A time-frequency formula for LMMSE filters for nonstationary underspread continuous-time stochastic processes

We study linear minimum mean square error (LMMSE) filters for estimating a nonstationary second-order continuous-time stochastic process from a noisy observation. The equation for the optimal filter is treated in the Weyl symbol domain, and the involved Weyl symbols are assumed to belong to certain modulation spaces. By discretizing this equation using a Gabor frame we transform it into a matrix equation and obtain a formula for the filter by matrix inversion. The inverse matrix has off-diagonal decay at a rate that increases the more underspread the process is.