Delay structure of wideband noises with application to filtering problems

Abstract

Filtering of random processes is one of the central subjects in the theory of random processes. Especially, Kalman filtering, originating from the famous works of Kalman and Bucy [1, 2], is one of the powerful methods of estimation and widely used in applications. The overwhelming majority of the results in filtering theory for linear and nonlinear systems has been obtained for a pair of independent or correlated white noises corrupting the state and observation systems in finite [3, 4] and infinite [5,6] dimensional spaces. However, real noises are only approximations to white noises. Fleming and Rishel [7] wrote that the real noises are wideband and white noises are an ideal case of wideband noises. Whenever the parameters of white and wideband noises are sufficiently close to each other, the wideband noise can be replaced by the white noise to make the respective mathematical model simpler. Therefore, in order to get a more adequate version of the filtering equations, a method of handling and working with wideband noises must be developed. In this paper, we aim to give a mathematical background for the wideband noises. We show that under general conditions a wideband noise can be modeled as a distributed delay of a white noise, demonstrating an important relationship between practical wideband noises and ideal white noises. We show that an equation, corrupted by a wideband noise, is indeed an infinite dimensional equation corrupted by a white noise. From this, we modify the Kalman filter and the nonlinear filtering equation for wideband-noise-driven systems.