The equation of optimal filtering, Kalman's filter and Theorem on normal correlation

TL;DR

This paper demonstrates that the optimal filtering equation for a Markov signal with Gaussian noise aligns with the classical Kalman's filter and the theorem on normal correlation, unifying different approaches under certain conditions.

Contribution

It proves the equivalence of Dobrovidov's optimal filtering equation with Kalman's filter and the normal correlation theorem for Gaussian Markov processes.

Findings

Optimal filtering equation coincides with Kalman's filter in Gaussian case.

Theorem on normal correlation provides the same conditional expectation.

Unifies different filtering approaches for Gaussian Markov signals.

Abstract

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. In this paper it is assumed that an unknown useful signal $(S_n)_{n\ge1}$ is Markov. 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 (so called equation of optimal filtering) of such signal has been received by A.V. Dobrovidov. 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, the equation of optimal filtering coincides with the classical Kalman's filter and the conditional expectation defined by the theorem on normal correlation.