Orthogonal LIP Nonlinear Filters

This chapter provides an overview of orthogonal linear-in-the-parameter (LIP) nonlinear filters defined in the real domain. After reviewing the classical theory of Volterra and Wiener filters, it presents different classes of real LIP nonlinear filters, whose basis functions are orthogonal for specific distributions of the input signal, under the unified framework of functional link polynomial (FLiP) filters. The class of FLiP filters includes non orthogonal and orthogonal nonlinear filters, as the Wiener nonlinear filters, the even mirror Fourier nonlinear filters, the Legendre and Chebyshev nonlinear filters. Under the same framework, many other families of orthogonal LIP nonlinear filters could be defined. All FLiP filters can arbitrarily well approximate any causal, time-invariant, finite-memory, continuous, nonlinear system according to the Stone-Weierstrass theorem. The orthogonality of the basis functions allows fast convergence of gradient descent adaptation algorithms and efficient identification of the nonlinear systems using the cross-correlation method, i.e., simply computing the cross-correlation between the output of the system and the basis functions. Moreover, perfect periodic sequences (PPSs), which are deterministic periodic sequences that guarantee the orthogonality of the basis functions over a period, can also be easily developed. Recent identification techniques for FLiP and LIP nonlinear filters are also reviewed within the chapter. In particular, we consider the identification of FLiP filters using PPSs and the multiple-variance identification method, which allows to contrast the fact that the estimated model approximates the unknown system well only at the same input variance of the measurement. Experimental results discussing the identification of real nonlinear devices illustrate the advantages provided by orthogonal LIP filters and by the novel identification techniques in applications.