Characterization of Weyl-Heisenberg frames via Poisson summation relationships

Fourier theorems and Poisson summation are applied to characterize the sampled ambiguity function, yielding facts about Weyl-Heisenberg frames. Relationships between related cross- and auto-ambiguity functions of a signal f and a window g over two complementary lattices L and L* in the time-frequency plane are derived and used to: characterize the frames {g|L }-translates of g over L-in terms of the behavior of g on L*; calculate a new, simple formula for the upper frame bound that is as tight as previously reported calculations for the Gaussian window; demonstrate the existence of a new class of tight frames; and provide error analysis of certain Riemann approximations to ambiguity integrals