SUMMARYThe paper presents a mathematical theory of handling and working with wideband noises. We demonstrate that a wideband noise can be represented as a distributed delay of a white noise. From this, we deduce that the behavior of a wideband noise is the same as the behavior of an infinite dimensional colored noise along the boundary line. All these are used to deduce a complete set of formulae for the Kalman-type optimal filter and also to derive nonlinear filtering equation for wideband-noise-driven linear and nonlinear systems.
The classic Kalman filtering equations for independent and correlated white noises are ordinary differential equations deterministic or stochastic with the respective initial conditions. Changing the noise processes by taking them to be more realistic wide band noises or delayed white noises creates challenging partial differential equations with initial and boundary conditions. In this paper, we are aimed to give a survey of this connection between Kalman filtering and boundary value problems, bringing them into the attention of mathematicians as well as engineers dealing with Kalman filtering and boundary value problems.
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