A second-order differential approach for underdetermined convolutive source separation

This paper concerns the underdetermined case of the convolutive source separation problem, i.e. the situation when the number of observed convolutively mixed signals is lower than the number of sources. We propose a criterion and associated algorithm which, unlike classical approaches, make it possible to perform the separation of a subset of these sources by exploiting their assumed non-stationarity properties. This approach uses the second-order statistics of the signals and adapts the filters of a direct separating system so as to cancel the ”differential cross-correlation” of signals derived by this system. This new method is related to the general differential source separation concept that we proposed. Its effectiveness is shown by means of numerical tests. 1. PROBLEM STATEMENT Blind source separation (BSS) methods aim at estimating a set of source signals from a set of observed signals , which are mixtures of these source signals [1],[2]. The mixed signals are typically provided by sensors, and in the so-called convolutive mixture model, each propagation path from source to sensor is represented by a filter, whose transfer function is denoted hereafter. In the domain, the overall relationship between the column vectors and of sources and observations then reads: ! " # $ &% (1) where the elements of the mixing matrix # are the transfer functions ' . Most investigations have been performed in the case when: i) , so that the matrix # is square, and ii) this matrix is invertible. The BSS problem then basically consists in determining an estimate of the inverse of ( , so as to restore the source signal vector from the observation vector in (1). Various methods have been proposed to this end, based on the assumed statistical independence or uncorrelation of the source signals (see e.g. the surveys, resp. in [1] for the specific case of linear intantaneous mixtures and in [2] for the convolutive case). As stated above, most of these investigations were performed under the assumption . In many practical situations however, only a limited number of sensors is acceptable, due e.g. to cost constraints or physical configuration, whereas these sensors receive a larger number of sources. This underdetermined situation corresponding to )* has been considered by a few authors, mainly in the restricted case of linear instantaneous mixtures (see e.g. [3] and references therein, [4]). In a previous paper [5], we also introduced a general ”differential BSS” concept for treating the underdetermined case. We defined a version of this concept by exploiting the assumed non-stationarity properties of the sources, i.e. the variations of the statistics of a subset of these sources over time. Using the ”differential statistics” of the latter sources then makes it possible to separate them exactly (whereas the other sources yield some additional ”noise” contributions in the outputs of the proposed system). We introduced a specific differential BSS criterion resulting from this general concept and intended for linear instantaneous mixtures. This specific approach is based on the optimization of the ”differential normalized kurtosis” that we introduced to this end. The current paper presents major extensions of our previous work on underdetermined BSS from two points of views. On the one hand, we here develop a practical approach intended for convolutive mixtures. On the other hand, the criterion and algorithm proposed below are based on our general differential BSS concept but are not mere convolutive extensions of the ones that we previously developed for linear instantaneous mixtures. Instead, they use other statistical signal parameters and adaptation rules. These differences stem from the well-known fact that linear instantaneous BSS can only be performed by resorting to higherorder statistics if no specific assumptions are made on the sources, whereas (strictly causal [6]) convolutive BSS may also be achieved by means of second-order statistics. By using the latter approach, we here introduce a method based on correlation (i.e. secondorder) parameters and their cancellation, which is to be contrasted with our previous approach based on normalized kurtosis (i.e. fourth-order parameter) and its min/maximization. 2. PROPOSED APPROACH 2.1. Redefining the classical approach The investigation reported in this paper is based on the classical decorrelation approach that has been used by various authors for solving the basic configuration of the convolutive BSS problem [6]-[9]. We therefore first redefine this classical method in a way which is suited to the approach that we will then use to extend it. The considered basic configuration involves two convolutive mixtures of two uncorrelated sources, defined in the domain as: ,+. /+0 213 +54 $ 647 (2) 4 . 4&+ $ + 1 4 &% (3) where #+548 and 4&+are strictly causal MA filters and their orders are (at most) equal to 9 . These mixed signals are provided to a separating system, which aims at restoring the source signals