Differential source separation for underdetermined instantaneous or convolutive mixtures: concept and algorithms

Abstract

This paper concerns the underdetermined or noisy case of the blind source separation (BSS) problem, i.e. the situation when the number of observed mixed signals is lower than the number of sources, which is of high practical interest. We first propose a general differential BSS concept to handle this case. This approach applies to linear instantaneous and convolutive mixtures. It uses optimization criteria based on differential parameters so as to achieve “partial BSS”, i.e. so as to make some sources invisible in these criteria and to perform the exact separation of the other sources only. In other words, each output signal is thus reduced to a mixture of (i) only one visible source and (ii) all invisible sources. Various BSS methods may be derived from this concept. We illustrate it by applying this concept to a specific criterion and associated algorithms, which exploit the assumed non-stationarity of some sources. The resulting approach applies to convolutive mixtures and uses the second-order statistics of the signals. It adapts the filters of a direct BSS system so as to cancel the “differential cross-correlation function” (introduced in this paper) of signals derived by this system. We analyze the stability of this approach, by using the ordinary differential equation method, and we show its performance by means of numerical tests.