Non-Stationary Stochastic Processes Estimation

10 Filtering of processes with periodically correlated increments

In this chapter, we deal with the filtering problem for stochastic processes with periodically correlated dth increments observed with a periodically stationary stochastic noise process.

By the filtering problem we understand the problem of the mean square optimal linear estimation of the functionals

$\begin{array}{c}\mathit{A}\mathit{\xi }=\underset{0}{\overset{\infty }{\int }}\mathit{a}\mathrm{(}\mathit{t}\mathrm{)}\mathit{\xi }\mathrm{(}-\mathit{t}\mathrm{)}\mathit{d}\mathit{t}\mathrm{,}{\mathit{A}}_{\mathit{N}\mathit{T}}\mathit{\xi }=\underset{0}{\overset{\mathrm{(}\mathit{N}+1\mathrm{)}\mathit{T}}{\int }}\mathit{a}\mathrm{(}\mathit{t}\mathrm{)}\mathit{\xi }\mathrm{(}-\mathit{t}\mathrm{)}\mathit{d}\mathit{t}\mathrm{,}\end{array}$

which depend on the unknown values of the stochastic process $\mathit{\xi }\mathrm{(}\mathit{t}\mathrm{)}$ with periodically correlated dth increments. Estimates are based on observations of the process $\mathit{\zeta }\mathrm{(}\mathit{t}\mathrm{)}=\mathit{\xi }\mathrm{(}\mathit{t}\mathrm{)}+\mathit{\eta }\mathrm{(}\mathit{t}\mathrm{)}$ at points $\mathit{t}\le 0$, where $\mathit{\eta }\mathrm{(}\mathit{t}\mathrm{)}$ is an uncorrelated with the process $\mathit{\xi }\mathrm{(}\mathit{t}\mathrm{)}$ periodically stationary stochastic process.

10.1 Hilbert space projection method of filtering ...