Time-frequency hop codes based upon extended quadratic congruences

Time-frequency hop codes are developed that can be used for coherent multiuser echolocation and asynchronous spread spectrum communication systems. They represent a compromise between Costas codes, which have nearly ideal autoambiguity but not so good cross-ambiguity properties, and linear congruential codes, which have nearly ideal cross-ambiguity but unattractive autoambiguity properties. Extended quadratic congruential code words are shown to have reasonably good autoambiguity and cross-ambiguity properties across the whole class of code sets considered. A uniform upper bound is placed on the entire cross-ambiguity function surface, and bounds are placed on the position and 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 the new codes     

The hit array: an analysis formalism for multiple access frequencyhop coding

A formalism is presented for the analysis of general frequency hop waveforms, such as those suitable for use in coherent active radar and sonar echolocation systems as well as multiple-access spread-spectrum communications. This formalism is based on the concept of coincidence, or `hit', between two frequency hopping patterns. The collection of all possible hits, together with their locations, is recorded in time-frequency space, which produces the high array associated with the two patterns considered. If the code length is sufficiently small with respect to the time-bandwidth product chosen, the hit array can be viewed as a digital representation of the corresponding ambiguity function. Salient properties of the hit array formalism are derived, including simple relationships between hit arrays resulting from basic symmetry-preserving transformations. These properties make it possible to predict the performance of a given set of frequency hop waveforms directly from the associated set of frequency hopping patterns     

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