Signal analysis, moment problems & uncertainty measures

Modern spectral estimation techniques often rely on second order statistics of a time-series to determine a power spectrum consistent with data. Such statistics provide moment constraints on the power spectrum. In this paper we study possible distance functions between spectra which permit a reasonable quantitative description of the uncertainty in moment problems. Typically, there is an infinite family of spectra consistent with given moments. A distance function between power spectra should permit estimating the diameter of the uncertainty family, a diameter which shrinks as new data accumulates. Abstract properties of such distance functions are discussed and certain specific options are put forth. These distance functions permit alternative descriptions of uncertainty in moment problems. While the paper focuses on the role of such measures in signal analysis, moment problems are ubiquitous in science and engineering, and the conclusions drawn herein are relevant over a wider spectrum of problems.