Separating More Sources Than Sensors Using Time-Frequency Distributions
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Separating More Sources Than Sensors Using Time-Frequency Distributions Nguyen Linh-Trung ´ Service Syst`eme T´el´ecommunications Spatiales, Centre National d’Etudes Spatiales, 18 avenue Edouard Belin, 31401 Toulouse, France Email: [email protected]
Adel Belouchrani ´ ´ D´epartement d’Electronique, Ecole Nationale Polytechnique, 10 avenue Hassen Badi, PB 182 EL Harrach, 16200 Algiers, Algeria Email: [email protected]
Karim Abed-Meraim ´ D´epartement Traˆıtement du Signal et des Images, Ecole Nationale Sup´erieure des T´el´ecommunications, 46 rue Barrault, 75634 Paris Cedex 13, France Email: [email protected]
Boualem Boashash College of Engineering, University of Sharjah, P.O. Box 27272, Sharjah, United Arab Emirates Email: boualem [email protected] Received 8 July 2004; Revised 24 March 2005; Recommended for Publication by Kostas Berberidis We examine the problem of blind separation of nonstationary sources in the underdetermined case, where there are more sources than sensors. Since time-frequency (TF) signal processing provides effective tools for dealing with nonstationary signals, we propose a new separation method that is based on time-frequency distributions (TFDs). The underlying assumption is that the original sources are disjoint in the time-frequency (TF) domain. The successful method recovers the sources by performing the following four main procedures. First, the spatial time-frequency distribution (STFD) matrices are computed from the observed mixtures. Next, the auto-source TF points are separated from cross-source TF points thanks to the special structure of these mixture STFD matrices. Then, the vectors that correspond to the selected auto-source points are clustered into different classes according to the spatial directions which differ among different sources; each class, now containing the auto-source points of only one source, gives an estimation of the TFD of this source. Finally, the source waveforms are recovered from their TFD estimates using TF synthesis. Simulated experiments indicate the success of the proposed algorithm in different scenarios. We also contribute with two other modified versions of the algorithm to better deal with auto-source point selection. Keywords and phrases: underdetermined blind source separation, spatial time-frequency distribution, time-frequency synthesis, unsupervised vector clustering, nonstationary sources.
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INTRODUCTION
Blind source separation (BSS) considers the estimation of multiple sources from multiple observations (mixtures) received by a set of sensors, where the observations have been linearly mixed by the transfer medium. The transfer medium between the sources and the mixtures forces each mixture to contain a combination of the sources. The term “blind” indicates that no a priori knowledge of both the sources and the structure of the transfer medium is available. To compensate for this lack of information, the sources are usually assumed to be statistically independent [1]. BSS is important
when precise modeling of the medium tra
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