The Kalman filter in the context of adaptive filter theory

Model-based adaptive algorithms are usually derived with the help of the Wiener-Hopf equation based on empirical statistics. They are often interpreted as an extension to their model-independent counterparts, i.e. the stochastic-gradient based adaptive filters. As a consequence, it is generally not considered worthwhile to show the analogy between Kalman filters and adaptive filters. This article pursues just these two goals. First, it tries to remove the notion that the Kalman filter is a complicated and unnecessary detour from the subject of adaptive filtering. Second, the advantage of a deeper insight into adaptive algorithms from Kalman's viewpoint emerges from our treatment. Based on a time-varying FIR filter model, the Kalman filter is completely derived and serves as a general framework for the special case of model-based adaptive filters.