Orthonormal functions for nonlinear signal processing and adaptive filtering

A systematic approach to constructing a nonlinear adaptive filter is presented. The approach is based on a signal dependent orthonormal expansion implemented in two stages: (i) a signal independent standard orthonormal expansion; (ii) scaling using an estimate of the vector probability density function (pdf). Further it is demonstrated that the standard orthonormal function set can also provide an estimate of the pdf when used in conjunction with an inverse Fourier transform. The orthonormality has two implications for adaptive filtering: (i) model order reduction is trivial because the size of a coefficient in the final linear combiner is directly related to its contribution to the overall mean squared error; (ii) consistent, rapid convergence of stochastic gradient algorithms is assured. A typical nonlinear adaptive algorithm is presented