  In the last twenty years modulation spaces, introduced by H. G. Feichtinger
in 1983, have been successfully addressed to the study of signal analysis,
PDE's, pseudodifferential operators, quantum mechanics, by hundreds of
contributions. In 2011 M. de Gosson showed that the time-frequency
representation Short-time Fourier Transform (STFT), which is the tool to define
modulation spaces, can be replaced by the Wigner distribution. This idea was
further generalized to $\tau$-Wigner representations in [9]. In this paper
time-frequency representations are viewed as images of symplectic matrices via
metaplectic operators. This new perspective highlights that the protagonists of
time-frequency analysis are metaplectic operators and symplectic matrices
$\mathcal{A} \in Sp(2d,\mathbb{R})$. We find conditions on $\mathcal{A}$ for
which the related symplectic time-frequency representation $W_\mathcal{A}$ can
replace the STFT and give equivalent norms for weighted modulation spaces. In
particular, we study the case of covariant matrices $\mathcal{A}$, i.e., their
corresponding $W_\mathcal{A}$ are members of the Cohen class.
  Finally, we show that symplectic time-frequency representations
$W_\mathcal{A}$ can be efficiently employed in the study of Schr\"{o}dinger
equations. This new approach may have further applications in quantum mechanics
and PDE's.
