Frequency estimation based on Hankel matrices and the alternating direction method of multipliers

TL;DR

This paper introduces a high-resolution frequency estimation method using Hankel matrices and the alternating direction method of multipliers, capable of handling missing data and achieving near-optimal accuracy.

Contribution

It presents a novel approach combining Hankel matrix rank constraints with ADMM for frequency estimation, improving robustness and performance over existing methods.

Findings

Achieves near Cramér-Rao bound accuracy

Handles missing data samples effectively

Simple and easy to implement algorithm

Abstract

We develop a parametric high-resolution method for the estimation of the frequency nodes of linear combinations of complex exponentials with exponential damping. We use Kronecker's theorem to formulate the associated nonlinear least squares problem as an optimization problem in the space of vectors generating Hankel matrices of fixed rank. Approximate solutions to this problem are obtained by using the alternating direction method of multipliers. Finally, we extract the frequency estimates from the con-eigenvectors of the solution Hankel matrix. The resulting algorithm is simple, easy to implement and can be applied to data with equally spaced samples with approximation weights, which for instance allows cases of missing data samples. By means of numerical simulations, we analyze and illustrate the excellent performance of the method, attaining the Cram\'er-Rao bound.