A Unified Approach to Time-Frequency Representations and Generalized Spectrogram

TL;DR

This paper introduces a unified framework for time-frequency representations using metaplectic Wigner distributions, encompassing classical spectrograms and their variations, with characterizations of their properties and boundedness.

Contribution

It develops a comprehensive theory linking various time-frequency representations through symplectic matrices, providing new characterizations and extending the understanding of their boundedness and structure.

Findings

Unified framework for time-frequency representations via symplectic matrices.

Complete characterization of generalized spectrograms from metaplectic Wigner distributions.

Results on the $L^p$-boundedness of these distributions and related operators.

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 $\tau$-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.