Analysis of the stability of time-domain source separation algorithms for convolutively mixed signals

In this paper, we investigate the self-adaptive source separation problem for convolutively mixed signals. The proposed approach uses a recurrent structure adapted by a generic rule involving arbitrary separating functions. We rst analyze the stability of this class of algorithms. We then apply these results to some classical rules for instantaneous and convolutive mixtures that were proposed in the literature but only partly analyzed. This provides a better understanding of the conditions of operation of these rules. Eventually, we de ne and analyze a normalized version of the proposed type of algorithms, which yields several attractive features. 1. PROBLEM STATEMENT AND CLASSICAL RESULTS Blind source separation is a generic signal processing problem which concerns e.g. antenna or microphone array processing [1]. In the model commonly used to represent it [2][4], two sensors provide measured signals y1(n) and y2(n), which are unknown convolutive mixtures of two unknown source signals x1(n) and x2(n), i.e. in the Z domain: Y1(z) = X1(z) + A12(z)X2(z) (1) Y2(z) = A21(z)X1(z) +X2(z); (2) where Aij(z) is the unknown transfer function of the channel that links source j to sensor i. The impulse response of this channel is denoted (aij(k))k 0 hereafter. In this paper, both mixing lters Aij are assumed to have a causal moving average (MA) structure with the same order M . Moreover, the sources x1(n) and x2(n) are assumed to be stationary, zero-mean and statistically independent. The source separation problem consists in estimating the source signals xj(n) from the measured signals yi(n) up to a permutation factor and a lter. This problem was initially investigated in the case of instantaneous mixtures, i.e. mixtures for which each MA lter Aij is restricted to the single coe cient aij(0). It was recognized that in this case, if no assumptions are made on the color of the source signals, these signals cannot be separated by resorting only to second-order statistics [1]. The rst solution to this problem was proposed by H erault and Jutten [1]. It is based on the recurrent structure shown in Figure 1, with each MA separating lter Cij restricted to the single coe cient cij(0). +