J = (1=pq) [jf(a)j � jjf(y)j � R j ].

Fig. 2. J and J surfaces over signal-space for 32-QAM.

Fig. 3. ISI traces for 32-QAM signaling.

Fig. 2(b). The optimum values of and in (7) are describedin Appendices A and B, respectively.

III. SIMULATION RESULTS

The CXA and SCXA were applied in a simulation of fouroutdoor wireless channel models. The microwave channelmodels “chan1,” “chan2,” “chan4,” and “chan5” in the SignalProcessing Information Base [13] were used with a -spaced40-tap FIR equalizer. The residual ISI traces (as defined in [14])were measured and compared as a performance parameter forCXA, SCXA, and the traditional constant modulus algorithm(CMA) on 32- and 128-QAM signaling for SNR dB.Each of the ISI traces was obtained by averaging 50 MonteCarlo experiments. Single-spike initialization was consideredin all cases such that the tap-20 was initialized as one and therest were set to zero. The simulation results are summarized inFigs. 3 and 4.

Notice that, irrespective of channels and QAM sizes, bothCXA and SCXA are outperforming CMA by offering much

ABRAR AND QURESHI: BLIND EQUALIZATION OF CROSS-QAM SIGNALS 747

Fig. 4. ISI traces for 128-QAM signaling.

Fig. 5. Cost for CXA and SCXA under phase-offset. The jagged appearanceof the cost for SCXA is the consequence of the use of sliced symbols within itscomputation.

faster convergence speed. In addition to fast ISI mitigation capa-bility, CXA and SCXA are also found capable of recovering thecorrect orientation of QAM constellation. The effect of phase-offset on CXA and SCXA cost functions is depicted in Fig. 5 fora cross-QAM, where SCXA can be noticed to be more sensitiveto the phase-offset than CXA.

We also compared the experimental excess mean square error(EMSE) and the theoretical EMSE for the CXA and SCXA asthe function of step-size in a noise-free environment and ide-ally equalized condition (refer to Appendix B for EMSE ex-pressions). The experimental values were generated as averagesover ten trials for 32-QAM using a 20-tap -spaced equalizer(with single-spike initialization). As depicted in Fig. 6, the resultdemonstrates that the theoretical values predicted by expression(14) and (16) match the experimental results reasonably well.

Fig. 6. Theoretical and experimental EMSE for CXA and SCXA.

TABLE IOPTIMUM VALUES OF k AND R FOR SOME RROSS-QAM

APPENDIX AOPTIMUM VALUES OF FOR CXA AND SCXA

We borrow from Goupil et al. [7] that the dispersion constantmust be selected in such a way that the perfect equalizer (in

the sense of the zero-forcing criterion) is the minimum of thecost criterion in a noiseless environment. The condition isexpressed by the relation [7]

(8)

where is the coefficient of the one-tap equalizer. It is describedin [7] that for a perfect channel, the coefficient should con-verge to 1. In other words, the equalizer must at least be ableto recover the power of the signal [7]. For CXA, we formu-late an optimization problem: find such that

. For , we have ,where

. We find. Solving , we get

(9)

For SCXA, we formulate: find such that. Dif-

ferentiating w.r.t. , we have. Solving and substituting , we

find , leading to

(10)

The values of (9), (10) are listed in Table I.

748 IEEE SIGNAL PROCESSING LETTERS, VOL. 13, NO. 12, DECEMBER 2006

Fig. 7. Area of regions II and IV are 0:5R (1� k) and 0:5R k, respectively,while the total area is R .

APPENDIX BOPTIMUM VALUES OF FOR CXA AND SCXA

We assume that the value of is optimum if it minimizesEMSE. The EMSE for a Bussgang-type algorithm with costfunction can be expressed as follows [15]:

(11)

where ,, and .

For CXA, with , and. After

some steps, we obtain. Taking a further

derivative w.r.t. , we obtain. Now, substituting

the values of and in (11), we obtain

(12)

where (refer to Fig. 7)

(Reg. )(Reg. )(Reg. )

(Reg. ).

(13)

Now we make use of the QAM orientation symmetries, whichenable us to evaluate for any one of the in-phase or quadra-ture components in one of the quadrants. We select the regions IIand IV in the first quadrant to evaluate . The value ofis equal to 4 and in regions II and IV, respectively. Using therelative area of different regions, we evaluate the expectation, asgiven by

Similarly, we compute the expectation in the numerator of (12),

leading to

(14)

where , ,, , and . The

optimum value of for CXA is obtained as follows:

CXA (15)

For SCXA, we findand .Next, we obtain

. We substitute the values ofand in (11) to get

(16)

where , ,, , and . The

optimum value of for SCXA is obtained as follows:

SCXA (17)

The values of (15), (17) are listed in Table I.

REFERENCES

[1] K. Wesolowski, “Self-recovering adaptive equalization algorithms fordigital radio and voiceband data modems,” in Proc. Eur. Conf. CircuitTheory Design, 1987, pp. 19–24.

[2] K. N. Oh and Y. O. Chin, “Modified constant modulus algorithm: blindequalization and carrier phase recovery algorithm,” in Proc. IEEE ICC,Jun. 1995, vol. 1, pp. 498–502.

[3] S. Abrar, A. Zerguine, and M. Deriche, “Soft constraint satisfactionmultimodulus blind equalization algorithms,” IEEE Signal Process.Lett., vol. 12, no. 9, pp. 637–640, Sep. 2005.

[4] J. Yang, J.-J. Werner, and G. A. Dumont, “The multimodulus blindequalization and its generalized algorithms,” IEEE J. Sel. AreasCommun., vol. 20, no. 5, pp. 997–1015, Jun. 2002.

[5] J.-J. Werner and J. Yang, “Technique for Improving the Blind Conver-gence of an Adaptive Equalizer Using a Transition Algorithm,” U.S.Patent 5809074, Sep. 1998.

[6] S. Abrar, “Compact constellation algorithm for blind equalization ofQAM signals,” in Proc. IEEE INCC, Jun. 2004, pp. 170–174.

[7] A. Goupil and J. Palicot, “Constant norm algorithms class,” in Proc.EUSIPCO, 2002.

[8] T. Thaiupathump, “New algorithms for blind equalization and blindsource separation/phase recovery,” Ph.D. dissertation, Univ. Pennsyl-vania, Philadelphia, PA, 2002.

[9] N. Benvenuto and U. Cherubini, Algorithms for Communications Sys-tems and Their Applications. New York: Wiley, 2002.

[10] T. Thaiupathump, L. He, and S. A. Kassam, “Square contour algo-rithm for blind equalization of QAM signals,” Signal Process. (Else-vier), February 16, 2006, accepted (galley proof is available online).

[11] S. Abrar and R. A. Axford, Jr., “Sliced multi-modulus blind equaliza-tion algorithm,” ETRI J., vol. 27, no. 3, pp. 257–266, Jun. 2005.

[12] J.-J. Werner and J. Yang, “Blind Equalization Algorithm With the JointUse of the Constant R and the Sliced Symbols,” U.S. Patent 6 493 381,Dec. 2002.

[13] [Online]. Available: http://spib.rice.edu/spib/microwave.html. SignalProcessing Information Base (SPIB), Rice Univ.

[14] O. Shalvi and E. Weinstein, “New criteria for blind deconvolution ofnonminimum phase systems (channels),” IEEE Trans. Inf. Theory, vol.36, no. 2, pp. 312–321, Mar. 1990.

[15] A. Goupil and J. Palicot, “A geometrical derivation of the excess meansquare error for Bussgang algorithms in a noiseless environment,”Signal Process., vol. 84, pp. 311–315, 2004.