On the Underspread/Overspread Classification of Random Processes

TL;DR

This paper examines how the underspread/overspread classification affects the spectral analysis of nonstationary processes, highlighting that underspread processes allow for effective time-varying spectral representations similar to stationary cases.

Contribution

It clarifies the applicability of time-varying power spectra for nonstationary processes based on their underspread or overspread classification.

Findings

Underspread processes permit the use of time-varying power spectra.

Most definitions of time-varying spectra are limited for general nonstationary processes.

The time-frequency-parametrized Wiener filter exemplifies the main conclusion.

Abstract

We study the impact of the recently introduced underspread/overspread classificationon the spectra of processes with square-integrable covariance functions. We briefly review the most prominent definitions of a time-varying power spectrum and point out their limited applicability for {\em general} nonstationary processes. The time-frequency-parametrized approximation of the nonstationary Wiener filter provides an excellent example for the main conclusion: It is the class of underspread processeswhere a time--varying power spectrum can be used in the same manner as the time--invariant power spectrum of stationary processes.