Spectral Logic and Its Applications for the Design of Digital Devices

PREFACE

Spectral logic is a mathematical discipline in the area of abstract harmonic analysis devoted to applications in engineering, primarily electrical and computer engineering.

Abstract harmonic analysis has evolved from classical Fourier analysis by replacing the real line R, which is a particular locally compact Abelian group, by an arbitrary locally compact Abelian or compact non-Abelian group. The exponential functions, which are group characters of R, are replaced by group characters of Abelian groups and group representations for non-Abelian groups.

Spectral techniques mainly deal with signals of compact groups and in most cases finite groups. In this way, when using transforms defined in terms of group characters, the origins of spectral techniques can be found in classical Fourier analysis (181). Switching (Boolean) functions are an example of functions defined on a particular finite group that is the group of binary-valued n-tuples under the componentwise addition modulo 2 (EXOR). This group is known as a finite dyadic group.

For functions of this group, the Fourier transform is defined in terms of the discrete Walsh functions, which are characters of the finite dyadic groups. The origins of spectral techniques in terms of the discrete Walsh functions can be found in the theory of Walsh analysis, which is defined in terms of continuous Walsh functions introduced in 1923 (638) and interpreted as group characters of the infinite dyadic group in 1949 (176). Initially, ...