An important objective of the classical processing of stationary
random sequences under nonparametric uncertainty is the problem of filtering
in case when the distribution of the underlying signal is unknown (see \cite{Dobrovidov:12}).
Let us have partly observable Markov random sequence $(S_n,X_n)_{n\geq1}$, where $S = (S_n)_{n\geq1}$ is unobservable and $X=(X_n)_{n\geq1}$ is observable.
The connection between them is given by the following expression
$X_n=\varphi(S_n,\eta_n)$, where $(\eta_n\in\mathbb{R})_{n\geq1}$ is an i.i.d random sequence and $\varphi$ is some function.
It is assumed that an unknown useful signal $(S_n)_{n\ge1}$ is Markovian. This allows
us to construct an estimate of the useful signal, expressed in terms
of the distribution density function of an observable random sequence $(X_n)_{n\ge1}$. The equation
of the optimal Bayesian estimation \cite{Stratonovich:66} (so called an optimal filtering equation) of such
signal has been received by \cite{Dobrovidov:1983}. Our main result is the following. It is proved that
when the unobservable Markov sequence is defined by a linear equation with the
Gaussian noise $\eta_n$, the equation of optimal filtering coincides both with the classical
Kalman's filter and the conditional expectation defined by the theorem on
normal correlation (see \cite{ShiryaevLiptser:2001}). As auxiliary result the explicit inversion of the covariance matrix is obtained.
The latter matrix is a Toeplitz matrix \citep{Trench:2001} with the shift on the diagonal elements.