Gabor Discretization Of The Weyl Product For Modulation Spaces And Filtering Of Nonstationary Stochastic Processes

摘要

We discretize the Weyl product acting on symbols of modulation spaces, using a Gabor frame defined by a Gaussian function. With one factor fixed, the Weyl product is equivalent to a matrix multiplication on the Gabor coefficient level. If the fixed factor belongs to the weighted Sjostrand space M_ω~(∞,1) , then the matrix has polynomial or exponential off-diagonal decay, depending on the weight ω. Moreover, if its operator is invertible on L~2 , the inverse matrix has similar decay properties. The results are applied to the equation for the linear minimum mean square error filter for estimation of a nonstationary second-order stochastic process from a noisy observation. The resulting formula for the Gabor coefficients of the Weyl symbol for the optimal filter may be interpreted as a time-frequency version of the filter for wide-sense stationary processes, known as the noncausal Wiener filter.