Symplectic Analysis of Time-Frequency Spaces

TL;DR

This paper introduces a symplectic perspective on time-frequency spaces, showing how metaplectic operators can represent classical time-frequency transforms and establishing conditions for these representations to characterize modulation and Wiener spaces.

Contribution

It demonstrates that shift-invertibility alone is insufficient for characterizing modulation spaces, requiring additional upper-triangularity conditions, and introduces new time-frequency representations for signal analysis.

Findings

Metaplectic operators can represent all classical time-frequency transforms.

Shift-invertibility alone does not suffice for modulation space characterization.

New families of time-frequency representations are proposed for signal decomposition.

Abstract

We present a different symplectic point of view in the definition of weighted modulation spaces $M^{p,q}_m(\mathbb{R}^d)$ and weighted Wiener amalgam spaces $W(\mathcal{F} L^p_{m_1},L^q_{m_2})(\mathbb{R}^d)$. All of the classical time-frequency representations, such as the short-time Fourier transform (STFT), the $\tau$-Wigner distributions and the ambiguity function, can be written as metaplectic Wigner distributions $\mu(\mathcal{A})(f\otimes \bar{g})$, where $\mu(\mathcal{A})$ is the metaplectic operator and $\mathcal{A}$ is the associated symplectic matrix. Namely, time-frequency representations can be represented as images of metaplectic operators, which become the real protagonists of time-frequency analysis. In [E. Cordero and L. Rodino (2022) "Characterization of Modulation Spaces by symplectic representations and applications to Schr\"odinger equations", arXiv:2204.14124], the authors suggest that any metaplectic Wigner distribution that satisfies the so-called "shift-invertibility" condition can replace the STFT in the definition of modulation spaces. In this work, we prove that shift-invertibility alone is not sufficient, but it has to be complemented by an upper-triangularity condition for this characterization to hold, whereas a lower-triangularity property comes in to play for Wiener amalgam spaces. The shift-invertibility property is necessary: Ryhaczek and and conjugate Ryhaczek distributions are not shift-invertible and they fail the characterization of the above spaces. We also exhibit examples of shift-invertible distributions without upper-tryangularity condition which do not define modulation spaces. Finally, we provide new families of time-frequency representations that characterize modulation spaces, with the purpose of replacing the time-frequency shifts with other atoms that allow to decompose signals differently, with possible new outcomes in applications.