Nadège Thirion-Moreau

A Nonunitary Joint Block Diagonalization Algorithm for Blind Separation of Convolutive Mixtures of Sources

This letter addresses the problem of the nonunitary joint block diagonalization of a given set of complex matrices whose potential applications stem from the blind separation of convolutive mixtures of sources and from the array processing. The proposed algorithm is based on the algebraic optimization of a least-mean-square criterion. One of its advantage is that a pre-whitening stage is no more compulsorily required when this algorithm is applied in the blind source separation context. Computer simulations are provided in order to illustrate its behavior in three cases: when exact block-diagonal matrices are built, then when they are progressively perturbed by an additive Gaussian noise and, finally, in the context of blind separation of convolutive mixtures of temporally correlated sources with estimated correlation matrices. A comparison with a classical orthogonal joint block diagonalization algorithm is also performed, and a new performance index is introduced to measure the performance of the separation.

Nonorthogonal joint diagonalization of spatial quadratic time-frequency matrices for source separation

The separation problem of an instantaneous mixture of deterministic or random source signals is addressed. We show that the separation can be realized through the nonorthogonal joint diagonalization of spatial quadratic time-frequency matrices. One advantage of the proposed method is that it does not require any whitening stage, and thus, it is intended to work even with a class of correlated signals. We also propose a general automatic time-frequency point selection procedure for the determination of the above matrices to be joint diagonalized. An analytical example and computer simulations are provided in order to illustrate the effectiveness of the proposed approach and to compare it with classical ones.

Computing the polyadic decomposition of nonnegative third order tensors

Computing the minimal polyadic decomposition (also often referred to as canonical decomposition or sometimes Parafac) amounts to finding the global minimum of a coercive polynomial in many variables. In the case of arrays with nonnegative entries, the low-rank approximation problem is well posed. In addition, due to the large dimension of the problem, the decomposition can be rather efficiently calculated with the help of preconditioned nonlinear conjugate gradient algorithms, as subsequently shown, if equipped with an algebraic calculation of the globally optimal stepsize in low dimension. Other algorithms are also studied (gradient and quasi-Newton approaches) for comparisons. Two versions of each algorithm are considered: the enhanced line search version (ELS), and the backtracking version alternating with ELS. Computer simulations are provided and demonstrate the good behavior of these algorithms dedicated to nonnegative arrays, compared to others put forward in the literature. Finally, applications in the context of data analysis illustrate various algorithms. The main advantage of the suggested approach is to explicitly take into account the nonnegative nature of the loading matrices in the problem parameterization, instead of enforcing positive entries by projection. According to the experiments we have run, such an approach also happens to be more robust with respect to possible modeling errors.

Gradient-based joint block diagonalization algorithms: Application to blind separation of FIR convolutive mixtures

This article addresses the problem of the non-unitary joint block diagonalization of a given set of complex matrices. Two new algorithms are provided: the first is based on a classical gradient approach and the second is based on a relative gradient approach. For each algorithm, two versions are provided: the fixed stepsize and the optimal stepsize version. Computer simulations are provided to illustrate the behavior of both algorithms in different contexts. Finally, it is shown that these algorithms enable solving the problem of the blind separation of finite impulse response (FIR) convolutive mixtures of (non-stationary correlated) sources. We focus on methods based on the use of spatial quadratic time-frequency spectra or distributions. The suggested approach main advantage is to enable the elimination of the spatial whitening of the observations which has been proven to establish a bound with regard to the best reachable performances in the blind sources separation context.

Generalized criteria for blind multivariate signal equalization

We consider the problem of blind multivariate signal equalization. Assuming that the input signals are i.i.d. and statistically mutually independent, we propose both a generalization of some available equalization criteria and a generalization of some source separation criteria to the convolutive case. Hence, we obtain a new generalized class of objective function for blind equalization.

Blind sources separation based on bilinear time-frequency representations: A performance analysis

In this communication, the problem of blind sources separation is considered. Many solutions have been brought to that problem, among which the method recently introduced in [1][2], that consists in joint-diagonalizing a combined set of “spatial t-ƒ distributions (stƒd)” matrices. In [4], we have introduced new criteria of selection of the t-ƒ points to use in the building of matrices sets to joint-diagonalize or to anti joint-diagonalize. As these criteria are no more sufficient in the noisy case, new constraints are introduced in the shape of thresholds (calculated both by Bayesian and Neyman-Pearson approaches) leading to new criteria of selection of the matrices sets. Performance studies on these new algorithms are proposed versus SNR and computer simulations are provided.

Algebraic Joint Zero-Diagonalization and Blind Sources Separation

This paper adresses the problem of the joint zero-diagonalization of a given set of matrices. We establish the identiflability conditions of the zero-diagonalizer, and we propose a new algebraical algorithm based on the reformulation of the initial problem into a joint-diagonalization problem. The zero-diagonalizer is not constrained to be unitary. Computer simulations illustrate the behavior of the algorithm. Moreover, as an application, we show that the blind separation of correlated sources can be performed applying this algorithm to a particular set of spatial quadratic time-frequency distribution matrices. In this case, computer simulations are also provided in order to illustrate the performances of the proposed algorithm and to compare it with other existing ones.

Convolutive Blind Signal Separation Based on Asymmetrical Contrast Functions

In this paper, we consider the blind signal separation problem for convolutive mixtures, in the real case. More precisely, we present a generalization of classical contrast functions to more flexible asymmetric forms. We provide several examples of these new criteria which are useful for sources having different high-order statistics. We also perform a statistical study of the proposed source separation approach, including both the consistency and the asymptotic normality aspects. These theoretical results are also confirmed by numerical simulations

Under-Determined Blind Identification of Cyclo-Stationary Signals with Unknown Cyclic Frequencies

In this communication, we propose a new method to blindly identify the mixing matrix of a possibly under-determined mixture of cyclostationary source signals. It is based on the use of a linear operator applied on the observations correlation matrix. Exploiting the properties of the above transformed matrix, a set of cyclic frequencies is first estimated. Then it is used to construct different estimations of the mixing matrix column vectors. Finally using a classification procedure, the mixing matrix is then estimated

Non unitary joint block diagonalization of complex matrices using a gradient approach

This paper addresses the problem of the non-unitary approximate joint block diagonalization (NU - JBD) of matrices. Such a problem occurs in various fields of applications among which blind separation of convolutive mixtures of sources and wide-band signals array processing. We present a new algorithm for the non-unitary joint block-diagonalization of complex matrices based on a gradient-descent algorithm whereby the optimal step size is computed algebraically at each iteration as the rooting of a 3rd-degree polynomial. Computer simulations are provided in order to illustrate the effectiveness of the proposed algorithm.