Uncertainty Bounds for Spectral Estimation

TL;DR

This paper develops methods to quantify the uncertainty in spectral estimates derived from finite data, providing explicit bounds and metrics to assess the range of possible true power spectra, especially in the presence of spectral discontinuities.

Contribution

It introduces a framework for measuring spectral uncertainty using weak topology metrics and computes explicit bounds for high-resolution spectral estimation techniques.

Findings

Explicit uncertainty bounds for spectral estimates are derived.

Weak topology metrics effectively quantify spectral uncertainty.

Filter-bank pre-processing can be tuned using these bounds for improved resolution.

Abstract

The purpose of this paper is to study metrics suitable for assessing uncertainty of power spectra when these are based on finite second-order statistics. The family of power spectra which is consistent with a given range of values for the estimated statistics represents the uncertainty set about the "true" power spectrum. Our aim is to quantify the size of this uncertainty set using suitable notions of distance, and in particular, to compute the diameter of the set since this represents an upper bound on the distance between any choice of a nominal element in the set and the "true" power spectrum. Since the uncertainty set may contain power spectra with lines and discontinuities, it is natural to quantify distances in the weak topology---the topology defined by continuity of moments. We provide examples of such weakly-continuous metrics and focus on particular metrics for which we can explicitly quantify spectral uncertainty. We then consider certain high resolution techniques which utilize filter-banks for pre-processing, and compute worst-case a priori uncertainty bounds solely on the basis of the filter dynamics. This allows the a priori tuning of the filter-banks for improved resolution over selected frequency bands.