Statistical and computational performance of a class of generalized Wiener filters (Corresp.)

Abstract

A class of suboptimal Wiener filters is considered, and their computational and statistical performances (and the trade-off between the two) are studied and compared with those for known classes of suboptimal Wiener filters. A general model of a suboptimal Wiener filter over a group is defined, which includes, as special cases, the known filters based on the discrete Fourier transform (DFT) in the case of a cyclic group and the Walsh-Hadamard transform (WHT) in the case of a dyadic group. Statistical and computational performances of various group filters are investigated. The cyclic and the dyadic group filters are known to be computationally the best ones among all the group filters. However, they are not always the best ones statistically and other (not necessarily Abelian) group filters are studied. Results are compared with those for the cyclic group filters (DFT), and the general problem of selecting the best group filter is posed. That problem is solved numerically for small-size signals <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(\leq 64)</tex> for the first-order Markov process and random sine wave corrupted by white noise. For the first-order Markov process with the covariance matrix <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">B^{(S,l)} = \rho^{|s-l|}</tex> as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\rho</tex> increases, the use of various non-Abelian groups results in improved statistical performance of the filter as compared to the DFT. Similarly, for the random sine wave with covariance matrix <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">B^{(s,l)} = \cos \lambda (s - l)</tex> as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\lambda</tex> decreases, non-Abelian groups result in a better statistical performance of the filter than the DFT does. However, that is compensated for by the increased number of computations to perform the filtering.