Frequency hop multiple access codes based upon the theory of cubiccongruences

The need for families of frequency-hop codes which have mutually small auto-ambiguity and cross-ambiguity functions is discussed. Current coding methods are reviewed. A new family of frequency-hop codes based upon the number-theoretic concept of cubic congruences is introduced. It is shown that for about 50% of the prime numbers, families of full codes exist which have at most two coincidences for any time-frequency shift in their auto-ambiguity functions and at most three coincidences in the set of mutual cross-ambiguity functions     

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     

Ambiguity properties of quadratic congruential coding

The ambiguity characteristics of multiple access frequency hop codes based on standard quadratic congruences are investigated in the light of results obtained for codes based on Costas arrays and extended quadratic congruences. While the autoambiguity properties are found to be very similar to those of Costa codes, i.e. nearly ideal, the cross-ambiguity properties of quadratic congruential codes are much better. These results are valid across the whole class of code sets considered, but they are obtained at some expense in the pulse compression characteristics of the codes. A uniform upper bound is placed on the entire cross-ambiguity function surface, and bounds are placed on the amplitude of spurious peaks in the autoambiguity function. These bounds depend on the time-bandwidth product and code length exclusively and lead naturally to a discussion of the design tradeoffs for these two parameters. Examples of typical autoambiguity and cross-ambiguity functions are given to illustrate the performance of quadratic congruential coding with respect to Costas coding