On Construction of Piecewise Constant Orthonormal Functions Based on Rescaling Cantor Set with Their Application in Orthogonal Multiplexing Systems

Abstract

The purpose of this paper is to construct a novel system of discontinuous piecewise constant orthogonal functions that is complete with respect to the measure on 4-adic-type Cantor-like sets, particularly on a rescaled Cantor set. The construction process is rigorously developed, and an accurate method for generating these functions is presented. This orthogonal function system is then applied within the framework of an orthogonal multiplexing scheme, providing a practical solution for communication systems. A numerical example illustrates the use of the proposed system as a communication carrier signal designed to reduce multiple access interference in communication channels. The input signal is approximated using these piecewise constant functions, which are naturally computed through a specific Fourier series expansion. Following a formal introduction of this Fourier series, the procedure for obtaining the corresponding Fourier coefficients of the input signal is detailed. These coefficients are then transmitted through the designed multiplexing system to enable efficient and interference-free communication.