Time-Frequency Hop Codes Based Upon Extended Quadratic Congruences

Abstract

Time-frequency hop codes are developed based upon an extension of the theory of quadratic congruences. These codes can be used for coherent multiuser echolocation and asynchronous spread spectrum communication systems. They represent a compromise between Costas codes, which have nearly ideal auto-but not so good cross-ambiguity properties, and linear congruential codes, which have nearly ideal cross- but unattractive auto-ambiguity properties. Extended quadratic congruential (EQC) code words are shown to have reasonably good auto- 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 auto-ambiguity function. These bounds depend on time-bandwidth product and code length exclusively, and lead naturally to a discussion of the design tradeoffs for these two parameters. Examples of typical auto- and cross-ambiguity functions are given to illustrate the performance of the new codes. © 1988 IEEE