Underdetermined blind separation of sparse sources with instantaneous and convolutive mixtures

We consider the underdetermined blind source separation problem with linear instantaneous and convolutive mixtures when the input signals are sparse, or have been rendered sparse. In the underdetermined case the problem requires solving three sub-problems: detecting the number of sources, estimating the mixing matrix, and finding an adequate inversion strategy to obtain the sources. This paper solves the first two problems. We assume that the number of sources is unknown, and estimate it by means of an information theoretic criterion (MDL). Then the mixing matrix is expressed in spheric coordinates and we estimate sequentially the angles and amplitudes of each column, and their order. The performance of the method is illustrated through simulations.