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Discrete Stochastic Processes and Optimal Filtering by Roger Ceschi, Jean-Claude Bertein

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7.1. Position of problem

The aim of the filtering that we are going to study consists of “best estimating”, in the sense of the classic criteria of least mean squares, a discrete process XK governed by an equation of the form:

images

This process (physical, biological, etc.) called the state process is what interests the user.

It represents for example the position, speed and acceleration of a moving object.

This process is inaccessible directly and it is studied by means of a process YK governed by an equation of the form:

images

YK is called the observation process.

NK and WK are the system noise and the measurement noise respectively and will be explained in more detail in what follows.

The Kalman filter, with its creation, brought into widespread use the optimal filter for non-stationary systems.

It is also recursive: the predicted images is obtained starting from the filtration at the preceding instant images and the filtration images from its predicted and from the measurement of the process YK+1 at the instant ...

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