  In this work, we extend Wigner's original framework to analyze linear
operators by examining the relationship between their Wigner and Schwartz
kernels. Our approach includes the introduction of (quasi-)algebras of Fourier
integral operators (FIOs), which encompass FIOs of type I and II. The symbols
of these operators reside in (weighted) modulation spaces, particularly in
Sj\"ostrand's class, known for its favorable properties in time-frequency
analysis. One of the significant results of our study is demonstrating the
inverse-closedness of these symbol classes.
  Our analysis includes fundamental examples such as pseudodifferential
operators and Fourier integral operators related to Schr{\"o}dinger-type
equations. These examples typically feature classical Hamiltonian flows
governed by linear symplectic transformations $S \in Sp(d, \mathbb{R})$. The
core idea of our approach is to utilize the Wigner kernel to transform a
Fourier integral operator $ T $ on $ \mathbb{R}^d $ into a pseudodifferential
operator $ K$ on $ \mathbb{R}^{2d}$. This transformation involves a symbol
$\sigma$ well-localized around the manifold defined by $ z = S w $.
