Ramanujan sums for signal processing of low frequency noise

TL;DR

This paper introduces a novel signal processing method using Ramanujan sums for analyzing low frequency noise and arithmetical sequences, addressing limitations of traditional Fourier analysis in this regime.

Contribution

The paper develops a Ramanujan-Fourier transform (RFT) tailored for low frequency noise analysis, providing a new tool for studying arithmetical sequences and experimental signals.

Findings

RFT effectively captures low frequency resonances in signals.

The method reveals new properties of arithmetical functions.

Application to experimental data demonstrates improved analysis of low frequency noise.

Abstract

An aperiodic (low frequency) spectrum may originate from the error term in the mean value of an arithmetical function such as M\"obius function or Mangoldt function, which are coding sequences for prime numbers. In the discrete Fourier transform the analyzing wave is periodic and not well suited to represent the low frequency regime. In place we introduce a new signal processing tool based on the Ramanujan sums c_q(n), well adapted to the analysis of arithmetical sequences with many resonances p/q. The sums are quasi-periodic versus the time n of the resonance and aperiodic versus the order q of the resonance. New results arise from the use of this Ramanujan-Fourier transform (RFT) in the context of arithmetical and experimental signals