Fourier nonlinear filters

• Two sub-classes of linear-in-the-parameters nonlinear filters, called Fourier nonlinear (FN) filters and even mirror Fourier nonlinear (EMFN) filters, are presented and discussed.
• The filters derive from the truncation of multidimensional generalized Fourier series.
• Like the Volterra filters, FN and EMFN filters are universal approximators for causal, time-invariant, finite memory continuous nonlinear systems.
• Differently from Volterra filters, FN and EMFN filters satisfy an orthogonality property for white uniform input signals.
• The orthogonality property guarantees fast convergence of gradient descent adaptation algorithms and efficient identification methods for nonlinear systems.

In this paper, two new sub-classes of linear-in-the-parameters nonlinear discrete-time filters, derived from the truncation of multidimensional generalized Fourier series, are presented. The filters, called Fourier nonlinear filters and even mirror Fourier nonlinear filters, are universal approximators for causal, time-invariant, finite-memory, continuous nonlinear systems, according to the Stone–Weierstrass approximation theorem. Their properties and limitations are discussed in detail. In particular, we show, by means of appropriate simulation examples, that an orthogonality property they satisfy for white uniform input signals is useful for improving the identification of nonlinear systems.