We perform Wigner analysis of linear operators. Namely, the standard time-frequency representation Short-time Fourier Transform (STFT) is replaced by the $\mathcal{A}$-Wigner distribution defined by $W_{\mathcal{A}}(f)=\mu (\mathcal{A})(f\otimes \overline{f})$, 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 τ-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 Sjöstrand class (in particular, in the Hörmander 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 Hörmander wave front set and identify the possible presence of a ghost region in the Wigner wave front.

In the second part of the paper applications to Fourier integral operators and Schrödinger equations will be given.

我们对线性算子进行 Wigner 分析。即，标准时频表示短时傅里叶变换(STFT) 被替换为$\mathcal{一个}$- Wigner 分布定义为$W_{\mathcal{一个}}(F)=\mu (\mathcal{一个})(F\otimes \overline{F})$， 在哪里$\mathcal{一个}$是一个$4d\times 4d$辛矩阵和$\mu (\mathcal{一个})$是关联元理算子。基本示例由所谓的τ -Wigner 分布给出。这种表示为调制空间提供了新的表征，当$\tau \in (0,1)$. 此外，它们可以有效地用于研究具有 Sjöstrand 类（特别是 Hörmander 类）符号的伪微分算子的非对角衰减${\mathrm{小号}}_{0,0}^0$）。新颖性依赖于通过元算子定义时频表示，开发概念框架并为对量化过程的新理解铺平道路。我们根据 Wigner 波前集推导出伪微分算子的微局部属性。最后，我们将 Wigner 与全局 Hörmander 波前集进行比较，并确定 Wigner 波前可能存在鬼区。

在论文的第二部分中，将给出傅里叶积分算子和薛定谔方程的应用。