# Blind Source Separation for Convolutive Mixtures: A Unified Treatment

## Abstract

Blind source separation (BSS) algorithms for time series can exploit three properties of the source signals: nonwhiteness, nonstationarity, and nongaussianity. While methods utilizing the first two properties are usually based on second-order statistics (SOS), higher-order statistics (HOS) must be considered to exploit nongaussianity. In this chapter, we consider all three properties simultaneously to design BSS algorithms for convolutive mixtures within a new generic framework. This concept derives its generality from an appropriate matrix notation combined with the use of multivariate probability densities for considering the time-dependencies of the source signals. Based on a generalized cost function we rigorously derive the corresponding time-domain and frequency-domain broadband algorithms. Due to the broadband approach, time-domain constraints are obtained which provide a more detailed understanding of the internal permutation problem in traditional narrowband frequency-domain BSS. For both, the time-domain and the frequency-domain versions, we discuss links to well-known and also to novel algorithms that follow as special cases of the framework. Moreover, we use models for correlated spherically invariant random processes (SIRPs) which are well suited for a variety of source signals including speech to obtain efficient solutions in the HOS case. The concept provides a basis for off-line, online, and block-on-line algorithms by introducing a general weighting function, thereby allowing for tracking of time-varying real acoustic environments.

## Keywords

Blind Source Separation Convolutive Mixtures Second-Order Statistics Higher-Order Statistics Time Domain Frequency Domain Broadband Approach Spherically Invariant Random Processes## Preview

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