Time-frequency representations of Wigner type and pseudo-differential operators

Abstract

We introduce a τ \tau -dependent Wigner representation, Wig τ \operatorname {Wig}_\tau , τ ∈ [ 0 , 1 ] \tau \in [0,1] , which permits us to define a general theory connecting time-frequency representations on one side and pseudo-differential operators on the other. The scheme includes various types of time-frequency representations, among the others the classical Wigner and Rihaczek representations and the most common classes of pseudo-differential operators. We show further that the integral over τ \tau of Wig τ \operatorname {Wig}_\tau yields a new representation Q Q possessing features in signal analysis which considerably improve those of the Wigner representation, especially for what concerns the so-called “ghost frequencies”. The relations of all these representations with respect to the generalized spectrogram and the Cohen class are then studied. Furthermore, a characterization of the L p L^p -boundedness of both τ \tau -pseudo-differential operators and τ \tau -Wigner representations are obtained.