What is a Kalman filter?

The Kalman filter is an algorithm, named after one of the primary developers of its theory. Given some measured variables with a certain inaccuracy, this filter can be used to determine the state of technical, scientific or economic systems. 

In simple terms, the Kalman filter is used to estimate the system state by combining several error-prone  variables. Both the mathematical structure of the underlying dynamic system and the measurement errors must be known.

Application

The Kalman filter can be used to determine the position of a vehicle. For this purpose, a faulty GPS signal is available, which partly jumps around the actual position. Another possibility to determine the position is the integration of the driving course, i.e. speed and steering position. In the long run, however, even small errors are integrated to a wrong position.

The Kalman filter combines both signals here, so that the position cannot jump through the GPS, but in the long run still does not move away from the real position.

Properties

The estimation of the mean value depends in a linear way on the observation, the Kalman filter is therefore a linear filter. As the number of iterations increases, the estimates for mean and variance come closer to the actual values as desired.

It is therefore a so-called unbiased and consistent estimator with minimal variance. The filter is a linear optimal filter, since the estimation properties lead to a minimization of the mean square error.

Even generalized nonlinear filters often do not yield better results with normally distributed variables. Unlike other linear estimators, which also minimize error squares, the Kalman filter also allows problems with correlated noise components to be treated, which are often encountered in practice.

Application areas

A special feature of the Kalman filter is its special mathematical structure, which enables it to be used in real-time systems in various technical fields.

These include the evaluation of GPS or radar signals for tracking moving objects as in the application example, but also the use in electronic control loops of ubiquitous communication systems such as radio and computers.

The Kalman filter can also be used to reduce measurement noise.

In contrast to the classical FIR and IIR filters of signal and time series analysis, the Kalman filter is based on a state space modelling, where an explicit distinction is made between the dynamics of the system state and the process of its measurement.

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