10.4.1 Introduction

A Kalman filter may be considered as a generalisation of the Wiener filter in two directions. Primarily, it does not require stationarity of the involved processes and therefore it can also be used to estimate nonstationary signals. Besides that does not a priori limit the information used for designing the filter to the last N samples as was the case with the Wiener Levinson filter, but it is capable of utilising all the available empirical information which the observed signal offers from the beginning of measurement or reception. The amount of information increases with every further sampling period so that, particularly in the special case of stationary signals (or signals with properties which change slowly in time), the estimate improves in the course of time. The structure of a Kalman filter is recursive and its coefficients are modified, based on the available information, in every sampling period in order to provide an optimal estimate of the original signal. This recursivity also means that the filter can have (and usually does have) an infinite impulse response so that it is more general than the Wiener Levinson filter.

On the other hand, the Kalman filter is restricted with respect to the Wiener filter in that it introduces an a priori model of the observed signal (based on Markov chains); the class of signal which can be restored is thus reduced. We shall see that, on the other hand, the order of the introduced model can...