Signal Processing 84 (2004) 1759–1776 www.elsevier.com/locate/sigpro Dierential source separation for underdetermined instantaneous or convolutive mixtures: concept and algorithms YannickDeville a; b; * ,MohammedBenali b ,Fr ed ericAbrard b a Laboratoire d’Astrophysique, Observatoire Midi-Pyrenees, Universite Paul Sabatier, 14 Av. Edouard Belin, 31400 Toulouse, France b Laboratoire d’Acoustique, M etrologie, Instrumentation, Universit e Paul Sabatier, Bˆ at. 3R1B2, 118, Route de Narbonne, 31062 Toulouse Cedex, France Received 5 May 2003; received in revised form 26 February 2004 Abstract This paper concerns the underdetermined or noisy case of the blind source separation (BSS) problem, i.e. the situation when the number of observed mixed signals is lower than the number of sources, which is of high practical interest. We rst propose a general dierential BSS concept to handle this case. This approach applies to linear instantaneous and convolutive mixtures. It uses optimization criteria based on dierential parameters so as to achieve “partial BSS”, i.e. so as to make some sources invisible in these criteria and to perform the exact separation of the other sources only. In other words, each output signal is thus reduced to a mixture of (i) only one visible source and (ii) all invisible sources. Various BSS methods may be derived from this concept. We illustrate it by applying this concept to a specic criterion and associated algorithms, which exploit the assumed non-stationarity of some sources. The resulting approach applies to convolutive mixtures and uses the second-order statistics of the signals. It adapts the lters of a direct BSS system so as to cancel the “dierential cross-correlation function” (introduced in this paper) of signals derived by this system. We analyze the stability of this approach, by using the ordinary dierential equation method, and we show its performance by means of numerical tests. ? 2004 Elsevier B.V. All rights reserved. Keywords: Blind signal separation; Dierential criterion and algorithm; Dierential correlation function; Instantaneous or convolutive mixture; Non-stationary source; Ordinary dierential equation; Partial source separation; Stability analysis; Underdetermined or noisy mixture 1. Problem statement Blind source separation (BSS) methods aim at restoringasetof N source signals x j (n)fromasetof P observed signals y i (n),whicharemixturesofthese source signals [5,7,12,17]. The mixed signals y i (n) are often provided by a set of sensors, and the mix- ing phenomenon then results from the simultaneous ∗ Corresponding author. E-mail address: ydeville@cict.fr (Y. Deville). propagationofallsignalsfromtheiremissionlocations to all sensors. This gives rise to two major classes of BSS problems, depending on the features of the con- sidered propagation. In the so-called linear instanta- neous mixture model, each propagation channel from source j to sensor i is represented by a scalar coef- cient a ij , which typically reects attenuation during propagation.Theoverallrelationshipbetweenthecol- umn vectors x(n) and y(n) of sources and observa- tions is then expressed in the discrete time domain as y(n)= Ax(n); (1) 0165-1684/$-see front matter ? 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2004.06.003