A New Interpretation of
the Wiener Filter



J. Scott Goldstein
MIT Lincoln Laboratory
244 Wood Street
Lexington, MA 02173-9108
tel: (781) 981-0459

I.S. Reed
University of Southern California

L.L. Scharf
University of Colorado

Abstract A new interpretation of the Wiener filter is introduced which demonstrates a low complexity and motivates a different method of reduced-rank Wiener filtering. The vector process observed by the Wiener filter is first decomposed by a sequence of transformations which project the data onto the cross-correlation vector and its orthogonal complement, in a stage-by-stage manner. The resulting data vector, at the output of this decomposition, is then processed by a nested chain of scalar Wiener filters to provide the minimum-mean square error solution.

The implementation of this Wiener filter is efficient, and can be realized in the voltage domain by a data matrix bidiagonalization technique or in the power domain using the Householder tridiagonalization. Rank reduction is obtained by simply pruning the tree-structure of the decomposition and does not require any eigen-analysis of the covariance matrix.

This new multistage Wiener filter provides the diagonalization of the covariance matrix in a different basis representation. This technique represents an extension of the cross-spectral metric in that the cross-spectral energy is compactly represented in few coefficients. A narrowband array processing example is presented to compare the performance of the multistage Wiener filter, the cross-spectral metric Wiener filter and the principal-components Wiener filter as a function of rank.



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