- Title
- Gabor discretization of the Weyl product for modulation spaces and filtering of nonstationary stochastic processes
- Creator
- Wahlberg, Patrik; Schreier, Peter J.
- Relation
- Applied and Computational Harmonic Analysis Vol. 26, Issue 1, p. 97-120
- Publisher Link
- http://dx.doi.org/10.1016/j.acha.2008.02.005
- Publisher
- Academic Press
- Resource Type
- journal article
- Date
- 2009
- Description
- 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 Sjöstrand 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 L², 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.
- Subject
- pseudodifferential calculus; Weyl calculus; modulation spaces; Gabor frames; nonstationary stochastic processes; optimal filtering
- Identifier
- http://hdl.handle.net/1959.13/806726
- Identifier
- uon:7202
- Identifier
- ISSN:1063-5203
- Language
- eng
- Reviewed
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