A new transform for solving the noisy complex exponentials approximation problem

TL;DR

This paper introduces a novel discrete transform and estimator for accurately approximating complex measures from noisy moments, improving solutions to the complex exponentials approximation problem.

Contribution

It proposes a new discrete transform and estimator specifically designed for noisy complex moments, enhancing the accuracy of complex measure approximation.

Findings

The new transform effectively estimates the unknown measure from noisy data.

Simulation results demonstrate the method's robustness and accuracy.

The approach provides a practical solution for noisy complex exponential problems.

Abstract

The problem of estimating a complex measure made up by a linear combination of Dirac distributions centered on points of the complex plane from a finite number of its complex moments affected by additive i.i.d. Gaussian noise is considered. A random measure is defined whose expectation approximates the unknown measure under suitable conditions. An estimator of the approximating measure is then proposed as well as a new discrete transform of the noisy moments that allows to compute an estimate of the unknown measure. A small simulation study is also performed to experimentally check the goodness of the approximations.