Adaptive Source Separation with Uniform Performance

This paper presents a family of adaptive algorithms for the blind separation of independent signals. Source separation consists in recovering a set of independent signals from some linear mixtures of them, the coeecients of the mixtures being unknown. In the noiseless case, thèhardness' of the blind source separation problem does not depend on the mixing matrix (see the companion paper 1]). It is then reasonable to expect adaptive algorithms to exhibit convergence and stability properties that would also be independent of the mixing matrix. We show that this desirable uniform performance feature is simply achieved by considering`serial updating' of the separating matrix. Next, generalizing from the gradient of a standard cumulant-based contrast function, we present a family of adaptive algorithms called`PFS', based on the idea of serial updating. The stability condition and the theoretical asymptotic separation levels are given in closed form and, as expected, depend only on the distributions of the sources. Performance is also illustrated by some numerical experiments. 1. Blind source separation Consider an array of m sensors receiving signals emitted by n statistically independent sources. The array output at time t is a m 1 vector x(t) modelled as x(t) = A s(t); (1) where the mn matrix A is called thèmixing' matrix and where the n source signals are collected in a n 1 vector denoted s(t). In the complex case, this is the familiar linear model used in narrow band array processing. Adaptive source separation consists in updating a nm matrix B(t), called the separating matrix, such that y(t) def = B(t) x(t) (2) is an estimate of the source signals, hence the terminology`source separation'. s(t)-A m n x(t)-B(t) n m-y(t) = ^ s(t) Figure 1. Blind source separation model We emphasize that B(t) should be updated without resorting to any information about the spatial mixing matrix A. This is in sharp contrast tòstandard' array processing and beamforming techniques where the columns of A or their dependence on the location of the sources is assumed to be known. Matrix A is supposed to be a xed matrix with full column rank but no other assumptions are made. The crucial property source separation relies on is the mutual statistical independence of the source signals. Note that under these assumptions, the n n global system matrix C(t) def = B(t) A (3) may only be identiied up to the product of a permutation and a diagonal …