I. Introduction
Designing signals with thumbtack ambiguity functions, i.e., functions whose absolute value has a graph with a strong peak at the origin over a broad shallow base, is a special case of the more general issue of designing signals with a prescribed ambiguity function. The many attacks on this difficult problem [14, p. 125] since the publication of Woodward's book have yielded a great deal of insight into the nature of the ambiguity function (see [4]), but no computationally practicable solution to the general syn problem has been provided. The elegant paper of Wilcox [13] provides a mathematically complete solution, but it should be borne in mind that the speed of convergence of his solution depends on the smoothness of the underlying functions. Since the ideal ambiguity function is a delta function, and hence a poor input to the Wilcox algorithm, the search for practical solutions to the synthesis problem remains a challenging problem. Algebraic properties of the mapping that maps a function into its ambiguity function were studied by Auslander and Tolimieri [3].