Filtering of periodically correlated processes (Q2850862)

The authors deal with the problem of the mean square optimal linear estimation of the linear functional \(A{\zeta}=\int_0^{\infty}{a}(t){\zeta}(-t) dt\) which depends on the unknown values of a periodically correlated stochastic process \({\zeta}(t)\). The estimate is based on observations of the process with additive noise \({\zeta}(t)+\theta(t)\) at points of time \(t<0\). Formulas for calculating the mean square error and the spectral characteristic of the optimal linear estimate of the functional \(A{\zeta}\) are proposed in the case of spectral certainty, where the spectral densities are known exactly. The minimax (robust) method of estimation is used in the case where the spectral densities are not known exactly but a set of admissible spectral densities is given. Formulas that determine the least favourable spectral densities and the minimax spectral characteristics of the optimal linear estimate of \(A{\zeta}\) are proposed for some special sets of admissible densities. For more results and references, see [\textit{M. Moklyachuk} and \textit{O. Masyutka}, Minimax-robust estimation technique for stationary stochastic processes. Saarbrücken: LAP Lambert Academic Publishing (2012; Zbl 1289.62001); \textit{M. P. Moklyachuk}, Robust estimates for functionals of stochastic processes. Kyïv: Vydavnycho-Poligrafichnyĭ\ Tsentr, Kyïvskyĭ\ Universytet (2008; Zbl 1249.62007)].