Recursive Algorithms for Adaptive Transversal Filters: Optimality and Time-Variance

This thesis presents a unified theory for the design and analysis of recursive algorithms for the adaptation of transversal digital filters. First, the widely used error minimization approach to algorithm design is investigated and it is shown not to allow a coherent derivation of practical algorithms from an optimality criterion. The reason is found in the incompatibility of the assumption of a time-invariant application environment for the optimality definition and of the practical demand on the adaptive filter for tracking in time-varying environments. The present proposal for a deterministic approach to algorithm design goes beyond mere error minimization in that the time variation of the coefficients of the adaptive filter is included as well. In the sequel a wealth of algorithms is shown to fulfil this novel unified description and several algorithm modifications, which often appear ad hoc, are derived without invoking approximations. This covers also the utilization of error filters which is common practice in the adaptation of recursive filter structures. The second part of the thesis is devoted to the application of the described algorithm class to the tracking of time-varying environments. As a result, the tracking behaviour can be described as a filtering operation on the time evolution of the coefficients of a reference model which is imposed through the environment but not directly observable. Due to the general structure of the adaptation algorithms, this learning filter is always linear and of first order. To facilitate the incorporation of prior knowledge about the expected time evolution, the algorithm structure needs to be extended with so-called coefficient filters. This allows to tailor the tracking behaviour in response to practical demands as linear higher-order filtering or nonlinear filtering. In conclusion, a series of topical proposals for such coefficient filters is discussed on the basis of the accordingly extended unified optimality criterion.