Zak transform and a new approach to waveform design

The ambiguity function A_f (τ, v) of a transmitted signal f(t) measures the uncertainty with which the returning echo distinguishes, simultaneously, both ranges and velocities of a target system. Generally speaking, A_f (τ, v) is desired to be of "thumbtack" shape, i.e., a function whose absolute value has a graph with a strong peak at the origin over a broad shallow base. The ambiguity function can be computed directly from the Zak transform Z_f (x,y) of the signal f(t), so waveforms with desirable ambiguity functions can be designed in the Zak domain. In the Zak domain, computation of A_f (τ, v) on the integer lattice is exceptionally simple, particularly for pulse train signals. For a pulse train, the Zak transform is obtained by multiplying the Zak transform of a rectangular pulse of duration 1 by a multivariate trigonometric polynomial whose coefficients are the coefficients defining the pulse train. Reversing this observation, one can start with such a trigonometric polynomial and construct a pulse train signal. We propose a systematic method for constructing such waveforms, which we illustrate in a particular case