  We perform Wigner analysis of linear operators. Namely, the standard
time-frequency representation \emph{Short-time Fourier Transform} (STFT) is
replaced by the $\mathcal{A}$-\emph{Wigner distribution} defined by
$W_{\mathcal A} (f)=\mu({\mathcal A})(f\otimes\bar{f})$, where ${\mathcal A}$
is a $4d\times 4d$ symplectic matrix and $\mu({\mathcal A})$ is an associate
metaplectic operator. Basic examples are given by the so-called $\tau$-Wigner
distributions. Such representations provide a new characterization for
modulation spaces when $\tau\in (0,1)$. Furthermore, they can be efficiently
employed in the study of the off-diagonal decay for pseudodifferential
operators with symbols in the Sj\"ostrand class (in particular, in the
H\"{o}rmander class $S^0_{0,0}$). The novelty relies on defining time-frequency
representations via metaplectic operators, developing a conceptual framework
and paving the way for a new understanding of quantization procedures. We
deduce micro-local properties for pseudodifferential operators in terms of the
Wigner wave front set. Finally, we compare the Wigner with the global
H\"{o}rmander wave front set and identify the possible presence of a ghost
region in the Wigner wave front. \par In the second part of the paper
applications to Fourier integral operators and Schr\"{o}dinger equations will
be given.
