Kalman Filters

Abstract

This chapter describes the Kalman filters that provide an optimal estimate of hidden signals through a linear combination of previous estimates of these signals and with the newest available measurement signals. The Kalman filters can be considered an extension of the Wiener filtering concept [1, 2], in the sense that it allows for an estimate of non-directly measurable state variables of dynamic systems. The Kalman filter has as objective the minimization of the estimation square errors of nonstationary signals buried in noise. The estimated signals themselves are modeled utilizing the so-called state–space formulation [3] describing their dynamical behavior. While the Wiener filter provides the minimum MSE solution for the hidden parameters, leading to the optimal solution for an environment with wide-sense stationary signals, the Kalman filter offers a minimum MSE solution for time-varying environments involving linear dynamic systems whose noise processes involved are additive Gaussian noises. In the latter case of Kalman filters, the parameters of the dynamic systems can be time-varying.