A necessary and sufficient condition for the blind extraction of the sparsest source in convolutive mixtures

This paper addresses sparse component analysis, a powerful framework for blind source separation and extraction that is built upon the assumption that the sources of interest are sparse in a known domain. We propose and discuss a necessary and sufficient condition under which the ℓ₀ pseudo-norm can be used as a contrast function in the blind source extraction problem in both instantaneous and convolutive mixing models, when the number of observations is at least equal to the number of sources. The obtained conditions allow us to relax the sparsity constraint of the sources to its maximum limit, with possibly overlapping sources. In particular, the W-disjoint orthogonality assumption of the sources can be discarded. Moreover, no assumption is done on the mixing system except invertibility. A differential evolution algorithm based on a smooth approximation of the ℓ₀ pseudo-norm is used to illustrate the benefits brought by our contribution.