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 metaplectic spectrograms, which contain the classical ones and their variations arising from the tau-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 boundedness of both A-Wigner distributions and related metaplectic pseudodifferential operators is provided.