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“Multichannel Blind Deconvolution: Fir Matrix Algebra and Separation of Multipath Mixtures”
by Russell H. Lambert
May 1996
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 algebra are given in the form of ``vector error" generalizations of the traditional tools of adaptive Wiener filtering.
Analysis in the FIR polynomial (frequency) representation has revealed three distinct expressions of the Bussgang property giving rise to three fundamental classes of blind (self-organizing) cost functions for separation, equalization or both. The classes are: the traditional Bussgang methods, the Direct Minimum Entropy Deconvolution (DMED), and the third Bussgang form (DBAC3), with the EASI-like property that sensor data is not directly used in the update equation. The latter two methods are shown to possess greater stability, insensitivity to eigenvalue disparity, and speed of convergence than the traditional Bussgang methods.