Blind Multichannel Deconvolution and Convolutive Extensions of Canonical Polyadic and Block Term Decompositions

Tensor decompositions such as the canonical polyadic decomposition (CPD) or the block term decomposition (BTD) are basic tools for blind signal separation. Most of the literature concerns instantaneous mixtures/memoryless channels. In this paper, we focus on convolutive extensions. More precisely, we present a connection between convolutive CPD/BTD models and coupled but instantaneous CPD/BTD. We derive a new identifiability condition dedicated to convolutive low-rank factorization problems. We explain that under this condition, the convolutive extension of CPD/BTD can be computed by means of an algebraic method, guaranteeing perfect source separation in the noiseless case. In the inexact case, the algorithm can be used as a cheap initialization for an optimization-based method. We explain that, in contrast to the memoryless case, convolutive signal separation is in certain cases possible despite only two-way diversities (e.g., space × time).