Nonlinear filters based on combinations of piecewise polynomials with compact support

When nonlinear filters are designed based on input-output observations, the descriptive nonlinear transformation between the input and the output signals is often unknown. Filters with generalized models are therefore required for a good characterization of these cases. In this paper, we introduce a class of filters obtained from the additive combination of multiple kernels. Each kernel constitutes a localized model for the desired nonlinear mapping and is defined as a multivariate continuous piecewise polynomial (CPP) function with compact support. By bringing together the efficiency of piecewise polynomial approximations and the flexibility of kernel combinations, we end up with models able to represent effectively a broad variety of nonlinearities. A localized threshold decomposition operator (LTD) is introduced to provide closed-form expressions for continuous nonsmooth kernels, whereas the method of splines is used to derive smooth functionals. Simple equalization problems involving nonlinear, nonminimum-phase, and non-Gaussian channels are used to examine the advantages of the proposed methods and compare them with other conventional alternatives.