What is a Kalman filter?
The Kalman filter is a filter named after its discoverer. Set of mathematical equations. This filter can be used to draw conclusions about the state of many technical, scientific or economic systems in the presence of faulty observations.
Put simply, the Kalman filter for estimating the system state by combining several error-prone measured variables. Both the mathematical structure of the underlying dynamic system and the measurement distortions must be known.
An application example
The Kalman filter can be used to determine the position of a vehicle. On the one hand, an error-prone GPS signal is available for this purpose, which sometimes jumps around the actual position. Another way to determine the position is to integrate the driving history, i.e. speed and steering position. In the long term, however, even small errors are integrated to give an incorrect position.
The Kalman filter combines both signals here so that the position can no longer jump due to the GPS, but still does not move away from the real position in the long term.
Properties
The estimate of the mean value depends linearly on the observation; the Kalman filter is therefore a linear filter. As the number of iterations increases, the estimates for the mean and variance approach the actual values with arbitrary precision.
It is therefore a so-called consistent estimators that are true to expectations with minimum variance. The filter is a linear optimal filter, as the estimation properties lead to a minimization of the mean square error.
Even generalized non-linear filters often do not provide better results with normally distributed variables. In contrast to other linear estimators, which also minimize squared errors, the Kalman filter also allows the Treatment of problems with correlated noise componentsas they are frequently encountered in practice.
Areas of application
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 areas.
These include the evaluation of GPS or radar signals to track the position of moving objects, as in the application example, but also the use in electronic control circuits of ubiquitous communication systems such as radio and computers.
The Kalman filter can also be used for Reduction of measurement noise can be used.
In contrast to the classic FIR and IIR filters, which are based on signal and time series analysis, the Kalman filter is based on a State space modelingwhere an explicit distinction is made between the dynamics of the system state and the process of measuring it.
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