Systems of Discrete Walsh-Like Sequential Functions

Abstract

In this paper the algorithm for constructing the discrete (0,1)-sequent functions constituting the whole symmetric systems of orthogonal equidistant functions on the example of the eighth-order systems developed. Discrete sequential functions form by replacing their piecewise constant values +1 or -1 in the time domain (from the original space) with numerical values 0 and 1 in the image space. We refer to Walsh-like functions as (0,1)-sequent functions in which the number of zeros and ones in each half of the definition interval is not necessarily the same as in classical Walsh functions. By the directed search method, each of the 30 formed whole groups of equidistant sequent functions unfolds, like the group of classical Walsh functions of the eighth order, into 28 symmetric systems of sequent functions. The main result achieved in this work should consider an expansion of the set of Walsh-like systems of the eighth order by more than an order of magnitude. The algorithm's simplicity for synthesizing such systems of sequential functions and the high speed of spectral processing of discrete signals provided by the proposed bases open the Walsh-like systems for broad prospects of application in various fields of science and technology.