Gauss-ramanujan functions: constructions, properties, and applications in communications and signal processing

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.