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Nonlinear Estimation in Continuous Time Systems.

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The nonlinear estimation of continuous time nonstationary signals contained in additive Gaussian white noise is considered in this study. The theory presented is more general than former studies and most previously known results are easily obtained as special cases, including the Kalman-Bucy theory and the Stratonovich-Kushner equations. A new approach to continuous time estimation is developed which is in the same spirit as the Bode-Shannon approach to Wiener filter theory. It is shown, for the first time, that nonstationary continuous time processes containing additive Gaussian white noise can be transformed causally into an innovation process, or equivalently, a Gaussian white noise. This innovation process contains all of the information of the original process and consequently nonlinear estimators can be designed to operate on the innovations rather than on the original observations. This approach leads to a number of new descriptions of nonlinear estimators the two most useful are a stochastic integral representation and an infinite orthogonal series representation. One of the important properties of the series description is that the series can be terminated after any specified number of terms, yielding a suboptimal nonlinear estimator and the remainder of the series can be summed and expressed in closed form. The innovation process approach is developed for nonstationary linear estimation as well as nonlinear estimation and a close correspondence between these two theories is demonstrated. Some new contributions to linear estimation theory are presented, including a proof of the causal invertibility of Kalman filters and a simple derivation of linear smoothing algorithms. Author

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