Mixed-Spectrum Signals -- Discrete Approximations and Variance Expressions for Covariance Estimates

TL;DR

This paper derives exact formulas for the variance of covariance estimates in mixed-spectrum signals, analyzing convergence, asymptotic behavior, and implications for signal approximation and direction of arrival estimation.

Contribution

It provides closed-form variance expressions for mixed-spectrum signals and explores their convergence and asymptotic properties, including non-ergodic cases and sinusoidal approximations.

Findings

Covariance estimates can converge for non-ergodic processes.

Asymptotic variance depends on time-frequency resolution.

Sinusoidal approximations exhibit distinct limiting behaviors.

Abstract

The estimation of the covariance function of a stochastic process, or signal, is of integral importance for a multitude of signal processing applications. In this work, we derive closed-form expressions for the variance of covariance estimates for mixed-spectrum signals, i.e., spectra containing both absolutely continuous and singular parts. The results cover both finite-sample and asymptotic regimes, allowing for assessing the exact speed of convergence of estimates to their expectations, as well as their limiting behavior. As is shown, such covariance estimates may converge even for non-ergodic processes. Furthermore, we consider approximating signals with arbitrary spectral densities by sequences of singular spectrum, i.e., sinusoidal, processes, and derive the limiting behavior of covariance estimates as both the sample size and the number of sinusoidal components tend to infinity. We show that the asymptotic regime variance can be described by a time-frequency resolution product, with dramatically different behavior depending on how the sinusoidal approximation is constructed. In a few numerical examples we illustrate the theory and the corresponding implications for direction of arrival estimation.