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A framework is introduced based on Wirtinger calculus for nonlinear complex-valued signal processing such that all computations are directed out in the complex domain. The main update rules for the maximum likelihood and maximization of non-Gaussianity approaches are derived, and their properties and density matching strategies are discussed.

New approaches to source separation that take advantage of nonstationarities are developed, based on principles of maximum likelihood and minimum mutual information. Efficient algorithms are used to exploit these principles in both the off-line case, using a new joint diagonalization procedure; and in the on-line case, via a Newton-like procedure.

A study of the statistical and geometrical structures of estimating functions applied to the problem of blind source separation found that some estimating functions are redundant. Calculations show that the unknown distribution of sources (r(s))affects the definition of the optimal estimating function and that a general form for defining admissible estimating function works well even if r(s) is misspecified. The theoretical framework developed through the study can be applied to multiterminal blind deconvolution.

A blind identification technique for temporally correlated sources based on a joint diagonalization of several covariance matrices is proposed. The technique capitalizes on the time coherence of the source signals. It allows the separation of Gaussian sources and is more robust due to its use of several covariances. Numerical experiments have proven the benefit of capitalizing on several covariance matrices in complex contexts.

A class of source separation algorithms for equivalent adaptive source separation via independence (EASI), in signal processing, is described. The EASI algorithms are based on the idea of serial updating and their performance does not depend on mixing matrix, making it possible to tune and optimize the algorithm once for all mixtures. The convergence rates, stability conditions and interference rejection levels depend on the distributions of the source signals.

Contracted quadricovariance and diagonal slice are two fourth-order cumulant matrices that enable the derivation of closed-form expressions for the asymptotic covariance in MUSIC-like direction-of-arrival estimates during narrow-band array processing. The matrices help determine the noise variance and dynamic source range effects and analysis of the asymptotic performance is conducted within a general framework.