Orthogonal Ramanujan Sums, its properties and Applications in Multiresolution Analysis

TL;DR

This paper introduces Orthogonal Ramanujan Sums (ORS), a new mathematical tool derived from Ramanujan sums, and demonstrates their application in signal representation and multiresolution analysis, including a connection to Haar transform.

Contribution

The paper proposes Orthogonal Ramanujan Sums (ORS) and introduces the Orthogonal Ramanujan Periodic Transform, extending Ramanujan sums for signal analysis and linking to Haar transform.

Findings

ORS provides a new signal representation method.

ORS-based multiresolution analysis generalizes Haar transform.

Haar transform is a special case of the proposed framework.

Abstract

Signal processing community has recently shown interest in Ramanujan sums which was defined by S.Ramanujan in 1918. In this paper we have proposed Orthog- onal Ramanujan Sums (ORS) based on Ramanujan sums. In this paper we present two novel application of ORS. Firstly a new representation of a finite length signal is given using ORS which is defined as Orthogonal Ramanujan Periodic Transform.Secondly ORS has been applied to multiresolution analysis and it is shown that Haar transform is a spe- cial case.