Gauss-Ramanujan Functions: Constructions, Properties, and Applications in Communications and Signal Processing

TL;DR

This paper introduces Gauss-Ramanujan functions constructed from Ramanujan sequences and Gaussian pulses, analyzing their properties and demonstrating their potential in advanced communication and signal processing applications.

Contribution

It presents a novel construction of Gauss-Ramanujan functions, analyzes their spectral and orthogonality properties, and explores their applications in new modulation schemes and wavelets for communication systems.

Findings

Exact and approximate overlap expressions for GP and DGP

Spectral and orthogonality properties of GauRam functions

Enhanced modulation schemes and wavelets for next-generation communication

Abstract

In this article, I construct a new set of functions based on Ramanujan sequences (RSEs), Gaussian pulse (GP), and its delayed Gaussian pulse (DGP). The motivation for this construction is based on the special properties of RSEs, GP, and DGP. First, I present a procedure for constructing Gauss-Ramanujan (GauRam) functions using selected RSEs. I develop an insightful analysis for deterministic and stochastic overlap between GP and DGP. Specifically, I present exact and closed form approximation expressions for delay-averaged GP and DGP overlap and then evaluate them numerically. Later, I derive and analyze the mathematical (spectral) properties of selected GauRam functions. I extend the analysis by analyzing the Hilbert transform of the first-order GauRam function and validating orthogonality and its usefulness in analytic signal representations. Furthermore, I present insightful applications of these functions in communications and signal processing. Specifically, I present the continuous-wave Gauss-Ramanujan modulation (GRM) scheme, Gauss-Ramanujan Shift Keying (GRSK) scheme, and Gauss-Ramanujan wavelets and their analysis and comparisons with benchmarking. The desirable properties of these novel modulation schemes and wavelets enable their use in next-generation hybrid and energy-efficient communication systems and signal processing.