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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.

Two fourth-order cumulant-based techniques for the estimation of the mixing matrix in underdetermined independent component analysis is studied. The number of sources that can be allowed is roughly quadratic in the number of observations.

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 statistical survey of the subspace estimation techniques involved in blind identification of single-input multiple-output finite impulse response systems provides evidence of the usefulness of minimal polynomial concepts in parameterizing the noise/signal aspects. A set of matrices exhibiting linear dependence on the unknown parameters may be exploited to derive several estimation procedures. A weighted least-square solution of overdetermined linear equations provides the estimation framework while an asymptotic evaluation shows the insignificance of weighting.

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.

A new technique based on subspace features is developed for application to blind identification of multichannel FIR filters. The new technique is an improved version of the earlier methods using conventional algorithms. The new technique is computationally efficient and does not require an 'a priori' knowledge about correlation of symbols emitted. The new technique can be applied for situations in practical digital communications where high speed convergence is essential.

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.

The covariance of sensor array data is used to derive a simple general formula that is subjected to a Gaussian model for the asymptotic covariance of direction-of-arrival (DOA) estimators. The formula is also applicable to a broad range of statistical models used as source signals. Hence the source signal distribution does not interfere with the asymptotic performance of high-resolution covariance-based DOA estimators.

Research is presented concerning the application of subspace-based estimates to the development of system identification and array processing. The equivalence between subspace matching and subspace fitting is discussed.

The performance of a least mean square dominant invariant subspace algorithm is analyzed. The algorithm is a stochastic gradient algorithm driving the columns of a matrix W to an orthonormal basis of a dominant invariant subspace of a correlation matrix based on an unconstrained minimization problem. The evolution of the matrix defined as WW(raised to the H power) and its asymptotic distribution are discussed.

In this paper, we perform a complete asymptotic performance analysis of the stochastic approximation algorithm (denoted subspace network learning algorithm) derived from Oja's learning equation, in the case where the learning rate is constant and a large number of patterns is available. This algorithm drives the connection weight matrix W to an orthonormal basis of a dominant invariant subspace of a covariance matrix. Our approach consists in associating to this algorithm a second stochastic approximation algorithm that governs the evolution of W[W.sup.T] to the projection matrix onto this dominant invariant subspace. Then, using a general result of Gaussian approximation theory, we derive the asymptotic distribution of the estimated projection matrix. Closed form expressions of the asymptotic covariance of the projection matrix estimated by the SNL algorithm, and by the smoothed SNL algorithm that we introduce, are given in case of independent or correlated learning patterns and are further analyzed. It is found that the structures of these asymptotic covariance matrices are similar to those describing batch estimation techniques. The accuracy or our asymptotic analysis is checked by numerical simulations and it is found to be valid not only for a 'small' learning rate but in a very large domain. Finally, improvements brought by our smoothed SNL algorithm are shown, such as the learning speed/misadjustment tradeoff and the deviation from orthonormality. Index Terms - Adaptive estimation, eigenvectors, Oja's learning equation, principal component analysis, subspace estimation.