The paper deals with the identification of nonlinear systems with adaptive filters. In particular, adaptive filters for functional link polynomial (FLiP) filters, a broad class of linear-in-the-parameters (LIP) nonlinear filters, are considered. FLiP filters include many popular LIP filters, as the Volterra filters, the Wiener nonlinear filters, and many others. Given the large number of coefficients of these filters modeling real systems, especially for high orders, the solution is often very sparse. Thus, an adaptive filter exploiting sparsity is considered, the improved proportionate NLMS algorithm (IPNLMS), and an optimal step-size is obtained for the filter. The optimal step-size alters the characteristics of the IPNLMS algorithm and provides a novel gradient descent adaptive filter. Simulation results involving the identification of a real nonlinear device illustrate the achievable performance in comparison with competing similar approaches.
