Metaplectic time-frequency representations
Gianluca Giacchi
TL;DR
The paper presents a unified, algebraically grounded framework for time-frequency representations based on metaplectic operators, recasting classical constructs like the Wigner distribution and the STFT as special cases of metaplectic Wigner distributions $W_\mathcal{A}$. It develops four core themes—Cohen's class membership, shift-invertibility criteria, generalized spectrograms, and uncertainty principles—and culminates with inversion formulas and Gabor-frame theory for metaplectic representations, linking reconstruction to modulation-space norms. By tying representation properties to the projection $\mathcal{A}$ and its blocks, the work clarifies when a representation behaves like a Cohen-type distribution or reduces to a scaled STFT, and it provides a principled path to discretization via metaplectic Gabor frames. This synthesis highlights how the symplectic structure governs the behavior and reconstruction of time-frequency representations, offering a comprehensive toolkit for analysis and signal processing applications within modulation spaces.
Abstract
Time-frequency representations stemmed in 1932 with the introduction of the Wigner distribution. For most of the 20th century, research in this area primarily focused on defining joint probability distributions for position and momentum in quantum mechanics. Applications to electrical engineering were soon established with the seminal works of Gabor and the researchers at Bell Labs. In 2012, Bai, Li and Cheng used for the first time metaplectic operators, defined in the middle of 20th century by Van Hove, to generalize the Wigner distribution and unify effectively the most used time-frequency representations under a common framework. This work serves as a comprehensive up-to-date survey on time-frequency representations defined by means of metaplectic operators, with particular emphasis on the recent contributions by Cordero and Rodino, who exploited metaplectic operators to their limits to generalize the Wigner distributions. Their idea provides a fruitful framework where properties of time-frequency representations can be explained naturally by the structure of the symplectic group.