Rational approximations of spectral densities based on the Alpha divergence

TL;DR

This paper introduces a method for approximating spectral densities using Alpha divergence, resulting in a family of rational solutions that generalize previous approaches and include solutions close to non-rational ones.

Contribution

It develops a new approximation framework based on Alpha divergence, extending prior work and providing a family of rational spectral density solutions.

Findings

The approach yields a family of rational solutions.

It generalizes the Kullback-Leibler based solution.

Numerical results show solutions close to non-rational densities.

Abstract

We approximate a given rational spectral density by one that is consistent with prescribed second-order statistics. Such an approximation is obtained by minimizing a suitable distance from the given spectrum and under the constraints corresponding to imposing the given second-order statistics. Here, we consider the Alpha divergence family as a distance measure. We show that the corresponding approximation problem leads to a family of rational solutions. Secondly, such a family contains the solution which generalizes the Kullback-Leibler solution proposed by Georgiou and Lindquist in 2003. Finally, numerical simulations suggest that this family contains solutions close to the non-rational solution given by the principle of minimum discrimination information.