Orthogonal transforms in digital logic design

• Main Author: Lloyd, A. M.
• Published: University of Bath 1980
• Subjects: 510; Pure mathematics
• Online Access: https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.276312

In this thesis an investigation is carried out into the uses and applications of orthogonal matrices in digital logic design. All but the first chapter are primarily concerned with the application of the Rademacher-Walsh matrix in this area. In the first chapter, it is shown how other matrices can be formed from the Walsh-Hadamard matrix, and comparisons are made between the properties of these matrices. Particular reference is made to where these other matrices have their major advantages and setbacks in logic design. The second chapter shows how the summation of certain coefficients of a given function, taken from the Rademacher-Walsh and other such complete spectra, to a specified maximum value, directly corresponds to a realisation in terms of MD, OR and majority gates. By the consideration of only a few coefficients, it is possible to define large factors of a logic function. The third chapter is devoted to the consideration of how to deal with the spectrum once the procedure described in chapter two has been carried out. Two design algorithms are described; one recursive and the other non-recursive, both of which incorporate spectral addition techniques. In chapter four, two associated design techniques are proposed, which make more optimal use of the information that the spectrum offers. The first of these considers the incorporation of don't care terms in the function specification, and in so doing overcomes some of the disadvantages of the techniques described in chapter three. The second technique is concerned with the design of multiplexer universal-logic- module circuits. In the final chapter, it is shown how spectral techniques can be applied to the diagnosis of faults in two-level combinational logic circuits. Throughout this thesis, an emphasis is placed on the practical rather than mathematical implications of the techniques.