Iterative execution of discrete and inverse discrete Fourier transforms with applications for signal denoising via sparsification

TL;DR

This paper introduces iterative algorithms involving Fourier transforms and sparsification for signal denoising, demonstrating convergence and effectiveness through simulations.

Contribution

It presents a novel iterative sparsification-based Fourier transform method with proven convergence and practical utility for denoising signals with Gaussian noise.

Findings

Convergence achieved when real domain sparsity stabilizes.

Effective recovery of periodic spike signals in noisy environments.

Outperforms some existing denoising methods in simulations.

Abstract

We describe a family of iterative algorithms that involve the repeated execution of discrete and inverse discrete Fourier transforms. One interesting member of this family is motivated by the discrete Fourier transform uncertainty principle and involves the application of a sparsification operation to both the real domain and frequency domain data with convergence obtained when real domain sparsity hits a stable pattern. This sparsification variant has practical utility for signal denoising, in particular the recovery of a periodic spike signal in the presence of Gaussian noise. General convergence properties and denoising performance relative to existing methods are demonstrated using simulation studies. An R package implementing this technique and related resources can be found at https://hrfrost.host.dartmouth.edu/IterativeFT.