Proceedings Article | 11 October 1994

Proc. SPIE. 2303, Wavelet Applications in Signal and Image Processing II

KEYWORDS: Fourier transforms, Mathematics, Convolution, Electronic filtering, Space operations, Time-frequency analysis, Lead, Tin, Bandpass filters, Bohrium

The Weyl correspondence is a convenient way to define a broad class of time-frequency localization operators. Given a region (Omega) in the time-frequency plane R<SUP>2</SUP> and given an appropriate (mu) , the Weyl correspondence can be used to construct an operator L((Omega) ,(mu) ) which essentially localizes the time-frequency content of a signal on (Omega) . Different choices of (mu) provide different interpretations of localization. Empirically, each such localization operator has the following singular value structure: there are several singular values close to 1, followed by a sharp plunge in values, with a final asymptotic decay to zero. The exact quantification of these qualitative observations is known only for a few specific choices of (Omega) and (mu) . In this paper we announce a general result which bounds the asymptotic decay rate of the singular values of any L((Omega) ,(mu) ) in terms of integrals of (chi) <SUB>(Omega</SUB> ) * <SUP>-</SUP>(mu) <SUP>2</SUP> and ((chi) <SUB>(Omega</SUB> ) * <SUP>-</SUP>(mu) )<SUP>^</SUP><SUP>2</SUP> outside squares of increasing radius, where <SUP>-</SUP>(mu) (a,b) equals (mu) (-a, -b). More generally, this result applies to all operators L((sigma) ,(mu) ) allowing window function (sigma) in place of the characteristic functions (chi) <SUB>(Omega</SUB> ). We discuss the motivation and implications of this result. We also sketch the philosophy of proof, which involves the construction of an approximating operator through the technology of Gabor frames--overcomplete systems which allow basis-like expansions and Plancherel-like formulas, but which are not bases and are not orthogonal systems.