Quantitative Error Analyses of Spectral Density Estimators Using Covariance Lags

TL;DR

This paper provides a quantitative analysis of errors in spectral density estimation using covariance lags, accounting for noise and sample size, with bounds derived for univariate and multivariate cases.

Contribution

It introduces a convex optimization-based spectral density estimator and derives tight error bounds considering practical noise and sample limitations.

Findings

Error bounds for spectral density estimation are established.

Covariance lags can be exactly matched in the estimator.

Results are extended from univariate to multivariate spectral density estimation.

Abstract

Spectral density estimation is a core problem of system identification, which is an important research area of system control and signal processing. There have been numerous results on the design of spectral density estimators. However to our best knowledge, quantitative error analyses of the spectral density estimation have not been proposed yet. In real practice, there are two main factors which induce errors in the spectral density estimation, including the external additive noise and the limited number of samples. In this paper, which is a very preliminary version, we first consider a univariate spectral density estimator using covariance lags. The estimation task is performed by a convex optimization scheme, and the covariance lags of the estimated spectral density are exactly as desired, which makes it possible for quantitative error analyses such as to derive tight error upper bounds. We analyze the errors induced by the two factors and propose upper and lower bounds for the errors. Then the results of the univariate spectral estimator are generalized to the multivariate one.