Data-driven Algorithms for signal processing with trigonometric rational functions

TL;DR

This paper introduces adaptive, noise-robust algorithms for reconstructing and processing periodic signals using trigonometric rational models, combining Prony's method and AAA algorithm, with a new MATLAB software implementation.

Contribution

It presents a novel framework for fitting trigonometric rational models to data that handles noise, missing data, and perturbations without manual tuning.

Findings

Algorithms effectively reconstruct signals from noisy and incomplete data.

The MATLAB system demonstrates practical utility in biomedical and acoustic applications.

Methods outperform traditional techniques in robustness and adaptability.

Abstract

Rational approximation schemes for reconstructing periodic signals from samples with poorly separated spectral content are described. These methods are automatic and adaptive, requiring no tuning or manual parameter selection. Collectively, they form a framework for fitting trigonometric rational models to data that is robust to various forms of corruption, including additive Gaussian noise, perturbed sampling grids, and missing data. Our approach combines a variant of Prony's method with a modified version of the AAA algorithm. Using representations in both frequency and time space, a collection of algorithms is described for adaptively computing with trigonometric rationals. This includes procedures for differentiation, filtering, convolution, and more. A new MATLAB software system based on these algorithms is introduced. Its effectiveness is illustrated with synthetic and practical examples drawn from applications including biomedical monitoring, acoustic denoising, and feature detection.