online sos-based multichannel blind equalization algorithm with noise

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0165-1684/$ - se

doi:10.1016/j.sig

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Foundation of

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Signal Processing 85 (2005) 1602–1610

www.elsevier.com/locate/sigpro

Online SOS-based multichannel blind equalizationalgorithm with noise$

Xizhong Shena,b,�, Xizhi Shia

aNational Key Laboratory for Vibration, Shock and Noise, Shanghai Jiao Tong University, Shanghai 200030, ChinabSchool of Mechanical and Automation Engineering, Shanghai Institute of Technology, Shanghai, 200235, China

Received 1 July 2004; received in revised form 22 November 2004

Abstract

Multichannel blind equalization is an important task for numerous applications such as speech separation,

dereverberation, communication, signal processing and control, etc. In this paper, only second-order statistics is used to

construct a cost function and thus a new online algorithm is derived with natural gradient search method for

multichannel blind equalization. The proposed algorithm can deal with observations with lower-level noise. Simulations

and comparisons indicate the efficiency and ability of the algorithm to perform blind equalization with simple

calculation, fast convergence speed and high accuracy, and also can carry out speech separation and dereverberation.

r 2005 Elsevier B.V. All rights reserved.

Keywords: Blind equalization; Second-order statistics (SOS); Natural gradient; Non-stationary

1. Introduction

Multichannel blind equalization (MBE) is animportant task for numerous applications such asspeech separation, dereverberation, communica-tion, signal processing and control, etc. Itsresearch has received extensive interests all over

e front matter r 2005 Elsevier B.V. All rights reserve

pro.2005.03.004

ch is supported by National Natural Science

China (NSFC), No: 60372075, and also

Shanghai Education, No: 03YQHB160, and

hnology Commission of Shanghai Municipality

04ZR14126, Shanghai, China.

ng author. Tel.: +8621 54788246.

ss: [email protected] (X. Shen).

the world, and many exciting results have beenreported [1–3 and references therein, 4–9, 11, 12].Its task is to recover the source signals given onlythe observations in the sense of uncertain scaling,permutation and delay.In general, the key assumption in MBE lies in

the mutual statistical independence of sources.When sources are temporally independent andidentical distributed (i.i.d.) non-Gaussian signals,it is necessary to use higher-order statistics (HOS)to achieve source blind deconvolution (BD) andMBE. Along this assumption, many MBE algo-rithms have been developed [1–3 and referencestherein]. In these algorithms, stationary sources

d.

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X. Shen, X. Shi / Signal Processing 85 (2005) 1602–1610 1603

were considered and HOS was proved to benecessary implicitly or explicitly.On the other hand, MBE is studied only with

second-order statistics (SOS) due to the computa-tional complexity of HOS. Hua and Tugnait [5]have proved the solvability under the assumptionthat the finite-impulse-response (FIR) system isirreducible and the input signals are spatiallyuncorrelated with distinct power spectra, and thengave blind deconvolution algorithm using SOS.But Ref. [5] did not give concrete algorithm.Mitusuru Kawamoto and Yujiro Inouye [6] alsoderived an algorithm in the frequency domainunder two conditions, one is a weaker conditionthat the FIR system is equalizable by means of theSOS of the outputs, and the other is that the inputsignals are spatially uncorrelated and have distinctpower spectra. They also gave a necessary andsufficient conditions for solving the blind decon-volution.In this paper, we extend the existing algorithm

[7] of blind signal separation of additive instanta-neous mixtures to the task of MBE (joint signalseparation and deconvolution) with only SOS. Weapply correlation matrix to MBE in time domain,and further developed the proposed algorithm in[12]. We reconstruct a cost function and thenderive the novel blind equalization algorithm withnatural gradient search method based only oncorrelation of the output signals. However, unlikethe existing algorithm [6], our algorithm is devel-oped in the time domain, and can be implementedonline easily. But the algorithm proposed in [6]must calculate the inverse of matrix, see Section 4for details, and it is not desirable. The proposedalgorithm can also treat observations with noise,for correlation matrix of the equalizer output isinsensitive to noise at ta0, which is applied in thispaper. Simulations and comparisons are imple-mented online, and they indicate better perfor-mance of the algorithm. This is mainly becausecalculations are simpler relative to other algo-rithms based on HOS and it does not calculateinverse of matrix. Further, it has a fasterconvergence and more converging accuracy thanthe ones specified in Section 4 in our simulations.Also the algorithm has the ability to performspeech separation and dereverberation.

2. Problem formulation

In multichannel blind deconvolution and equaliza-tion, an n-dimensional vector of received discrete-timesignals xðtÞ ¼ ½x1ðtÞ;x2ðtÞ; . . . ; xnðtÞ�

T is assumed tobe produced from an m-dimensional vector of sourcesignals sðtÞ ¼ ½s1ðtÞ; s2ðtÞ; . . . ; smðtÞ�

T, using the mix-ture model [1–3]

xðtÞ ¼Xþ1

p¼�1

Hpsðt � pÞ þ nðtÞ, (1)

where Hp 2 Cnm is an n m-dimensional matrix ofmixing coefficients at lag p, and nðtÞ is n-vector ofnoise. And mpn, i.e., the number of observationsmust equal to or greater than that of the sourcesignals. The goal is to calculate possibly scaled,permutated and delayed estimates of the sourcesignals sðtÞ from the received signals xðtÞ only usingthe approximate knowledge of SOS of the estimate ofthe sources sðtÞ.In this paper, we estimate the source signals

directly using a truncated version of a doublyinfinite multichannel equalizer of the form [1–3]

yðtÞ ¼Xþ1

p¼�1

WpðtÞxðt � pÞ, (2)

where yðtÞ ¼ ½y1ðtÞ; y2ðtÞ; . . . ; ymðtÞ�T is an m-di-

mensional vector of outputs andWpðtÞ;�1pppþ1, is a sequence of m n-dimensional double-infinite coefficient matrices. Inpractice, we use a truncated version with certainlength. In operator form, the input and output ofthe equalizer satisfy

xðtÞ ¼ HðzÞ½sðtÞ� þ nðtÞ, (3)

yðtÞ ¼ Wðz; tÞ½xðtÞ� ¼ Cðz; tÞ½sðtÞ� þWðz; tÞnðtÞ,

(4)

where Wðz; tÞ ¼Pþ1

p¼�1 WpðtÞz�p, Hðz; tÞ ¼Pþ1

p¼�1 Hpz�p and Cðz; tÞ ¼ Wðz; tÞHðzÞ are thez-transforms of the equalizer, channel, and com-bined channel-plus-equalizer impulse responses,respectively. In the formulations, z�1 is the delayoperator, z�p½siðtÞ� ¼ siðt � pÞ. Then, the goal of

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X. Shen, X. Shi / Signal Processing 85 (2005) 1602–16101604

MBE task is to adjust Wðz; tÞ such that the zero-forcing condition [3]

limt!1

Cðz; tÞ ¼ PDKðzÞ (5)

is satisfied, where P is an m m-dimensionalpermutation matrix with a single unity entry inany of its rows or columns, D is a regular diagonalmatrix with m m dimension, and KðzÞ is an m

m regular diagonal matrix with monic monomialsas diagonal entries being. When t ! 1, the wholesystem is transparent [6], which means that thesystem can be blindly identified up to a permuta-tion P, a scaling D, and a delay KðzÞ. Thecomposite system of the two systems is illustratedin the schematic diagram in Fig. 1.We discuss MBE of MIMO system with SOS in

this paper. Some researches have proved thatMBE is possible with only SOS [3,5,6]. Thus, ourwork is based on the previous work, and our aim isto further develop them. We first give twoassumptions A1 and A2, and one definition D1with their main property P1 to perform MBEabout the model (1), and all the given content iscited by the following text.D1. Correlative matrix Ryðt; tÞ of yðtÞ

Ryðt; tÞ ¼ hyðtÞyTðt � tÞi (6)

and h�i denotes the ensemble average of *.D1 is used in constructing cost function in

Section 3, and it is trivial in defining.A1. Each source signal vector sðtÞ is nonsta-

tionary and its components are statistically un-correlated with each other with zero mean, and itscorrelation matrix is different from each other.

s(t)

n(t)

H(t)

c(z,t)

W(z,t) y(t)

Fig. 1. The composite system of the two systems, the unknown

system and the equalizer.

A2. The components of nðtÞ are mutuallyuncorrelated with correlation matrix of nðtÞ

diagonal, and Rnðt; tÞ ¼ 0; ta0.Assumption A1 is the same as is [6] except that

the input sequence sðtÞ is a zero-mean stationaryvector process in [6]. Here we assume that thesource signal vector sðtÞ is nonstationary becausethe source signals are nonstationary in test 2 of thesimulations, and the test shows the proposedalgorithm is valid for nonstationary speech signals.Assumption A2 is used to deal with system (1) withnoise, and it indicates the ability of MBE of theproposed algorithm.P1. The correlation matrix Rsðt; tÞ ¼ hsðtÞsTðt �

tÞi of source signals vector sðtÞ is diagonal and is afunctional of ðt; tÞ.

3. Algorithm of MBE

3.1. Cost function

For MBD and MBE, the typical cost function isderived from the minimal mutual information(MMI), maximal entropy (ME), independentcomponent analysis (ICA) or maximal likelihood(ML). The cumulative distribution function isneeded in all of the methods. However, since theprobability density function (pdf) is unknown inadvance, most algorithms rely on a score function[1–3] adapting to the type of sources or estimationof pdf directly [9]. The derived algorithms are hardto meet the online demands due to the problem oftheir computational complexity.It has been proved that only SOS is sufficient to

perform blind deconvolution and equalizationunder the assumption A1 [3,5,6]. For MBE ofnonstationary signals, Kawamoto and Inouyehave shown that only SOS is sufficient and (5) issatisfied in frequency domain, and also give costfunction and algorithm, which diagonalizes thepower spectral matrix SyðzÞ of equalizer outputyðtÞ [6]. Based on these foundation theories, wereconstruct equalizer not in frequency domain butin time domain, i.e., we utilize the diagonalizationof the correlation matrix of yðtÞ directly. Inaddition, when Hp ¼ WpðtÞ ¼ 0;8pa0 in Eqs. (1)and (2), that mixture is instantaneous, Choi

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Seungjin et al. [7] have derived the algorithm withnatural gradient. We extend it to the general case,MBE, with the same idea in [12], but in differentmethods in reconstructed cost function.According to the theory proposed by Kawar-

aoto and Inouye [6], the whole system is transpar-ent when diagonalizing the power spectra matrixof the output of the equalizer. Since the corre-sponding correlation matrix is the inverse Fouriertransform of power spectra matrix, the wholesystem is also transparent when diagonalizing itscorrelation matrix.Therefore, we give the following cost function:

J1ðWðtÞ; tÞ ¼Xt

kF1ðt; tÞk2F, (7)

where

F1ðt; tÞ ¼ Ryðt; tÞ � Kðt; tÞ (8)

and Kðt; tÞ ¼ diag½Ryðt; tÞ�. Here, ‘‘diag’’ meansthe diagonal matrix, and k � kF is Frobeniousnorm.By Hadamard inequality [10], one can easily

obtain that when the cost function is minimized,the correlation matrix function Ryðt; tÞ approachesto be diagonalized, and the system is transparent.In practice, we find that (8) can be changed as

Fðt; tÞ ¼ Ryðt; tÞ � KðtÞ, (9)

where KðtÞ ¼ diag½Ryðt; 0Þ�, and it works betterthan (8). Therefore, we take (9) in the followingtext without strictly mathematical derivation. Inthat case, Eq. (7) can be rewritten as

JðWðtÞ; tÞ ¼Xt

kFðt; tÞk2F. (10)

For simplicity, we take the note of JðWðtÞ; tÞ as J

without any confusion. By (9), we obtain

J ¼Xta0

XN

i;j¼1

ðRijðt; tÞÞ2þ

XN

i;j¼1

ðRijðt; 0Þ � KijðtÞÞ2,

(11)

where Rijðt; tÞ is the element of Ryðt; tÞ.From (11), we can see that when (7) is

minimized, Ryðt; tÞ at t ¼ 0 approaches to bediagonal when minimizing the second term in(11), and jRijðt; tÞj;8i; j;8ta0, is also minimized.

However, the strict mathematical proof is still onstudy.Since correlation matrix Ryðt; tÞ is insensitive to

noise at ta0, which is easily obtained byAssumption A2, the following algorithm can treatobservations with noise, see more in Section 4. AsRyðt; tÞ; ta0 is less affected by nðtÞ, seen from A2,the cost function (7) can be used to deal withobservations contaminated by lower level noise.

3.2. Natural gradient-based algorithm

Gradient descent learning is a popular methodin blind signal processing to derive a learningalgorithm for the purpose of minimizing a givencost functions. When a parameter space (on whicha cost function is defined) is a Euclidean space withan orthogonal coordinate system, the conventionalgradient gives the steepest descent direction.However, if a parameter space is a curvedmanifold (Riemannian space), an orthonormallinear coordinate system does not exist and theconventional gradient does not give the steepestdescent direction, and thus the nature gradient isefficient. The natural gradient was also shown tobe efficient in on-line learning. See Amari’s paperin 1998 and references therein for more details ofnatural gradient and related work [1,2,11,13].We now derive the algorithm with nature gradient.

First define a modified differential matrix as

dXðz; tÞ ¼Xþ1

p¼�1

dXpðtÞ ¼ dWðz; tÞWðz; tÞ�1. (12)

From (4), it is easy to obtain an important relation:

dyðtÞ ¼ dWðz; tÞ½xðtÞ� ¼ dXðz; tÞ½yðtÞ�. (13)

We then obtain from (10) with (12) and (13),

dJ

dXpðtÞ¼

Xt

Fðt; tÞh½yðt � tÞyðt � pÞT

þ yðtÞyðt � t� pÞT�i. ð14Þ

Detailed derivation can be easily obtained as in Amariand Cichocki [13]. The differential in (14) is in termsof the modified coefficient differential matrix dXðz; tÞin (12). Note that dXðz; tÞ is a linear combinationof the coefficient differentials dW ijðz; tÞ in thematrix polynomial dWðz; tÞ. As long as Wðz; tÞ is

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nonsingular, dXðz; tÞ represents a valid search direc-tion to minimize (7), because dXðz; tÞ spans the sametangent space of matrices as spanned by dWðz; tÞ[8,11]. For these reasons, an alternative stochasticgradient search method is used in the following form:

Wpðt þ 1Þ ¼ WpðtÞ � mðtÞdJðWðz; tÞÞ

dXpðtÞ

� �Wðz; tÞ,

(15)

where the right-side operator Wðz; tÞ acts on thegradient term in brackets only in the time dimensionp. The search direction employed by (15) is nothingmore than the natural gradient search direction usingthe Riemannian metric tensor of the space of allmatrix filters of the form of Wðz; tÞ.Incorporating (14) in (15), we update coefficient

matrix as follows:

Wpðt þ 1Þ ¼ WpðtÞ þ mðtÞXt

Fðt; tÞh½yðt � tÞupðtÞT

þ yðtÞupðt � tÞT�i, ð16Þ

where mðtÞ is the learning rate which is selected byexperience and

upðtÞ ¼XL

q¼0

WTq ðtÞyðt � p þ qÞ. (17)

3.3. Practical implementation

In practice, the double-infinite noncausal equal-izer cannot be implemented, and the FIR causalequalizer is used to approximate the double-infinite one, which is given by

yðtÞ ¼XL

p¼0

WpðtÞxðt � pÞ, (18)

where L is the length of equalizer. However, evenwith this restriction, the pth coefficient matrixupdating in (16) depends on future equalizeroutputs yðt � p þ qÞ, 0pqpL, when p ¼ 0through the definition of upðtÞ in (17) for atruncated equalizer. Instead of using approxima-tion and data storage to estimate upðtÞ, the lastterm in (16) is delayed by L samples. This delayedupdate maintains the same statistical relationshipsbetween the signals in the updates and provides

similar performance mainly on convergence to (16)for the small step sizes [11].In addition, when t is big enough, the parameter

WpðtÞ of the equalizer converges to a steady value.Then we can assume that WpðtÞ � Wpðt � 1Þ �� Wpðt � 2LÞ, such that we get from (17)

upðtÞ � u0ðt � pÞ; p ¼ 0; 1; . . . ;L. (19)

With these changes, the proposed algorithm in thegeneral case of signal is

Wpðt þ 1Þ ¼ WpðtÞ þ mðtÞ

Xt

Fðt � L; tÞh½yðt � t� LÞuðt � pÞT

þ yðt � LÞuðt � t� pÞT�i, ð20Þ

where uðtÞ ¼PL

q¼0WTL�qðtÞyðt � p � qÞ.

The assemble average in (20) and the onlineestimate of (6) is important to the performance ofthe algorithm, mainly on convergence and fluctua-tion of the algorithm. We set the online estimate as

Ryðt; tÞ ¼ ð1� aÞRyðt � 1; tÞ þ ayðtÞyTðt � tÞ,

(21)

where a is affecting factor, selected by experience ingeneral, such as a ¼ 0:1� 0:001. We also can dealwith the assemble average in (20) exactly as in (21).At last, the algorithm derived by natural gradient

is characterized with equivariant performance [8,11]in general. Although equivariance indicates theuseful convergence behavior of the algorithm, itdoes not guarantee that Wðz; tÞ is adequate forequalizing the channel, nor does it guarantee goodperformance when the equalizer filter has a poorinitialization. In fact, initializations can causeconvergence problems [8]. For this reason, acenter-tap initialization scheme is employed, i.e.,

Wðz; 0Þ ¼ Iz�j ; j ¼ intL

2

� �, (22)

where int means taking the integer of the number.

4. Simulation and comparison

We present two tests about our algorithm inthis section. One is simulation in comparisonwith other algorithms, and the other is speech

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separation with dereverberation. To write simply,we name our algorithm (20) as second-orderstatistics blind deconvolusion (SOSBD). To ex-plain the performance of the algorithm, we useintersymbol interference (ISI) as the evaluatingindex. Its definition is as follows.D2 ISI

ISIiðtÞ ¼

Pj;l jcijðl; tÞj

2 �maxj;l jcijðl; tÞj2

maxj;ljcijðl; tÞj2

, (23)

where cijðl; tÞ is lth term of ði; jÞ entry of thechannel-plus-equalizer systematic filter matrixCðz; tÞ at time t. ISI expresses the degree ofdemixing and deconvolving, and better the per-formance, the lower ISI.

Comparison test 1 (MBE of simulating data).First we give the two algorithms to compare withSOSBD. Ref. [6] offers the algorithm (K–IAlgorithm) in frequency domain as

Wpðt þ 1Þ ¼ WpðtÞ þmðtÞ

L þ 1,

XLf

k¼0

ej2pf kpfSyðf kÞ�1

� ½diagSyðf kÞ��1g

Wðf kÞSxðf kÞ, ð24Þ

where Wðf kÞ is the discrete Fourier Transform ofWpðtÞ on p at time t, and details can be seen in [6].K–I algorithm is implemented in frequency do-main, and involved in calculating the inverse ofpower spectra matrix Syðf kÞ. Thus, it is not easy toperform MBE online and is computationallycomplex. Ref. [8] proposes an algorithm (AAalgorithm) in time domain for stationary sourcesignal

Wpðt þ 1Þ ¼ WpðtÞ þ mðtÞðWpðtÞ

� fðyðt � LÞÞu0ðt � pÞÞ, ð25Þ

where fð.Þ is score function with its element definedby f iðyiÞ ¼ p0ðyiÞ=pðyiÞ, and pðyiÞ is the pdf of theequalizer output yiðtÞ and p0ðyiÞ is its derivation onyiðtÞ; see [8] for details. We can see that AAalgorithm use the score function fð.Þ, which isunknown. Since the pdf is also unknown and itsestimate is difficult to calculate, it is hard toimplement online and is complex in computation.Therefore, our algorithm SOSBD is simple in the

meaning of calculating complexity and in solvingfor inverse matrix.We adopt the simulating data in Ref. [6] to do

experiment, i.e., we consider a two-input three-output FIR system, HðzÞ ¼ HI ðzÞUKðzÞ, where

HI ðzÞ ¼

1:0þ 0:15z�1 þ 0:1z�2 0:65þ 0:25z�1 þ 0:15z�2

0:2þ 0:15z�1 þ 0:1z�2 1:0þ 0:25z�1 þ 0:1z�2

0:3þ 0:2z�1 þ 0:05z�2 0:5þ 0:2z�1 þ 0:1z�2

2664

3775,

U ¼

cosp12

� sinp12

sinp12

cosp12

264

375; LðzÞ ¼

1 0

0 z�1

� �

and HI ðzÞ is irreducible [6]. The two input sourcesignal is defined as

s1ðtÞ ¼ n1ðtÞ; s2ðtÞ ¼n2ðtÞ

1� 0:5z�1,

where n1ðtÞ and n2ðtÞ are white Gaussian randomsignals with zero mean and unit variance, and thesource signals are also independent of each other.All the experiments were carried out with artificialmixing. To further explain our algorithm, we setnoise at observations with variance 0.2.In the simulations, we select L ¼ 15, a ¼ 0:1 in

all three algorithms for comparison. The result isdepicted in Figs. 2 and 3. Fig. 2 shows thecombined channel-plus-equalizer FIR defined in(4), and Fig. 3 shows ISI of the whole system. Weadjust the learning rates and other parameters inall the simulations so that they work with the bestperformance. We select mðtÞ ¼ 0:001 in K–I algo-rithm, mðtÞ ¼ 0:01 in SOSBD, and mðtÞ ¼ 0:001 inAA algorithm. From Figs. 2 and 3, we can see thatSOSBD and AA algorithm have the same perfor-mance. Both SOSBD and AA can make thealgorithm convergence, and the accuracy and thespeed of convergence are higher compared to K–Ialgorithm. Fig. 4 shows the cost function (7) varieswith the iterative number. From Figs. 3 and 4, thecombined channel-plus-equalizer FIR asymptoti-cally approaches the demand of diagnolization (5)when the cost function is statistically minimized.When the additional noise has variance 0.5, K–I

algorithm does not converge, and SOSBD and AA

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Fig. 2. Combining channel-plus-equalizer impulse response

with lower noise found after 2000 iterations at t ¼ 2000.

Fig. 3. Intersymbol interference of the whole system.

Fig. 4. Cost function (7).

Fig. 5. Comparison of K–I, SOSBD and AA algorithms by

showing the combining channel-plus-equalizer impulse re-

sponse with higher noise.


algorithms have almost the same performance inconvergence, see Fig. 5.

Speech test 2 (Speech separation and derever-

beration): We also present simulation results for

speech separation and dereverberation. Here, wegive the unknown system (1) in impulse responseHp. A two-input system two-output system isconsidered with impulse response length 3, which is

h11 ¼ ½0:5 0:2 0:2�; h12 ¼ ½0:2 0:2 0:1�,

h21 ¼ ½0:1 0:2 � 0:1�; h22 ¼ ½0:5 0:2 0:1�.

Two input source signals are digital voice signalsspoken in Chinese, which is sampled at 8 kHz andhas lasted about 10 s. In SOSBD algorithm, we

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Fig. 6. Combined channel-plus-equalizer filter impulse response at about 10 s. (a) Initial elements cijðnÞ, (b) elements cijðnÞ after

running.

Fig. 7. Inter-symbol interference of test 2. Fig. 8. Cost function.


set the learning rate mðtÞ ¼ 0:01 and a ¼ 0:1.The simulation results are depicted in Figs. 6–8.Fig. 6 shows the combined channel-plus-equalizer filter impulse response, (Fig. 6a) showsthe initial value of cijðnÞ before running, and(Fig. 6b) shows the value after running about10 s. Fig. 7 shows the intersymbol interference.Fig. 8 shows the cost function defined in (10).It is easily seen that the cross-talk betweenchannels is reduced greatly and reverberation isdecreased. Therefore, the proposed algorithm has

the ability to perform blind dereverberation tosome degree.

5. Summary and conclusions

Only second-order statistics for MBE is used toconstruct a cost function and thus a new onlinealgorithm is derived with natural gradient searchmethod for multichannel blind equalization. Si-mulations show the efficiency of the proposed

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algorithm, and also indicate the ability to performspeech separation and dereverberation simulta-neously. By comparison, the proposed has thefollowing benefits:

�
SOSBD algorithm has less computational com-plexity than AA algorithm and K–I algorithm inthe sense that SOSBD algorithm does notestimate the score function difficultly, and doesnot calculate the inverse matrix.
�
SOSBD algorithm converges faster with highaccuracy compared to K–I algorithm, and hasalmost the same performance as the AA algo-rithm.
�
SOSBD algorithm can be used to deal withobserved signals contaminated by lower levelnoise.
Acknowledgements

The authors thank all the reviewers for theuseful comments.

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online sos-based multichannel blind equalization algorithm with noise

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