Optimum Frequency Hop Patterns Constructed on a Golomb Costas Array Using Two-Dimensional Circular Shift

This study presents a comprehensive mathematical model for analyzing optimum frequency hop (FH) patterns (OFHPs) derived from two-dimensional circular shifts applied to a Golomb Costas array. The investigation focuses on evaluating the autocorrelation and crosscorrelation properties of these patterns, proposing a structure for a FH pattern with one-gap row and one-gap column. Through systematic research involving theoretical derivation, computational analysis, graphical illustration, and discussion of examples, we delve into the theory behind OFHPs based on Golomb Costas arrays. Using the definitions of delay distance and Doppler distance between OFHPs, we provide an in-depth explanation of the properties of OFHPs and the OFHP family. Findings, supported by proven theorems, enhance understanding of the core nature of these patterns, demonstrating their ideal autocorrelation and crosscorrelation properties. The study also conducts simulation analyses of the ambiguity and cross-ambiguity functions for signals coded by the respective OFHPs, indicating that a signal coded by an OFHP exhibits a thumbtack ambiguity function and ideal cross-ambiguity with respect to other signals coded by different OFHPs. In addition, the study elaborates on the application of two-dimensional circular shift for designing OFHPs, drawing valuable insights from Golomb Costas arrays. This research significantly contributes to understanding OFHPs, their formation using Golomb Costas arrays, and the practical applicability of their ideal autocorrelation and crosscorrelation properties.