Two new gradient based non-unitary joint block-diagonalization algorithms

This paper addresses the problem of the non-unitary joint block diagonalization (NU - JBD) of a given set of matrices. Such a problem arises in various fields of applications among which blind separation of convolutive mixtures of sources and array processing for wide-band signals. We present two new algorithms based respectively on (absolute) gradient and relative gradient descendent approaches. The main advantage of the proposed algorithms is that they are more general (the real, positive definite or hermitian assumptions about the matrices belonging to the considered set are no more necessary and the found joint block diagonalizer can be either a unitary or non-unitary matrix). These algorithms also outperform the JBD algorithm based on an optimal step size but “approximate gradient” approach that we had previously suggested in [12]. In fact, here, the exact calculus of the complex gradient matrix is performed whereas it was approximated in [12]. Finally, by ensuring the invertibility of the estimated matrix, the relative gradient approach makes the proposed NU - JBD algorithm more stable and consequently more robust. Computer simulations are provided in order to illustrate the effectiveness of the proposed approaches in two cases: when exact block-diagonal matrices are considered and when they are perturbed by an additive Gaussian noise. A comparison with the method presented in [12] is also performed, emphasizing the good behavior of the proposed algorithms.