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.