Non-Unitary Joint Block Diagonalization of matrices using a Levenberg-Marquardt algorithm

This communication addresses the problem of the Non-Unitary Joint Block Diagonalization (NU - JBD) of a given set of complexmatrices. This problemoccurs in various fields of applications, among which is the blind separation of convolutive mixtures of sources. We present a new method for the NU - JBD based on the Levenberg-Marquardt algorithm (LMA). Our algorithm uses a numerical diagram of optimization which requires the calculation of the complex Hessian matrices. The main advantages of the proposed method stem from the LMA properties: it is powerful, stable and more robust. Computer simulations are provided in order to illustrate the good behavior of the proposed method in different contexts. Two cases are studied: in the first scenario, a set of exactly block-diagonal matrices are considered, then these matrices are progressively perturbed by an additive gaussian noise. Finally, this new NU - JBD algorithm is compared to others put forward in the literature: one based on an optimal step-size relative gradient-descent algorithm [1] and one based on a nonlinear conjugate gradient algorithm [2]. This comparison emphasizes the good behavior of the proposed method.