Spectral Factorization, Whitening- and Estimation Filter -- Stability, Smoothness Properties and FIR Approximation Behavior

TL;DR

This paper analyzes the stability and approximation properties of Wiener filters based on spectral factorization, emphasizing the role of spectral density smoothness in filter stability and FIR approximation accuracy.

Contribution

It provides a detailed investigation of the properties of whitening and estimation filters within Wiener filters, linking spectral density smoothness to stability and approximation behavior.

Findings

Smooth spectral densities ensure Wiener filter stability.

Smooth spectral densities allow arbitrary FIR approximation.

Less smooth spectral densities may lead to unstable Wiener filters.

Abstract

A Wiener filter can be interpreted as a cascade of a whitening- and an estimation filter. This paper gives a detailed investigates of the properties of these two filters. Then the practical consequences for the overall Wiener filter are ascertained. It is shown that if the given spectral densities are smooth (Hoelder continuous) functions, the resulting Wiener filter will always be stable and can be approximated arbitrarily well by a finite impulse response (FIR) filter. Moreover, the smoothness of the spectral densities characterizes how fast the FIR filter approximates the desired filter characteristic. If on the other hand the spectral densities are continuous but not smooth enough, the resulting Wiener filter may not be stable.