Filtering of periodically correlated processes

TL;DR

This paper develops formulas for optimal linear estimation of periodically correlated processes with noise, including robust minimax spectral characteristics, enhancing estimation accuracy under spectral uncertainty.

Contribution

It introduces explicit formulas for optimal estimation, mean square error, and robust spectral characteristics for periodically correlated processes with known or uncertain spectral densities.

Findings

Formulas for optimal linear estimates and mean square errors.

Explicit solutions for least favorable spectral densities.

Development of minimax robust spectral characteristics.

Abstract

The problem of optimal linear estimation of a linear functional depending on the unknown values of periodically correlated stochastic process from observations of the process with additive noise is considered. Formulas for calculating the mean square error and the spectral characteristic of the optimal linear estimate of the functional are proposed in the case where spectral densities are exactly known. Formulas that determine the least favorable spectral densities and the minimax (robust) spectral characteristics are proposed for a given class of admissible spectral densities.