Hammerstein uniform cubic spline adaptive filters: Learning and convergence properties

•We propose a nonlinear filtering approach based on uniform spline nonlinear functions.•The proposed approach is able to solve the identification of Hammerstein nonlinear systems.•The proposed approach outperforms other approaches based on adaptive polynomial filters.•We derive an upper bound on the choice of the learning rate.•We derive a lower bound on the excess mean square error.

In this paper a novel class of nonlinear Hammerstein adaptive filters, consisting of a flexible memory-less function followed by a linear combiner, is presented. The nonlinear function involved in the adaptation process is based on a uniform cubic spline function that can be properly modified during learning. The spline control points are adaptively changed by using gradient-based techniques. This new kind of adaptive function is then applied to the input of a linear adaptive filter and it is used for the identification of Hammerstein-type nonlinear systems. In addition, we derive a simple form of the adaptation algorithm, an upper bound on the choice of the step-size and a lower bound on the excess mean square error in a theoretical manner. Some experimental results are also presented to demonstrate the effectiveness of the proposed method in the identification of high-order nonlinear systems.