Optimal neural network realizations of nonlinear FIR and IIR filters

A rigorous framework is presented for the design of nonlinear digital and analog filters. The approach followed is based on a Generalized Fock (GF) space framework developed by the Principal Investigator and T.A.W. Dwyer, III. A GF space is a reproducing kernel Hilbert space of discrete or continuous Volterra series with a problem-dependent weighted inner product. The optimal nonlinear filter structure is obtained by an orthogonal projection of the desired filter into the subspace spanned by the representers of interpolating, smoothing, and other design constraint functionals in the appropriate GF space. One of the attractive features of this approach is that the solutions to the filter design problem appear naturally as feedforward (FIR) or recurrent (IIR) artificial neural networks. These results are derived for a GF space F(E^N) on a finite-dimensional Euclidian space E^N. Generalization to functional FIR and IIR nonlinear filters follows immediately from replacing F(E^N) by F(L₂(I)), where L₂(I) is the space of square integrable functions on an interval I of the real line