On a Property of Phase Correlation and Possibilities to Reduce the Walsh Function System

Abstract

Data processing algorithms are used for business, industry and public sector to filter input data, calculate values, detect abrupt changes, acquire information from data or to ensure signal consistency. It is an important research area for Big Data processing and processing of data received from Internet of Things. Typically, classical algorithms are exploited, i.e. statistical procedures, data mining techniques and computational intelligence algorithms. Referring to the area of signal processing, applications of mathematical transformation (e.g. Fourier Transform, Walsh–Fourier Transform) of input signals from either domain to the other are promising. They enable to perform complementary analyses and to consider additional signal components, in particular cyclic (periodic) ones (sin- and cos-components). The Walsh function system is a multiplicative group of Rademacher and Gray functions. In its structure, it contains discrete-harmonic, sin-components of the Rademacher functions, and cos-components of the Gray function, as well as discrete-irregular components of the Walsh function. In the paper, the phase interdependence property has been defined, in pairs of a complete Walsh function system. Odd (sin-components) and even (cos-components) Walsh function subsystems were extracted as theoretical and numerical processing databases. A perspective concerning the processing efficiency and digital signal processing is outlined.