Asymptotic singular value decay of time-frequency localization operators
Christopher E. Heil, Jayakumar Ramanathan, Pankaj Topiwala · Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE · 1994
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 R2 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) (Omega ) * -(mu) 2 and ((chi) (Omega ) * -(mu) )^2 outside squares of increasing radius, where -(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) (Omega ). 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.