  The Wigner distribution is a milestone of Time-frequency Analysis. In order
to cope with its drawbacks while preserving the desirable features that made it
so popular, several kind of modifications have been proposed. This
contributions fits into this perspective. We introduce a family of phase-space
representations of Wigner type associated with invertible matrices and explore
their general properties. As main result, we provide a characterization for the
Cohen's class. This feature suggests to interpret this family of
representations as linear perturbations of the Wigner distribution. We show
which of its properties survive under linear perturbations and which ones are
truly distinctive of its central role.
