Overview
ABSTRACT
In many technological fields (telecommunication, remote sensing, geolocalisation, industrial control, seismology), the useful information is not directly accessible as it is buried in the observed signal; this issue requires the development of hidden information methods. The Kalman filter, based upon a linear state model puts into equation the evolution of the useful signal and its relationship to the signal measured from a series of incomplete or noisy measurements. This article introduces elements of statistical estimations where the variable or the process to be estimated are hidden. It describes the dynamic state model, composed of the equation of the state process that the Kalman filter tries to estimate and the measuring process. The sequential estimation of the hidden process in order to deduce the Kalman filter is then presented.
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Yves DELIGNON: Professor at TELECOM/TELECOM Lille 1
INTRODUCTION
In many applications, such as telecommunications, remote sensing, geolocation, industrial control, seismology and biomedical engineering, the signal carrying the hidden information is not directly accessible and is embedded in the observed signal. The development of hidden information methods is therefore an important issue for various applications, and has given rise to a wealth of scientific literature in signal processing. In the case of a digital communication system, for example, the receiver receives a version of the transmitted signal degraded by noise and interference of various kinds (inter-symbol interference, co-channels or multiple access channels). The receiver, whose role is to extract the transmitted symbols from the received signal, is made up of successive processes, which together form an estimator. In another example, the tracking of a vehicle by a radar system is obtained by estimating the vehicle's speed and position from a radar signal.
The problem is how best to estimate the useful signal, based on observation and assumptions about the system. In 1949, Wiener was the first to propose a solution in the case where the signals involved are stationary. He developed a linear estimator minimizing the mean square error obtained by solving the Wiener-Hopf equation.
In 1960, Kalman proposed , an alternative to the Wiener filter that does away with the stationarity of the observed and hidden processes. The Kalman filter is based on a linear state model that equates the evolution of the useful signal and its relationship to the measured signal, and on an optimization criterion that exploits all observations, from the initial to the current instant. The filter obtained by Kalman is recursive: its response at each instant is in fact only a function of the signal observed at its input and its response at the previous iteration (figure 1 ). In this way, the Kalman filter does not require all past data to produce an estimate at the current instant. It therefore requires no data storage or reprocessing. This advantage makes it possible...
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