On the causality problem in time-domain blind source separation and deconvolution algorithms

Using a recently presented generic framework for multichannel blind signal processing for convolutive mixtures, we investigate the problem of incorporating acausal delays which are necessary with certain geometric constellations. Starting from a generic update equation which is applicable to blind source separation (BSS), multichannel blind deconvolution (MCBD), and multichannel blind partial deconvolution (MCBPD) for dereverberation of speech signals, two formulations of the natural gradient are derived. It is shown that one expression is applicable to mere causal filters whereas the other also allows an implementation of noncausal filters. Moreover, proper initialization methods for both cases are given. For the implementation of these algorithms, cross-relation estimation techniques, known from linear prediction, are discussed. Based on these results, relationships between traditional MCBD algorithms can be established. Experimental results of different acoustic scenarios show the applicability of the presented algorithms.