Double Preconditioning for Gabor Frames

Abstract

Abstract-We present an application of the general idea of preconditioning in the context of Gabor frames. While most (iterative) algorithms aim at a more or less costly exact numerical calculation of the inverse Gabor frame matrix, we propose here the use of "cheap methods" to find an approximation for it, based on (double) preconditioning. We thereby obtain good approximations of the true dual Gabor atom at low computational costs. Since the Gabor frame matrix commutes with certain timefrequency shifts it is natural to make use of diagonal and circulant preconditioners sharing this property. Part of the efficiency of the proposed scheme results from the fact that all the matrices involved share a well-known block matrix structure. At least, for the smooth Gabor atoms typically used, the combination of these two preconditioners leads consistently to good results. These claims are supported by numerical experiments in the second part of the paper. For numerical evaluations we introduce two new matrix norms, which can be calculated efficiently by exploiting the structure of the frame matrix.Index Terms-Block matrices; efficient algorithm; Gabor frame matrices; approximated dual windows; time-frequency analysis; matrix norms; discrete transforms; matrix inversion; EDICS : DSP-WAVL Wavelets theory and applications; DSP-FAST Fast algorithms for digital signal processing

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