Time-frequency formulation, design, and implementation of time-varying optimal filters for signal estimation

This paper presents a time–frequency framework for optimal linear filters (signal estimators) in nonstationary environments. We develop time–frequency formulations for the optimal linear filter (time-varying Wiener filter) and the optimal linear time-varying filter under a projection side constraint. These time–frequency formulations extend the simple and intuitive spectral representations that are valid in the stationary case to the practically important case of underspread nonstationary processes. Furthermore, we propose an approximate time–frequency design of both optimal filters, and we present bounds that show that for underspread processes, the time-frequency designed filters are nearly optimal. We also introduce extended filter design schemes using a weighted error criterion, and we discuss an efficient time–frequency implementation of optimal filters using multiwindow short-time Fourier transforms. Our theoretical results are illustrated by numerical simulations.