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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 44, NO. 12, DECEMBER 1996 3017 signals St - Equivariant Adaptive Source Separation output \ signals Mixing Separating matrix . Yt Xt A Bt \, Jean-Frangois Cardoso, Member, ZEEE, and Beate Hvam Laheld Abstract-Source separation consists of recovering a set of independent signals when only mixtures with unknown coeffi- cients are observed. This paper introduces a class of adaptive algorithms for source separation that implements an adaptive version of equivariant estimation and is henceforth called equi- variant adaptive separation via independence (EASI). The EASI algorithms are based on the idea of serial updating: This specific form of matrix updates systematically yields algorithms with a simple structure for both real and complex mixtures. Most importantly, the performance of an EASI algorithm does not depend on the mixing matrix. In particular, convergence rates, stability conditions, and interference rejection levels depend only on the (normalized) distributions of the source signals. Closed- form expressions of these quantities are given via an asymptotic performance analysis. The theme of equivariance is stressed throughout the paper. The source separation problem has an underlying multiplicative structure: The parameter space forms a (matrix) multiplicative group. We explore the (favorable) consequences of this fact on im- plementation, performance, and optimization of EASI algorithms. I. INTRODUCTION LIND separation of sources is receiving some attention in B the recent signal processing literature, sometimes under different names: blind array processing, signal copy, indepen- dent component analysis, waveform preserving estimation. . . In all these instances, the underlying model is that of n statistically independent signals whose m (possibly noisy) linear combinations are observed; the problem consists of recovering the original signals from their mixture. The ‘blind’ qualification refers to the coefficients of the mixture: No a priori information is assumed to be available about them. This feature makes the blind approach extremely versatile because it does not rely on modeling the underlying physical phenomena. In particular, it should be contrasted with standard narrowband array processing where a similar data model is considered, but the mixture coefficients are assumed to depend in a known fashion on the location of the sources. When the propagation conditions between sources and sensors, the sensor locations, or the receivers characteristics are subject to unpredictable variations or are too difficult to model with accuracy (think of multipaths in an urban environment), it may Manuscript received November 3, 1994; revised June 17, 1996. This work was supported by a grant from the Norwegian government. The associate editor coordinating the review of this paper and approving it for publication was Dr. Tyseer Aboulnasr.
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