Recovery of rational functions via Hankel pencil method and sensitivities of the poles

TL;DR

This paper presents a novel method for recovering rational functions by exploiting the exponential structure of Fourier coefficients using Hankel matrix pencils and ESPRIT, including sensitivity analysis of pole recovery errors.

Contribution

It introduces a new approach combining Hankel pencil methods with sensitivity analysis for rational function recovery, enabling independent pole reconstruction inside and outside the unit circle.

Findings

Effective pole recovery using Hankel pencils and ESPRIT.

Derived formulas for pole sensitivities under perturbations.

Numerical experiments validate the method and sensitivity analysis.

Abstract

In this paper, we introduce a new approach for the recovery of rational functions. The concept we propose is based on using the exponential structure of the Fourier coefficients of rational functions and the reconstruction of this exponential structure in the frequency domain. We choose ESPRIT as a method for the exponential recovery. The matrix pencil structure of this approach is the reason for its selection, as it makes our method suitable for the sensitivity analysis. According to our method, poles located inside and outside the unit circle are reconstructed independently as eigenvalues of some special Hankel matrix pencils. Furthermore, we derived formulas for sensitivities of poles of rational functions in case of unstructured and structured perturbations. Finally, we consider several numerical experiments and, using sensitivities, explain the recovery errors for poles.