Nonlinear system identification and prediction using orthogonal functions

We describe a systematic scheme for the nonlinear adaptive filtering of signals that are generated by nonlinear dynamical systems. The complete filter consists of three sections: a signal-independent standard orthonormal expansion, a scaling derived from an estimate of the vector probability density function (PDF), and an adaptive linear combiner. The orthonormal property of the expansions has two significant implications for adaptive filtering: first, model order reduction is trivial since the contribution of each term to the mean squared error is directly related to the coefficient in the final linear combiner; and second, consistent and rapid convergence of stochastic gradient algorithms is assured. A technique based on the inverse Fourier transform for obtaining a PDF estimate from the characteristic function is also presented. The prediction and identification performance of this nonlinear structure is examined for a number of signals, and it is contrasted with common radial basis function and linear networks.