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We study a class of quadratic time-frequency representations that, roughlyspeaking, are obtained by linear perturbations of the Wigner transform. Theysatisfy Moyal's formula by default and share many other properties with theWigner transform, but in general they do not belong to Cohen's class. Weprovide a characterization of the intersection of the two classes. To any suchtime-frequency representation, we associate a pseudodifferential calculus. Weinvestigate the related quantization procedure, study the properties of thepseudodifferential operators, and compare the formalism with that of the Weylcalculus.

Linear perturbations of the Wigner transform and the Weyl quantization