Generalized-marginal time-frequency distributions

We introduce a family of time-frequency (TF) distributions with generalized-marginals, i.e., beyond the time-domain and the frequency-domain marginals, in the sense that the projections of a TF distribution along one or more angles are equal to the magnitude squared of the fractional Fourier transforms of the signal. We present a necessary and sufficient condition for a TF distribution in the Cohen's class to satisfy generalized-marginals. We then modify the existing well-known TF distributions in the Cohen's class, such as Choi-Williams, Page distributions, so that the modified ones have generalized marginals. Numerical examples are presented to show that the proposed TF distributions have the advantages of both Wigner-Ville and other quadratic TF distributions which only have the conventional marginals. Moreover, they also indicate that the generalized-marginal TF distributions with proper marginals are more robust than the Wigner-Ville and the Choi-Williams distributions when signals contain additive noise.