Distances between power spectral densities

TL;DR

This paper introduces new distance measures between spectral density functions of random processes, motivated by filtering problems, and explores their mathematical properties and implications for spectral manifold geometry.

Contribution

It proposes novel spectral distance measures based on predictor performance degradation and smoothing, and characterizes their geometric structure and geodesics.

Findings

Defined a spectral distance based on predictor error ratios

Characterized geodesics on the spectral density manifold

Computed explicit distances for specific spectral density functions

Abstract

We present several natural notions of distance between spectral density functions of (discrete-time) random processes. They are motivated by certain filtering problems. First we quantify the degradation of performance of a predictor which is designed for a particular spectral density function and then it is used to predict the values of a random process having a different spectral density. The logarithm of the ratio between the variance of the error, over the corresponding minimal (optimal) variance, produces a measure of distance between the two power spectra with several desirable properties. Analogous quantities based on smoothing problems produce alternative distances and suggest a class of measures based on fractions of generalized means of ratios of power spectral densities. These distance measures endow the manifold of spectral density functions with a (pseudo) Riemannian metric. We pursue one of the possible options for a distance measure, characterize the relevant geodesics, and compute corresponding distances.