A Unified Approach to Time–Frequency Representations and Generalized Spectrograms

To overcome the impossibility of representing the energy of a signal simultaneously in time and frequency, many time–frequency representations have been introduced in the literature. Some of these are recalled in the Introduction. In this work, we propose a unified approach to the previous theory by means of metaplectic Wigner distributions WA, with A a symplectic matrix in Sp(2d,R), which were introduced by Cordero and Rodino (Appl Comput Harmon Anal 58:85–123, 2022) and then widely studied in subsequent papers. Namely, the short-time Fourier transform and the most popular members of Cohen’s class can be represented via metaplectic Wigner distributions. In particular, we introduce A-metaplectic spectrograms, which contain the classical ones and their variations arising from the τ-Wigner distributions of Boggiatto et al. (Trans Am Math Soc 362(9):4955–4981, 2010). We provide a complete characterization of those A-Wigner distributions which give rise to generalized spectrograms. This characterization is related to the block decomposition of the symplectic matrix A. Moreover, a characterization of the Lp-boundedness of both A-Wigner distributions and related metaplectic pseudodifferential operators is provided.