A general tool for multichannel and multipath problems is given in FIR matrix algebra. With Finite Impulse Response (FIR) filters (or polynomials) assuming the role played by complex scalars in traditional matrix algebra, we adapt standard eigenvalue routines, factorizations, decompositions, and matrix algorithms for use in multichannel /multipath problems. Using abstract algebra/group theoretic concepts, information theoretic principles, and the Bussgang property, methods of single channel filtering and source separation of multipath mixtures are merged into a general FIR matrix framework. Techniques developed for equalization may be applied to source separation and vice versa. Potential applications of these results lie in neural networks with feed-forward memory connections, wideband array processing, and in problems with a multi-input, multi-output network having channels between each source and sensor, such as source separation. Particular applications of FIR polynomial matrix alg...

'Bussgang' deconvolution techniques for blind digital channels equalization rely on a Bayesian estimator of the source sequenc defined on the basis of channel/equalizer cascade model which involves the definition of deconvolution noise. In this paper we consider four `Bussgang' blind deconvolution algorithms for uniformly distributed source signals and investigate their numeric, performance as well as some of their analytic features. Particularlz, we show...

In this paper sparseness measures are reviewed, extended and compared. Special attention is paid on measuring sparseness of noisy data. We review and extend several definitions and measures for sparseness, including the # , # norms. A measure based on order statistics is also proposed. The concept of sparseness is extended to the case where a signal has a dominant value other than zero. The sparseness measures can be easily modified to correspond to this new definition. Eight different measures are compared in three examples. It turns out that different measures may give complete opposite results if the distribution does not have a unique mode at zero. As conclusion, we suggest that the kurtosis should be avoided as a sparseness measure and recommend tanh-functions for measuring noisy sparseness.

Abstract—This paper reports a study on the problem of the blind simultaneous extraction of specific groups of independent components from a linear mixture. This paper first presents a general overview and unification of several information theoretic criteria for the extraction of a single independent component. Then, our contribution fills the theoretical gap that exists between extraction and separation by presenting tools that extend these criteria to allow the simultaneous blind extraction of subsets with an arbitrary number of independent components. In addition, we analyze a family of learning algorithms based on Stiefel manifolds and the natural gradient ascent, present the nonlinear optimal activations (score) functions, and provide new or extended local stability conditions. Finally, we illustrate the performance and features of the proposed approach by computer-simulation experiments. Index Terms—Blind-signal extraction, blind signal separation, independent component analysis, negentropy and minimum entropy, projection pursuit. I.

. The aim of this paper is to present some remarks on the cost functions for blind channel equalization by `Bussgang' algorithms recently discussed in the literature; also, some possible associated non-Bayesian estimators are considered with details, and the effects of the choice of such estimators in relation to `Bussgang' filtering are investigated.

We examine the similarity and difference between sparsity and statistical independence in image representations in a very concrete setting: use the best basis algorithm to select the sparsest basis and the least statistically-dependent basis from basis dictionaries for a given dataset. In order to understand their relationship, we use synthetic stochastic processes (e.g., spike, ramp, and generalized Gaussian processes) as well as the image patches of natural scenes. Our experiments and analysis so far suggest the following: 1) Both sparsity and statistical independence criteria selected similar bases for most of our examples with minor differences; 2) Sparsity is more computationally and conceptually feasible as a basis selection criterion than the statistical independence, particularly for data compression; 3) The sparsity criterion can and should be adapted to individual realization rather than for the whole collection of the realizations to achieve the maximum performance; 4) The importance of orientation selectivity of the local Fourier and brushlet dictionaries was not clearly demonstrated due to the boundary effect caused by the folding and local periodization. These observations seem to encourage the pursuit of sparse representations rather than that of statistically independent representations.

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