Filtering of periodically correlated processes
Iryna Dubovets'ka, Mykhailo Moklyachuk
TL;DR
This work tackles the problem of optimally estimating a linear functional $A\zeta=\int_{0}^{\infty} a(t)\zeta(-t)\,dt$ of a mean-square continuous periodically correlated process from noisy observations $\zeta(t)+\theta(t)$, where $\theta$ is PC and uncorrelated with $\zeta$. It first derives explicit formulas for the spectral characteristic $h(e^{i\lambda})$ and mean-square error when spectral densities $f(\lambda)$ and $g(\lambda)$ are exactly known, using canonical factorizations. The paper then extends to a minimax (robust) framework for uncertain spectral data, defining least favorable densities $(f^0,g^0)$ and minimax spectral characteristics $h^0$, and solving associated conditional extremum problems, particularly for the class $D_{0,0}$ with moment constraints. The results provide a rigorous, implementable approach for robust estimation of linear functionals of PC processes, including procedures to identify least favorable densities and compute 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.