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The purpose of this dissertation is to demonstrate that lattice theoretic methods can be used to decompose and unify computational structures over a variety of processing systems. The unification arguments provide a better understanding of the intricacies of the development of processing system decomposition. Since abstract algebraic techniques are used, the decomposition process is systematized which makes it conducive to the use of computers as tools for decomposition. A general algorithm using the lattice theoretic method is developed to examine the structures and therefore decomposition properties of integer and polynomial rings. Two fundamental representations, the Sino-correspondence and the weighted radix representation, are derived for integer and polynomial structures and are shown to be a natural result of the decomposition process. They are used in developing systematic methods for decomposing discrete Fourier transforms and discrete linear systems. That is, fast Fourier transforms and partial fraction expansions of linear systems are a result of the natural representation derived using the lattice theoretic method. The discrete Fourier transform is derived from a lattice theoretic base demonstrating its independence of the continuous form and of the field over which it is computed. The same properties are demonstrated for error control codes based on polynomials. Partial fraction expansions are shown to be independent of the concept of a derivative for repeated roots and the field used to implement them. |
