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

We introduce a $\tau$-dependent Wigner representation, $\operatorname{Wig}_\tau$, $\tau\in[0,1]$, 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 $\tau$ of $\operatorname{Wig}_\tau$ yields a new representation $Q$ 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 $L^p$-boundedness of both $\tau$-pseudo-differential operators and $\tau$-Wigner representations are obtained.