Walsh-functions: Early Ideas on Their Application in Signal Processing and Communication Engineering

Spectral techniques based on sinusoidal functions have a long and successful tradition in Signal Processing and in Communication Engineering. A mathematical reason is the fact that they constitute the eigenfunctions of linear operators which model by means of ordinary linear differential equations important physical building blocks for components and systems which are of practical importance there. Examples are given by electrical cables, RLC-circuits of different kind and other electrical equipment. From a mathematical point of view the system of sinusoidal functions is a special case of a complete system of orthogonal functions. For this reason we find in the past in Signal Processing and Communications Engineering efforts, to make use of other specific systems of orthogonal functions which show in their application similar desirable properties as they exist for sinusoidal functions One such a system is given by the Walsh Functions (as they were first studied in 1923 by Joseph Walsh) and their different possible variations and generalisations.