ON-LINE SEQUENTIAL MULTICHANNEL BLIND DECONVOLUTION: A DEFLATIONAPPROACH

Seungjin CHOI

School of Electrical and Electronics EngineeringChungbuk National University

48 Kaeshin-dong, CheongjuChungbuk 361-763, KOREAschoi@engine.chungbuk.ac.kr

Andrzej CICHOCKI

Lab for Open Information SystemsBrain Science Institute, RIKEN

2-1 Hirosawa, Wako-shiSaitama 351-01, JAPAN

cia@brain.riken.go.jp

ABSTRACT

We present a simple but efficient and powerful extension ofstandard Bussgang-type blind equalization algorithms thatis able to extract multiple source signals from their unknownconvolutive mixtures. The extraction of source signals oneby one using a deflation approach is proposed. A new adap-tive deflation algorithm which can cancel the contribution ofalready extracted source signals, is derived. This approachcan adopt any blind equalization algorithm (which can ex-tract a single source). Furthermore, it can be also applied tothe case when we do not know the number of source signalsin advance. Extensive computer simulation results confirmthe validity and high efficiency of our proposed method.

1. INTRODUCTION

Multichannel blind deconvolution has a variety of applica-tions in wireless communications, image processing, arrayprocessing, and some biomedical applications. In multi-channel blind deconvolution, an m dimensional vector ofreceived signals x�k� � �x��k� � � �xm�k��T is assumed tobe generated from an n dimensional vector of spatially in-dependent, temporally i.i.d. unknown source signals s�k� ��s��k� � � � sn�k��

T using the multi-variate linear time invari-ant filters, i.e.,

x�k� �

�Xp���

Hps�k � p� �H�q���s�k�� (1)

or equivalently in the scalar form

xi�k� �

�Xp���

nXi��

hji�psi�k � p�� (2)

Portion of this work was supported by Korea Science and Engineer-ing Foundation under the contract 981-0913-063-1 and by Brain ScienceInstitute, RIKEN, Japan

whereH�q��� �P�

p���Hpq�p (H�q��� is an unknown

(m � n) polynomial matrix with m � n, and q�i is de-lay operator such that q�is�k� � s�k � i�) represents thecharacteristics of FIR channel. The task of multichanneldeconvolution is to recover the source signals s�k� from thereceived signals x�k�, up to a scaled, permuted, and de-layed version of source signals, i.e., the estimates of sources, y�k� � �s�k� � P�D�q���s�k�, where P � IRn�n is apermutation matrix,� � IRn�n is a nonsingular scaling di-agonal matrix, and D�q��� is a diagonal matrix whose ithdiagonal element is given by q�di .

2. EXTRACTION OF SINGLE SOURCE SIGNAL

Let us consider an FIR equalizer whose the ith node outputyi�k� is described by

yi�k� ��X

p���

mXj��

wij�pxj�k � p�� (3)

where fwij�pg are FIR equalizer coefficients and fxj�k�gare the jth sensor output. For the sake of simplicity, we as-sume that source signals si�k� have unit constant modulus,i.e., jsi�k�j � �. A single source can be extracted fromthe minimization of the extension of the Constant Modulus(CM) criterion [7] which is described by

J ��

�

mXi��

E�fjyi�k�j� � �g��� (4)

Taking stochastic gradient descent, one can have the updat-ing rule for for FIR equalizer coefficients w ij�p which hasthe form of

wij�p�k � �� � wij�p�k�� �k�J

�wij�p

� wij�p�k� � �kf�yi�k��x�

j �k � p��(5)

ToNextProcessingUnit

x�

x�

x�

x�

xm

xm

x����

x����

x���m

W��

W��

W�m

U��

U��

Um�

y��k�

�

�

�

��

�

�

equalization unit deflation unit

Figure 1: The structure of mutichannel blind equalizer mod-ule

where complex conjugate is denoted by �, �k � is a learn-ing rate, and the nonlinear activation function f�y i�k�� isgiven by

f�yi�k�� � yi�k�� yi�k�jyi�k�j�� (6)

For a doubly-infinite FIR channel, the only existing min-ima of (4) correspond to points where a single source isextracted provided that the source signal is sub-Gaussian(negative kurtosis)1 [8] whenever the equalizer is doubly-infinite. For an finite order FIR channel, there also exista finite order FIR equalizer under mild conditions on FIRchannel [1, 8] (in this case, the number of sensors should begreater than the number of sources) and using (4) one canextract a single source successfully. However, with this di-rect extended CM criterion (4), the same sources might beextracted at different outputs. To avoid this problem, Inouye[4], Papadias and Paulraj [5] introduced a auxiliary con-straints and extra processing which spatio-temporally decor-relate the extracted signals. This leads to a (relatively com-plicated) iterative algorithm and it requires to know the num-ber of source signals in advance.

3. EXTRACTION OF MULTIPLE SOURCES

We present an efficient on-line approach to extract multiplesource signals one by one using the cascaded connectionof module (see Figure 1) which consists of a equalizationunit (a processing unit extracting a single source) and a de-flation unit (a processing unit eliminating the contributionof the already extracted signal from mixtures). Althoughour approach was motivated from [6, 2] where the observa-tion x�k� was restricted to the instantaneous mixtures of thesource signals s�k�, our proposed method is more generalin that it is dealing with the convolutive mixtures of s�k�.

1For super-Gaussian signals (positive kurtosis), the updating rule is de-rived from the maximization of (4) instead of minimization

In the �st module, the extracted output y��k� is describedby

y��k� �Xp

mXj��

w�j�pxj�k � p�� (7)

The equalization unit coefficients fw�j�pg can be updatedby CM algorithm,

w�j�p�k � �� � w�j�p�k� � �kf�yi�k��x�

j �k � p�� (8)

where f�yi�k�� is given in (6).Without loss of generality, we can assume that the first

extracted signal y��k� corresponds to the �st source signals��k�, i.e., y��k� � ��s��k � d��. The deflation unit co-efficients fui��pg are updated to minimize the energy (cost)function given by

� ��

mXi��

jx���i �k�j�� (9)

where

x���i �k� � xi�k��

Xp

ui��p�k�y��k � p�� (10)

Taking the stochastic gradient descent, the updating rule forfui��pg is given by

ui��p�k � �� � ui��p�k�� �kx���i �k�y���k � p�� (11)

In order to show that the learning algorithm (11) is ableto eliminate the contribution of the first extracted signal y��k�which is given by

y��k� � ��s��k � d��� (12)

from the observationx�k�, we investigate the stationary pointsof the averaged version of (11). If the learning algorithm(11) approaches steady state, we have

Efx���i �k�y���k � p�g

� Efxi�k�y�

��k � p�g � ui��p�k�Efjy��k � p�j�g

� � (13)

Then, ui��p�k� is given by

ui��p�k� �Efxi�k�y

�

��k � p�g

Efjy��k � p�j�g� i � �� � � � �m� (14)

Using the fact that y��k� is the first extracted source signal,we have

Efxi�k�y�

��k � p�g

� EfXl

mXj��

hij�lsj�k � l���s�

��k � p� d��g(15)

� ��hi��p�d���s��

where ��s� � Efjs��k�j�g. Using the result (15), the ui��p�k�

defined in (14) becomes

ui��p�k� �hi��p�d�

��� i � �� � � � �m� (16)

Thus, after a deflation processing is done, the input to theequalization unit in next module, fx���i �k�g

x���i �k� � xi�k��

Xp

hi��p�d�

����s��k � p� d��

� xi�k��Xq

hi��qs��k � q�� (17)

Therefore, we can see that the deflation algorithm given in(11) can eliminate the contribution due to the first sourcesignal s��k� from the received signals fxi�k�g. The deflated

mixture x���i �k� is fed into the next module in order to ex-tract the nd source signal, and produce the mixture witheliminating the contribution of nd extracted signal. Bycontinuing this procedure until the output of module con-verges to small value pre-specified (which means all sourcesignals are extracted), we can successfully extract all sourcesignals. We should emphasize that any other blind equaliza-tion algorithm instead of CM algorithm, can be applied toour proposed approach.

Note that the similar deflation approach to multichan-nel blind deconvolution has been introduced by Tignait [8].In [8], the batch-type deflation processing using the equa-tion (14) was applied to cancel the contribution due to thealready extracted signals. In this paper, we proposed an on-line deflation algorithm from the optimization of (9).

4. COMPUTER SIMULATIONS

4.1. Simulation 1

In this simulation, three different source signals s�k� waschosen as binary pulse-amplitude-modulated (PAM) signalswhich consist of random variables that are uniformly dis-tributed over the binary set f�����g. The main reason wechose binary sources is that the learning process can be eas-ily observed. Five sensor outputs x�k� were generated by

x�k� � H�s�k� �H�s�k � �� �H�s�k � �

� H�s�k � �� �H�s�k � ��

� Hs�k � �� (18)

where

H� �

������

��� ���� ����������� ����� ����������� ����� ������� ����� ��� ������ ���� ����

�������

H� �

������

��� ������ ���������� ���� ��� ����� �� � ���������� �� � ������������ ���� �� �

�������

H� �

������

����� ���� ���������� ���� � ������ � ���� ���� �� �� ����� ��������� ��� ����

�������

H� �

������

����� ���� ���������� ���� � ������ � ���� ���� �� �� ����� ��������� ��� ����

�������

H� �

������

��� ����� �� � ����� ��� � �� �� ���� ������������ ���� � ������ � � ���

�������

H �

������

������ ����� ������ �� ��� �� ����� �� � ��� ���� ����� ������� � ��� � �����

������� (19)

Since we have five sensor outputs, the cascaded connectionof five modules as shown in Figure 1 was constructed. Thenumber of sources, channel transfer function with its order,and sources itself were assumed to completely “unknown”.The learning rate for both equalization unit (with lengthL � � � ��) and deflation unit (with length L� � � �)was chosen as a constant value, �k � ��. The outputof each module is shown in Figure 2. It can observed thatthree source signal are well extracted, the 4th output con-verge to very small value close to zero, which implies thatthere are only three sources. Thus, the extraction processcan be stopped at the 4th module. To avoid non-causalityin deflation processing unit when we use a finite order FIRequalizer, the 2nd module is delayed by L and the 3rd mod-ule is delayed by L and so on. From the other hand, eachmodule is trained in unsupervised manner, simultaneously.

4.2. Simulation 2

The same channels given in Simulation 1 were used to gen-erate the observation vector x�k� with three 4 quadrature-amplitude-modulated (QAM) source signals. The learningrate �k � �� was used in this simulation. The outputof each module is shown in Figure 3. Three source sig-nals were successfully extracted by the first three modulesand the output of the 4th module was clustered around zero,which implies that only three source signals existed in the

observations.

5. CONCLUSIONS

We have presented a new approach to on-line multichannelblind deconvolution. Constant modular array was proposedfor multichannel blind deconvolution. It has been shownthat each module which consists of a a equalization unitand a deflation unit, can successfully extract a source signalfrom convolutive mixtures of multiple unknown source sig-nals and eliminate a contribution due to the extracted signalfrom mixtures. Cascaded connection of modules was ableto extract all source signals even when we do not have aknowledge of the number of unknown source signals.

6. REFERENCES

[1] S. Choi and A. Cichocki, “Blind signal deconvo-lution by spatio-temporal decorrelation and demix-ing,” in Neural Networks for Signal Processing VII,J. Principe, L. Gile, N. Morgan, and E. Wilson, eds.,pp. 426-435, 1997.

[2] A. Cichocki, R. Thawonmas and S. Amari, “Se-quential blind signal extraction in order specifiedby stochastic properties”, Electronics Letters, vol.33,No.1, pp.64-65, 1997.

[3] S. Haykin, ed., Blind Deconvolution, EnglewoodCliffs, NJ: Prentice Hall, 1994.

[4] Y. Inouye, “Blind deconvolution of multichannel lin-ear time-invariant systems of nonminimum phase,” inStatistical Methods in Control and Signal Processing,T. Katayama and S. Sugimoto, eds., Marcel Dekker,1997.

[5] C. B. Papadias and A. J. Paulraj, “A constant modulusalgorithm for multiuser signal separation in the pres-ence of delay spread using antenna arrays,” IEEE Sig-nal Processing Lett., vol. 4, pp. 178-181, June 1997.

[6] J. J. Shynk and R. P. Gooch, “Convergence propertiesof the multistage CMA adaptive beamformer,” in Proc.27th Asilomar Conf. on Signals, Systems, Computers,pp. 622-626, 1993.

[7] J. R. Treichler, B. G. Agee, “A new approach to mul-tipath correction of constant modulus signals”, IEEETrans. ASSP, vol. 31, pp. 459-472, 1983.

[8] J. K. Tugnait, “Blind spatio-temporal equalization andimpulse response estimation for MIMO channels us-ing a Godard cost function,” IEEE Trans. Signal Pro-cessing, vol. 45, no. 1, pp. 268-271, Jan. 1997.

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y 4

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Figure 2: The extrated signals from each modular network:(a) y��k�; (b) y��k�; (c) y��k�; (d) y�k�.

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Im[y

4(k)]

(c) (d)

Figure 3: The extrated signals from each modular network:(a) y��k�; (b) y��k�; (c) y��k�; (d) y�k�.