Non-Stationary Stochastic Processes Estimation

11 Filtering problem when signal and noise have periodically correlated increments

In this chapter, we present the results of our investigation of the filtering problem in the case where both the signal and the noise processes are processes with periodically stationary dth increments. Recall that solution of the filtering problem with periodically stationary noise is described in Chapter 10. In the present chapter we will show that additional conditions are required for the function $\mathit{a}\mathrm{(}\mathit{t}\mathrm{)}$, $\mathit{t}\ge 0$, determining the functional $\mathit{A}\mathit{\xi }$. In particular, the filtering problem for the functional ${\mathit{A}}_{\mathit{N}\mathit{T}}\mathit{\xi }$ defined on the bounded interval $[0\mathrm{,}\mathrm{(}\mathit{N}+1\mathrm{)}\mathit{T}]$ cannot be solved with the proposed approach in the general case.

11.1 Hilbert space projection method of filtering ...