11.1 Concept of Adaptive Filtering

In the previous chapter, when designing restoration filters which should provide optimal estimates of original signals based on their observed noisy and distorted versions, we used explicit information obtained by a priori identification of signal-source properties. Apart from identifying the deterministic distortion (e.g. by the relevant impulse response), it was about measuring or estimating probability or statistical characteristics of related stochastic processes, such as correlation functions or matrices. These can reasonably be provided in advance in many practically important cases, namely when modelled by stationary processes. Alternatively, the stationary signal-source parameters could also be estimated from the signal itself providing that a suitable signal-generation model has been introduced. Consequently, it is possible to design an optimal or suboptimal restoration system; in the case of stationary processes, the system is, or aims at, a time-invariant filter, such as the classical Wiener filter.

Nevertheless, if the filter is to work in an unknown environment (either because the identification is impossible or the environment is time varying in an unpredictable way), it must be capable of adapting to such a situation. We shall therefore deal in this chapter with adaptive filters that are able to learn from a given environment, i.e. they are capable of providing the necessary information estimates of the needed quantities in the course of their work, with-out any a priori information. It is then possible to expect that such filters will be able to react (with a certain delay)...