On the Well-Posedness of a Parametric Spectral Estimation Problem and Its Numerical Solution

TL;DR

This paper demonstrates that a parametric formulation of a spectral estimation problem is well-posed, enabling continuous dependence on priors and facilitating a numerical solution via continuation methods with proven convergence.

Contribution

It establishes the well-posedness of a spectral estimation problem in a parametric setting and introduces a continuation-based numerical algorithm for its solution.

Findings

The problem is well-posed when formulated parametrically.

The solution parameter depends continuously on the prior function.

The proposed algorithm converges effectively in numerical tests.

Abstract

This paper concerns a spectral estimation problem in which we want to find a spectral density function that is consistent with estimated second-order statistics. It is an inverse problem admitting multiple solutions, and selection of a solution can be based on prior functions. We show that the problem is well-posed when formulated in a parametric fashion, and that the solution parameter depends continuously on the prior function. In this way, we are able to obtain a smooth parametrization of admissible spectral densities. Based on this result, the problem is reparametrized via a bijective change of variables out of a numerical consideration, and then a continuation method is used to compute the unique solution parameter. Numerical aspects such as convergence of the proposed algorithm and certain computational procedures are addressed. A simple example is provided to show the effectiveness of the algorithm.