  We introduce new frames, called \textit{metaplectic Gabor frames}, as natural
generalizations of Gabor frames in the framework of metaplectic Wigner
distributions. Namely, we develop the theory of metaplectic atoms in a
full-general setting and prove an inversion formula for metaplectic Wigner
distributions on $\mathbb{R}^d$. Its discretization provides metaplectic Gabor
frames.
  Next, we deepen the understanding of the so-called shift-invertible
metaplectic Wigner distributions, showing that they can be represented, up to
chirps, as rescaled short-time Fourier transforms. As an application, we derive
a new characterization of modulation and Wiener amalgam spaces. Thus, these
metaplectic distributions (and related frames) provide meaningful definitions
of local frequencies and can be used to measure effectively the local frequency
content of signals.

Analysis of PDEs;Functional Analysis