最优滤波方程、卡尔曼滤波和正态相关定理
The equation of optimal filtering, Kalman's filter and Theorem on normal correlation
摘要：
非参数不确定性下平稳随机序列的经典处理的一个重要目标是在潜在信号的分布未知的情况下的滤波问题。本文假设未知有用信号$(S_n)_{n\ge1}$是马尔可夫的。这使我们能够构建有用信号的估计，用可观测随机序列$(X_n)_{n\ge1}$的分布密度函数表示。A.V. Dobrovidov给出了这种信号的最优贝叶斯估计方程（所谓的最优滤波方程）。我们的主要结果如下。证明了当不可观测的马尔可夫序列由一个线性方程定义时，其最优滤波方程与经典Kalman滤波器和由正态相关定理定义的条件期望是一致的。
英文摘要：
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.
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