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...

this paper, we introduce a neural network that can be used for both source separation and the estimation of the basis vectors of ICA. The remainder of the paper is organized as follows. The next section presents the necessary background on ICA and source separation. In the third section, we introduce and justify the basic neural network learning algorithms for signal separation. The fourth section provides mathematical analysis justifying the separation ability of the nonlinear PCA type learning algorithm. The fifth section then introduces the ICA neural network, a three-layer network whose layers perform input data whitening, separation, and ICA basis vector estimation, respectively. In the sixth section, we present experimental results. In the last section, the conclusions of this study are presented, and some possibilities for extending the data model are outlined.

The notion of equivariance is relevant to source separation because multiplication of mixed signals is equivalent to changing the unknown parameter (the mixing matrix) into another mixing matrix. Elaborating on this observation, a wide class of batch estimators of the mixing matrix is first shown to offer uniform performance: quality of separation does not depend on the hardness of the mixture. Equivariance is next extended to adaptive algorithms, by the device of `serial updating'. Adaptive separators based on such a learning rule also exhibit uniform performance in a strong sense. I Introduction: Source separation Source separation, blind array processing, signal copy, independent component analysis, waveform preserving estimation: : : : these keywords refer to a signal model which is receiving increasing attention in both signal processing and neural network literature since the seminal paper [1]. This model is that of n statistically independent signals whose m (possibly noisy) l...

Nonlinear extensions of one-unit and multi-unit Principal Component Analysis (PCA) neural networks, introduced earlier by the authors, are reviewed. The networks and their nonlinear Hebbian learning rules are related to other signal expansions like Projection Pursuit (PP) and Independent Component Analysis (ICA). Separation results for mixtures of real world signals and images are given. 1 Introduction Principal Component Analysis (PCA) is a widely used technique in data analysis. Mathematically, it is defined as follows: let C = Efxx T g be the covariance matrix of L-dimensional zero mean input data vectors x. The ith principal component of x is x T c(i), where c(i) is the normalized eigenvector of C corresponding to the ith largest eigenvalue (i). The subspace spanned by the principal eigenvectors c(1); : : : ; c(M) (M ! L) is called the PCA subspace (of dimensionality M ). PCA is used in many applications because of its optimality properties in data compression and inform...

: Independent Component Analysis (ICA) is a recently developed, useful extension of standard Principal Component Analysis (PCA). The associated linear model is used mainly in source separation, where only the coefficients of the ICA expansion are of interest. In this paper, we propose a neural structure related to nonlinear PCA networks for estimating the basis vectors of ICA. This ICA network consists of whitening, separation, and estimation layers, and yields good results in test examples. We also modify our previous nonlinear PCA algorithms so that their separation capabilities are greatly improved. 1. Introduction Currently, there is a growing interest among neural network researchers in unsupervised learning beyond PCA, often called nonlinear PCA. Such methods take into account higher-order statistics, and are often more competitive than standard PCA when realized neurally [9, 10]. Nonlinear PCA type methods can be developed from various starting points, usually leading to mutua...

The capabilities of self-organizing maps (SOMs) in parametrizing data manifolds qualify them as candidates for blind separation algorithms. We study the virtues and problems of the SOM-based approach in a simple example. Also numerical simulations of more general cases have been performed. It shows that the performance is unquestionable in the case of a linear mixture only if the observed data are prewhitened and inhomogeneities in the input data are compensated. The algorithm is robust with respect to deviations from linearity, although may fail for complex non-linearly distorted signals. Due to computational restrictions only mixtures from a few sources can be resolved. Under certain conditions it is possible to separate more sources than sensors using a dimension-increasing map. 1 Introduction The problem of extracting independent sources from sensor signals arises in many areas in science and technology. Particularly, it is an important issue in the processing of sensor informati...