IEEE COMMUNICATIONS LETTERS, VOL. 13, NO. 8, AUGUST 2009 609

Adaptive Blind Equalization forMIMO Systems under α-Stable Noise Environment

Li Sen, Qiu Tianshuang, and Zha Daifeng

Abstract—The presence of non-Gaussian impulsive noise inwireless system can degrade the performance of existing equal-izers and signal detectors. In this paper, the problem ofblind source separation and equalization for MIMO systemsin heavy-tailed impulsive noise is studied. A generalized multi-user constant modulus cost function by employing the frac-tional lower-order statistic of the equalizer input signal isproposed. The associated adaptive blind equalization algorithmbased on a stochastic gradient descent method is defined asfractional lower-order multi-user constant modulus algorithm(FLOS MU CMA). Computer simulations are presented to il-lustrate the performance of the new algorithm

Index Terms—α-stable distribution; constant modulus algo-rithm; fractional lower-order statistics; MIMO equalization.

I. INTRODUCTION

THE blind source separation and equalization problemfor MIMO systems arises in many applications, such

as spatial division multiple access (SDMA) where multipledigital signals that originate from different users are transmit-ted through the linear channel and received by an array ofantennas to exploit spatial diversity. The signals at the outputof the antennas are corrupted by both intersymbol interference(ISI) and interuser interference (IUI). To compensate forthe ISI, and also to separate different signals, a multi-userconstant modulus algorithm (MU CMA) was proposed underGaussian noise assumption [1], [2]. In order to improve theconvergence properties, the quasi-Newton multi-user constantmodulus algorithm was formulated in references [3]–[5]. How-ever, in many practical problems, the noise encountered ismore impulsive in nature than that predicted by a Gaussiandistribution. Examples are underwater acoustic noise, lowfrequency atmospheric noise, and many types of man-madenoise [6]. There exists a class of distributions called α-stabledistributions that can be used to model these types of noises[7]. In order to handle robustly α-stable noise and interferencein the data, the authors of [8-9] proposed a FLOS CMA

Manuscript received November 22, 2008. The associate editor coordinatingthe review of this letter and approving it for publication was S. Buzzi.

L. Sen is a Ph.D. candidate with the Electronic and Information EngineeringCollege, Dalian University of Technology, China. She is also a lecturer withInformation Science Technology College, Dalian Maritime University, China(e-mail: listen@newmail.dlmu.edu.cn).

Q. Tianshuang is with the Electronic and Information Engineering College,Dalian University of Technology, China (e-mail: qiutsh@dlut.edu.cn).

Z. Daifeng is with the College of Electronic Engineering, JiuJiang Univer-sity, China (e-mail: zhadaifeng@163.com).

This work was supported in part by the National Science Foundation ofChina under Grant 60772037 and 60872122, the Natural Science Foundationof Liaoning Province under Grant 20082142, and the Science ResearchProgram of the Education Department of Liaoning Province under Grant2008S027.

Digital Object Identifier 10.1109/LCOMM.2009.081982

algorithm based on the fractional lower-order constant mod-ulus property of the signal of interest, and studied the lockand capture properties of the FLOS CMA. In this paper, wegeneralize the MU CMA cost function firstly by employingthe fractional lower-order constant modulus property of theequalizer input signals like the FLOS CMA, and secondlyby utilizing fractional lower-order cross-correlation insteadof cross-correlation because of the second-order moments ofstable distribution is non-existence. The associated adaptiveblind equalization algorithm based on a stochastic gradientdescent method is defined as fractional lower-order multi-userconstant modulus algorithm (FLOS MU CMA). Computersimulations are presented to illustrate the performance of thenew algorithm.

II. α-STABLE DISTRIBUTION

The α-stable distribution does not have closed form ofp.d.f. of unified form. It can be conveniently described by itscharacteristic function as

ϕ (t) = e{jat−γ|t|α[1+jβsgn(t)�(t,α)]} (1)

where �(t, α) = tan απ2 ,if α �= 1;�(t, α) = 2

π log |t|,ifα = 1. α (0 < α ≤ 2)is the characteristic exponent thatmeasures the thickness of the tails of the distribution andthe smaller α is, the thicker the tails are. γ is the dispersionparameter. β is the symmetry parameter. a is the locationparameter. If β = 0, the distribution is symmetric and theobservation is referred to as the SαS (symmetry α -stable)distribution. When α = 2 and β = 0, the α-stable distributionbecomes a Gaussian distribution. An important differencebetween the Gaussian and the α-stable distribution is thatonly moments of order less than α exist for the α-stabledistribution. As the non-existence of the second moments ofstable distribution, the cross-correlation does not make sense.Let us now consider two stable random variables η1 and η2,their qth fractional lower-order cross-correlation is defined as[10], [11] rq

η1,η2= E

{η1η

<q−1>2

}, where η<q> = |η|q−1

η∗.As the reason above, we can get that the value of q is restrictedin q < α. For q = 2, the qth fractional lower-order cross-correlation gives a regular second order cross-correlation. Thenotations ()T and ()∗ denote transpose and complex conjugate,respectively.

1089-7798/09$25.00 c© 2009 IEEE

610 IEEE COMMUNICATIONS LETTERS, VOL. 13, NO. 8, AUGUST 2009

III. FRACTIONAL LOWER-ORDER MULTI-USER

CONSTANT MODULUS ALGORITHM

Let us consider an n-input m-output MIMO-FIR system .The received signal at the jth receiving antenna is defined by

xj (k) =n∑

i=1

si (k) ⊗ hji + vj (k) , j ∈ {1, · · · , m} (2)

Where ⊗ denotes the convolution of the vector sequences,si (k)is the input vector form the ith source, hji is the channelimpulse response from the ith source to the jth receivingantenna of order L. vj (k) is the additive α-stable noise at thejth receiving antenna. Each equalizer output can be writtenas

yl (k) =m∑

j=1

xj (k) ⊗ wlj (k) , l ∈ {1, · · · , n} (3)

Where wij (k) is the equalizer impulse response fromthe jth receiving antenna to the ith equalizer output,xj (k) is the input vector at the jth equalizer. Definethe following vectors Wl(k) = [wl1(k) · · ·wl1(k − N +1) · · ·wlm(k) · · ·wlm(k−N +1)]T , X(k) = [x1(k) · · ·x1(k−N +1) · · ·xm(k) · · ·xm(k−N +1)]T , where N is the lengthof equalizer. Then equation (3) can be described as

yl (k) = XT (k)Wl (k) (4)

The MU CMA cost function for the lth equalizer is definedby [1]

J lMU CMA = E

{(|yl (k)|2 − R2

)2}

+ κ

l−1∑m=1

N+L∑δ=−(N+L)

|rm,δ (k)|2 (5)

The first part of cost function (5) is the constant moduluscost, and the second part rm,δ (k) = E {yl (k) y∗

m (k − δ)} isthe cross-correlation between the lth equalizer and the mthequalizer output with lag δ. κ ∈ R+ is the mixing parameter.From the definition, we can see that the cost function (5)involves fourth-order moments of the measurements, and asthe use of second- or higher-order statistics in effect amplifiesthe noise in the presence of α-stable noise, we use thefractional lower-order constant modulus cost instead of theconstant modulus cost like reference [8-9]. At the same timethe cross-correlation rm,δ (k) in cost function (5) is also asecond order statistics, so in this paper, we use fractionallower-order cross-correlation instead of the cross-correlation.Then the cost function of the introduced fractional lower-ordermulti-user constant modulus algorithm (FLOS MU CMA)has the expression

J lFLOS MU CMA = E

{(|yl (k)|p − Rp)

2}

+ κ

l−1∑m=1

N+L∑δ=−(N+L)

∣∣∣rpm,δ (k)

∣∣∣2 (6)

where Rp = E{|s|2p

}/E {|s|p} is the fractional lower-order

dispersion constant, rpm,δ (k) = E

{yl (k) y<p−1>

m (k − δ)}

isthe pth fractional lower-order cross-correlation between the

−2 −1 0 1 2−2

−1

0

1

2

Re

Im

(a)

−2 −1 0 1 2 −2

−1

0

1

2

Im

Re

(b)

Fig. 1: Retrieved signal of MU CMA.(a) the output of equal-izer 1 (b) the output of equalizer 2.

−2 −1 0 1 2−2

−1

0

1

2

Re

Im

(a)

−2 −1 0 1 2−2

−1

0

1

2

Re

Im

(b)

Fig. 2: Retrieved signal of FLOS MU CMA.(a) the output ofequalizer 1 (b) the output of equalizer 2.

lth equalizer and the mth equalizer output with lag δ. It canbe seen that the cost function (6) involves 2p–order momentsof the measurements and as the reason that only momentsof order less than α exist for the α–stable distribution, thusp < α/2. When p = 2 the FLOS MU CMA algorithm is rightthe MU CMA algorithm, so the FLOS MU CMA algorithmcan be regarded as a generalized MU CMA algorithm. Theassociated adaptive update equation based on a stochasticgradient descent method is described by

Wl (k + 1) = Wl (k) − μf le (k)X∗ (k) (7)

where

f le (k) = p (|yl (k)|p − Rp) |yl (k)|p−2 yl (k)

+κl−1∑m=1

N+L∑δ=−(N+L)

r̂pm,δ (k) |ym (k − δ)|p−2

ym (k − δ)

and r̂pm,δ (k) is the estimator of rp

m,δ (k), with λ ∈ (0, 1]it canbe recursively computed as

r̂pm,δ (k) = λr̂p

m,δ (k − 1) + (1 − λ) yl (k) y<p−1>m (k − δ)

(8)

IV. COMPUTER SIMULATIONS

The effectiveness of the new algorithm is tested by sim-ulating a 2-input and 4-output FIR system, which impulseresponse is defined in reference [2]. The input sequences are

SEN et al.: ADAPTIVE BLIND EQUALIZATION FOR MIMO SYSTEMS UNDER α-STABLE NOISE ENVIRONMENT 611

0 5 10 150

1

k

|C11

(k)|

(a)

0 5 10 150

1

k

|C12

(k)|

(b)

0 5 10 150

1

k

|C21

(k)|

(c)

0 5 10 150

1

k |C

22(k)

|

(d)

Fig. 3: The impulse response of combined channel andFLOS MU CMA equalizer system after 80000 iterations.

0 2 4 6 8−25

−20

−15

−10

−5

0

samples

FLOS_MU_CMAMU_CMA

TI 1

(dB

)

X104

(a)

0 2 4 6 8−15

−10

−5

0

5

10

15

samples

TI

FLOS_MU_CMAMU_CMA

X10 4

2 (

dB

)

(b)

Fig. 4: Convergence properties of the MU CMA andFLOS MU CMA methods (a) equalizer 1 (b) equalizer 2.

independent with a 4-QAM modulation scheme, the charac-teristic exponent of the stable noise is α = 1.85. As the char-acteristic of the stable distribution makes the standard SNRmeaningless, then a new SNR measure, generalized signal-to-noise ratio (GSNR), is defined as GSNR = 10log10σ

2s

/γv ,

where σ2s is the variance of the signal, γvis the dispersion

parameter of the stable noise. The equalizer length is N = 9,and the other equalizer parameters are λ = 0.99, GSNR =20dB and κ = p = 0.875. The constellation diagrams ofthe output of the MU CMA and FLOS MU CMA equalizerare shown in Fig. 1 and Fig. 2, respectively. We can see thatthe signal constellation of MU CMA is not as well defined,but the retrieved signal of FLOS MU CMA method is tightlyclustered about the four constellation points.

The impulse response of the combined channel and equal-izer system from the ith source to the jth equalizer outputis

Cij (k) =m∑

q=1

hqi ⊗ wjq (k) (9)

and it satisfies [12]

|Cij (k)| ={

δ (k − d) , if j = i0, if j �= i

(10)

which means that each equalizer is able to capture the inputwith a time-delay d . In Fig. 3, the impulse response of

combined channel and FLOS MU CMA equalizer system isgiven. It was shown that the two sources are separated and thefirst equalizer output retrieves the first source and the secondoutput retrieves the second sources.

To see how the method adjusts the equalizer parametersadaptively to suppress both the ISI and IUI, define totalinterference (TI ) as

TIj =

∑i,k

|Cij (k)|2 − maxi,k |Cij (k)|2

maxi,k |Cij (k)|2 (11)

Fig. 4 illustrates the TI of the MU CMA andFLOS MU CMA methods, it is apparent that theFLOS MU CMA has a smoother convergence behaviorthan the MU CMA. The figures demonstrate that occurrenceof noise outliers during the adaptation procedure have anadverse affect to the converge performance of the MU CMAbut do not affect as much the proposed FLOS MU CMA.

V. CONCLUSION

A new cost function is proposed to solve the problem ofblind source separation and equalization for MIMO systemsin stable noise based on the fractional lower-order statistic ofthe equalizer input signal. The associated adaptive algorithmFLOS MU CMA is derived based on a stochastic gradient de-scent method. Simulation results have illustrated the effectivesource separation properties of the new algorithm.

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