Conal Distances Between Rational Spectral Densities

TL;DR

This paper extends the Thompson and Hilbert metrics to rational spectral densities, creating a complete, filtering-invariant metric space with explicit geodesics, useful for engineering applications.

Contribution

It introduces a generalized metric framework for spectral densities that maintains rationality and can be efficiently computed, with explicit geodesic paths.

Findings

The metric space is complete and has a Finsler manifold structure.

Distances are filtering invariant and computationally efficient.

Geodesic paths preserve rational spectral densities.

Abstract

The paper generalizes Thompson and Hilbert metric to the space of spectral densities. The resulting complete metric space has the differentiable structure of a Finsler manifold with explicit geodesics. The resulting distances are filtering invariant, can be computed efficiently, and admit geodesic paths that preserve rationality; these are properties of fundamental importance in many engineering applications.