On cross-ambiguity properties of Welch-Costas arrays

We discuss cross-ambiguity properties of a specific family of Costas arrays called Welch-Costas (W-C) arrays. These properties are of interest in multiuser radar and sonar system, especially since Costas arrays are known to possess ideal auto-ambiguity functions. The theory of W-C arrays is reviewed. It is then proved that only pairs of W-C arrays can have at most two hits in their cross-ambiguity function (best possible case). The maximum number of hits in the cross-ambiguity functions of a family of W-C arrays is shown to be a function of the number of W-C arrays in the family. The upper bound on the number of hits in the cross ambiguity functions for a family of W-C arrays is also derived. Specific examples of how reducing the number of W-C arrays improves the cross-ambiguity properties are given for various types of prime numbers.>