Proceedings of International Conference on Applied Innovation in IT
2025/06/27, Volume 13, Issue 2, pp.331-338

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


Bahram Mohammadzadeh, Inas Hasan Abed and Saad Naji


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 of 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 in the context of an orthogonal multiplexing scheme, offering a practical solution for communication systems. A numerical example is provided to demonstrate the application of the proposed system as a communication carrier signal, specifically designed to mitigate multiple access interference in communication channels. The input signal is approximated using the piecewise constant functions, which are naturally computed through an -Fourier series expansion. After formally defining the -Fourier series, we detail the process of recovering the corresponding Fourier coefficients of the input signal, which are then transmitted through the designed multiplexing system for efficient and interference-free communication.

Keywords: Rescaling Cantor Set (R.C.S.), Adic-Type Cantor-Like Sets, Orthogonal Functions, Fourier Series, Hartley Transform.

DOI: 10.25673/120453

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