Fixed-Point Algorithms for the Blind Separation of Arbitrary Complex-Valued Non-Gaussian Signal Mixtures
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Research Article Fixed-Point Algorithms for the Blind Separation of Arbitrary Complex-Valued Non-Gaussian Signal Mixtures Scott C. Douglas Department of Electrical Engineering, School of Engineering, Southern Methodist University, P.O. Box 750338, Dallas, newline TX 75275, USA Received 1 October 2005; Revised 10 May 2006; Accepted 22 June 2006 Recommended by Andrzej Cichocki We derive new fixed-point algorithms for the blind separation of complex-valued mixtures of independent, noncircularly symmetric, and non-Gaussian source signals. Leveraging recently developed results on the separability of complex-valued signal mixtures, we systematically construct iterative procedures on a kurtosis-based contrast whose evolutionary characteristics are identical to those of the FastICA algorithm of Hyvarinen and Oja in the real-valued mixture case. Thus, our methods inherit the fast convergence properties, computational simplicity, and ease of use of the FastICA algorithm while at the same time extending this class of techniques to complex signal mixtures. For extracting multiple sources, symmetric and asymmetric signal deflation procedures can be employed. Simulations for both noiseless and noisy mixtures indicate that the proposed algorithms have superior finitesample performance in data-starved scenarios as compared to existing complex ICA methods while performing about as well as the best of these techniques for larger data-record lengths. Copyright © 2007 Hindawi Publishing Corporation. All rights reserved.
1.
INTRODUCTION
Both blind source separation (BSS) and independent component analysis (ICA) are concerned with m-dimensional linear signal mixtures of the form x(k) = As(k),
(1)
where A is an unknown (m × m) mixing matrix and s(k) = [s1 (k) · · · sm (k)]T is a vector-valued signal of sources. In most treatments of either task in the scientific literature, the sources {si (k)} are assumed to be statistically independent and real-valued, and the matrix A is assumed to be full rank. If certain additional separability conditions are met, it is possible to compute a demixing matrix B such that y(k) = Bs(k)
(2)
contains independent elements that are possibly scaled and shuffled with respect to the sources in s(k). Separation or extraction of the independent components is considered successful in such cases, as demixing of the mixed sources has been achieved. Numerous algorithms have been developed for separating real-valued mixtures, including maximum-likelihood information-theoretic approaches
[1–4], contrast-based approaches [5–7], and decorrelationbased approaches [8–10]. Among these methods, the FastICA procedure in [7] has a number of nice features, including fast convergence, global convergence for kurtosisbased contrasts, and the lack of any step-size parameter. For a kurtosis-based measure of negentropy, the FastICA algorithm employs a separation criterion similar to other approaches involving cumulant-based contrasts [5, 6], although the optimization method employed by the FastICA algorithm is quite different f
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