Logical manipulations and design of tributary networks in the arithmetic spectral domain

Abstract

Formulae to represent composite arithmetic spectra of switching functions for basic logic connectives of such functions are shown. In contrast to the Walsh spectral domain, no complex dyadic convolution is involved in the calculation of composite arithmetic spectra, and the reintroduction of the transformation matrix has been avoided in the final formulae. Other important operations used in classification and optimisation of standard and tributary logical network have also been analysed in the arithmetic spectral domain. These operations include spectral decomposition, input and output negations, permutations of input variables, substitution of an input variable by a logical operation with some input variables or by the output of the function and the variable itself. Based on the introduced formulae, a new method to design tributary networks through operations on arithmetic spectra is shown.