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.
Categories: math.AP math.FA