Abdeldjalil Aïssa-El-Bey
email: abdeldjalil.aissaelbey@telecom-bretagne.eu

Karim Abed-Meraim
email: karim.abed@telecom-paristech.fr

Yves Grenier
email: yves.grenier@telecom-paristech.fr

Yingbo Hua
email: yhua@ee.ucr.edu

A General Framework for Second Order Blind Separation of Stationary Colored Sources

This paper focuses on the blind separation of stationary colored sources using the second order statistics of their instantaneous mixtures. We first start by presenting a brief overview of existing contributions in that field. Then, we present necessary and sufficient conditions for the identifiability and partial identifiability using a finite set of correlation matrices. These conditions depend on the autocorrelation function of the unknown sources. However, it is shown here that they can be tested directly from the observation through the decorrelator output. This issue is of prime importance to decide whether the sources have been well separated. If that's not the case then, further treatments will be needed. We then propose an identifiability testing based on resampling (jackknife) technique that is validated by simulation

Introduction

Source separation aims to recover multiple sources from multiple observations (mixtures) received by a set of linear sensors. The problem is said to be 'blind' when the observations have been linearly mixed by the transfer medium, while having no a priori knowledge of the transfer medium or the sources. Blind source separation (BSS) has applications in several areas, such as communication, speech and audio processing, biomedical engineering, geophysical data processing, etc [START_REF] Cichocki | Adaptive Blind Signal and Image Processing[END_REF]. BSS of instantaneous mixtures has attracted so far a lot of attention due to its many potential applications and its mathematical tractability that leads to several nice and simple BSS solutions [START_REF] Cichocki | Adaptive Blind Signal and Image Processing[END_REF][START_REF] Pham | Blind source separation of instantaneous mixtures of nonstationary sources[END_REF][START_REF] Belouchrani | A blind source separation technique using second-order statistics[END_REF][START_REF] Cardoso | A blind beamforming for non-gaussian signals[END_REF][START_REF] Abed-Meraim | A general framework for blind source separation using second order statistics[END_REF].

Assume that m narrow band signals impinge on an array of n ≥ m sensors. The measured array output is a weighted superposition of the signals, corrupted by additive noise, i.e.

x(t) = y(t) + η(t) = As(t) + η(t) ,

where s(t) = [s 1 (t), • • • , s m (t)] T is the m × 1 complex source vector, η(t) =

[η 1 (t), • • • , η n (t)] T is the n×1 complex noise vector, A is the n×m full column rank mixing matrix, and the superscript T denotes the transpose operator. The source signal vector s(t) is assumed to be a multivariate stationary complex stochastic process.

In this paper, we only consider the second order BSS methods. Hence, hence the component processes s i (t), 1 ≤ i ≤ m are assumed to be temporally coherent and mutually uncorrelated, with zero mean and second order moments:

S(τ ) def = E (s(t + τ )s (t)) = diag[ρ 1 (τ ), • • • , ρ m (τ )] , (2)
where ρ i (τ ) def = E(s i (t + τ )s * i (t)), the expectation operator is E, and superscripts * and denote the conjugate of a complex and the complex conjugate transpose of a vector, respectively. The additive noise η(t) is modeled as a white stationary zero-mean complex random process of covariance E(η(t)η (t)) = σ 2 Q. The latter matrix is proportional to identity, (i.e.

E(η(t)η (t)) = σ 2 I) when the noise is spatially white. Under these assumptions, the observed data correlation matrices are given by : R x (τ ) = AS(τ )A + δ(τ )σ 2 Q .

From this expression, one can observe that the noise free correlation matrices are 'diagonalizable' under the linear transformation B = A # where the superscript (•) # is the Moore-Penrose's pseudo-inversion operator, i.e. BR x (τ )B H are diagonal ∀τ = 0. Hence, the source separation is achieved by decorrelating the signals at different time lags.

Before all, note that complete blind identification of separating (demixing) matrix B (or the equivalently mixing matrix A) is impossible in the blind context, because the exchange of a fixed scalar between the source signal and the corresponding column of A leaves the observations unaffected. Also note that the signals numbering is immaterial. It follows that the best that can be done is to determine B up to a permutation and scalar shifts of its columns [START_REF] Belouchrani | A blind source separation technique using second-order statistics[END_REF],

i.e., B is a separating matrix if and only if :

By(t) = P Λs(t) (3)
where P is a permutation matrix and Λ a non-singular diagonal matrix.

Under the above assumptions, we provide a general framework for the BSS including the study of the identifiability and its testing as well as the introduction of a simple but efficient decorrelation method and its performance analysis. The paper is organized as follows. Section 2 reviews the principal contributions to the BSS problem using the second order statistics. Section 3 states the necessary and sufficient second order identifiability conditions.

We then propose an identifiability testing based on resampling technique in section 4. Section 5 proposes a blind source separation algorithm using relative gradient technique. Section 6 is devoted to the performance analysis of the considered decorrelation method and the validation of the identifiability testing technique. Section 7 is for concluding remarks.

The work using the second order statistics to achieve the blind source separation was initiated by L. Féty et al. [START_REF] Féty | News methods for signal separation[END_REF]. Féty's method is based on the simultaneous diagonalization of the correlation matrices R x (0) and R x [START_REF] Cichocki | Adaptive Blind Signal and Image Processing[END_REF]. Independently, L. Tong et al. proposed a similar technique, namely the AMUSE technique (Algorithm for Multiple Unknown Signals Extraction) [START_REF] Tong | Amuse: A new blind identification algorithm[END_REF][START_REF] Tong | Indeterminacy and identifiability of blind identification[END_REF], that achieves the BSS by the simultaneous diagonalization of two symmetric matrices R x (0) and (R x (τ k ) + R x (τ k ))/2 with τ k = 0. This method has been extended in [START_REF] Tomé | Blind source separation using a matrix pencil[END_REF] where a generalized eigenvalue decomposition of a matrix pen-

cil (R x (τ 1 ), R x (τ 2 )) is considered.
Later on, A. Belouchrani et al. proposed the SOBI (Second-Order Blind Identification) algorithm [START_REF] Belouchrani | A blind source separation technique using second-order statistics[END_REF] that generalizes the previous methods to the case where more than two correlation matrices are used. In the SOBI algorithm, the separation is achieved in two steps; the first step is the whitening of the observed signal vector by linear transformation. The second step consists of applying a joint approximate diagonalization algorithm to a set of different time-lag correlation matrices of the whitened signal vector. A variant of SOBI has been presented by D. Nuzillard et al. in [START_REF] Nuzillard | Second-order blind source separation in the fourier space of data[END_REF] allowing direct signal separation from frequency domain data by exploiting the source correlation properties expressed in the time domain. This algorithm is referred to as f-SOBI, standing for "frequency domain SOBI".

Numerous approaches have been proposed in recent years both for the formulation of the diagonalization criterion and for the algorithms considered for its minimization. One of the most popular and computationally appealing approach for the joint diagonalization of a set of matrices M 1 , • • • , M K is the unitary Approximate Joint Diagonalization (J.F. Cardoso et al. [START_REF] Cardoso | Jacobi angles for simultaneous diagonalization[END_REF]), which minimizes the criterion

K k=1 off(BM k B ) (4)
with respect to B, subject to the unitary constraint B B = I, where

off(P ) = i =j |P ij | 2 . ( 5
)
The unitary constraint implies the assumption of a unitary mixing matrix.

Hence, in the general case, a pre-processing "spatial hard-whitening" stage is required, in which the non-unitary factor of the overall demixing matrix is found and applied to the data.

In [START_REF] Abed-Meraim | A general framework for blind source separation using second order statistics[END_REF][START_REF] Ziehe | A fast algorithm for joint diagonalization with non-orthogonal transformations and its application to blind source separation[END_REF] an iterative algorithm using relative gradient technique has been considered for the minimization of (4) without unitary constraint. An alternative approach for non-unitary Approximate Joint Diagonalization has been proposed by Yeredor (the "AC-DC" algorithm [START_REF] Yeredor | Non-orthogonal joint diagonalization in the least-squares sense with application in blind source separation[END_REF]), which minimizes

K k=1 M k -AD k A 2 (6)
without constraining A to be unitary. In [START_REF] Féty | News methods for signal separation[END_REF],

D k , k = 1, • • • , K represent diagonal matrices.
While computationally efficient in small-scale problems, this algorithm has been observed to exhibit extremely slow convergence in large-scale problems. This criterion is also considered in [START_REF] Wu | A unifying criterion for blind source separation and decorrelation : Simultaneous diagonalization of correlation matrices[END_REF] where a gradient descent technique is used for its minimization and in [START_REF] Vollgraf | Quadratic optimization for simultaneous matrix diagonalization[END_REF] where quadratic optimization is used.

A computationally efficient unconstrained minimization algorithm was proposed by D.T. Pham et al. [START_REF] Pham | Joint approximate diagonalization of positive definite matrices[END_REF], whose target criterion is the Kullback-Leibler divergence between the n × n operand and the diagonal matrix with the same diagonal as the operand :

K k=1 λ k [log det diag(BM k B ) -log det(BM k B )] (7)
where λ k , k = 1, • • • , K are positive scalar factors. This approach requires all the target matrices to be positive definite, which limits its applicability as a generic BSS tool. Another class of BSS techniques based on the second order statistics is the one using the maximum likelihood principle. This method uses the Gaussian asymptotic property of the discrete Fourier transform of the second order stationary observations [START_REF] Pham | Séparation aveugle de sources temporellement corrélées[END_REF][START_REF]Blind separation of mixture of independent sources through a quasimaximum likelihood approach[END_REF][START_REF] Dégerine | Separation of an instantaneous mixture of gaussian autoregressive sources by the exact maximum likelihood approach[END_REF].

There are multiple potential applications of blind source separation using second order statistics. Among others, one may cite the work of G.

(0), R x (1)} (resp. {R x (0), (R x (τ k ) + R x (τ k ))/2}) are all distinct.
In [START_REF] Belouchrani | A blind source separation technique using second-order statistics[END_REF], the authors have shown that BSS using SOBI algorithm with the correlation matrix set {R x (τ 1 ), • • • , R x (τ K )} is possible only if the vectors

ρ i = [ρ i (τ 1 ), • • • , ρ i (τ K )], i = 1, • • • , m
are pairwise linearly independent. This result has been generalized in [START_REF] Abed-Meraim | Generalized second order identifiability condition and relevant testing technique[END_REF] to establish a necessary and sufficient identifiability condition with a finite set of correlation matrices. It is recalled and further developed in section 3.

3 Second Order Identifiability

Necessary and sufficient conditions of identifiability

In [START_REF] Tong | Indeterminacy and identifiability of blind identification[END_REF], Tong et al. showed that sources are blindly separable based on (the whole set) of second order statistics only if they have different spectral density functions. In practice we achieve the BSS using only a finite set of correlation matrices. Therefore, the previous identifiability result was generalized to that case in [START_REF] Belouchrani | A blind source separation technique using second-order statistics[END_REF][START_REF] Abed-Meraim | Generalized second order identifiability condition and relevant testing technique[END_REF] leading to the necessary and sufficient identifiability conditions given by the following theorem :

Theorem 1 Let τ 1 < τ 2 < • • • < τ K be K ≥ 1 time lags, and define ρ i = [ρ i (τ 1 ), ρ i (τ 2 ), • • • , ρ i (τ K )] and ρi = [ e(ρ i ), m(ρ i )]
where e(x) and m(x) denote the real part and imaginary part of x, respectively. Taking advantage of the indetermination, we assume without loss of generality that the sources are scaled such that ρ i = ρi = 1, for all i 1 . Then, BSS can be achieved using the output correlation matrices at time lags τ 1 , τ

s(t) = [s 1 (t), s2 (t), s 3 (t), • • • , s m (t)] T with [ã 1 , ã2 ] = [a 1 , a 2 ]T and         s1 (t) s2 (t)         = T -1         s 1 (t) s 2 (t)         (9)
with

T =         cos θ sin θ -sin θ cos θ         , if = +1 T =         cosh θ sinh θ sinh θ cosh θ         , if = -1
be detected (and a fortiori could not be estimated) from the considered set of correlation matrices. This hypothesis will be held in the sequel.

verifies

R x(τ k ) = R x (τ k ) and S(τ k ) = S(τ k ) for k = 1, • • • , K, where S(τ k ) def = E(s(t + τ k )s (t)). 2
Interestingly, we can see from condition (8) that BSS can be achieved from only one correlation matrix R x (τ k ) provided that the vectors [ e(ρ i (τ k )), m(ρ i (τ k ))] and [ e(ρ j (τ k )), m(ρ j (τ k ))] are pairwise linearly independent for all i = j.

Note also that from [START_REF] Tong | Indeterminacy and identifiability of blind identification[END_REF], BSS can be achieved if at most one temporally white source signal exists. Similarly, recall that when using higher order statistics, BSS can only be achieved if at most one Gaussian source signal exists.

Under the condition of Theorem 1, the BSS can be achieved by decorrelation according to the following result:

Theorem 2 Let τ 1 , τ 2 , • • • , τ K be K time lags and z(t) = [z 1 (t), • • • , z m (t)] T
be an m×1 vector given by z

(t) = Bx(t). Define r ij (τ k ) def = E(z i (t+τ k )z * j (t)).
If the identifiability condition holds, then B is a separating matrix (i.e. By(t) = P Λs(t) for a given permutation matrix P and a non-singular diagonal matrix Λ) if and only if

r ij (τ k ) = 0 and K k=1 |r ii (τ k )| > 0 ( 10
)
for all 1 ≤ i = j ≤ m and k = 1, 2, • • • , K.
Proof: see the proof of Theorem 3 in Appendix A.

Note that, if one of the time lags is zero, the result of Theorem 2 holds only under the noiseless assumption. In that case, we can replace the condition

K k=1 |r ii (τ k )| > 0 by r ii (0) > 0, for i = 1, • • • , m.
On the other hand, if all the time lags are non-zero and if the noise is temporally white (but can be spatially colored with unknown spatial covariance matrix) then the above result holds without the noiseless assumption.

Partial Identifiability

It is generally believed that when the identifiability conditions are not met, the BSS cannot be achieved. This is only half the truth as it is possible to partially separate the sources in the sense that we can extract those which satisfy the identifiability conditions. More precisely, the sources can be separated in blocks each of them containing a mixture of sources that are not separable using the considered set of statistics. For example, consider a mixture of 3 sources such that ρ1 = ρ2 while ρ1 and ρ3 are linearly independent. In that case, source s 3 can be extracted while sources s 1 and s 2 cannot. In other words, by decorrelating the observed signal at the considered time lags, one obtain 3 signals one of them being s 3 (up to a scalar constant) and the two others are linear mixtures of s 1 and s 2 .

This result can be mathematically formulated as follows: assume there are d distinct groups of sources each of them containing d i source signals with same (up to a sign) correlation vector ρi

, i = 1, • • • , d (clearly, m = d 1 + • • • + d d ).
The correlation vectors ρ1 , • • • , ρd are pairwise linearly independent. We write

s(t) = [s T 1 (t), • • • , s T d (t)]
T where each sub-vector s i (t) contains the d i source signals with correlation vector ρi .

Theorem 3 Let z(t) = Bx(t) be an m × 1 random vector satisfying equation

(10) for all 1 ≤ i = j ≤ m and k = 1, • • • , K.
Then, there exists a permutation matrix P such that z(t)

def = P z(t) = [z T 1 (t), • • • , z T d (t)] T where z i (t) = U i s i (t), U i being a d i × d i non-singular matrix.
Moreover, sources belonging to the same group, i.e., having same (up to a sign) correlation vector ρi can not be separated using only the correlation matrices R

x (τ k ), k = 1, • • • , K.
This result shows that when some of the sources have same (up to a sign) correlation vectors then the best that can be done is to separate them per blocks and this can be achieved by decorrelation. However, this result would be useless if we cannot check the linear dependency of the correlation vectors ρi and partition the signals per groups (as shown above) according to their correlation vectors. This leads us to the important problem of testing the identifiability condition that is discussed next.

4 Identifiability testing

Theoretical result

The necessary and sufficient identifiability condition (8) depends on the correlation coefficients of the source signals. The latter being unknown, it is therefore impossible to a priori check whether the sources are 'separable' or not from a given set of output correlation matrices. However, it is possible to check a posteriori whether the sources have been 'separated' or not. We have the following result:

Theorem 4 Let τ 1 < τ 2 < • • • < τ K be K distinct time lags and z(t) = Bx(t).
Assume that B is a matrix such that z(t) satisfies2 equation (10) for all

1 ≤ i = j ≤ m and k = 1, • • • , K.
Then there exists a permutation matrix P

such that for k = 1, • • • , K. E(z(t + τ k )z (t)) = P T S(k)P (11)
In other words the entries of z(t) def = P z(t) have the same correlation coefficients as those of s(t) at time lags

τ 1 , • • • , τ K , i.e. E(z i (t + τ k )z * i (t)) = ρ i (τ k ) for k = 1, • • • , K and i = 1, • • • , m. Proof: see Appendix B.
From Theorem 4, the existence of condition ( 8) can be checked by using the

approximate correlation coefficients r ii (τ k ) def = E(z i (t+τ k )z * i (t)). It is important
to point out that even if equation ( 10) holds, it does not mean that the source signals have been separated. Three situations may happen:

(1) For all pairs (i, j), ρi and ρj (computed from z(t)) are pairwise linearly independent. Then we are sure that the sources have been separated and that z(t) = s(t) up to the inherent indeterminacies of the BSS problem. In fact, testing the identifiability condition is equivalent to pairwise testing the angles between ρi and ρj for all 1 ≤ i = j ≤ m. The larger the angle between ρi and ρj , the better the quality of source separation (see performance analysis in [START_REF] Belouchrani | A blind source separation technique using second-order statistics[END_REF]).

(2) For all pairs (i, j), ρi and ρj are linearly dependent. Thus, the sources haven't been separated and z(t) is still a linear combination of s(t).

(3) A few pairs (i, j) out of all pairs satisfy ρi and ρj linearly dependent.

Therefore the sources have been separated in blocks.

Now, having only one signal realization at hand, we propose to use a resampling technique to evaluate the statistics needed for the testing.

Testing using resampling techniques

Note that in practice the source correlation coefficients are calculated from noisy finite sample data. Due to the joint effects of noise and finite sample size, it is impossible to obtain the exact source correlation coefficients to test the identifiability condition. The identifiability condition should be tested using a certain threshold α, i.e., decide that ρi and ρj are linearly independent if

| ρi ρT j | -1 > α.
To find α we use the fact that the estimation error of ρi ρT j is asymptotically

Gaussian3 and hence one can build the confidence interval of such a variable according to its variance. This algorithm can be summarized as follows:

(1) Estimate a demixing matrix B and z(t) def = Bx(t) using an existing second order decorrelation algorithm (e.g. SOBI [START_REF] Belouchrani | A blind source separation technique using second-order statistics[END_REF]).

(2) For each component z i (t), estimate the corresponding normalized vector ρ i .

(3) Calculate the scalar product R(i, j) = | ρi ρT j | for each pair (i, j).

(4) Estimate σ(i,j) the standard deviation of R(i, j) using resampling technique (see section 4.3).

(5) Choose α (i,j) according to the confidence interval. e.g. to have a confidence interval equal to 99.7% we choose α (i,j) = 3σ (i,j) , and compare | R(i, j)-1|

to α (i,j) to test whether sources i and j have been separated or not.

Resampling techniques

In many signal processing applications one is interested in forming estimates of a certain number of unknown parameters of a random process, using a set of sample values. Further, one is interested in finding the sampling distribution of the estimators, so that the respective means, variances, and cumulants can be calculated, or in making some kind of probability statements with respect to the unknown true values of the parameters.

The bootstrap [START_REF] Zoubir | The bootstrap and its application in signal processing[END_REF] was introduced by Efron [START_REF] Efron | The jackknife, the bootstrap and other resampling plans[END_REF] as an approach to calculate confidence intervals for parameters in circumstances where standard methods cannot be applied. The bootstrap has subsequently been used to solve many other problems that would be too complicated for traditional statistical analysis.

In simple words, the bootstrap does with a computer what the experimenter would do in practice, i.e. if it were possible: he or she would repeat the experiment. With the bootstrap, the observations are randomly reassigned, and the estimates recomputed. These assignments and recomputations are done hundreds or thousands of times and treated as repeated experiments.

The jackknife [START_REF] Miller | The jackknife -a review[END_REF] is another resampling technique for estimating the standard deviation. As an alternative to the bootstrap, the jackknife method can be thought of as drawing T samples of size (T -1) each without replacement from the original sample of size T [START_REF] Miller | The jackknife -a review[END_REF].

Suppose we are given the sample X = {X 1 , X 2 , . . . , X T } and an estimate, θ, from X . The jackknife method is based on the sample delete-one observation at a time,

X (i) = {X 1 , X 2 , . . . , X i-1 , X i+1 , . . . , X T }
for i = 1, 2, . . . , T , called the jackknife sample. This i th jackknife sample consists of the data set with the i th observation removed. For each i th jackknife sample, we calculate the i th jackknife estimate, θ(i) of ϑ, i = 1, 2, . . . , T . The jackknife estimate of the standard deviation of θ is defined by

σ = T -1 T T i=1   θ(i) - 1 T T j=1 θ(j)   2 ( 12
)
The jackknife is computationally less expensive if T is less than the number of replicates used by the bootstrap for standard deviation estimation because it requires computation of θ only for the n jackknife data sets. For example, if L = 25 resamples are necessary for standard deviation estimation with the bootstrap, and the sample size is T = 10, then clearly the jackknife would be computationally less expensive than the bootstrap. In order to test the separability of the estimated signals, we have used a jackknife method to estimate the variance of the scalar product quantities R(i, j) for i, j = 1, 2, . . . , m. This is done according to the following steps:

(1) From each signal z i = [z i (0), . . . , z i (T -1)] T , generate T vectors such as z (j)

i = [z i (0), . . . , z i (j -1), z i (j +1), . . . , z i (T -1)] T and j = 0, 1, . . . , T -1.

(2) For each vector z (j)

i , estimate the corresponding vector ρ

(3) Estimate R such as its (i, j) th entry is

R(i, j) = 1 T T -1 k=0 ρ (k) i , ρ (k) j ρ (k) i ρ (k) j
where •, • denotes the scalar product and • is the euclidian norm.

(4) Estimate the standard deviation of R(i, j) by:

σ(i,j) = T -1 T T -1 k=0   ρ (k) i , ρ (k) j ρ (k) i ρ (k) j - 1 T T -1 l=0 ρ (l) i , ρ (l) j ρ (l) i ρ (l) j   2

Discussion

Some useful comments are provided here to get more insight onto the considered testing method and its potential applications and extensions.

• The asymptotic performance analysis of SOBI derived in [START_REF] Belouchrani | A blind source separation technique using second-order statistics[END_REF], shows that the separation performance of two sources s i and s j depends on the angle between their respective correlation vectors ρ i and ρ j . Hence, measuring this angle gives a hint on the interference rejection level of the two considered sources.

As a consequence, one can use the measure of this angle not only to test the separability of the two sources but also to guarantee a target (minimum) separation quality. Choosing the threshold α (i,j) accordingly is an important issue that deserves further investigation.

• The testing method can be incorporated into a two stage separation procedure where the first stage consists in a second order decorrelation method (e.g. SOBI). The second stage would be an HOS-based separation method applied only when the testing indicates a failure of separation at the first step.

• In many practical situations, one might be interested in only one or few source signals. This is the case for example in the interference mitigation problem in [START_REF] Belouchrani | Interference mitigation in spread spectrum communications using blind source separation[END_REF] or in the power plants monitoring applications [START_REF] D'urso | Blind identification methods applied to electricite de france's civil works and power plants monitoring[END_REF]. In this situation, the partial identifiability result is of high interest as it proves that the desired source signal can still be extracted even if a complete source separation cannot be achieved.

5 Separation algorithm using relative gradient

SOS-based separation criteria

Based on Theorem 2, we can define different objective functions for signal decorrelation. In [START_REF] Kawamoto | Blind separation of sources using temporal correlation of the observed signal[END_REF], the following criterion4 was used

G(z) = K k=1 log det (diag(R z (τ k ))) -log det (R z (τ k )) (13)
where diag(A) is the diagonal matrix obtained by zeroing the off diagonal entries of A. Another criterion used in [START_REF] Abed-Meraim | A general framework for blind source separation using second order statistics[END_REF] is

G(z) = K k=1 1≤i<j≤m [|r ij (τ k )+r ji (τ k )| 2 +|r ij (τ k )-r ji (τ k )| 2 ]+ m i=1 | K k=1 |r ii (τ k )|-1| 2 . (14)
The last term in ( 14) is introduced to avoid trivial (zero-valued) solutions.

Equations ( 13) and ( 14) are non-negative functions which are zero if and only

if R z (τ k ) = E(z(t + τ k )z (t)) are diagonal for k = 1, • • • , K or equivalently if (10
) is met. Hence, one can achieve the BSS through signal decorrelation by minimizing one of the previous cost functions.

Iterative Decorrelation Algorithm (IDA)

The separation criteria we have presented take the form:

B is a separating matrix ⇐⇒ G(z(t)) = 0 ( 15
)
where z(t) = Bx(t) and G is a given objective function. An efficient approach to solve (15) is the one proposed in [START_REF] Abed-Meraim | A general framework for blind source separation using second order statistics[END_REF][START_REF]Blind separation of mixture of independent sources through a quasimaximum likelihood approach[END_REF]. It is a block technique based on the processing of T received samples and consists of searching the zeros of the sample version of [START_REF] Vollgraf | Quadratic optimization for simultaneous matrix diagonalization[END_REF]. Solutions are obtained iteratively in the form:

B (p+1) = (I + (p) )B (p) (16)
z (p+1) (t) = (I + (p) )z (p) (t) (17)
At iteration p, a matrix (p) is determined from a local linearization of

G(Bx(t)).
It is an approximate Newton technique with the benefit that (p) can be very simply computed (no Hessian inversion) under the additional assumption that B (p) is close to a separating matrix. This procedure is illustrated as follows:

Using (17), we have:

r (p+1) ij (τ k ) = r (p) ij (τ k )+ m q=1 * (p) jq r (p) iq (τ k )+ m l=1 (p) il r (p) lj (τ k )+ m l,q=1 (p) il * (p) jq r (p) lq (τ k ) (18)
where

r (p) ij (τ k ) def = E z (p) i (t + τ k )z * (p) j (t) (19)
≈ 1 T -τ k T -τ k t=1 z (p) i (t + τ k )z * (p) j (t) (20)
Under the assumption that B (p) is close to a separating matrix, it follows that

| (p) ij | 1 and |r (p) ij (τ k )| 1 for i = j
and thus, a first order approximation of r (p+1) ij (τ k ) is given by:

r (p+1) ij (τ k ) ≈ r (p) ij (τ k ) + * (p) ji r (p) ii (τ k ) + (p) ij r (p) jj (τ k ) (21)
similarly, we have:

r (p+1) ji (τ k ) ≈ r (p) ji (τ k ) + * (p) ij r (p) jj (τ k ) + (p) ji r (p) ii (τ k ) (22)
From ( 21) and ( 22), we have:

r (p+1) ij (τ k )+r (p+1) ji (τ k ) ≈ 2r (p) jj (τ k ) e( (p) ij )+2r (p) ii (τ k ) e( (p) ji )+(r (p) ij (τ k )+r (p) ji (τ k )) r (p+1) ij (τ k )-r (p+1) ji (τ k ) ≈ 2r (p) jj (τ k ) m( (p) ij )-2r (p) ii (τ k ) m( (p) ji )+(r (p) ij (τ k )-r (p) ji (τ k ))
with  = √ -1. By replacing the previous equation into [START_REF] Wu | A unifying criterion for blind source separation and decorrelation : Simultaneous diagonalization of correlation matrices[END_REF], we obtain the following least squares (LS) minimization problem min r

(p) jj , r (p) ii E (p) ij + 1 2 (r (p) ij + r (p) ji ), 1 2 (r (p) ij -r (p) ji )
where

E (p) ij def =         e( (p) ij ) m( (p) ij ) e( (p) ji ) -m( (p) ji )         (23) r (p
) ij = [r (p) ij (τ 1 ), • • • , r (p) ij (τ K )] T (24)
A solution to the LS minimization problem is given by:

E (p) ij = -r (p) jj , r (p) ii # 1 2 (r (p) ij + r (p) ji ), 1 2 (r (p) ij -r (p) ji ) (25)
where A # denotes the pseudo-inverse of matrix A. Equations ( 23) and [START_REF] Efron | The jackknife, the bootstrap and other resampling plans[END_REF] provide the explicit expression of (p) ij for i = j. For i = j, the minimization of ( 14) using the first order approximation leads to :

K k=1 r (p) ii (τ k ) 1 + 2 e( (p) ii ) -1 = 0. ( 26
)
Without loss of generality, we take advantage of the phase indeterminacy to assume that ii are real-valued and hence e( ii ) = ii . Consequently, we obtain :

(p) ii = 1 - K k=1 |r (p) ii (τ k )| 2 K k=1 |r (p) ii (τ k )| (27)
In the case of real-valued signals, the LS minimization becomes : min

H (p) ij e (p) ij + ψ (p) ij
where

H (p) ij =         1 1         ⊗ r (p) jj , r (p) ii (28)
e

(p) ij = (p) ij , (p) ji T (29)
ψ (p) ij =         r (p) ij r (p) ji         (30)
and ⊗ denotes the Kronecker product. A solution to the LS minimization problem is given by:

e (p) ij = -H (p)# ij ψ (p) ij (31)
Remark: A main advantage of the above algorithm is its flexibility and easy implementation in the adaptive case. This is the focus of the next subsection.

This algorithm can Also be extended to deal with BSS of convolutive mixtures as shown in [START_REF] Aïssa-El-Bey | Iterative blind source separation by decorrelation algorithm: algorithm and performance analysis[END_REF].

Adaptive implementation

To derive an adaptive version of the above batch algorithm, we replace in the above formulae the iteration index p by the time index t and estimate adaptively the correlation coefficients r (t) ij (τ k ). The adaptive algorithm can be summarized as follows : At time instant t

• Update the correlation matrices, i.e., R z (τ k ), k = 1, . . . , K, using the fol-lowing averaging technique :

z(t) = B (t-1) x(t) R (t) z (τ k ) = λR (t-1) z (τ k ) + (1 -λ)z(t)z (t -τ k )
where 0 < λ < 1 is a positive forgetting factor. Note that r

(t) ij (τ k ) is the (i, j)-th entry of R (t) z (τ k ).
• Estimate (t) using equations ( 25) and ( 27) and the updated correlation coefficients r

(t) ij (τ k ).
• Update the value of the separating matrix, the correlation matrices R z (τ k ), k = 1, . . . , K, and the estimated sources z(t + 1 -τ k ), k = 1, . . . , K :

B (t) = (I + (t) )B (t-1) R (t) z (τ k ) = (I + (t) )R (t) z (τ k )(I + (t) ) z(t + 1 -τ k ) = (I + (t) )z(t + 1 -τ k ).
Besides its computational simplicity, this algorithm has the advantage of uniform performance (i.e. it has, in noiseless case, the same asymptotic performance whatever the mixing matrix is) and stability [START_REF] Xiang | Adaptive blind source separation by second order statistics and natural gradient[END_REF].

6 Performance analysis

Theoretical performance analysis

In this section, asymptotic (i.e. for large sample sizes) performance analysis results of the previous separation method is given. We consider the case of instantaneous mixture with i.i.d complex-valued sources satisfying, in addition to the identifiability condition k∈Z |ρ i (τ k )| < +∞ for i = 1, . . . , m. The noise is assumed Gaussian with variance σ 2 I. Assuming that the permutation indeterminacy is P = I, one can write:

BA = I + δ ( 32
)
and hence, the separation quality is measured in our simulations from the mixing matrix A and the decorrelation matrix B using the mean rejection level criterion [START_REF] Belouchrani | A blind source separation technique using second-order statistics[END_REF] defined as:

Iperf def = 1≤p =q≤m E (|(BA) pq | 2 ) ρ q (0) E (|(BA) pp | 2 ) ρ p (0) (33) = 1≤p =q≤m E |δ pq | 2 ρ q (0) ρ p (0) ( 34
)
Our performance analysis consists in deriving the closed-form expression of the asymptotical variance errors: lim

T →+∞ T E |δ pq | 2 (35)
By using a similar approach to that in [START_REF]Blind separation of mixture of independent sources through a quasimaximum likelihood approach[END_REF] based on the central-limit and continuity theorems, one obtains the following result :

Theorem 5 Vector δ ij def = [ e(δ ij ) e(δ ji ) m(δ ij ) -m(δ ji )] T is asymptot-
ically Gaussian distributed with asymptotic covariance matrix

∆ ij def = lim T →+∞ T E δ ij δ T ij (36) = 1 4 H # ij Ψ ij H # ij ( 37
)
where

H ij = I 2 ⊗ ρ T i ρ T j ( 38
)
Ψ ij =         Ξ (ij) 11 Ξ (ij) 12 Ξ (ij) 21 Ξ (ij) 22         (39) with Ξ (ij) 11 = Γ (ij) 11 + Γ (ij) 12 + Γ (ij) 21 + Γ (ij) 22
(40)

Ξ (ij) 12 =  Γ (ij) 11 -Γ (ij) 12 + Γ (ij) 21 + Γ (ij) 22
(41)

Ξ (ij) 21 = - Γ (ij) 11 + Γ (ij) 12 -Γ (ij) 21 + Γ (ij) 22
(42)

Ξ (ij) 22 = Γ (ij) 11 -Γ (ij) 12 -Γ (ij) 21 + Γ (ij) 22 ( 43
)
and

Γ (ij) 11 (k, k ) = τ ∈Z r ii (τ k + τ )r jj (τ k + τ ) (44) Γ (ij) 22 (k, k ) = τ ∈Z r ii (τ k + τ )r jj (τ k + τ ) (45) Γ (ij) 12 (k, k ) = τ ∈Z r ii (τ k + τ )r jj (τ k -τ ) (46) r ii (τ k ) = ρ i (τ k ) + δ(τ k )σ 2 b i b T i . ( 47
)
Note that Γ

(ij) 21 = Γ (ij)T 12
and b i represents the i th row of B = A # .

In the case of real-valued signals, the preceding result becomes :

∆ ij = H # ij Ψ ij H #T ij ( 48
)
where

H ij =         1 1         ⊗ ρ T i ρ T j ( 49
)
Ψ ij =         Γ (ij) 11 Γ (ij) 12 Γ (ij) 21 Γ (ij) 22         (50) 2 -1 2 S i (f )e 2πτ f df ),
the previous result can be rewritten by using the normalization assumption of Theorem 1 ( ρ i = 1 for all i) as :

∆ ij = 1 ρ 2 ij -1 2         1 -ρ ij -ρ ij 1         D         1 -ρ ij -ρ ij 1         (51)
where

ρ ij = ρ i ρ T j ( 52
)
and

D = 1 2 -1 2 S i (f )S j (f )V ij V T ij df . ( 53
)
Note that

V ij = [ e ( i (f )) e ( j (f ))] T , (54)
with

l (f ) = K k=1 ρ l (τ k ) exp(-2πτ k f ) for l ∈ {i, j} (55)
and

S l (f ) = S s l (f ) + σ 2 b l 2 , for l ∈ {i, j} (56)
where S s l (f ) is the power spectral density of the l th source. By replacing (56) in (53), we obtain :

D = 1 2 -1 2 S s i (f )S s j (f )V ij V T ij df + σ 4 b i 2 b j 2 1 2 -1 2 V ij V T ij df +σ 2 b i 2 1 2 -1 2 S s j (f )V ij V T ij df + σ 2 b j 2 1 2 -1 2 S s i (f )V ij V T ij df
From the above expression, we notice that in the noiseless case the performance of the considered BSS method is independent from the mixture matrix (i.e. equivariance property). In that case, the performance limit is essentially function of the "non-collinearity" of vectors ρ i and ρ j (one can see it mainly from the term ρ 2 ij -1 2 that appears in the denominator of equation ( 51)).

Simulation-based performance analysis

We present in this section some numerical simulations to evaluate the performance of the previous separation algorithm. We consider in our simulation an array of n = 4 sensors with half wavelength spacing, receiving two signals in the presence of stationary real temporally white noise. The two source signals are generated by filtering real white Gaussian processes by an autoregressive (AR) model of order 1 with coefficient a 1 = 0.95e 0.5 and a 2 = 0.5e 0.7 (except for Figure 6). The sources have direction of arrivals (DOA) φ 1 = 30 and φ 2 = 45 degrees respectively. Note that the simulation results shown here do not depend on this explicit choice of the source DOAs but rather on the angle difference φ 2 -φ 1 , as illustrated by Figure 7. The number of time lags is K = 5

(except for Figure 8). The signal to noise ratio is defined as SNR = 10 log 10

σ 2 s σ 2 n ,
where σ 2 n and σ 2 s are the noise variance and signal variance respectively. The mean rejection level is estimated over 1000 Monte-Carlo runs.

In Figure 1, we compare the separation performance obtained by the decorrelation algorithm with SOBI algorithm.

Figure 2 shows the mean rejection levels against the signal to noise ratio SNR.

We compare the IDA algorithm with the SOBI algorithm which is based on a joint diagonalization of a set of covariance matrices [START_REF] Belouchrani | A blind source separation technique using second-order statistics[END_REF]. The additive noise is temporally white but spatially colored. The noise covariance is assumed to be of the form 0.9 |i-j| (i.e.

E(η(t)η (t)) = nσ 2 n QQ H / Q 2 , where Q is given by Q ij =
η(t) = √ nσ n Q Q η(t)
where η(t) is a unit norm white Gaussian noise). In this case, IDA performs much better than SOBI at low SNR. This is an important advantage of the IDA method over existing SOS methods that often assume the noise covariance matrix known up to a scalar constant.

In Figure 3 totic closed form expressions of the rejection level are pertinent from snapshot length of about 100 samples. In the plots (4,4) T respectively. This means that asymptotic conditions are reached for short data block size. Figure 4 shows the mean rejection level against the signal to noise ratio SNR.

E (|δ ij | 2 ) and E (|δ ji | 2 ) are replaced by ∆ ij (1,1)+∆ ij (3,3) T and ∆ ij (2,2)+∆ ij
We compare the empirical performance with theoretical performance for T = 1000 sample size. Figure 5 shows the mean rejection level versus the number of sensors using the theoretical formulation for T = 1000 sample size. We observe that, the more the number of sensors, the lower the rejection level is in the low SNR case.

For high SNRs the number of sensors has negligible effect on the separation performance (in accordance with the uniform performance property).

Figure 6 shows Iperf versus the spectral shift δθ. The spectral shift δθ represents the spectral overlap of the two sources. In this figure, the noise is assumed to be spatially white and its level is kept constant at 10dB and 30dB. We let a 1 = 0.7e 0.5 and a 2 = 0.5e (0.5+δθ) . The plot evidences a significant increase in rejection performance by increasing δθ. sample size T = 1000, two AR sources with coefficient a 1 = 0.95e 0.5 and a 2 = 0.5e 0.7 are considered. Their respective DOAs are φ 1 = 30 degrees and φ 2 = φ 1 +δφ. From the plots, we observe a significant performance degradation when δφ is close to zero. Indeed in that case, the identifiability condition w.r.t. the mixture matrix A is ill-satisfied. Also, we can observe that in the absence of noise or equivalently when the noise is negligible (e.g. for SNR=50dB), the method has uniform performance independent from the mixture matrix A and its numerical conditioning. The plots in Figure 8 illustrate the effect of the number of time lags K for different SNRs. In this simulation the sources arrive from the directions φ 1 = 10 and φ 2 = 13 degrees. This choice is to have an ill-conditioned mixture matrix and hence a difficult separation context. In that case, the increase of the number of correlation matrices is needed to improve the separation quality.

Otherwise, in a 'good' context, increasing the number of time lags would not significantly affect the performance of the considered algorithm. From this figure, we also observe that a large increase of the number of correlation matrices leads to a degradation of the separation performance. The reason is that, the correlation coefficients of considered sources decrease exponentially towards zero and consequently the signal term in the large time lags correlation matrices is negligible and their estimations are too noisy.

Performance assessment of the testing technique

We present in this section some simulation results to illustrate the performance of the separability testing method. In the simulated environment we consider uniform linear array with n = 2 sensors receiving the signals from m = 2 unitpower first order autoregressive sources (with coefficients a 1 = 0.95e 0.5 and a 2 = 0.5e 0.7 ) in the presence of stationary complex temporally white noise.

The considered sources are separable according to the identifiability result, i.e. their respective correlation vectors ρ 1 and ρ 2 are linearly independent. The time lags (delays) implicitly involved are τ 0 , • • • , τ 9 (i.e., K = 10). We use SOBI algorithm [START_REF] Belouchrani | A blind source separation technique using second-order statistics[END_REF] to obtain the decorrelated sources. The statistics in the curves are evaluated over 2000 Monte-Carlo runs. We first present in Figure 9 a simulation example where we compare the rate of success of the testing procedure (success means that we decide the 2 sources size is T = 1000 and the SNR = 25dB. Figure 12 shows the rate of success versus the spectral shift δθ. As we can see, small values of δθ lead to high rates of 'non-separability' decision by our testing procedure. Indeed, when δθ is close to zero the two vectors ρ 1 and ρ 2 are close to 'linear dependency'.

That means that the separation quality of the two sources is poor in that case which explains the observed testing results. In the last figure, we assume there exist three sources. The first two sources are complex white Gaussian processes (hence ρ 1 = ρ 2 ) and the third one is an autoregressive signal with coefficient a 3 = 0.95e 0. 5 . The plots in Figure 13 compares the average values of scalar products for ρ i and ρ j (i, j = 1, 2, 3)

with their corresponding threshold values 1-α (i,j) versus the SNR. The sample size is fixed to T = 500 and the number of sensors is n = 3. This example illustrates the situation where two of the sources (here sources 1 and 2) cannot be separated (this is confirmed by the testing result) while the third one is extracted correctly (the plots show clearly that R(1, 3) < 1 -α (1,3) and R(2, 3) < 1 -α (2,3) ).

Conclusion

The second order blind source separation of stationary colored sources is a simple decorrelation problem. The signal decorrelation for a finite set of time lags leads to source separation under certain conditions that are fully detailed in this paper. The separability conditions depend on the unknown sources and hence cannot be verified (tested) directly. However, we present in this paper a testing procedure that uses the output signals of the decorrelator to verify a posteriori whether the sources have been correctly separated or not. The signal decorrelation can be achieved in many different ways. We have presented in this work, one of them that has the advantage of simplicity and efficiency both in block and adaptive processing. Performance analysis using computer simulations and validation of the different theoretical results are provided at the end of the paper. Testing IDA in real world data is one of the perspectives of this work.

A Proof of Theorem 3

Before proceeding, note that Theorem 3 is a generalization of Theorem 2 and hence its proof is implicitly a proof of Theorem 2. Also, note that the result of Theorem 2 is a proof for the sufficiency of the identifiability condition of Theorem 1. Indeed, since A is full column rank, we know that a decorrelation matrix exist (e.g. B = A # ). Now, Theorem 2 result demonstrates that under condition (8), a decorrelation matrix at the considered time lags achieves the desired blind source separation.

For proof of Theorem 3, let us notice that equation ( 10) is equivalent to 5

CS(τ k )C H diagonal for k = 1, • • • , K C( K k=1 S(τ k ) H C H CS(τ k ))
i (x) = K k=1 e(ρ i (τ k ))x k +x K K k=1 m(ρ i (τ k ))
x k and P i (y), i = 1, • • • , m, respectively. Under the assumption of Theorem 2, we have 6 P i (y) ≡ 0 for all i P i (x)P j (y) ≡ P i (y)P j (x) if i and j belong to two distinct groups P i (x)P j (y) ≡ P i (y)P j (x) if i and j belong to the same group 5 The noise term is neglected. However, this term equals zero if all the time lags are non-zero and if the noise is temporally white. 6 The equalities below are understood in terms of equalities of polynomials.

37

Therefore, P (x, y) def = d i=1 P i (y) 1≤i<j≤d (P i (x)P j (y) -P j (x)P i (y)) ≡ 0 (here i and j denote the group index not the source index) and thus there exist infinite values of (x, y) such that P (x, y) = 0. Let (x 0 , y 0 ) be such a value and let S 1 = and (ii).

To complete the proof, note that

M 1 def = K k=1 x k 0 C e(S(τ k ))C H + x k+K 0 C m(S(τ k ))C H = CS 1 C H is diagonal and M 2 def = K k=1 y k 0 C e(S(τ k ))C H + y k+K 0 C m(S(τ k ))C H = CS 2 C H is diagonal and non-singular. It follows that, M 1 M -1 2 = CS 1 S -1 2 C -1 is
a diagonal matrix with d distinct eigenvalues (condition (ii)). Using standard spectral theory, e.g., [START_REF] Horn | Matrix Analysis[END_REF], we conclude that C = P Λ for a given permutation matrix P and a given non-singular block-diagonal matrix Λ, i.e.,

Λ = diag(U 1 , • • • , U d )
where U i is a d i × d i non-singular matrix. Finally, the fact that sources belonging to the same group cannot be separated from the considered set of statistics is simply a direct consequence of the necessity of condition [START_REF] Tong | Indeterminacy and identifiability of blind identification[END_REF]. We propose here to derive the expression of the asymptotic covariance error.

Note that the asymptotic Gaussianity of the error δ comes from the central limit theorem as shown in [START_REF] Belouchrani | A blind source separation technique using second-order statistics[END_REF][START_REF]Blind separation of mixture of independent sources through a quasimaximum likelihood approach[END_REF]. The expression of δ can be obtained by replacing in equations ( 17) and ( 23) z (p) (t) and (p) by s(t) and δ respectively.

Hence, from the vectorized version of equation ( 25) the expression of matrix δ ij δ T ij can be written as :

δ ij δ T ij = 1 4 H # ij ψ ij H # ij (C.1)
where z i (t + τ k )z j (t)z i (t + τ k )z j (t ) (C.8)

H ij = I 2 ⊗ [r ii r jj ] (C.2)
ψ ij =         ξ (ij)
γ (ij) 22 (k, k ) = 1 (T -τ k )(T -τ k ) T -τ k t=1 T -τ k t =1
z j (t + τ k )z i (t)z j (t + τ k )z i (t ) (C.9)

γ (ij) 12 (k, k ) = 1 (T -τ k )(T -τ k ) T -τ k t=1 T -τ k t =1
z i (t+τ k )z j (t)z j (t +τ k )z i (t ) (C.10)

γ (ij) 21 (k, k ) = 1 (T -τ k )(T -τ k ) T -τ k t=1 T -τ k t =1
z j (t+τ k )z i (t)z i (t +τ k )z j (t ) (C.11) Therefore, using the index τ = t -t , we obtain

Fig. 1 .Fig. 2 .

 12 Fig. 1. Comparison between IDA and SOBI for spatially white noise: Mean Rejection Level in dB versus SNR for 2 autoregressive sources 4 sensors and T = 1000.

Fig. 3 .

 3 Fig. 3. Mean Rejection Level in dB versus the sample size T for 2 autoregressive sources 4 sensors and SNR = 40dB.

Fig. 4 .

 4 Fig. 4. Mean Rejection Level in dB versus the SNR for T = 1000 : (a) 4 sensors and 2 AR sources with DOAs φ 1 = 30 and φ 2 = 45 degrees; (b) 7 sensors and 5 AR sources with DOAs φ 1 = 15, φ 2 = 30, φ 3 = 45, φ 4 = 60 and φ 5 = 120 degrees respectively.

Fig. 5 .

 5 Fig. 5. Mean Rejection Level in dB versus the number of sensors n for 2 autoregressive sources and T =1000.

Figure 7 Fig. 6 .

 76 Figure 7 assesses the performance of IDA versus the angle difference δφ = φ 2 -φ 1 for different values of the SNR. The number of sensors is n = 4, the

Fig. 7 .

 7 Fig. 7. Mean Rejection Level in dB versus the angle difference δφ for 2 autoregressive sources, 4 sensors and T =1000.

Fig. 8 .

 8 Fig. 8. Mean Rejection Level in dB versus the number of time lags K for 2 autoregressive sources, 4 sensors and T =1000.

Fig. 9 .

 9 Fig. 9. Rate of success versus SNR for 2 autoregressive sources and 2 sensors and β = 99.7%: comparison of the performance of our testing algorithm for different sample sizes T .

Fig. 13 .

 13 Fig.[START_REF] Yeredor | Non-orthogonal joint diagonalization in the least-squares sense with application in blind source separation[END_REF]. Average values of the |R(i, j)| and thresholds 1 -α (i,j) versus SNR for 3 sources and 3 sensors : 2 sources are complex white Gaussian processes and the third one is an autoregressive signal.

  C H > 0 where C = BA, H denotes the matrix conjugate transpose, and the inequality A > 0 stands for A positive definite. This latter inequality implies in particular that C is full rank. Note that CS(τ k )C H is diagonal if and only if both C e(S(τ k ))C H and C m(S(τ k ))C H are diagonal. Then we shall show that there exist linear combinations of e(S(k)) and m(S(k)), i.e., S 1 = K k=1 α k e(S(τ k )) +α k m(S(τ k )) and S 2 = K k=1 β k e(S(τ k )) + β k m(S(τ k )), such that (i) S 2 is non-singular and (ii) the diagonal entries of S 1 S -1 2 take d distinct values of multiplicity d 1 , • • • , d d , respectively. A simple way to prove it is to consider linear combinations of the form α k = x k , α k = x k+K and β k = y k , β k = y k+K (x and y being real scalars). The diagonal entries of S 1 and S 2 are P

K k=1 x k 0

 0 e(S(τ k )) + x k+K 0 m(S(τ k )) and S 2 = K k=1 y k 0 e(S(τ k )) + y k+K 0 m(S(τ k )). S 1 and S 2 satisfy conditions (i)

2 B Proof of Theorem 4 2 C

 242 According to the above development, we haveC = P diag(U 1 , • • • , U d ). On the other hand we have CS(τ k )C H diagonal for k = 1, • • • , K. This leads to U i S i (τ k )U H i diagonal ∀i = 1, • • • , d, ∀k = 1, • • • , K. Now since we have grouped the sources in such a way each group corresponds to sources with (up to a sign) correlation vector, i.e. S i (τ k ) = ρ i (τ k )Σ i where Σ i are diagonal matrices with diagonal entries equal to ±1, the previous equation becomesU i Σ i U H i diagonal for i = 1, • • • , d. Moreover,since Σ i is real diagonal and since we normalized to unity the correlation vectors (both estimated and exact), then U i Σ i U H i is real diagonal with diagonal entries equal to ±1. Finally, since the signature (i.e., the number of positive eigenvalues and the number of negative eigenvalues) of a quadratic form Q is invariant by multiplication left and right by a non-singular matrix and its conjugate transpose, respectively, then U i Σ i U H i has the same signature (i.e., number of +1s and number of -1s) as Σ i . In other words, there exists a d i × d i permutation matrix P i such thatU i Σ i U H i = P i Σ i P T i . It follows then that, for k = 1, • • • , K E(z(t + τ k )z (t)) = CS(τ k )C H = P S(τ k )P T (B.1) for a given permutation matrix P . Equation (B.1) means that the entries z i (t) of z(t) have the same correlation coefficients as s P (i) (t) for i = 1, • • • , m, where P (1), • • • , P (m) are the images of 1, • • • , m by the permutation P . This verifies the conclusion of Theorem 4. Proof of Theorem 5

  are K × K matrices given by ∀1 ≤ k, k ≤ K : τ k )(T -τ k )

TTT

  , k ) = lim T →+∞ T (T -τ k -|τ |) (T -τ k )(T -τ k ) T -τ k τ =-T +τ k r ii (τ k + τ )r jj (τ k + τ ) (T -τ k -|τ |) (T -τ k )(T -τ k ) T -τ k τ =-T +τ k r ii (τ k + τ )r jj (τ k + τ ) (T -τ k -|τ |) (T -τ k )(T -τ k ) T -τ k τ =-T +τ k r ii (τ k + τ )r jj (τ k -τ ) (T -τ k -|τ |) (T -τ k )(T -τ k ) T -τ k τ =-T +τ k r ii (τ k -τ )r jj (τ k + τ ) = Γ (ij) 21 (k, k )whereΓ (ij) 11 (k, k ), Γ (ij) 22 (k, k ), Γ (ij) 12 (k, k ) and Γ (ij)21 (k, k ) are given by equations (44), (45), (46) and (47). 2

  2 , • • • , τ K if and only if for all 1 ≤ i = j ≤ m :

	ρi and ρj are (pairwise) linearly independent	(8)
	Proof: The proof of sufficiency is given in Appendix A. Here we here only prove
	that (8) is necessary to achieve BSS using correlation functions R x (τ k )	def =
	E(x(t + τ k )x (t)), k = 1, • • • , K. In fact, if two sources, say s 1 and s 2 , have
	correlation coefficients such that ρ1 = ρ2 where = ±1, then any 'virtual'
	signal of the form x(t) = Ãs(t) + w(t) where à = [ã 1 , ã2 , a 3 , • • • , a m ] and

We implicitly assume here that ρ i = 0, otherwise the source signal could not

Because of the inherent indetermination of the BSS problem, we assume without loss of generality that the exact and estimated sources are similarly scaled, i.e., ρi = 1.

More precisely, one can prove that the estimation error √ T δ( ρi ρT j ) is asymptotically, i.e. for large sample size T , Gaussian with zero mean and finite variance.

In that paper, only the case where τ 1 = 0 was considered.

have been separated) to detect the sources separability for different sample sizes versus the SNR in dB. The confidence interval is fixed to β = 99.7%.

One can observe from this figure that the performance of the testing procedure degrades significantly for a small sample size due to the increased estimation errors and the fact that we use the asymptotic normality of considered statistics. In Figure 10, we present a simulation example where we compare the rate of success according to the sample size for different confidence intervals. The SNR is set to 25dB. Clearly, the lower the confidence interval is, the higher is the rate of success of the testing procedure. Also, as observed in Figure 9, the rate of success increases rapidly when increasing the sample size.

In Figure 11, we present a simulation example where we plot the rate of success The simulation example presented in Figure 12 assumes two source signals with parameters a 1 = 0.5e 0.5 and a 2 = 0.5e (0.5+δθ) , where δθ represents the spectral overlap of the two sources. The number of sensors is n = 5, the sample


John J Murphy

Prof Adrien Quintard

Patrick Mcardle

Dr Alexandre Alexakis
email: alexander.alexakis@unige.ch

John C Stephens
email: john.stephens@nuim.ie

Scheme

Dr ] J J Murphy

Prof P Mcardle

Asymmetric Organocatalytic 1,6-Conjugate Addition of Aldehydes to Dienic Sulfones**

Keywords: asymmetric synthesis, conjugate addition, dienes, organocatalysis, sulfones

Building enantiopure complex molecules simply, in a minimum number of operations, and in an environmentally friendly approach is one of the greatest challenges for synthetic chemistry, and particularly, for chemical industry. [START_REF] Breuer | For reviews on industrial approaches for the synthesis of chiral molecules[END_REF] Taking into account the requirements for the industrial application of an laboratory-scale reaction (functional group/H 2 O tolerance, simple procedures, no extreme temperatures), enamine catalysis has recently appeared as a method of choice to fulfil this ideal goal of reaction efficiency. [2,3] In this field, the asymmetric conjugate addition to activated alkenes has been extensively studied, as evidenced by the large number of publications on the subject. [START_REF] Sulzer-Mossø | For reviews on enamine 1,4-addition[END_REF] In contrast, the analogous asymmetric 1,6-addition to extended conjugated systems remains underdeveloped. [START_REF] Hayashi | For some recent examples of metal-catalyzed 1,6-conjugate additions[END_REF][START_REF] Bernardi | Jørgensen has reported a phase-transfer-catalyzed 1,6-conjugate addition of b-ketoesters. Regioelectivity issues can probably be accounted for in this reaction thanks to the unsubstituted 6-position of the acceptor[END_REF] Several groups reported the 1,4-addition to activated dienes in enamine catalysis without any traces of vinylogous 1,6addition. [START_REF] Csµkþ | For a recent review on conjugate addition reactions to electrondeficient dienes[END_REF] This higher reactivity of the b position compared to the d position seems to be a general trend difficult to overcome (Scheme 1). It probably arises from the poor propagation of the electronic effect through the conjugated system. This problem of charge delocalization is in contrast to the principle of vinylogy where the reactivity is in theory extended through the p-p system. [8] In our continuing efforts toward the development of new approaches for the stereoselective construction of enantiopure synthetically useful buildings blocks, we thought about expanding the scope of enamine Michael reactions to 1,6-addition.

We hypothesized that a suitably designed Michael acceptor would be able to promote exclusively the 1,6addition. To this purpose, we have focused our attention on unsaturated sulfones. The sulfonyl group is known for its electron-withdrawing ability together with high synthetic versatility. [START_REF]For some selected reviews on sulfones[END_REF] It has been shown that a vinyl sulfone with a single activating sulfone group was not sufficiently reactive to promote intermolecular enamine attack and generate a 1,4conjugate addition. Instead a second sulfone was required to generate the Michael-type addition. [10] As a result, 1,3-bis-(sulfonyl) butadiene (Scheme 1), should be able to promote exclusively the 1,6-addition by the insertion of an appropriately placed second electron-withdrawing group. [START_REF] Masuyama | For the synthesis and application of sulfonyl butadiene[END_REF] This butadiene should serve as an exciting application of the exceptional potential of charge delocalization in vinylogous reactions. The sulfone in the a position would not sufficiently activate the b-carbon atom toward enamine addition but would be expected to sufficiently delocalize the charge of the d-carbon atom thanks to the cooperative effect of the second sulfonyl group, thus promoting the single 1,6-addition (Scheme 1). Herein, we present our results on this unprecedented asymmetric 1,6-addition that leads, in operationally simple conditions, to exceptional levels of diastereo-and enantioselectivities for the formation of highly attractive chiral dienes.

We began our study by synthesizing 1,3-bis(sulfonyl) butadiene substrates 1 by a high-yielding, four-step reaction sequence. [12] To evaluate the feasibility of the asymmetric 1,6addition, we subjected the sulfonyl diene 1 to the addition of butanal 2 a using 30 mol % of the organocatalyst (R)-diphenylprolinol silyl ether 3. Chloroform was chosen as it easily solubilized the 1,3-bis(sulfonyl) butadiene. As we expected from our proposal (Scheme 1), only the 1,6-addition product was obtained in an excellent yield of 91 % using only 2 equivalents of aldehyde (Table 1,entry 1). The observed high reactivity and regioselectivity is in total agreement with our preliminary hypothesis of charge delocalization. The intermediate linear product was not observed and spontaneously cyclized to form the conjugated diene 4. It must be pointed out here that the cyclized product could be isolated from the crude reaction mixture by a very simple procedure. After evaporation of the solvent, the solid was only triturated with ice-cold methanol to directly obtain the pure compound. More remarkably, an exceptional diastereo-and enantiocontrol was observed in this reaction to furnish the 1,6-adduct in an astonishingly high 99 % ee and 99:1 d.r. while performing the reaction at room temperature. Decreasing the catalyst loading to 10 mol % led to the same excellent stereoselectivities (99 % ee, 99:1 d.r.) but as expected, a prolonged reaction time was needed to obtain 100 % conversion (120 h vs. 24 h, result not shown).

We explored the scope and limitations of this reaction by testing 1,3-bis(phenylsulfonyl)butadiene 1 a with a variety of different sterically demanding aldehydes 2 a-f (Table 1). Gratifyingly, all reactions gave the 1,6-addition product exclusively with no trace of the 1,4-adduct. Furthermore, the products were all isolated as virtually pure stereoisomers. The unbranched aldehydes 2 a-c underwent a fast 1,6addition in excellent yields, diastereoselectivities, and enantioselectivities (Table 1, entries 1-3). Perhaps most notable, branched aldehydes isovaleraldehyde 2 d and citronellal 2 e reacted efficiently even though longer reaction times where required to reach completion (40 h and 144 h respectively; Table 1, entries 5 and6). This lower reactivity is consistent with the higher steric hindrance of the substrates. Again we were happy to see that the expected compounds were still formed with perfect stereocontrol even though they required a longer reaction time (compound 4 d and 4 e were isolated as single stereoisomers). Furthermore, this protocol could also be applied for unsaturated phenylacetaldehyde 2 f, that underwent a high-yielding reaction with excellent diastereoselectivity and enantioselectivity (Table 1,entry 7). This attractive synthon should lead to an enantiomerically pure C 2 symmetric diene by sulfone removal.

To fully explore this remarkable transformation, we then continued to investigate the scope of the reaction by testing the 1,6-conjugate addition of valeraldehyde 2 b to a family of 1,3-bis(phenylsulfonyl)butadienes 1 a-e in the presence of 30 mol % of organocatalyst in chloroform (Table 2). A variety of different aryl substituents with a range of electronic properties could be used without affecting the overall selectivity of the reaction. All reactions gave greater than 99% conversion by 1 H NMR spectroscopy. The yields of the isolated 1,6-addition products were slightly lower in all cases (71 to 81 % yield vs. 98 % for the phenyl). This outcome probably arises from an increase in the solubility of the final compounds and results in product loss during workup. When the electron-withdrawing properties of the substituents were increased from F, Cl, to Br the reactions were slightly accelerated. The lower electron density of the acceptor 5 f containing a nitro substituent resulted in an impressive increase in reactivity (4 h vs. 24 h to obtain a full conversion; Table 2, entry 6 vs. entry 2). This result is in agreement with a Michael 1,6-addition mechanism and should indicate that the CÀC bond formation and not the cyclization is the ratedetermining step. This finding is consistent with the fact that no traces of the noncyclized product could be observed when monitoring the reaction by 1 H NMR spectroscopy. d.r.

(syn/anti) [b] ee [%] [c] ( R)-3 144 (S)-citronellal 89 [e] (4 e) 1:99 -7

( R)-3 24 Ph 96 (4 f) 1:99 99

[a] Yield of isolated product. [b] Determined by 1 H NMR spectroscopy and HPLC analysis. [c] Determined by HPLC on a chiral stationary phase for the anti products. [d] Opposite S,S enantiomer of the product formed. [e] Isolated as a single diastereoisomer as determined by 1 H NMR spectroscopy. TMS = trimethylsilyl.

Table 2: Scope of the bis(arylsulfonyl) butadienes.

Entry

t [h] X Yield [%] [a]
d.r. (syn/anti) [b] ee [%] [c] 1 24 OMe 75 ( 5 [a] Yield of isolated product. [b] Determined by 1 H NMR spectroscopy and HPLC analysis. [c] Determined by HPLC on a chiral stationary phase for the anti products.

In addition, the absolute and relative configuration of both the R,R adduct and S,S adduct of 4 b could be determined by X-ray crystallography (Figure 1 and the Supporting Information). [START_REF]R)-4b) and[END_REF] Despite the high synthetic potential of the disclosed reaction, it is also highly interesting in terms of mechanism. Although further experimentation is needed to have a complete understanding of the reaction mechanism, a plausible stepwise mechanism can be proposed (Scheme 2). The absolute configuration of the products is consistent with previous results obtained in 1,4-addition to other Michael acceptors catalyzed by catalyst 3. [START_REF] Mielgo | [END_REF] The acyclic synclinal transition-state model, as described by Seebach and Golin ´ski, could be applied to the 1,6-conjugate addition of aldehydes to 1,3-bis(sulfonyl) butadienes and explains the observed high levels of stereoselectivities. [15] Steric repulsion away from the bulky groups of the pyrrolidine ring promotes the selective attack of the Re face of the E-trans enamine and the Re face of the Michael acceptor forming 6. Previous studies have described a similar model for the 1,4-addition of aldehydes to vinyl sulfones, nitroolefins, [16] and vinyl phosphonates. [17] Even though this classical model can rationalize the observed stereoselectivity, several aspects still need to be addressed. The question on whether the catalyst is involved in the cyclization and in promoting the elimination is still not clear. This is more plausible given the fact that no linear product is observed, which implies that the cyclization/elimination steps are fast with the catalyst still involved. Despite the preliminary evidence for a 1,6-addition, a possible [4+2] cycloaddition cannot be ruled out and further investigations should shed light on these interesting problems. [START_REF] Masuyama | For the synthesis and application of sulfonyl butadiene[END_REF]18] The products obtained through the 1,6-conjugate addition/condensation reaction are highly interesting building blocks. To illustrate the synthetic utility of this method the adduct 4 a was converted into 8 in excellent 97 % yield through another conjugate addition of methyllithium (Scheme 3).

Perfect regioselectivity and good levels of diasteroselectivities (4:1 d.r.) were obtained for the subsequent creation of two new stereogenic centers in this final molecule containing four contiguous stereocenters. After a simple recrystallization, compound 8 was isolated as a 12:1 mixture of two diasterosiomers in 99 % ee among the 16 possible stereosiomers. The addition anti to the propyl group on the adjacent carbon atom was confirmed using NOE studies and 1 H, [START_REF]R)-4b) and[END_REF] C, DEPT, and HSQC spectra. This result highlights the great potential of the obtained dienes for further transformations by indicating the most electrophilic position in 4 b.

In conclusion we have developed an unprecedented enamine 1,6-addition by exploiting the properties of charge delocalization in 1,3-bis-(sulfonyl) butadienes. By appropriately designing a Michael acceptor, unique reactivities were obtained for the formation of highly valuable dienes containing two versatile vinyl sulfones. This remarkable reaction should find its applications in total synthesis thanks to its operational simplicity and to the exceptional levels of stereoselectivities of the final products (typically 99 % ee and 99:1 d.r.). We are convinced that this activation principle by charge delocalization through the addition of a second electron-withdrawing group should serve as a keystone for the development of new powerful 1,6- addition reactions. Full mechanistic studies as well as further investigations employing ketones and additional 1,6-acceptors are currently being pursued in our laboratories and will be published in due course.

Experimental Section

Typical procedure for the organocatalytic 1,6-addition reaction: Diene (0.2 mmol, 1 equiv) was added to a sample vial containing (R)-diphenylprolinol silyl ether (19,5 mg, 0.06 mmol, 0.3 equiv) dissolved in chloroform (0.5 mL), followed by addition of the aldehyde (0.4 mmol, 2 equiv). The reaction mixture was then stirred at RT until the reaction was complete (as evident by TLC). The reaction mixture was concentrated under reduced pressure and triturated with ice-cold methanol (2 3 mL) to yield the solid pure product.

Figure 1 .Scheme 2 .Scheme 3 .

 123 Figure 1. ORTEP drawing of (R,R)-4 b with ellipsoids at 20 % probability.

Table 1 :

 1 Scope of aldehydes for the 1,6-addition.

	Entry Cat.	t [h] R	Yield
			[%] [a]

www.angewandte.org 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim Angew. Chem. Int. Ed. 2011, 50, 5095 -5098

Angew. Chem. Int. Ed. 2011, 50, 5095 -5098 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim www.angewandte.org

www.angewandte.org 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim Angew. Chem. Int. Ed. 2011, 50, 5095 -5098

We thank the Irish Research Council for Science, Engineering & Technology (IRCSET) and NUI Maynooth for funding. We also thank Dr. Padraig McLoughlin of Teagasc and the Ashtown Food Research Centre, Dublin for some NMR analysis.