Optimal separation of independent narrow-band sources: Concept and performance 1

Signal Processing 73 (1999) 27—47

Optimal separation of independent narrow-band sources: Concept and performance Pascal Chevalier* Thomson-CSF-Communications, UTTC/TSI/LTA, 66 rue du Fosse& Blanc, 92231 Gennevilliers, France Received 22 September 1997; received in revised form 14 September 1998

Abstract Since the beginning of the eighties, the blind second and higher order Narrow-Band (NB) source separation methods have received much interest. However, despite this fact, relatively few papers have been devoted to the performances evaluation of these methods in arbitrary noisy contexts and to the comparison of these performances to the optimal ones. The main reason for this may be that the concept of optimal NB source separation in noisy situations does not seem to be very clear to most of the research workers working on blind source separation, which may be explained by the fact that the concept of NB source separator performance is not clear. Assuming instantaneously mixed NB sources, the purpose of this paper is to fill the gaps previously mentioned by introducing the concept of NB source separator performance, by presenting the optimal NB source separator, by comparing the latter to the Weighted Least Square and to the Least Square source separators, by giving a Maximum Likelihood interpretation of the latter and by computing and discussing the steady-state performance of these three separators in arbitrary noisy contexts  1999 Published by Elsevier Science B.V. All rights reserved. Zusammenfassung Seit dem Anfang der Achtzigerjahre haben blinde Schmalband-(Narrow Band, NB)-Quellentrennungsmethoden basierend auf Statistiken zweiter oder ho¨herer Ordnung gro{e Beachtung gefunden. Nichtsdestoweniger behandeln relativ wenige Arbeiten die Evaluation der Leistungsfa¨higkeit dieser Methoden in beliebigen rauschbehafteten Umgebungen und den Vergleich mit der optimalen Leistungsfa¨higkeit. Mo¨glicherweise ist ein Hauptgrund dafu¨r darin zu finden, da{ das Konzept der optimalen NB-Quellentrennung in rauschbehafteten Situationen fu¨r die meisten der im Bereich der blinden Quellentrennung arbeitenden Forscher relativ unklar zu sein scheint, was damit erkla¨rt werden kann, da{ das Konzept der Leistungsfa¨igkeit einer NB-Quellentrennung um nichts klarer ist. Der Zweck dieser Arbeit ist es, die oben erwa¨hnten Lu¨cken zu schlie{en, wobei instantan gemischte NB-Quellen angenommen werden. Insbesondere wird das Konzept der Leistungsfa¨higkeit einer NB-Quellentrennung eingefu¨hrt, es wird der optimale NB-Quellentrenner pra¨sentiert und mit den Weighted-Least-Square- und Least-Square-Quellentrennern verglichen, fu¨r den letzteren wird eine Maximum-Likelihood-Interpretation angegeben, und die Leistungsfa¨higkeit dieser drei Trennmethoden im eingeschwingenen Zustand wird in beliebigen rauschbehafteten Umgebungen berechnet und diskutiert.  1999 Published by Elsevier Science B.V. All rights reserved.

* Tel.: #33 146 13 2698; fax: #33 1 46 13 25 55; e-mail: [email protected]  This work was supported by the Direction des Recherches et Etudes Techniques (DRET) of the french Defence Office. 0165-1684/99/$ — see front matter  1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 1 6 5 - 1 6 8 4 ( 9 8 ) 0 0 1 8 3 - 2

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P. Chevalier / Signal Processing 73 (1999) 27–47

Re´sume´ Depuis le de´but des anne´es 80, les me´thodes de se´paration aveugle de sources a` bande e´troite, aussi bien a` l’ordre deux qu’aux ordres supe´rieurs a` deux, ont fait l’objet de nombreux travaux. Toutefois, relativement peu de papiers ont e´te´ consacre´s a` l’e´valuation des performances de ces me´thodes dans des contextes arbitrairement bruite´s et a` la comparison de ces performances aux performances optimales. Ce constat peut probablement s’expliquer par le fait que le concept de se´parateur de sources optimal en situation bruite´e ne paraıˆ t pas tre`s clair pour la plupart des chercheurs travaillant sur la se´paration aveugle de sources, ce qui peut eˆtre lie´ au fait que le concept de performances d’un se´parateur de sources semble tout aussi obscur. En restreignant le proble`me aux me´langes instantane´s de sources a` bande e´troite, l’objet du papier est de combler les lacunes pre´ce´demment de´crites en introduisant le concept de performances d’un se´parateur de sources, en pre´sentant le se´parateur de sources optimal, en comparant ce dernier aux se´parateurs des moindres carre´s et des moindres carre´s ponde´re´s, en donnant une interpre´tation au sens du maximum de vraisemblance de ces derniers et enfin, en calculant et en discutant les performances de ces trois se´parateurs en re´gime e´tabli dans des contextes arbitrairement bruite´s.  1999 Published by Elsevier Science B.V. All rights reserved. Keywords: Separator performance; Signal-to-interference-plus-noise ratio; Optimal source separation; Spatial matched filter; Weighted least square ; Optimal interference canceller; Maximum likelihood

1. Introduction Up to the middle of the eighties, the implementation of most of the array filtering techniques [47] required the prior knowledge of an information about the signals of interest for the receiver, such as the direction of arrival (DOA) [1,37] (radar) or a reference signal [65] (radiocommunications). However, for some practical applications such as passive listening, reference signals are not available and the sources DOA estimation may be either difficult to obtain, (in particular in the presence of mutual coupling between the sensors), or costly to implement, due for example to the necessity of calibrating the array. In such situations of practical interest, it may be very useful to implement blind array filtering techniques, which do not require any prior information about the sources or any array calibration. Among these techniques, the blind source separation methods are of great interest and seem to be very promising for many applications. Contrary to the classical array filtering problem, for which only one source is of interest for the receiver and the others are considered as jammers, the source separation problem is clearly a multidimensional one in the sense that all the received sources are of interest for the receiver. As a consequence, while a classical spatial filter is characterized by a complex vector w which generates only

one scalar output y(t)"wRx(t), where x(t) is the input vector and R means transposition and conjugation, an NB source separator is characterized by a matrix ¼ and generates multiple scalar outputs or, equivalently, an output vector y(t)" ¼Rx(t). The initial works on blind multidimensional array filtering, in the sense defined previously, have been presented in the open literature in the seventies for instantaneous mixtures of sources. The associated methods exploit only the information contained in the second order statistics of the observed signals, assumed NB for the array, and aim at decorrelating the latter. The first method, introduced by White [63,64] with a Nolen’s idea [52], resolves the observed signals into their principal or eigenvector components. The second method, introduced by Brennan et al. [6] and presented in [47], achieves a Gram-Schmidt (GS) orthogonalization procedure on the data. Unfortunately, the GS and the Nolen’s blind NB source separators perform well only for decorrelated sources which are disparate in power and/or angle of arrival as shown in [28] and [40] respectively. Some other works were presented a few years after by Bar-Ness [2] in the same spirit. However, the problem of blind separation of multiple independent sources has been clearly described for the first time only in 1985 by Herault

P. Chevalier / Signal Processing 73 (1999) 27—47

et al. in [41], where the use of the higher order (HO) statistics of the data were considered. Since these pioneering works, there has been an increasing interest for the blind source separation problem and many other methods have been developed for instantaneous mixtures of NB sources in particular [4,9,12,13,23,24,27,29,30,33,36,42,43,48,53,54,58,59, 62], although some others process convolutive mixtures [51,60] or wide-band [8,50] sources. More precisely, limiting, in this paper, the analysis to instantaneous mixtures of NB sources, some of these blind methods still exploit the information contained in the second-order (SO) statistics of the data, assuming decorrelated sources, and aim at generating outputs which are decorrelated to each other either at the same time [2] or at different times [4,33,53,58,62]. However, most of the recently developed blind techniques also exploit the information contained in the HO statistics of the observed signals, assuming non-Gaussian and statistically independent sources. In particular, some of these HO techniques aim at generating outputs minimizing or nulling cross-moments of non-linear functions of the latter [3,33,41,42,54], whereas most of the others aim, by some means or the other, at minimizing or nulling some functions of the fourth-order (FO) cross-cumulant of the outputs [12,13,24,27,29,30,36,43,48,59] or at maximizing some HO criteria or contrast functions [13,24,25, 27,46,48,49,66]. However, despite this great development of blind methods, relatively few papers have been devoted to the performance evaluation of these methods in arbitrary noisy contexts, which obviously limits their use in operational situations. More precisely, some of these papers are devoted to the convergence [45] and the stability [31,34,56] analysis of some particular methods [42] whereas some other papers [5,10,11,13,67] develop analytical computations of particular performance criteria at the output of some FO cumulant-based methods. Recently, a parametrical analysis of both steadystate and convergence performance of several HO methods [13,24,27] in arbitrary noisy contexts, has been investigated in [7,15—17,26], essentially by computer simulations. In particular, the conditions under which the previous blind methods blindly implement, in noisy situations, the optimal NB

29

source separator have been analysed. However, this analysis concerns only three particular HO methods and gives very scarce analytical results about the output performance. To make possible the analytical performance comparison of all the blind source separation methods with respect to the optimal one in an arbitrary noisy context, it is necessary to be aware of the optimal performance that can be reached at the output of a source separator in this arbitrary noisy context, which does not seem to be well known actually. In other words, no upper bound on the steady-state performance of the blind methods in an arbitrary noisy context is actually available. One reason for this may be the fact that the concept of optimal NB source separation in a noisy situation does not seem to be very clear to most of the research worker working on blind source separation, who are often used to neglecting the influence of noise in their work. Another reason may be that the concept of NB source separator performance is not clear. For example, what is the meaning of ‘‘the separator 1 is better than the separator 2’’ when several sources are impinging on the array? The purpose of this paper is to fill the gaps previously mentioned by introducing the concept of NB source separator performance, by presenting the optimal NB source separator, by comparing the latter to the Weighted Least Square and to the Least Square source separators, by giving a Maximum Likelihood interpretation of the latter and by computing and discussing the steady-state performance of these three separators in arbitrary noisy contexts. The SO and FO statistical properties at the output of these separators jointly with the situations for which the developed SO and FO blind methods may blindly implement the optimal one are presented in [20]. Note that researchers who may consider some of the results presented in the paper as already known can benefit from the tutorial aspect of this paper and the unified presentation of the subject.

2. Hypotheses and problem formulation Consider an array of N NB sensors and let us call x(t) the vector of the complex amplitudes of the

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signals at the output of these sensors. Each sensor is assumed to receive a noisy and instantaneous mixture of P stationary and statistically independent NB sources. Under these assumptions, the observation vector x(t) can be written as follows: . x(t)" m (t)a #b(t)_Am(t)#b(t), (1) G G G where b(t) is the noise vector, assumed stationary, m (t) and a are the complex envelope and the steerG G ing vector of the source i respectively, m(t) is the vector whose components are the m (t) and A is the G (N;P) matrix of the sources steering vectors a . On G the other hand, under the previous hypotheses, the correlation matrix of the observation vector, R _E[x(t)x(t)R], can be written as follows: V . R " n a aR#g B_AR AR#g B, (2) V G G G  K  G where g is the mean power of the noise per sensor,  B is the noise’s spatial coherence matrix such that Tr[B]"N where Tr means trace, n is the power of G the source i received by an omnidirectional sensor, the matrices R _n a aR,R _E[m(t)m(t)R] and QG G G G K R _AR AR are the correlation matrices of the Q K source i, of the vector m(t) and of all the sources respectively. Considering only linear structures of filtering, the problem of NB source separation, under the previous hypotheses, is to find an (N;P) matrix ¼, called separator, whose output vector, y(t), is, to within a diagonal matrix K and a permutation matrix P, an estimate, of m(t), the vector of the complex envelopes of the sources. The separator ¼ is defined to within a diagonal and a permutation matrix since neither the value of each output power of the separator nor the order in which the outputs are arranged change the estimation quality of the sources. This indetermination defines equivalence classes of separators [32]. Thus, the NB source separation problem which is addressed in this paper can be summarized by the following expression: y(t)_¼Rx(t)"PRKRmL (t).

(3)

Note that for non-Gaussian, non-circular or cyclostationary sources, the source separation quality

can be improved by using non-linear [22,57], widely-linear [19,55] or poly-periodic [18,21] structures of filtering respectively, but this problem is not addressed in this paper.

3. NB source separator performance To be able to quantitatively evaluate the restitution’s quality of each source by an NB source separator ¼ and to be able to say that a separator ¼ is  better than a separator ¼ for the restitution of  a given source k in a given context, it is necessary to define the concept of NB source separator performance, presented for the first time in [15]. On the other hand, for instantaneously mixed NB sources, the performance at the output of a separator ¼, for the detection, the estimation or the demodulation of a given source, are, under some classical Gaussian noise assumption, directly related to the associated output signal-to-total-noise ratio [61]. For this reason, we define the concept of NB source separator performance from the signal-to-totalnoise-ratio of each source at the separator’s outputs. To this end, we call interference for the source k, all the sources j with jOk. Thus, for each output i of the separator ¼ and for each source k, we define the concepts of signal-to-noise ratio (SNR) and signal-to-interference-plus-noise ratio (SINR) for the source k at the output i of the separator ¼, by the ratios between the source k power and the noise power, and between the source k power and the sum of the noise and the interferences power for the source k at this output, respectively. More precisely, the SNR and the SINR of the source k at the output i of the separator ¼, whose columns are the vectors w , are denoted by SNRk[w ] and G G SINRk[w ], respectively, and are defined, using (2), G by n "wRa " SNRk[w ]_ I G I , (4) G g wRBw  G G "wRa G I , SINRk[w ]_n (5) G I wRR w G @I G where the matrix R , defined by R _R !R , is @I @I V QI the correlation matrix of the noise plus interferences for the source k.

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With these definitions, for each source k received by the array, we define the maximum SINR of the source k at the output of the separator ¼, denoted by SINRMk[¼], by the maximum value of the quantities SINRk[w ] when i varies from 1 to P. G Finally, we define the performance, in terms of output signal-to-interference-plus-noise ratio, of an (N;P) NB source separator ¼, by the P-uplet P(¼), defined by P(¼)_(SINRM1[¼],SINRM2[¼],2, SINRMP[¼]).

(6)

We verify that ¼ and ¼KP have the same performance, which means that all the separators of a given equivalence class have the same performance. Moreover, we will say that a separator ¼ is better than a separator ¼ for the restitution   of the source k in a given context if SINRMk[¼ ]  ' SINRMk[¼ ] for this context. 

The optimal NB source separator is the separator which gives the best performance for the restitution of each source. It maximizes, over all the possible separators ¼, the quantities SINRMk[¼] for each source k.

4.1. Presence of noise In the presence of noise, the optimal NB source separator is unique, for a given context, to within a diagonal and a permutation matrix, which defines the optimal equivalence class of NB source separators. It is easy to show [1,47] that it corresponds to the separator ¼ whose columns are the spa  tial matched filters (SMF) [14] associated to the different sources (w "aR\a ), which equivaG  V G lence class is defined by ¼ _R\AKP ,   V

The structure of ¼ , whose implementation   requires prior knowledge or estimation of the matrix A, is depicted in Fig. 1, where R\ is the V inverse of a R square root matrix and where V R\R_(R\)R. This figure shows that the optimal V V reception of the sources is obtained by beamforming in the direction of each source (ARR\R) after V a data decorrelation stage (R\"R\ B\, V V where R _B\R B\R) composed of a noise V V decorrelation stage (B\) followed by a data decorrelation stage in spatially white noise (R\). V Introducing expression (1) into Eq. (8), the output of the optimal NB source separator can be written as y (t)"PRKR[ARR\Am(t)#ARR\b(t)],   V V

ARR\A"g\ARB\A V 

(7)

(8)

(10)

We deduce from Eq. (9) that in the presence of noise, the exact restitution of the sources is not possible. Moreover, in the latter situation, expression (10) shows that the matrix ARR\A has no V reason to be a diagonal matrix in the general case. This means that in most situations, an interference residue exists at each output of the optimal separator ¼ , which thereby does not completely nul  lify the interferences. In fact, the interferences are completely nulled at the output of ¼ when the   matrix ARR\A is a diagonal matrix, which, in V the presence of noise, occurs in particular when the matrix ARB\A is diagonal as shown by Eq. (10), i.e. when the vectors B\a are orthogonal to each G other. In this latter case, we deduce from Eqs. (9)

and whose output can be written as y (t)_¼R x(t)"PRKRARR\x(t).     V

(9)

where, applying a matrix inversion formula to expression (2), the matrix ARR\A takes the form V ;[I!(g R\#ARB\A)\ARB\A].  K

4. Optimal NB source separator

31

Fig. 1. Optimal NB source separator structure.

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and (10) that ¼ takes the particular form   ¼ _B\AKP, (11)   which shows that the optimal reception of the sources is obtained by beamforming in the direction of each source (ARB\†) after a noise decorrelation stage (B\), as depicted in Fig. 2. Finally, from Eq. (5)—(7), we easily deduce the expression of the optimal NB source separator performance, given by P(¼ )_(n aR R\a ,n aR R\a ,2,n aRR\a )     @    @  N N @N N (12)

4.2. Absence of noise In the absence of noise, the matrix R is reduced V to the matrix of the sources R , which is not invertQ ible when N'P, and ¼ is necessarily a solution   of the equation R ¼ "AKP. This equation Q   has an infinite of solutions but the minimumnorm solution (with respect to the norm ""¼""_Tr[¼R¼]) is given by ¼ "R\AKP, (13)   Q where R\ is the pseudo-inverse of R , defined by Q Q R\_º K\ºR, where K is the matrix of the Q Q Q Q Q P non zero eigenvalues of R and º is the matrix of Q Q the P associated eigenvectors. Using a square root decomposition of R\, given by R\_ Q Q (R\)(R\)R and choosing (R\)R" Q Q Q K\ºR, the matrix º _(R\)RAR is Q Q   Q K a (P;P) unitary matrix (ºR º "º ºR "         I , where I is the (P;P) identity matrix) and the N N structure of ¼ takes, in this case, the form depic  ted in Fig. 3. Note that the matrix (R\)Ris also Q the pseudo-inverse of R, denoted R\. Q Q Using the fact that R _ARR A, it is easy to Q K show, after some algebraic manipulations, that

Fig. 2. Optimal NB source separator structure when ARB\A is diagonal.

Fig. 3. Optimal NB source separator structure in the absence of noise.

¼ can also be written, in the absence of noise, as   ¼ "A(ARA)\KP, (14)   which means that the output of ¼ corresponds,   in this case, to the ideal output, given by y (t)"PRKRm(t), (15)   and that the performance of ¼ becomes infinite   for each source since the interferences are completely nulled: P(¼ )_(#R,#R,2,#R).  

(16)

5. Weighted least square and least square NB source separators Before analysing the performance of the optimal NB source separator, we present the Weighted Least Square (WLS) and the Least Square (LS) NB source separators, which correspond to the optimal one in some cases and which play an important roˆle in the behaviour interpretation of some HO blind source separation methods as shown in [20].

5.1. The weighted least square source separator The WLS NB source separator is the separator whose output y (t) is the WLS estimate of the   vector m(t), i.e. the estimate which minimizes the WLS criterion C [m(t)]_(x(t)!Am(t))RB\(x(t)!Am(t)), (17)   and we say that a separator ¼ belongs to the equivalence class of the WLS separator, called ¼¸S equivalence class, if the output of ¼ is the WLS estimate of a vector PRKRm(t) where P and K are a permutation and a diagonal matrix respectively. It is easy to show, from the optimization of (17), that the WLS equivalence class of separators is

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defined by the separators ¼ which can be writ  ten as ¼ _B\A(ARB\A)\KP  

(18)

and whose output vectors are given by y (t)_¼R x(t)     "PRKR(ARB\A)\ARB\x(t).

(19)

Introducing expression (1) into Eq. (19), the output of the WLS source separator can be written as y (t)"PRKR[m(t)#(ARB\A)\ARB\b(t)], (20)   which shows that contrary to the optimal separator, the WLS source separator nulls completely the interferences whatever the mixture matrix A, the spatial coherence B and the power g of the  noise. Moreover, we deduce from Eq. (18) that for each column i (1)i)P), w , of the matrix ¼ , G  UJQ there exists a source number k (1)k)P), such that w "B\A(ARB\A)\f , G  I

(21)

where f (k-th column of K) is a (P;1) vector whose I components are zero except for the kth which is an arbitrary non-zero complex value f . This expresI sion shows [47] that w is the vector which G  minimizes the quadratic form wRB w under the linear constraint ARw"f . Using expression (2), we I deduce from this result that w is also the vector G  which minimizes the output power wRR w under V the linear constraint ARw"f . In other words, I w is, for the source k, the canceller of interferG  ence which maximizes the output SINR for the source k. Thus the WLS source separator implements, for each source k, the Optimal Interference Canceller (OIC) for this source k. It corresponds to the OIC source separator which can also be written as ¼ _R\A(ARR\A)\KP. (22)   V V As a consequence of this result, the WLS source separator corresponds to the optimal one, ¼ ,   whenever the latter nulls completely the interferences, i.e. when the vectors B\a are orthogonal to G each other or in the absence of noise (Section 4).

33

On the other hand, defining the matrix R _B\R B\R, we can easily verify from Q Q Eq. (18) that R BR¼ "B\AR KP, (23) Q   K and as it has been shown previously that BR¼   is the minimum norm (""BR¼ """   Tr[¼R B¼ ]) solution of Eq. (23), we obtain an     other expression of ¼ , alternative to Eqs. (18)   and (22), given by ¼ "B\RR\B\AKP, (24)   Q where R\ is the pseudo-inverse of R . Using Q Q Eq. (24) in Eq. (19), this expression allows a simple description of the WLS source separator structure, depicted in Fig. 4, where, using a square root decomposition of R\, given by R\_ Q Q (R\)(R\)R, the matrix º _(R\)RB\ Q Q   Q ;AR is a (P;P) unitary matrix. K This figure shows that the WLS estimate of the sources is obtained by beamforming in the direction of each source (ARB\R(R\)) after a noise Q decorrelation stage (B\) and an orthogonalization of the sources steering vectors (R\)R. Q Note that in the case where the matrix A is invertible, which occurs, for linearly independent sources steering vectors, when P"N, the WLS separator, defined by Eq. (18), is equal to ¼ "A\RKP and does no longer depend on the   matrix B, which does not mean that its output performance is independent of B. This result, which, in this case, holds for all the Interference Canceller (IC) source separators, is directly related to the absence of extra degrees of freedom available to process the noise. Indeed, an (N; P) source separator has, for the restitution of each source, N degrees of freedom among which, for an IC separator, P!1 are used for the nulling of the P!1 interferences for this source, one is used to prevent the associated useful source to be nulled and N!P degrees of freedom are available for the minimization of the output noise power. However when

Fig. 4. WLS source separator structure.

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N"P, no degree of freedom is available to minimize the output noise, which, in this case, plays no role in the IC separator computation. Finally, from Eqs. (5), (6) and (21), we deduce the expression of the WLS source separator performance, given by P(¼ )_(SINRM1[¼ ],SINRM2[¼ ],2,       SINRMP[¼ ]), (25)   where n " f R(ARB\A)\ARB\a " I . SINRMk[¼ ]" I I   g f R(ARB\A)\f  I I (26)

5.2. The Least Square source separator The LS NB source separator is the separator whose output y (t) is the LS estimate of the vector  m(t), i.e. the estimate which minimizes the LS criterion C [m(t)]_(x(t)!Am(t))R(x(t)!Am(t)). (27)  The LS source separator corresponds to the WLS source separator when the noise is spatially white (B"I). Defining the ¸S equivalence class in the same manner as the WLS equivalence class has been defined, it is easy to show that the LS equivalence class of separators is defined by the separators ¼ which can be written as  ¼ _ A(ARA)\KP, (28)   and whose output vectors are given by y (t)_¼R x(t)"PRKR(ARA)\ARx(t)   "PRKR[m(t)#(ARA)\ARb(t)]. (29) Thus, the LS source separator is also an IC source separator whatever the mixture matrix A, the spatial coherence B and the power g of the  noise. Optimal in the absence of noise, since it coincides in this case with Eq. (14), the LS source separator corresponds to the OIC for a spatially white noise and remains a sub-optimal IC in the other cases. Moreover, we deduce from Eqs. (13), (14) and (28) that ¼ can also be written as  ¼ "R\AKP, (30)  Q

and using Eq. (30) in Eq. (29), this allows a simple description of the LS source separator structure, depicted in Fig. 5, where the matrix º _  (R\)RAR is a (P;P) unitary matrix. Q K This figure shows that the LS estimate of the sources is obtained by beamforming in the direction of each source (AR(R\)) after an orthoQ gonalization of the sources steering vectors (R\)R. Q Finally, from Eqs. (5), (6) and (28), we deduce the expression of the LS source separator performance, given by P(¼ )_(SINRM1[¼ ],SINRM2[¼ ],2,    SINRMP[¼ ]),  where

(31)

n " f R(ARA)\ARa " I I SINRMk[¼ ]" I .  g f R(ARA)\ARBA(ARA)\f  I I (32)

6. Maximum likelihood interpretation of the SMF, the WLS and the LS NB source separators We present in this section a Maximum Likelihood (ML) interpretation of the optimal, the WLS and the LS source separators, which allows the understanding of the WLS and LS source separators sub-optimality.

6.1. ML interpretation of the optimal source separator For each source k, let us compute the ML estimate of the complex envelope m (t) of the source k, I assuming that the steering vector a is known I and that the total noise for the source k is zeromean, stationary, known and Gaussian. Noting

Fig. 5. LS source separator structure.

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p(x(t)/m (t)) the probability density of the observaI tion x(t) conditionally to m (t), the ML estimate of I m (t), under the previous assumptions, maximizes I the following expression: p(x(t)/m (t)) I 1 exp[!(x(t)!m (t)a )R\(x(t) " I I @I n,det[R ] @I !m (t)a ], (33) I I and is given by mL (t)_(aRR\a)\aRR\x(t) I @I I @I I+* "(aRR\a )\aRR\x(t). (34) I V I I V Defining mL (t)_(mL (t), mL (t), ,mL (t))2, +* +* +* 2 .+* the vector whose components are the ML estimates of each source, we obtain mL (t)_Diag[(aRR\a )\]ARR\x(t), (35) +* I V I V where Diag[(aRR\a )\] is the (P;P) diagonal I V I matrix whose elements are the quantities (aRR\a )\. Thus, expression (35) shows that the I V I vector mL (t) is the output of a separator belong+* ing to the optimal equivalence class of separators. In other words, the separator which generates an ML estimate of each source belongs to the optimal equivalence class of separators.

6.2. ML interpretation of the WLS source separator Let us now compute the ML estimate of the vector m(t) whose components are the complex envelope of the sources, assuming that the matrix A is known and that the background noise is zeromean, stationary, known and Gaussian. Noting p(x(t)/m(t)) the probability density of the observation x(t) conditionally to m(t), the ML estimate of m(t), under the previous assumptions, maximizes the following expression: 1 p(x(t)/m(t))" exp[!(x(t) n,det[B] !Am(t))RB\(x(t)!Am(t))],

which corresponds to the WLS estimate of m(t). In other words, the separator which generates an ML estimate of the source vector m(t) belongs to the WLS equivalence class of separators. Thus, comparing Eqs. (37) and (35), we deduce that in the general case, the components of mL (t), the ML +* estimate of m(t), do not correspond to the ML estimates of the components of m(t).

6.3. ML interpretation of the LS source separator Assuming that the matrix A is known and that the background noise is zero-mean, stationary, spatially white and Gaussian, the ML estimate of the vector m(t) maximizes Eq. (36) with B"I and is given by mL (t)_(ARA)\ARx(t), (38) +* which corresponds to the LS estimate of m(t). In other words, the separator which generates an ML estimate of the source vector m(t), assuming a spatially white noise, belongs to the LS equivalence class of separators.

6.4. Summary The previous results show that the SMF, the WLS and the LS source separators are all related to an ML criterion but under different assumptions about the noise and the sources. In particular, contrary to the WLS and the LS source separators, the SMF source separator is the only one which takes into account the second-order statistics of the sources in addition to that of the noise and to the knowledge of the matrix A, which, in fact, explains its optimality. The position of the three previous source separators with respect to optimality is summarized in Fig. 6 for several situations.

7. Steady-state performance of the SMF, the WLS and the LS NB source separators (36)

and is given by mL (t)_(ARB\A)\ARB\x(t), +*

35

(37)

We present, in this section, the steady-state performance of the optimal, the WLS and the LS source separators firstly for the two sources case

36

P. Chevalier / Signal Processing 73 (1999) 27–47

and where a , such that (0)"a ")1), is the   spatial correlation coefficient [44] of the sources 1 and 2 for the inner product (u,€) _uRB\€, defined by

Fig. 6. SMF, WLS and LS separators with respect to optimality.

and then for the general P (P'2) sources case such that (P)N). This analysis is very useful in particular to understand how the different physical parameters such as the SNR and the angular separation of the sources, the noise spatial coherence, the number of sources and sensors or the array geometry control the output steady-state performance of the three source separators. Moreover, let us recall that the performance of the optimal NB source separator ¼ corresponds to an upper-bound for the   performance of all the blind NB source separators ¼. Besides, some of the following results extend to arbitrary noise spatial coherence matrix B some results which have only been partially obtained for spatially white noise in the context of adaptive array steady-state performance analysis [38,39,44,68]. Note that the case P'N is not analysed in this paper but some results about this case can be found in [14,35]. 7.1. Two sources case 7.1.1. SMF source separator performance Using expression (2) into Eq. (12) with P"2, the SINRM of the two sources 1 and 2 at the output of the optimal source separator ¼ can be com  puted and after some elementary algebraic manipulations, we obtain





e SINRM1[¼ ]"e 1!  "a " , (39)     1#e  where n e _ IaRB\a (1)k)2) I g I I 

(40)

aR B\a   a _ . (41)  (aR B\a )(aR B\a )     This coefficient is a function of the noise spatial coherent matrix B, the angular separation between the sources, the array geometry, the number and the kind of sensors, the polarization of the sources and sensors. Note that the quantity SINRM2[¼ ] is obtained from Eq. (39) by ex  changing the indices 1 and 2. Expression (39) shows all the physical parameters which control the optimal source separator performance. More precisely, this expression shows that the SINRM1 at the output of ¼ is always   proportional to the input SNR of the source 1. Besides, this SINRM1 is maximum and equal to e in the absence of interference (e "0) or when   the two sources are B\ -orthogonal (a "0). In  these latter situations the output SINRM1 is, for spatially white noise and identical sensors, proportional to the number of sensors N. Nevertheless, in the general case, the presence of a second source (interference) degrades the performance and the output SINRM1 becomes a decreasing function of e and "a ", which means, in particular, that   the output SINRM1 decreases when the input SNR of the source 2 increases or when the angular separation of the two sources decreases beyond the 3 dB beamwidth of the array (for sensors not diversely polarised). In particular when "a " is maximum  and equal to one, which is the case when the two sources come from the same direction and have the same polarization (if the sensors are diversely polarised), the output SINRM1 is equal to the input one and the source separator is inoperative. On the other hand, for a strong interference source (e 1), as long as "a " is not too close   to unity, expression (39) can be written approximately as SINRM1[¼ ]+e [1!"a "], "a "O1,      (42) which does no longer depend on the input SNR of the interference source since the latter is rejected

P. Chevalier / Signal Processing 73 (1999) 27—47

under the output background noise level which solely controls the output total noise power, as it is shown hereafter. Thus, in the presence of a strong interference not too close to the source 1, the SINRM1 at the output of ¼ is mainly controlled   by the maximal SINRM1 that can be reached at the output of ¼ , e , and by the square modulus of    the spatial correlation coefficient a which gets  together all the physical parameters controlling the optimal source separator performance for the source 1, except the input SNR1 which controls e .  To understand more precisely the behaviour of the optimal source separator and to explain the previous results, it is useful to compute the SNR of the sources 1 and 2 at the output of the SMF for the source 1, denoted by w . Using Eqs. (7) and (2) QKD in Eq. (4) for P"2, we obtain, after some algebraic manipulations, SNR1[w ]   "e

 




e 1!  1#e



"a



 

"





(43)



(44)

\ 1 ; 1! 1! "a " , (1#e )   SNR2[w ]   e  " (1#e



"a " ) 






\ 1 ; 1! 1! "a " (1#e )  

The latter expression shows that generally, the interference is not completely nulled by the SMF except when the sources 1 and 2 are B\ -orthogonal (a "0) or in the absence of background  noise (e P#R), which confirms the results of  Section 4. In these two particular situations, we verify that the SNR and SINR of the source 1 at the output of w coincide.   Furthermore, for a weak interference source (e 1), the SNR of the source 2 at the output of  w can be written approximately as QKD SNR2[w ]+e "a ",    

(45)

37

which is proportional to e and to "a " and   which shows in particular that the rejection of a weak interference is similar to that obtained with a conventional beamformer pointing in the direction of the source 1. Thus, a weak interference is generally weakly rejected by the SMF since, in this case, the main source of performance degradation is the background noise, which is minimized by the SMF so as to maximize the output SINR. In this latter situation, we verify that the SNR and the SINR of the source 1 at the output of w co  incide and are approximately equal to e .  However, for a strong interference (e 1), as  long as "a " is not too close to unity, expression  (44) becomes "a "  , "a "O1, (46)  1!"a "   which is inversely proportional to e and which is  an increasing function of "a ". Thus, a strong  interference is all the more well rejected by the SMF than its input SNR is high and than the spatial correlation coefficient between the two sources is weak. In these situations, the residual background noise becomes the main source of performance degradation at the output of the SMF, which explains why the SNR and SINR of the source 1 at the output of w coincide and do not   depend on the input SNR of the interference as shown by Eq. (42). All the previous results are illustrated in Section 7.1.4.

SNR2[w ]+   e

1

7.1.2. WLS source separator performance Using expression (2) into Eq. (26) with P"2, the SINRM of the two sources 1 and 2 at the output of the WLS source separator ¼ can be computed   and after some elementary algebraic manipulations, we obtain SINRM1[¼ ]"e [1!"a "], (47)     whereas the quantity SINRM2[¼ ] is obtained   from Eq. (47) by exchanging the indices 1 and 2. Comparing Eq. (47) to Eq. (39), we verify that in all cases SINRM1[¼ ](SINRM1[¼ ] except     for B\-orthogonal sources or in the absence of noise where the two SINRM coincide, which confirms the results of Section 5. More precisely,

38

P. Chevalier / Signal Processing 73 (1999) 27–47

expressions (39) and (47) show that the degradation of performance, with respect to the optimal ones, obtained at the output of the OIC for the source 1, w , is an increasing function of "a " and a de   creasing function of e . This performance degra dation is very small and SINRM1[¼ ]+   SINRM1[¼ ] for a strong interference or when   the spatial correlation coefficient between the two sources is very weak. However, it becomes higher for weak interference whose angular separation with source 1 is lower than the 3 dB beamwidth of the array, as it is illustrated in Section 7.1.4. In fact, these results can be explained by the fact that contrary to the optimal source separator, the WLS source separator is a purely geometric one which adapts itself only to the spatial characteristics of the sources by nulling completely the interference whatever the input SNR of the latter, which is obviously sub-optimal for a weak interference which does not constitute the main source of performance degradation. As a summary, ¼ and   ¼ have approximately the same performance   for strong or well angularly separated sources while ¼ becomes all the more sub-optimal when the   sources become weak and get closer to each other. 7.1.3. LS source separator performance For a spatially white noise (B"I), the performance of the LS source separator corresponds to that of the WLS one. However, for a spatially coloured noise, the performance of the LS source separator is, for each source, weaker than those of the WLS one. Using expression (2) in Eq. (32) with P"2, the SINRM of the two sources 1 and 2 at the output of the LS source separator ¼ can be com  puted and after some elementary algebraic manipulations, we obtain [1!"a "] ' SINRM1[¼ ]"e ,  ' f

and where the symbol I corresponds to the identity matrix. Note that the quantity SINRM2[¼ ] is  obtained from Eq. (48) by exchanging the indices 1 and 2. 7.1.4. Performance illustrations The results obtained in Section 7.1 for the two sources case are illustrated in this section for a spatially white noise (B"I) and for a Uniformly spaced Linear Array (ULA) of N omnidirectional sensors equispaced half a wavelength apart. Under these assumptions, the WLS and the LS source separators coincide and the spatial correlation coefficient between the sources 1 and 2, a , is simply ' denoted by a . The elevation angle of the two  sources is assumed to be zero whereas their azimuthal angles with respect to the broadside are denoted by h and h , respectively.   Fig. 7 shows the variations of "a " as a function  of h for h "0° and for several values of the   number of sensors. Fig. 8 shows the variations of the SINRM1 (which is here equal to the SINRM2) at the output of ¼ as a function of h for h "0° and for     several values of the number of sensors, assuming that the input SNR of the sources is equal to 10 dB. Note the increasing performance as the number of sensors increases and the decreasing performance as the two sources get beyond the 3 dB beamwidth of the array.

(48)

where aR Ba aR Ba f_  #  "a " aR a aR a '     !2






aR Ba aR Ba      Re[a a ], '  aR a aR a    

(49)

Fig. 7. "a " as a function of h , h "0°, N"2,3,4,8.   

P. Chevalier / Signal Processing 73 (1999) 27—47

39

Figs. 9 and 10 show the variations of the SINRM1 at the output of ¼ and ¼ as a func   tion of h for h "0°, N"2 and for a weak   and a strong interference, respectively. Note the increasing performance degradation with respect to optimality at the output of ¼ as "a "   increases and as the input SNR of the interference decreases. Fig. 11 shows the variations of the SNR1 and SNR2 at the output of the SMF for the source 1 as

a function of h for h "0°, N"4 and assuming   that the input SNR of the sources is equal to 10 dB. Note the increasing value of the output SNR2, which remains below 0 dB as long as "a " is not in  the vicinity of unity, and the decreasing value of the output SNR1, which is much higher than 0 dB as long as "a " does not approach unity, as "a "   increases. Note also the complete rejection of the interference (source 2) when the two sources are orthogonal (h "$30°). 

Fig. 8. SINRM1[¼ ] as a function of h , h "0°, n /g "       n /g "10 dB, N"2,4,8.  

Fig. 10. SINRM1[¼ ] and SINRM1[¼ ] as a function of    h , h "0°, N"2, n /g "10 dB, n /g "15 dB.      

Fig. 9. SINRM1[¼ ] and SINRM1[¼ ] as a function of h ,     h "0°,N"2, n /g "10 dB, n /g "!5 dB.     

Fig. 11. SNR1[w ] and SNR2[w ] as a function of h ,      h "0°, N"4, n /g "n /g "10 dB.     

40

P. Chevalier / Signal Processing 73 (1999) 27–47

Fig. 12 shows the variations of the SNR2 at the output of the SMF for the source 1 as a function of the input SNR of the source 2 (interference) for h "0°, h "10° and for several values of the num  ber of sensors. Note, for a given value of N, the increasing value of the output SNR2 with the input SNR2 as the latter is weak and the decreasing value of the output SNR2 when the input SNR2 increases as the latter is high. Besides, note, for high values of the input SNR2, the increasing value of the output SNR2 as N decreases Eq. (46), whereas, for weak values of the input SNR2, the increasing (N "4) or the decreasing (N"8) values of the output SNR2 as N increases, depending on the value of N"a ",  Eq. (45).

7.2. P (P'2) sources case 7.2.1. SMF source separator performance In the general case of P sources (P)N), the computation of the optimal output performance directly from expressions (2) and (12) is complicated and it is more interesting to derive the performance from the spectral decomposition of the correlation matrices R _B\R B\R (1)k)P), where @I @I it is recalled that R , defined in Section 3 is the @I total noise (noise plus interferences) matrix for the source k. In particular, the spectral decomposition

Fig. 12. SNR2[w ] as a function of n /g , h "0°, h "       10°, N"2,4,8.

of the matrix R

@

is given by

. R _ n (B\a )(B\a )R#g I @ G G G  G , _ j u uR , (50) H H H H where the N vectors u are the orthonormalized H eigenvectors of R associated to the eigenvalues @ j . It is well-known, due to the particular algeH braic structure of R that the P!1 largest @ eigenvalues of R are such that j "c #g @ H H  (1)j)P!1), whereas the N!P#1 lowest eigenvalues are equal to g , where the quantities  c (1)j)P!1), correspond to the P!1 nonH zero eigenvalues of R when g "0. From these @  results, we easily obtain, after some algebraic manipulations, a useful expression of R\ given by @ .\ c H u uR , (51) R\ "g\ I! H H @  g #c H H  and using the latter expression and the fact that R\"B\RR\ B\, we deduce from Eq. (12) @ @ the expression of the SINRM1 at the output of ¼ in the general case of P sources, given by   SINRM1[¼ ]   "aR B\†u " .\ c  H H "e 1! . (52)  (aR B\a ) g #c   H H  Note that the SINRMk[¼ ] for kO1 is ob  tained from Eq. (52) by replacing the index 1 by k. Expression (52) is the extension to the P sources case of expression (39) valid for P"2. It shows all the physical parameters which control the optimal source separator performance in the presence of an arbitrary number of sources such that P)N. In particular, the SINRM1 at the output of ¼ is   maximum and equal to e in the absence of inter ference (c "0,1)j)P!1) or when the source H 1 is B\-orthogonal to all the other sources, since span+u 1)j)P!1,"span+B\a ,2) H  G i)P,, which occurs in particular when ARB\A is a diagonal matrix. In all the other situations, the presence of several sources degrades the performance and this degradation is an increasing function of the number of sources. Indeed, if we denote









P. Chevalier / Signal Processing 73 (1999) 27—47

by SINRM1[¼ ,P,N] the SINRM1 obtained at   the output of ¼ in the presence of P sources and   for a given array of N sensors, it is shown in Appendix A that the addition of a new source to the other gives, for the same array of sensors, a new SINRM1 at the output of ¼ , denoted by   SINRM1[¼ ,P#1,N] and such that   SINRM1[¼ ,P#1,N])SINRM1[¼ ,P,N].     (53) In the same manner, it is shown in Appendix A that, for a given number of sources P, the addition of a new sensor to the previous ones gives a new SINRM1 at the output of ¼ , noted   SINRM1[¼ ,P,N#1] and such that   SINRM1[¼ ,P,N])SINRM1[¼ ,P,N#1],     (54) which means that an increase in the number of sensors can never decrease the performance at the output of the optimal source separator. Let us now consider the case where the interferences for the source 1 (i.e. the sources k such that kO1) are B\-orthogonal to each other. In these conditions, it is easy to verify that for 1)j)P!1, u "(aR B\a )\B\a H H> H> H> and c "n aR B\a , which gives the folH H> H> H> lowing expression for the SINRM1[¼ ]:   . e G "a " , 1! SINRM1[¼ ]"e G    1#e G G (55)





where a is the spatial correlation coefficient beG tween the sources 1 and i, defined by Eq. (41) with the index 2 replaced by the index i. Comparing Eq. (55) with Eq. (39), we deduce that in the presence of B\-orthogonal interference sources, the performance degradation of the SINRM1[¼ ]   due to the presence of several sources is the sum of the degradation due to each interference considered separately. Thus, B\-orthogonal interference sources are processed by the SMF separately and their contributions add to each other. On the other hand, for strong interference sources (e 1,2)i)P), we obtain (c g ,1) G H  j)P!1) and as long as [1!SINRM1[¼ ]/e ]   

41

is not in the vicinity of unity, expression (52) can be approximated by





.\ "aR B\Ru " H SINRM1[¼ ]+e 1!  ,    (aR B\a )   H (56) which no longer depends on the input SNR of the interference sources and which corresponds to expression (42) if P"2. It is shown in Appendix B that the term under the summation appearing in Eq. (56) has a physical meaning and can be written as .\ "aR B\†u " H ""a ",  (57) ' (aR B\a )   H where a corresponds to what we call the spatial ' correlation coefficient between the source 1 and its associated interferences (all the other sources), for the inner product (u, €) , quantity which has initially been introducted in [44] for a spatially white noise. This coefficient is defined by the normalized classical inner product between the vector B\a  and its orthogonal projection, denoted by (B\a ) , on the hyperplane, H _B\J , span &   ned by the vectors B\a (2)i)P), where J is G  the hyperplan spanned by the steering vectors of the interferences for the source 1. The coefficient a corresponds to the cosine of , where ' '

is the angle between B\a and the hyper'  plane B\J , and can be written as  aR B\†[(B\a ) ]  &  a _ ' (aR B\a )([(B\a ) ]R[(B\a ) ])    &  & _cos . (58) ' Then, using Eq. (57) in Eq. (56), the SINRM1 at the output of ¼ can be written, for strong interfer  ence sources, as SINRM1[¼ ]+e [1!"a "] , "a "O1.    ' ' (59) Thus, in the presence of multiple strong interferences not too close to the source 1, the SINRM1 at the output of ¼ is mainly controlled by the   maximal SINRM1 that can be reached at the output of ¼ ,e , and by the square modulus of the   

42

P. Chevalier / Signal Processing 73 (1999) 27–47

spatial correlation coefficient a which gets to'' gether all the physical parameters controlling the optimal source separator performance, except the input SNR1 which controls e . Note that for  P"2, "a " is reduced to "a " and Eq. (59) '  corresponds to Eq. (42), whereas in the presence of P"3 sources, we can easily show that "a "#"a "!2Re(a a a )     . "a ""  ' 1!"a "  (60) Besides, for P!1 B\-orthogonal interferences for the source 1, we obtain . "a "" "a " (61) ' G G and expression (59) is reduced to Eq. (55) where the quantities e (2)i)P) are assumed to be infinite. G In this latter case of P!1 B\-orthogonal interference for the source 1, the SNR of the source 1 and its associated interferences at the output of the SMF for the source 1, can be easily computed from Eqs. (8), (2) and (5) and we obtain, after some algebraic manipulations, SNR1[w ]  

 



 . e H 1! "a " H 1#e H H \ . 1 , (62) )"a " ; 1! (1! (1#e ) H H H SNRk[w ]   e I " "a " (1#e ) I I \ . 1 ; 1! (1! )"a " , kO1, (1#e ) H H H (63) "e





 

which generalize, for P!1 B\-orthogonal interferences, expressions (43) and (44) obtained for the 2 sources case. We still verify that B\-orthogonal interferences are completely nulled by the SMF only when they are B\-orthogonal to the source 1 (a "0) or in the absence of background noise I

(e P#R). In the other cases, expression (63) I shows that the output SNR of each interference increases, i.e. the rejection of the latter decreases with the number of interferences. Furthermore, for weak sources (e 1,1) I k)P), the SNR of the source k at the output of w can be written approximately as   SNRk[w ]+e "a ", (64)   I I which is proportional to e and to "a " and I I which shows in particular that the rejection of a weak interference is similar to that obtained with a conventional beamformer pointing in the direction of the source 1, a result similar to that obtained in the 2 sources case. However, for strong interferences (e 1, kO1), I as long as "a " is not too close to unity, expres' sion (63) becomes "a " I , 1!"a " I ' "O1, kO1,

SNRk[w ]+   e

1

"a (65) ' which is inversely proportional to e and which is I an increasing function of "a " and "a ". All the I ' previous results are illustrated in Section 7.2.3. 7.2.2. WLS source separator performance Developing expression (26) for the P sources case, it is shown in Appendix C that the SINRM of the source 1 at the output of the WLS source separator ¼ can be written as   SINRM1[¼ ]"e [1!"a "], (66)    ' whereas the quantities SINRMk[¼ ] for kO1   are obtained from Eq. (66) by replacing the indice 1 by k. Comparing Eq. (66) with Eq. (52), we verify that in all cases SINRM1[¼ ](   SINRM1[¼ ] except when the source 1 is B\  orthogonal to the interferences or in the absence of noise, where the two SINRM coincide, which confirms the results of Section 5. In particular, expressions (66) and (52) show that the degradation of performance, with respect to the optimal ones, obtained at the output of the OIC for the source 1, w , increases when the ratios c /g decrease. In   H  fact, this performance degradation is very small and

P. Chevalier / Signal Processing 73 (1999) 27—47

43

SINRM1[¼ ]+SINRM1[¼ ] for strong in    terferences or when the spatial correlation coefficient between the source 1 and the interferences is very weak. However, it becomes higher for weak interference when angular separation with the source 1 is lower than the 3 dB beamwidth of the array, as it is illustrated in Section 7.2.3. 7.2.3. Performance illustrations The results obtained in Section 7.2 for the P (P'2) sources case are illustrated in this section for a spatially white noise (B"I) and for a Uniformly spaced Linear Array (ULA) of N omnidirectional sensors equispaced half a wavelength apart. The elevation angle of the sources is assumed to be zero whereas their azimuthal angles with respect to the broadside are denoted by H ,1)i)P. G Fig. 13 shows the variations of the SINRM1 at the output of ¼ as a function of h for several    values of the number of interferences and for N"4, assuming that the input SNR of all the sources is equal to 20 dB. The sources 2, 3 and 4 are such that h "0°, h "30° and h "90°. Note the decreas   ing performance as the number of sources increases. Fig. 14 shows the variations of the SNR of the source 2 at the output of the SMF for the source 1 as a function of the input SNR of the source 2, for several values of the number of sources and for N"4, assuming that the input SNR of the sources

Fig. 13. SINRM1[¼ ] as a function of h , P"2,3 and 4,    N"4, h "0°, h "30°, h "90°, n /g "20 dB (1)i)4).    G 

Fig. 14. SNR2[w ] as a function of n /g , P"2,3 and 4,     N"4, h "60°, h "0°, h "90°, h "30°, n /g "20 dB (1     G  )i)4).

k, with kO2, is equal to 20 dB. The sources are such that h "60°, h "0°, h "90° and h "30°.     Note the decreasing rejection of the interfences as the number of sources increases.

8. Conclusion In this paper, the concept of NB source separator performance has been introduced. This tool allows the quantitative evaluation of the restitution’s quality of each source by a given separator in a given environment and makes possible the performance comparison of two different separators for arbitrary sources and noise. From this performance criterion, the optimal NB source separator has been introduced and compared with the WLS and the LS source separators. A Maximum Likelihood interpretation of these informed separators has been recalled. The structure of these informed separators have then been described and the performance of the latter have been computed in a general noisy situation, for an arbitrary number of sources. The results, which correspond to upper-bounds for the performance of all the blind NB source separators, show the influence on the performance of different physical parameters such as the SNR and the DOA

44

P. Chevalier / Signal Processing 73 (1999) 27–47

of the sources, the number of sources and sensors, the array geometry or the noise spatial coherence. These results may be very useful in comparing optimality of performance of blind separators.

To show the validity of expression (57), where a is defined by Eq. (58), let us recall that the ' vector B\a can be written as  B\a _(B\a ) #(B\a ) , ,   &  &

Appendix A. To show the validity of expression (53), let us denote by R the correlation matrix of the total @. noise for the source 1 in the presence of P statistically independent sources. In the presence of one more source, the correlation matrix of the total noise for the source 1, denoted by R , is given @.> by R _R #n a aR , (A.1) @.> @. .> N> N> where n and a are the input power and the .> N> steering vector of the source P#1, respectively. Applying the Inversion Lemma to Eq. (A.1) we obtain n N> R\ "R\ ! @.> @. 1#n aR R\ a N> N> @. N> ;R\ a aR R\ , (A.2) @. N> N> @. and using the fact that SINRM1[¼ ,P,N]_   n aR R\ a , denoted by SINRM1[P, N], we ob  @.  tain SINRM1[P#1,N] "SINRM1[P,N]! 1#n

Appendix B.

n n  N> aR R\ a N> N> @. N> (A.3)

;"aR R\ a "  @. N> which demonstrates inequality (53). On the other hand, the ((N#1);1) filter w _[w2 , 0]2, where it is recalled that w is ,>     the (N;1) SMF associated to the source 1, has no reason to correspond to the ((N#1);1) SMF associated to this source. As a consequence of this result, the SINR at the output of w , which in ,> fact corresponds to the SINR at the output of w , is always less than or equal to the SINR at   the output of the ((N#1);1) SMF for the source 1, which demonstrates the expression (54).

(B.1)

where (B\a ) , is the orthogonal projection of  & B\a on the space H,, which corresponds to the   space orthogonal to H . Using Eq. (B.1) in Eq. (58),  we deduce that [(B\a ) ]R[(B\a ) ]  &  & . "a "" ' aR B\a  

(B.2)

Since H is the hyperplane spanned by the vec tors B\a , (2)i)P), it is also spanned by the G orthonormal vectors u (1)j)P!1), and thus H the vector (B\a ) can be written as  &



.\ (B\a ) " u uR  & H H H



B\a . 

(B.3)

Finally using Eq. (B.3) in Eq. (B.2), we obtain Eq. (57).

Appendix C. We show in this appendix that the quantity SINRM1[¼ ], defined by Eq. (26) for k"1, is   given by expression (66). First note that since the hyperplane spanned by the vectors B\a (2)i)P), corresponds to the one spanned G by the orthonormal vectors u (1)j)P!1), H there exists a ((P!1);(P!1)) full rank matrix ¹ such that B\A"[B\a ,º ¹],  

(C.1)

where º is the (N;(P!1)) matrix whose col umns are the vectors u (1)j)P!1). Using H Eq. (C.1), the (P;P) matrix ARB\A can be computed and can be written as



ARB\A"

b m







mR  , ¹R¹

(C.2)

P. Chevalier / Signal Processing 73 (1999) 27—47

where b _aR B\a and l _¹RºR B\a . Then,       the matrix (ARB\A)\ is given by

 

(ARB\A)\"

b mR m

C

,

(C.3)

where b_1/(b !mR (¹R¹)\m ), m_!b*(¹R¹)\m     and C is a particular ((P !1);(P!1)) matrix which is useless for what follows. From the previous expressions, we easily deduce expressions (C.4) and (C.5), given by f R (ARB\A)\ARB\a   "b[aR B\a !aR B\Rº ºR B\a ],      

(C.4)

f R (ARB\A)\f "b   "[aR B\a !aR B\Rº ºR B\a ]\.       (C.5) Finally, the quantity SINRM1[¼ ], defined by   Eq. (26) for k"1, can be computed from Eq. (C.4) and (C.5) and using Eq. (57) we find the expression (66).

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