The Meaning of Distances in Spectral Analysis

Abstract

The analysis of signals into constituent harmonics and the estimation of their power distribution are considered fundamental to systems engineering. Due to its significance in modeling and identification, spectral analysis is in fact a "hidden technology" in a wide range of application areas, and a variety of sensor technologies, ranging from radar to medical imaging, rely critically upon efficient ways to estimate the power distribution from recorded signals. Robustness and accuracy are of at most importance, yet there is no universal agreement on how these are to be quantified. Thus, in this talk, we will motivate the need for ways to compare power spectral distributions. 

Metrics, in any field of scientific endeavor, must relate to physically meaningful properties of the objects under consideration. In this spirit, we will discuss certain natural notions of distance between power spectral densities. These will be motivated by problems in prediction theory and related properties of time-series. Analogies will be drawn with an old subject of a similar vein, that of quantifying distances between probability distributions, which has given rise to information geometry. The contrast and similarities between metrics will be highlighted by analyzing mechanical vibrations, speech, and visual tracking.