Blind Separation of Cyclo-stationary Signals using Generalized EigenvalueApproach

Tarik Aoufi1, M’hamed Bakrim1,2, Driss Aboutajdine1

{1} GSCM-LRIT, Faculté des Sciences, av. Ibn Battouta, BP 1014, Rabat, Maroc{2} LP2M2E, Faculté des Sciences et Techniques, Marrakech, Maroc

E-mail: tarik.aoufi@gmail.com, bakrim@fstg-marrakech.ac.ma, aboutaj@fsr.ac.ma

Abstract— This communication addresses the problem of the blindseparation of cyclo-stationary source signals in the case of aninstantaneous linear mixture. We show that the separation canbe realized by the generalized eigenvectors that simultaneouslydiagonalize the cyclic correlation matrices of the observation signals.The mixture matrix is estimated by the eigenvectors of the pencilmatrix constructed with two different cyclic correlation matricesof the observation signals. We also discuss the extension of someexisting second-order blind source separation methods. The mainadvantages of the proposed method are that a pre-whitening stage isdropped, the cyclic frequencies are unknown and it is consistent innoisy case. Numerical simulation results are presented to illustratethe performance of the proposed method in digital communicationscontext.

Index Terms— Blind sources separation, instantaneous linear mix-ture, simultaneous diagonalization, generalized eigenvalue problem,cyclo-stationary signals.

I. INTRODUCTION

Blind source separation (BSS) is a fundamental problem en-countered in many practical applications such as telecommuni-cations, radar, sonar, speech processing, biomedical, mechanics,image processing, etc... The aim of the BSS is to recover theunobserved input signals called sources from their observedunknown mixtures.

In the signal processing community, many solutions have beenproposed to solve this problem and several of them are basedon second-order statistics (SOS) because they are sufficientto achieve a successful source separation. In the literature,the well-known approaches are AMUSE [12] and SOBI [3]algorithms applied to stationary sources, and pencil approach[4] [5] [6] exploiting the non-stationarity and temporal structureof sources to estimate the mixing matrix (or the demixingmatrix) in the presence of spatially correlated but temporallywhite noise (not necessarily Gaussian). Such a problem has beenalready considered in [1] [2] [10] [11] for the cyclo-stationarysignals in order to solve the interferences problem in digitalcommunication. The theory of linear and quadratic processingof cyclostationary signal has been first introduced in [8].

In this paper we present a novel method which exploit theparticular structure of cyclic correlation matrix of sources toestimate the mixing matrix in the presence of stationary whitenoise. Also, we show that the blind sources separation can bereformulated as a generalized eigenvalue problem. Therefore,the mixture matrix will be estimated by the eigenvectors ofthe pencil matrix constructed by two different cyclic correlationmatrices of the observation signals. When the cyclic frequenciesare unknown, we use the estimator proposed in [7] to estimatethem. Our method consists of estimating in the first time the

cyclic frequencies and in the second step, we use a matrixpencil formed by output cyclic correlation matrices at differentcyclic frequencies to solve BSS problem. We also show that itis possible to solve the blind source separation problem by thenon unitary simultaneous diagonalization by using the algorithmproposed in [13].

The rest of this paper is organized as follows. The problemstatement, assumptions and definitions are presented in sectionII. In the section III, we present the existing methods based onsecond order statistic and related to our proposed methods. Insection IV we describe the proposed methods. Finaly, numericalsimulations are provided in the section V to illustrate theeffectiveness of the proposed methods.

II. DATA MODEL, ASSUMPTIONS AND DEFINITIONS

In the simple form, the BSS problem can be formulated asfollows : several linear instantaneous mixture of n source signalsemitted are received by m sensors. The input/output relationshipof the mixing system is given by :

x(t) = As(t) + b(t) (1)

where A, s(t), x(t) and b(t) represent the m×n mixing matrix,the n × 1 source signals vector, the m × 1 observation signalsvector and the m × 1 noise signals vector respectively. Ourproposed methods are based on the following assumptions :

1) The components of source signals are unobservable, statis-tically independent, zero-mean and cyclostationary. Thus,their autocorrelation functions Rsi(t, τ) = E{si(t)s

∗i (t−

τ)} are Ti-periodic for all i = 1, ..., n. E{.} standsexpectation operator and Ti stands the cyclic-period of thei-th source signal si(t). The cyclic correlation coefficientfunction of si(t) at the cyclic frequency α (α ∈ { k

Ti, k ∈

Z}) is defined as

R(α)si

(τ) = limTi→∞

1

Ti

∫ Ti/2

−Ti/2

Rsi(t, τ) exp(−2ıπαt)dt

(2)where ı =

√−1.2) A is a full rank m× n matrix ( the number of sensors is

bigger than the number of sources) so A is invertible.3) The components of noise signals vector b(t) are stationary

white zero-mean random signals, mutually uncorrelatedand independent from the source signals. The correlationfunction of bj(t) for all j = 1, ..., m is

Rbj (τ) = σ2bj

δ(τ) (3)

where σ2bj

stands the variance of noise signal bj(t) andδ(τ) is the dirac function. We show that the cycliccorrelation coefficient function of bj(t) at cyclic frequency

1-4244-1378-8/07/$25.00 ©2007 IEEE. 689

α is written as

R(α)bj

(τ) = Rbj (τ) limTj→∞

1

Tj

∫ Tj/2

−Tj/2

exp(−2ıπαt)dt

(4)Hence, if α �= 0 then the correlation coefficient functionof bj(t) is null.

4) The cyclic frequencies of source signals are all distinct(∀i, j = 1, ..., n, i �= j, 1

Ti�= 1

Tj).

The overall objective of BSS is to obtain an estimate A of A,up to the standard BSS indeterminacies on ordering, scale andphase. Once A is known, the sources are estimated by :

s(t) = A�x(t) = PDs(t) (5)

Where (.)� denotes pseudo-inverse matrix, P is a permutationmatrix (corresponding to an arbitrary order of restitution of thesources) and D is a diagonal matrix (corresponding to arbitraryattenuations for the restored sources). Thus the condition ofseparation can be put in the following form :

A�A = PD (6)

III. SOS-BASED SOURCE SEPARATION

In this section, we briefly present the existing SOS basedmethods to solve the BSS problem having a link with theproposed methods.

A. AMUSE method

The method AMUSE (Algorithm for Multiple UnknownSignals Extraction) introduced by Tong in [12], is realized by thediagonalization of two symmetric matrices M0 and M1, whichone, say M0 is invertible. This method requires a whiteningstage. Thus, the orthogonal matrix V is estimated in order tocheck :

V M0VT = I and V M1V

T = D (7)

where I is the identity matrix and D is a diagonal matrix. Thematrix V is obtained by V = UB with BM1B

T = UT D1U , Band U are respectively the whitening and orthogonal matrices.We note that the matrices M0 and M1 are the correlationmatrices.

B. SOBI method

The SOBI method [3] extends AMUSE to several matrices.This method requires two steps (whitening followed by anorthogonal transformation) :

– First, the whitening matrix is estimated by B = D−12 UT

where D and U are respectively the eigenvalue and eigen-vector matrices of the Rx(0) (Rx is the correlation matrixof observation signals)

Rx(0) = UDUT (8)

– Second, the orthogonal matrix V = BA is estimated by theorthogonal simultaneous diagonalization of the correlationmatrices at different time delays.

C. Generalized eigenvalue problem

Let consider two n×n matrices M1 and M2, the generalizedeigenvalue problem consist of finding the scalar λ and the vectorx such as x �= 0 (or to find the matrix V ) and :

M1x = λM2x (or M1V = M2V D) (9)

where D is diagonal matrix.

Theorem 1 :

Let Λ1, D1 ∈ Cn×n be diagonal matrices with positive diagonal

entries and Λ2, D2 ∈ Cn×n be diagonal matrices with non-

zero diagonal entries. We assume that G satisfies the followingdecompositions :

D1 = GΛ1GH , D2 = GΛ2G

H (10)

where (.)H stands the complex conjugate, transpose.

Then the matrix G is the generalized permutation matrix, i.e.,G = PD (P is a permutation matrix and D is a diagonal matrix)if D1D

−12 and Λ1Λ

−12 have distinct diagonal entries.

Proof : see [6] for proof.

The linear transformation G which satisfies (10) is the eigen-vector matrix of Λ1Λ

−12 . In other words, the matrix G is the

generalized eigenvector matrix of the matrix Λ2 − λΛ1 calledpencil matrix, where λ is the generalized eigenvalue vector.

Theorem 2 (pp. 468 in [9] :

If M2 − λM1 is hermitian-definite, then there exists a nonsin-gular matrix U such as

UHM1U = D1 (11)

UHM2U = D2 (12)

where D1 and D2 are diagonal matrices.

Among the SOS-based source separation methods, a matrixpencil method [4] employs only two time-delayed correlationmatrices to estimate the mixing matrix in the presence oftemporally white noise. In this method, the choice of time delaysis very important in order to estimate the mixing matrix A.

In the next section, we propose to use the cyclic correlationcoefficient because it is insensitive to noise (for all time delay,the cyclic correlation coefficient of noise is null if the cyclicfrequency is different to zero).

IV. PROPOSED METHODS

A. principle

The cyclic correlation coefficient matrix of observation si-gnals vector is given by :

R(α)x (τ) = AR(α)

s (τ)AH + R(α)b (τ) (13)

If α �= 0 (see the assumption (3)) then

R(α)x (τ) = AR(α)

s (τ)AH (14)

Our aim is to find the matrix W such as W HA = I .We pose

M1 = R(αi)x (0) = AR(αi)

s (0)AH , ∀i = 1, ..., n (15)

and

M2 =

n∑i=1

R(αi)x (0) = A

n∑i=1

R(αi)s (0)AH (16)

where the αi represents the cyclic frequency of the source signalsi(t),

∑ni=1 R

(αi)s (0) is diagonal invertible matrix and R

(αi)s (0)

is diagonal matrix, thus we constitute a generalized eigenvalue

690

problem as follows ; we multiply the equations (15) and (16)with W

M1W = AR(αi)s (0)AHW

= AR(αi)s (0) (17)

M2W = A

n∑i=1

R(αi)s (0)AHW (18)

= An∑

i=1

R(αi)s (0) (19)

By replacing A = M2W{∑n

i=1 R(αi)s (0)

}−1

in (17) ;we obtain

M1W = M2WΛ (20)

where Λ ={∑n

i=1 R(αi)s (0)

}−1

R(αi)s (0).

Since Λ = diag{λ1, . . . , λn} is a diagonal matrix then theequation (20) represents the generalized eigenvalue problemdescribed in section III.

The matrix of the form M2 − λM1 (λ ∈ R) is said to be apencil. This property is frequently encountered when both M1

and M2 are hermitian and invertible (see III-C). Pencils of thiskind of matrices are referred to as hermitian-definite pencils [9].

For the requirement of hermitian matrices , we can use

M1 =1

2

{M1 + MH

1

}(21)

M2 =1

2

{M2 + MH

2

}(22)

Hence the matrix W is the generalized eigenvector matrix ofthe pencil M2 − λM1.

When the cyclic frequencies are unknown, we estimate them bymaximizing the following criteria [7] :

J(α) =

τm∑τ=−τm

∣∣∣R(α)x (τ)

∣∣∣2 (23)

J(α) = 0 if α �∈ {−αi, 0, αi}, i = 1, ..., n (αi is the cyclicfrequency of si(t)) else J(α) > 0, thus the cyclic frequency αi

is estimated by αi = argmax]0, 12 [ {J(α)}

B. Summary of the proposed methods

The first proposed method based on matrix pencil approachcan by summarized in the following five points :

1) Estimate the cyclic correlation coefficient of observationsignals,

2) Estimate the cyclic frequencies by the procedure presentedin the precedent subsection,

3) Calculate the two matrices defined in (21) and (22),4) Find the generalized eigenvectors matrix W which satis-

fies the equation (20)5) Estimate source signals by s(t) = W Hx(t).

To solve the BSS problem, we also propose to diagonalizesimultaneously the set matrices of cyclic correlation coefficientat different time delays and for all cyclic frequencies. In this

respect, we use the non unitary simultaneous diagonalizationalgorithm proposed in [13]. We summarize this second proposedapproach as follows

1) Estimate the cyclic correlation coefficient of observationsignals,

2) Estimate the cyclic frequencies by the procedure presentedin the precedent subsection,

3) Calculate the set matrices M for each estimated cyclicfrequency αi : M =

{R

(αi)x (0), ..., R

(αi)x (τmax)

},

4) Diagonalize simultaneously the set matrices M by thealgorithm proposed in [13],

5) Use the demixing matrix to estimate source signals.

V. COMPUTER SIMULATED EXPERIMENTS

We present simulations to illustrate the effectiveness of theproposed methods in the BSS context and to compare themwith the SOBI method.In fact, the SOBI method needs a pre-whitening stage. Thus weapply the simultaneous diagonalization of the whitening corre-lation matrix of observation signals. Our methods, SimultaneousDiagonalization of the set Matrices (SDM) based on non-unitarysimultaneous diagonalization and the method based on pencilapproach (PM) are directly applied to the cyclic correlationcoefficient matrices of the observations. We consider m = 2mixtures of n = 2 digital communication source signals definedas

s(t) =∑k∈Z

a(k)h(t − kT ) (24)

where a(t) is an i.i.d. zero-mean complex random binary se-quence, T represents a period symbol and h(t) is a deterministicwaveform signal. The cyclic frequency of the first source signal(resp. the second source signal) is 1

4(resp. 1

6).

These sources are mixed by the following mixture matrix :

A =

(1 0.25

0.5 1

),

Firstly, we estimate the cyclic frequencies of the source signalsfrom their mixtures in the case SNR = 1 dB and SNR =30 dB, using the estimator presented in the section IV. Theresults are presented in Fig. 1.

Secondly, for the proposed method using non unitary simulta-neous diagonalization, we estimate the set matrices of cycliccorrelation coefficient matrices at different estimated cyclicfrequencies and for all time delays τ ∈ {0, · · · , 10}, but for thefirst proposed, we use only M1 (21) and M2 (22) to estimatethe mixing matrix. In this case, the demixing matrix W use thematrix pencil method (20).

In order to measure the performance of our two proposedmethods, we use the performance index (PI) [14] defined by

PI =1

n(n − 1)

n∑

i=1

n∑

k=1

|gik|max

j|gij | − 1

+n∑

i=1

n∑

k=1

|gki|max

j|gji| − 1

691

−0.1 0 0.1 0.2 0.3 0.4 0.5 0.60

5

10

15

20

25

30

35

Estimated frequencies

SNR=1 dB

−0.1 0 0.1 0.2 0.3 0.4 0.5 0.60

5

10

15

20

25

30

35

Estimated frequencies

SNR=30 dB

Figure 1. below : The estimated cyclic frequencies for SNR = 1 dB,down : The estimated cyclic frequencies for SNR = 30 dB

where gij is the (i, j)-element of the global system matrixG = W HA. The term maxjgij represents the maximum valueamong the elements in the i − th row vector of G. Whereasthe term maxjgji designate the maximum value among theelements in the i − th column vector of G. When the perfectseparation is achieved, the performance index is zero. In prac-tice, a performance index around 10−2 indicates quite a goodperformance. On the Fig. 2, the obtained performance index PIis displayed versus the SNR. The result show that our proposedSDM and PM methods have better performances than the SOBImethod.

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 310

0.0005

0.0025

0.0045

0.0065

0.0085

0.0105

0.0125

0.0145

0.0165

0.0185

0.02

SNR (dB)

Per

form

ance

inde

x

SOBISDMPM

Figure 2. PI versus the SNR

VI. CONCLUSION

In this paper, we have proposed a new approach to blindcyclostationary sources separation in instantaneous mixturescontext by considering the generalized eigenvalue problem.We have also proposed to use the non unitary simultaneousdiagonalization to set cyclic correlation coefficient matrices atdifferent time delays. Numerical experiments show that our twomethods are less sensitive to noise and more efficient than SOBImethod based on whitening stage.

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