This thesis is concerned with the properties of a particular discrete transform, and its applications to the classification of multi-valued ("m-ary" ) logic functions and m-ary combinatorial logic analysis and synthesis. The transform used is composed of a complete set of orthogonal functions, namely Chrestenson Functions, and the methods developed are applicable for all m, m = 2, 3, ... . The definition of multi-valued systems and some examples of multivalued circuits are given in chapter 1. The necessity of a generalised design method which is not based on a particular algebra is considered and the scope of the thesis is stated. Chapter 2 introduces the algebraic notation, and continues to show the expansions of fully specified m-ary functions in (i) Lagrange form, (ii) generalised Reed-Muller form, and (iii) as polynomials' over the field of real numbers. Chapter 3 is an application of the mathematical developments covered in the previous chapter. Based on generalised Reed-Muller coefficients a realisation of m-ary functions using Universal-Logic-Modules is described. The realisation in this case is restricted to m being a power of a prime. The complex polynomial expansion of m-ary functions is considered in chapter 4. The coefficient set obtained is termed the "spectrum" of the given function. The effects of various operations in the function domain on the spectral values are investigated, and a classification of m-ary functions is described. Applications of spectral properties developed for m-ary combinatorial logic design are shown in examples. The implementation of any m-ary function involves some form of decomposition using physically available logic functions. The spectral properties developed in chapter 4 are further pursued in chapter 5 with an investigation into the relationships between the spectra of the logic functions involved in such a decomposition, and the spectrum of the overall function being realised. With the development of these spectral decomposition relationships, the range of tools for the spectral analysis of m-ary combinatorial logic is completed. Throughout this thesis emphasis is placed on the generality of techniques developed, such that these techniques may be applicable to whatever higher-valued logic microelectronic circuit realisations may evolve in the future.
|Date of Award||1980|