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THIRD ANNUAL
ASAP '95 WORKSHOP

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Rank Reduction for
Minimum Mean-Square
Error Signal Processing

J. Scott Goldstein and Douglas B. Williams
Digital Signal Processing Laboratory
School of Electrical and Computer Engineering
Georgia Institute of Technology
Atlanta, GA 30332-0250
email: scott@eecom.gatech.edu

Abstract
The minimum mean-square error (MMSE) performance measure is the most popular metric for optimal signal processing with applications including spectral estimation, noise cancellation, adaptive control, system identification, signal modeling, image processing, sensor array processing, and filter design. The MMSE performance measure results in the Wiener-Hopf equation, which yields the optimal Wiener filter as a function of the statistics that describe the particular problem at hand. In many cases, the rank of the required Wiener filter is too large to be realized. Of particular concern are those situations in which the statistics are unknown a priori and must be estimated. In such cases, the resulting Wiener filter is found adaptively, and the computational complexity of the adaptive processor is a function of the Wiener filter rank.

A common misconception is that selecting the eigen-subspace corresponding to the largest eigenvectors of the data covariance matrix provides the best low rank subspace for reduced rank processing. In this presentation we will present an alternative performance index based on a cross-spectral (C-S) metric for selecting the reduced rank subspace for MMSE statistical signal processing. We will show that this metric results in the subspace yielding the lowest MMSE, which is upper bounded by that of the eigen-subspace. Thus, the Wiener filter in the subspace selected by the C-S metric provides a direct minimization of the MMSE at each rank, whereas the subspace selected by those eigenvectors corresponding to the largest eigenvalues only optimizes the low rank representation of the covariance matrix.

This presentation provides examples of reduced rank MMSE signal processing for both sensor array processing and the dual-use application of signal modeling with Prony's method.

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