This paper systematically expounded the theory of optimal frequency hopping patterns based on Welch Costas arrays with 1-gap row, a theory established after studying the properties and mathematical models of the frequency hopping patterns obtained by making two-dimensional cyclic shift to Welch Costas arrays. The algebraic construction and the autocorrelation and cross-correlation properties of the Welch Costas arrays with 1-gap row were studied and certain theorems were proved. The optimal frequency hopping patterns will have ideal autocorrelation and cross-correlation when designed by means of Welch Costas arrays with 1-gap row.

This study presents a comprehensive mathematical model for analyzing optimum frequency hop patterns (OFHPs) derived from two-dimension circular shifts applied to a Golomb Costas array. The investigation focuses on evaluating the autocorrelation and cross-correlation properties of these patterns, proposing a structure for a frequency hop pattern with 1-gap row and 1-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 cross-correlation 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 crossambiguity with respect to other signals coded by different OFHPs. Additionally, the study elaborates on the application of two-dimension 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 cross-correlation properties.

This paper explores the autocorrelation and cross-correlation properties of Welch Costas arrays and introduces a method for constructing optimal frequency hopping patterns with ideal autocorrelation and cross-correlation through two-dimensional cyclic shift on a Welch Costas array. Additionally, it proposes orthogonal pattern division multiple access (OPDMA) utilizing these optimal patterns. The paper examines the relationship between the design parameters of these patterns and cellular system parameters, tailoring frequency hopping patterns for cellular system users. Furthermore, it conducts computations and analyses on the autocorrelation and cross-correlation of the designed patterns, demonstrating their ideal properties. It also discusses the role of delay distance and Doppler distance in achieving ideal cross-correlation between the optimal frequency hopping patterns. OPDMA emerges as a novel two-dimensional orthogonal multiple access technology offering signal immunity, effectively overcoming interference from multipath propagation, Doppler shift, and inter-user interference. This simplifies receiver design and enhances overall system performance, presenting potential applications in future cellular mobile communications, frequency hopping communications, radar, and sonar systems.

This paper proposes a specific algebraic structure and demonstrates its nature as an extension field, enabling the construction of Golomb Costas (GC) arrays. It provides detailed instructions and examples for constructing GC arrays using this extension field, along with a corresponding flowchart. Additionally, the paper conducts a thorough analysis, incorporating calculations and comparisons, to evaluate the autocorrelation of a GC array derived from the extension field compared to that of a diagonal frequency hopping array. The analysis reveals the superior autocorrelation properties of GC arrays based on the extension field. Furthermore, the paper establishes a mathematical model for the signal coded by the frequency hopping array and subsequently simulates and compares the ambiguity function of the signal coded by a GC array with that of a signal coded by a diagonal frequency hopping array. This comparison underscores the thumbtack ambiguity function of frequency hopping signal coded by a GC array. Moreover, the paper thoroughly investigates the relationship between the correlation function of GC arrays and the roots of an algebraic equation in a finite field, and strictly proves the ideal autocorrelation properties of Golomb Costas arrays.

Costas arrays have been widely applied in wireless communication, radar and cryptography as they have the ideal auto-ambiguity properties. Costas arrays are constructed mostly using the algebraic method, or they can be found by exhaustive search. This paper first introduces the Costas arrays and their properties and then proposes the vector-based algorithm to make depth-first search for Costas arrays since the exhaustive search algorithm based on check matrix is flawed. The experiment results indicate that the algorithm based on vectors has higher search speed on a single CPU when compared with the exhaustive searching algorithm based on check matrix.