BLIND WAVE SEPARATION BY SPATIAL OVERSAMPLING

Abstract

International audienceThis paper is a contribution to the problem of the separation of propagating source signals recorded simultaneously by a s e t of receivers. We propose to use a small-sized sensor array so that the waves are spatially oversampled. Sensors are assumed to be directional, to have the same complex frequency response and to be diierently oriented in space. Under these assumptions, sources are received on each sensor with diierent attenuations and with diierent time delays. When the dimensions of the array are chosen so that time delays are small in comparison with the coherence time of each source, we s h o w that the array outputs can be approximated to a particular model of instantaneous mixtures involving the sources and their rst derivative with respect to time. Because sources are statistically dependent to their rst derivative, this problem does not appear as a classical Blind Source Separation (BSS) problem. We present then a matched second-order blind identiication algorithm in order to estimate this particular mixing system. The validity of the proposed model and of our algorithm is connrmed by computer simulations in the case of audio sources. 1. GENERAL MODEL It is assumed that a set of N independent colored signals x 1 (t) : : : x N (t) are propagating in an echo-free environment. These signals are recorded on M sensors without any additive noise (presence of noise will be treated in the full paper). The observation satisfy the equation model below: y 1 (t) = x 1 (t) + x 2 (t) + : : : + x N (t) y i (t) = j=N X j=1 c iij x j (t ; iij) i = 2 : : : M (1) where iij and c iij represent respectively the relative delay and the relative amplitude of source x j (t) observed on the i th sensor versus the rst observation y 1 (t). We'll show in the full paper that in case of a compact sensor array, delays are suuciently small when: 2 iij << 1 2 2 2 M 8ii jj where M is the maximum frequency present in the observations. In this case, an approximation for the observations y i (t) (i = 2 : : : M) using an order one Taylor expansion can be considered: y i (t) c ii1 x 1 (t) ; c ii1 ii1 dx 1 (t) dt + c ii2 x 2 (t) ; c ii2 ii2 dx 2 (t) dt + : : : + c iiN x N (t) ; c iiN iiN dx N (t) dt : (2) Let consider the observation vector y(t) = y 1 (t) y 2 (t) : : : y N (t)] T. Using approximation (2) in (1), the set of equations (1) can be rewritten as: y(t) M 1 x(t) + M 2 _ x(t): where