Temporal and time-frequency correlation-based blind source separation methods. Part I: Determined and underdetermined linear instantaneous mixtures

Abstract

We propose two types of correlation-based blind source separation (BSS) methods, i.e. a time-domain approach and extensions which use time-frequency (TF) signal representations and thus apply to much more general conditions. Our basic TF methods only require each source to be isolated in a tiny TF area, i.e. they set very limited constraints on the source sparsity and overlap, unlike various previously reported TF-BSS methods. Our approaches consist in identifying the columns of the (scaled permuted) mixing matrix in TF areas where these methods detect that a source is isolated. Both the detection and identification stages of these approaches use local correlation parameters of the TF transforms of the observed signals. Two such Linear Instantaneous TIme-Frequency CORRelation-based BSS methods are proposed, using Centered or Non-Centered TF transforms. These methods, which are resp. called LI-TIFCORR-C and LI-TIFCORR-NC, are especially suited to non-stationary sources. We derive their performance from many tests performed with mixtures of speech signals. This demonstrates that their output SIRs have a low sensitivity to the values of their TF parameters and are quite high, i.e. typically 60 to 80 dB, while the SIRs of all tested classical methods range about from 0 to 40 dB. We also extend these approaches to achieve partial BSS for underdetermined mixtures and to operate when some sources are not isolated in any TF area.

Introduction

Blind source separation (BSS) methods aim at restoring a set of unknown source signals from a set of observed signals which are mixtures of these source signals [1], [2], [3]. Most of the approaches that have been developed to this end concern linear instantaneous mixtures and are based on independent component analysis (ICA). They assume the sources to be random stationary statistically independent signals, and they recombine the observed signals so as to obtain statistically independent output signals. The latter signals are then equal to the sources, up to some indeterminacies and under some conditions (especially, at most one source may be Gaussian for such methods to be applicable if no additional constraints are set on the sources).

In addition to ICA, a few other general concepts have been used for achieving BSS. This includes the class of approaches based on time-frequency (TF) analysis, which is the main framework considered in this paper. TF tools have been used in various ways in the BSS methods reported so far, as may be seen e.g. in [4], [5], [6], [7], [8], [9], [10], [11]. Among the trends which emerge from these papers, the following ones should especially be mentioned. A first set of methods is composed of approaches based on ratios of TF transforms of observed signals [4], [5], [6]. Some of these methods require the sources to have no overlap in the TF domain [4], which is quite restrictive. On the contrary, only slight differences in the TF representations of the sources are required by the type of methods that we introduced and extended in [5], [6]. Another general concept that has been proposed for achieving BSS consists in exploiting the sparsity of the sources in an adequate representation of these signals. The representation used in some of these approaches is based on a TF transform of the signals. This yields a second set of TF-BSS methods, which especially contains the approaches proposed in [7]. This second set of methods has some relationships with the above-mentioned first set, in the sense that the different constraints on the TF overlap between sources in the first set of methods may be considered as various conditions on the degree of sparsity of these transformed sources. All the approaches presented in the papers [4], [5], [6], [7] which compose the above two sets use the same TF transform, i.e. the short-time Fourier transform (STFT), which is a linear transform. On the contrary, other approaches use quadratic TF transforms (see e.g. [8], [9], [10], [11]), thus forming a third set of methods. This set especially includes TF-BSS methods which are significantly related to classical BSS approaches, as they consist of TF adaptations of previously developed joint-diagonalization methods, with subsequent modifications. It should be noted that, unlike classical ICA-based BSS methods, TF-based BSS approaches are intrinsically well-suited to non-stationary signals (and set no restrictions on the Gaussianity of the sources). They are therefore e.g. especially attractive for speech sources.

This first part of our paper mainly describes original linear TF-BSS approaches applicable to linear instantaneous mixtures. These approaches use STFTs, like the above-mentioned two sets of linear methods, but they rely on other types of parameters, which are based on the local correlations of the observed signals in the TF domain. Before describing these TF-BSS methods, we present a purely temporal version of such approaches, which only applies to more restrictive conditions, as shown below.

The remainder of this first part of our paper is therefore organized as follows. In Section 2 we define the first configuration, based on determined linear instantaneous mixtures, that we consider and the resulting goal of our investigation. We then present the associated temporal BSS method in Section 3 and we introduce its TF extensions in Section 4. Section 5 is devoted to the versions of these approaches intended for more complex configurations, especially underdetermined mixtures. Section 6 reports on a detailed analysis of the experimental performance achieved by all our temporal and TF approaches for artificial mixtures of real speech sources. It also contains a comparison to the performance of various BSS methods from the literature and of a somewhat related TF approach that we proposed in a previous paper. Section 7 contains a discussion of the features of the proposed methods, as compared to classical BSS approaches. This section also presents the conclusions drawn from this first part of our overall investigation and outline extensions of the proposed methods. In addition, specific topics are detailed in the appendices.