Novel classes of linearly independent transforms for ternary logic functions

Abstract

Eight new classes of ternary linearly independent trans- forms being the bases of temary polynomial expansions over GF(3) have been introduced here. All transforms are classified according to their mathematical relation- ship and various properties of forward transform matrices with their respective inverse transform matrices. Compu- tational costs to calculate all eight classes of linearly inde- pendent transforms over GF(3) are also discussed. Abstract harmonic analysis has been a specialized area of signal processing and resulted in various transforms such as Walsh, Reed-Muller and their possible modifications that have many attractive features and are useful in many applications. these transforms form the foundation of a new theory of nonlinear signal and image processing (I) and can be also used in many other areas of applied math- ematics (2,3). The algebra of linearly independent transformations constructed on the basis of different two-valued Boolean functions has been developed in (8). Linearly independent logic bas proved to be not only of great theoretical value, but also of practical value, to the design of fine-grain and cellular automata types of Field Programmable Gate Ar- rays (FPGAs) and different Programmable Logic Devices (PLDs) with XOR gates. Another important development in further consideration of different linearly independent functions as the efficient tools in the designing of mod- em FPGAs and PLDs was the identification of those lin- early independent expansions for which there exist fast forward and inverse transformations for efficient calcula- tion of their spectral coefficients (5) and hence finding the final hardware implementations. In this article, we have developed new types of temary linearly independent transforms over GF(3) that are differ- ent than recently introduced multiple-valued transforms for ternary case (6) with some special properties of gen- erating of their matrices that may be used not only in logic synthesis and analysis of properties of temary func- tions but also for the development of new types of codes and matching and checking similarities between different temary functions. The computational costs of the eight new classes of transforms are also discussed.