Uncertainty in time-frequency representations on finite Abelian groups and applications

Classical and recent results on uncertainty principles for functions on finite Abelian groups relate the cardinality of the support of a function to the cardinality of the support of its Fourier transform. We obtain corresponding results relating the support sizes of functions and their short-time Fourier transforms. We use our findings to construct a class of equal norm tight Gabor frames that are maximally robust to erasures. Also, we discuss consequences of our findings to the theory of recovering and storing signals with sparse time-frequency representations.