A Unified Approach to Time-Frequency Representations and Generalized Spectrogram
Elena Cordero, Gianluca Giacchi, Luigi Rodino
TL;DR
The paper develops a unified framework for time-frequency representations by treating them as metaplectic Wigner distributions $W_\text{A}$ with $\text{A}\in Sp(2d,\mathbb{R})$, showing that the STFT and Cohen's class representations are instances of this scheme. It provides a complete block-decomposition criterion on $\text{A}$ to characterize which $W_\text{A}$ yield generalized spectrograms, and it establishes $L^p$-boundedness results for $W_\text{A}$ and the related metaplectic pseudodifferential operators $Op_\text{A}(a)$. The paper introduces $\mathcal{A}$-metaplectic spectrograms and the more general $(\mathcal{A},\mathcal{B})$-spectrograms, unifying Cohen's class representations and recapturing classical results such as the fact that $W_\tau$ is a generalized spectrogram only for $\tau=0$ or $\tau=1$. In the one-dimensional setting it further shows a clean dichotomy: shift-invertible distributions and generalized spectrograms are disjoint yet together exhaust all Cohen-class metaplectic Wigner distributions, clarifying the landscape of time-frequency representations. This framework enables a systematic analysis of representations and their associated operators, with implications for signal processing and mathematical analysis.
Abstract
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 of the previous theory by means of metaplectic Wigner distributions $W_{\mathcal{A}}$, with $\mathcal{A}$ symplectic matrix in $Sp(2d,\mathbb{R})$, which were introduced by Cordero, Rodino (2022) and then widely studied in subsequent papers. Namely, the short-time Fourier transform and the most popular members of the Cohen's class can be represented via metaplectic Wigner distributions. In particular, we introduce $\mathcal{A}$-metaplectic spectrograms which contain the classical ones and their variations arising from the $τ$-Wigner distributions of Boggiatto, De Donno, and Oliaro (2010). We provide a complete characterization of those $\mathcal{A}$-Wigner distributions which give rise to generalized spectrograms. This characterization is related to the block decomposition of the symplectic matrix $\mathcal{A}$. Moreover, a characterization of the $L^p$-boundedness of both $\mathcal{A}$-Wigner distributions and related metaplectic pseudodifferential operators is provided.