  We exhibit the connection between the Wigner kernel and the Gabor matrix of a
linear bounded operator T : $\mathcal{S}(\mathbb{R}^d) \to \mathcal{S}'
(\mathbb{R}^d)$. The smoothing effect of the Gabor matrix is highlighted by
basic examples. This connection allows a comparison between the classes of
Fourier integral operators defined by means of the Gabor matrix and the Wigner
kernel, showing the nice off-diagonal decay of the Gabor class with respect to
the Wigner kernel one and suggesting further investigations. Modulation spaces
containing the Sj\"ostrand class are the symbol classes of this study.
