2850 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 48, NO. 10, OCTOBER 2000 Nonlinear Filters Based on Combinations of Piecewise Polynomials with Compact Support Edwin A. Heredia, Member, IEEE, and Gonzalo R. Arce, Fellow, IEEE Abstract—When nonlinear filters are designed based on input–output observations, the descriptive nonlinear transforma- tion 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 contin- uous 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. Index Terms—Approximation theory, channel equalization, non- linear filters, piecewise polynomial models, polynomial filters. I. INTRODUCTION I N OPTIMAL filter design, a pair of signals, which are often referred to as the input and desired signals, serves as the empirical data necessary for adjusting the filter parameters. For linear systems, the transformation relating the input and output signals is a linear combination and the parameters to be opti- mized are, in effect, the combination coefficients. The optimiza- tion, in this case, is based on minimizing a measure of the error between the desired output signal and the actual filter output. The least-squares (LS) error and the mean squared error (MSE) are typically used for this purpose. The theoretical and practical details of LS and MSE methods as well as the algorithms devel- oped for solving the optimization problem are well understood and have been extensively applied [14]. Optimal filter design using input–output observations is also possible in the nonlinear case. A nonlinear filter is described by a nonlinear input–output transformation and as in the linear case, the transformation parameters can be optimized following least-squares procedures [14], [18]. However, there are two is- sues that need to be examined. First, depending on the transfor- mation, solving for the optimal parameters may require the so- Manuscript received May 5, 1997; revised April 19, 2000. The associate ed- itor coordinating the review of this paper and approving it for publication was Dr. Ali H. Sayed. E. A. Heredia is with the Multimedia Technology Center, Samsung Elec- tronics, San Jose, CA 95134 USA (e-mail: EHeredia@sisa.samsung.com). G. R. Arce is with the Department of Electrical Engineering, University of Delaware, Newark, DE 19716 USA (e-mail: arce@ee.udel.edu). Publisher Item Identifier S 1053-587X(00)06681-2. lution of a nonlinear optimization problem. Second, the nature of the nonlinear transformation is not typically known a priori and therefore, flexible and efficient transformation models are required, in general, for the description of nonlinear systems. The solution to nonlinear optimization problems is often more difficult if not impossible [14]. Iterative search ap- proaches exist for this purpose, but depending on the problem, they typically exhibit slow convergence and with the risk of ending in local minima. It seems appropriate to avoid fully nonlinear least-squares optimizations unless, of course, all other options have been exhausted. Volterra filters are an example of nonlinear systems where the parameters are still obtained through linear search procedures [17], [18]. However, their fixed polynomial structure may be a limiting factor in several cases since it constrains the flexibility required for the input–output transformation model. A one-dimensional (1-D) nonrecursive nonlinear filter that relates the input signal with the output signal can be represented as (1) where window length; nonlinear transformation; random time series that represents modeling errors and noise. For Volterra filters, is a polynomial of a certain degree in variables. More flexible representations have been proposed for , including piecewise linear filters [7], [11], [16], piecewise Volterra filters [12], filters based on radial basis functions [4], [10], [13], filters based on neural networks [3], [10], and others. Volterra, piecewise linear, and piecewise Volterra filters pro- vide global mappings between and and, therefore, are re- ferred to as global models. The nodes of radial basis functions (RBF) and neural networks, on the other hand, are examples of local models. The approximation of in this case is achieved through multiple combinations of the nodes. In this paper, we use the term combination filters (CF) to describe those sys- tems that represent using an additive combination of local models. In general, a combination filter is represented as (2) where is the input signal window. are a set of multivariate functions referred to as the kernels, and are kernel locations. The pur- pose of each kernel is to describe the behavior of locally rather than globally, and it is through the multiplicity of kernels 1053–587X/00$10.00 © 2000 IEEE