  We study a class of quadratic time-frequency representations that, roughly
speaking, are obtained by linear perturbations of the Wigner transform. They
satisfy Moyal's formula by default and share many other properties with the
Wigner transform, but in general they do not belong to Cohen's class. We
provide a characterization of the intersection of the two classes. To any such
time-frequency representation, we associate a pseudodifferential calculus. We
investigate the related quantization procedure, study the properties of the
pseudodifferential operators, and compare the formalism with that of the Weyl
calculus.
