IEEE SIGNAL PROCESSING LETTERS, VOL. 17, NO. 9, SEPTEMBER 2010 791

Adaptive Solution for Blind Equalization andCarrier-Phase Recovery of Square-QAM

Shafayat Abrar, Graduate Student Member, IEEE, and Asoke Kumar Nandi, Senior Member, IEEE

Abstract—In this letter, we adaptively optimize the equalizeroutput energy to obtain a joint blind equalization and car-rier-phase recovery solution. The resulting (multimodulus) updatealgorithm possesses a particular zero-memory Bussgang-typenonlinearity. We provide evidence of good performance, in com-parison to existing adaptive methods, like RCA, MMA and CMA,through computer simulations for higher-order quadrature am-plitude modulation signalling on symbol- and fractionally-spacedchannels.

Index Terms—Adaptive equalizers, blind equalization.

I. INTRODUCTION

A BLIND EQUALIZER optimizes, through the choice ofits coefficients , a certain cost , such that its

output provides an estimate of the source up to some inde-terminacies [1]. Historically, the first-ever cost for this purposewas developed by Allen and Mazo [2] in 1974. They analyt-ically showed that the optimization of energy , whileanchoring the leading tap, is capable of inverting the channel.The potential of this idea remained unexplored until 1990, whenFeyh and Klemt [3] studied the problem

(1)

using eigenvalue analysis and discussed its inadmissibility. In1993, using order statistics, Satorius and Mulligan [4] solved1

(2)

Manuscript received March 09, 2010; revised June 16, 2010; accepted June24, 2010. Date of publication July 01, 2010; date of current version July 15,2010. The work of S. Abrar work was supported by the ORSAS (UK), the Uni-versity of Liverpool (U.K.) and the COMSATS Institute of Information Tech-nology (Pakistan). The associate editor coordinating the review of this manu-script and approving it for publication was Dr. Ricardo Merched.

S. Abrar is with the Department of Electrical Engineering and Elec-tronics, University of Liverpool, Liverpool L69 3GJ, U.K., on leave fromthe Department of Electrical Engineering, COMSATS Institute of Informa-tion Technology, Islamabad, Pakistan (e-mail: shafayat@liverpool.ac.uk;sabrar@comsats.edu.pk).

A. K. Nandi is with the Department of Electrical Engineering and Electronics,University of Liverpool, Liverpool L69 3GJ, U.K. (e-mail: aknandi@liver-pool.ac.uk).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/LSP.2010.2055853

1Notations � and � denote respectively the real and the imaginary partsof � � at time index �. For a sequence �� �, we define �������� ��

��

���� �������� ��, where ���� �� and ����� ��, respectively,denote the maximum and the minimum value in �� �.

and reported better results than those obtained from Shalvi–We-instein algorithm [5]. Recently, Meng et al. [6] formulated

(3)as a quadratic programming problem for blind equalization ofsquare-QAM and reported several impressive results. The pa-rameter denoted the largest quadrature component of .

Note that 1) costs (1)–(3) optimize subject to dif-ferent constraints, 2) costs (2) and (3) are capable of recoveringcarrier-phase, 3) cost (3) is able to restore true signal-energy,4) costs (1)–(3) have been optimized originally in block-pro-cessing manner, and 5) due to nondifferentiable constraints in(2) and (3), it is not trivially possible to use gradient-basedadaptive optimization. Motivated by these considerations, in thisletter, we present a pertinent deterministic and differentiablecost, and obtain an adaptive blind equalization and carrier-phaserecovery solution for square-QAM.

II. PROPOSED COST AND ADAPTIVE ALGORITHM

2Suppose is a cost function to be optimized forblind equalization purpose and that depends solely on the statis-tics of . The stochastic gradient-based adaptive optimizationof is obtained as [7, Ch. 6],

(4)

where the sign or is required, respectively, for maxi-mization or minimization of the cost with respectto , the asterisk denotes the complex conjugate ofthe base entity and is a small positive step-size. Notethat , and

, where .Defining to be an error-function andassuming a maximization scenario, we obtain

(5)

In principle, the satisfies Bussgang condition upon suc-cessful convergence [1], i.e., .

Based on the discussion in Section I, we present a determin-istic version of (3) involving instantaneous constraints, viz

2The baseband transmission of square-QAM symbols �� � is considered inthe presence of additive white Gaussian noise � through a moving-averagechannel �� �. An � -tap adaptive blind equalizer is employed to combat theintersymbol interference (ISI) caused by the channel. The received and equal-ized signals are � � � �� and � ��� , respectively, where��� � � � � � � � � � and � � � � � � � . Superscripts and � denote, respectively, transpose and conjugate-transpose.

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792 IEEE SIGNAL PROCESSING LETTERS, VOL. 17, NO. 9, SEPTEMBER 2010

(6)

where, for some , is defined as

ifif .

(7)

The constellation of an -alphabet square-QAM (free fromdistortion, noise and rotation) is contained in a square region ,centered at origin with perimeter . If the equal-ized sample falls inside region , then both constraints incost (6) are satisfied; we simply need to maximize . How-ever, when lies outside region , then depending on where

is residing at, either one or both of the constraints is/are vi-olated. In such a case, an ideal update would ensure thatthe resulting a posteriori output comes inside region

and the energy stays close to .The is differentiable. For some , we can show

(8)

where is the standard function. Next employingthe Lagrangian multipliers, and , we obtain

(9)Differentiating (9) with respect to , we obtain

(10a)

(10b)

where denotes either or . Solving (5) and (10), we get

(11)

If , then and the constraint is satisfied. On theother hand, the condition yields ;here, we suggest to compute such that the Bussgang condi-tion is satisfied. This consideration leads to

(12)The evaluation of (12) can be simplified by assuming that theupdate (11) is in the vicinity of an open-eye solution [1, Sect.2.8]. As a result, the output is the sum of delayed sourcesignal and convolutional noise [1]. For , (12) issatisfied due to the identical-and-independent distribution prop-erty exhibited by and ; for , however, the constraintsmay be satisfied by assuming

(13)

where the negative sign is used to update in the opposite di-rection to bring the equalized symbol inside or close to thecorner-points of region , and the is introduced to limit the

growth of . Due to the four-quadrant symmetry of QAMsignal (i.e., and , note that a singlevalue of is required to be computed for both and .

Denoting as one of the components of , we write. The is considered to be zero-mean Gaussian with

variance and pdf , where is the variance of .Now combining (12) and (13), we get

(14)Further, and

are evaluated as

where and .3

Note that the has an asymptotic value for small noise. Con-sidering a square-QAM, , andassuming and small , we evaluate (14) to get

(15)

where . Assuming a diminishing noise, we get

(16)

In Fig. 1, we demonstrate that when .We summarize our algorithm as follows:

ifif

(17)

Note that the polarity of variable determines the directionof adaptation such that the dispersion in is minimized awayfrom four corner points and this property has a majorrole in carrier-phase recovery. In [10], a term multimodulus was

3For the evaluation of �, we have considered � � ����� , where � is thevariance of additive noise and � � � is an empirically determined constant. Infuture, we aim to explore the feasibility of established adaptive methods for theestimation of convolutional noise � , like those addressed in [8] and [9].

ABRAR AND NANDI: ADAPTIVE SOLUTION FOR BLIND EQUALIZATION 793

Fig. 1. Parameter � versus ��� for some square-QAM.

Fig. 2. Carrier-phase recovery capability of ����.

coined for the algorithm which can jointly solve blind equal-ization and carrier-phase recovery; we use this terminology todenote (17) as -multimodulus algorithm .

Interestingly, the update (17) can be obtained by minimizingthe following cost:4

(19)

where the flags and are as defined in (17). Assumingand , we depict the sensitivity of cost

(19) to phase-offset in Fig. 2. Clearly, the exhibitscarrier-phase recovery capability.

Finally, note that another adaptive realization is possible ifconstraints are imposed on a posteriori output .Taking conjugate-transpose of (11) and post-multiplying it with

, we get . If, then the requirement yields

, where . Note that isfavorably negative for , and is similar to (13).

4Cost (19) is similar to the following cost of traditional MMA [10]:

��� � �� � �� (18)

where � �� �� �� � is termed as dispersion constant.

Fig. 3. Plots of ISI convergence. Each of the traces has been obtained by takingaverage of 300 Monte-Carlo realizations with independent generation of noiseand data samples. All traces in a given subplot exhibit same point of convergenceas marked by a vertical dashed line.

III. SIMULATION RESULTS AND CONCLUSIONS

We simulate adaptive equalizers implementing reduced-con-stellation algorithm (RCA) [11], constant modulus algorithm(CMA) [12], multimodulus algorithm (MMA) [10], and the pro-posed one , while considering square-QAM transmis-sion over complex-valued baud and fractionally spaced (nor-malized) channels, evaluating (transient) ISI [5, Eq.(50)] and(steady-state) symbol-error rate (SER) performances.

Experiment A: Firstly, we consider symbol-spaced equaliza-tion (TSE) of a voiceband channel [13]. A seven-tap equalizer isemployed with central spike initialization. The converging ISItraces are summarized in Fig. 3(a)–(b) for 16- and 64-QAM, re-spectively. Note that the is providing much lower ISIfloor than all others while the traditional MMA is performingbetter than RCA and CMA.

Secondly, we consider fractionally-spaced equalization(FSE) of a -spaced microwave radio channel (channel-1,SPIB [14]). We follow the multichannel equalizer architecture

794 IEEE SIGNAL PROCESSING LETTERS, VOL. 17, NO. 9, SEPTEMBER 2010

Fig. 4. Plots of SER versus ��� . Step-sizes have been selected such that(a) for TSE, both MMA and ���� acquired steady-state around 3000th,6000th and 20,000th iteration for 16-, 64- and 256-QAM, respectively, and (b)for FSE, both MMA and ���� acquired steady-state around 1500th, 3000thand 10,000th iteration for 16-, 64- and 256-QAM, respectively. The product of� and � is kept constant for ���� to maintain its convergence rate for all��� values.

as described in [12] and implement a 42-tap equalizer, wherewe have 21 taps each in even and odd sets of coefficientswith central spike initialization in even-set of coefficients. Theconverging ISI traces are summarized in Fig. 3(c)–(d) for 16-and 64-QAM, respectively. Note that the is providingremarkably much lower ISI floor than all others while the

MMA is again performing better than RCA and CMA. We haveused and 2.0, respectively, for TSE and FSE.

Experiment B: The evaluation of SER can provide the per-formance comparison over a range of SNR values and it canincorporate any degradation due to imperfect restoration of car-rier-phase and/or signal-energy. Here we simulate MMA and

over the same two channels as we used in Experiment A.In Fig. 4(a), we depict SER performances for 16/64/256-QAMover the voiceband channel. Both MMA and success-fully restored the 45 phase-offset, introduced by the channel,in all independent simulation runs.

Observe that at lower SNR values, both MMA andperformed almost identical; but, for higher SNR values,outperformed MMA for all QAM sizes. In Fig. 4(b), we depictSER results over the microwave radio channel. Again, we ob-serve that is yielding much lower SER than MMA.

We have proposed a deterministic cost for joint blindequalization, carrier-phase recovery and energy restoration ofsquare-QAM signals. We have experimentally showed that theresulting new algorithm can give better solution interms of removing ISI and low SER, under the presence ofnoise, than some existing adaptive algorithms, like RCA, CMAand MMA.

REFERENCES

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[14] [Online]. Available: http://spib.rice.edu/spib/microwave.html. SPIB,Rice Univ. Houston, TX