Conceptual Bases of Gray Transformations and Discrete Vilenkin-Crestenson Functions Systems

Abstract

The article is devoted to the theoretical foundations of the construction of generalized Gray codes. These include the classical "left-side" and the proposed "right-side" Gray's codes. Left-side codes develop from left to right and right-side codes from right to left. Left- and right-handed Gray's codes augmented with reverse permutation operators form a subset of composite Gray's codes. Such codes have found constructive application in solving synthesis and analysis problems of discrete systems of Vilenkin-Crestenson functions (VCF). Particular cases of VCF systems are systems of discrete exponential functions and Walsh functions. The so-called indicator matrices (IM), bijective connected to the VCF systems, form the basis for VCF systems synthesis. The order of IMs is determined by the logarithmic dependence on the order of the VCF system. Unique Walsh-Cooley function systems, the only ones in the set of Walsh function systems that provide linear connectivity of the frequency scales of DFT processors, are developed. Examples of trees of VCF systems give. The orders of IMs are related by logarithmic dependence with the orders of VCF systems. Using IM: (1) estimates of the number of symmetric VCF systems in an unbounded range of changes of their parameters are obtained, (2) structures are determined and (3) rules of the interconnection of the systems established. The fundamental axioms and lemmas corresponding to the VCF systems formulating and directions for further research are outlined.