A study about Chebyshev nonlinear filters

• We study a novel family of nonlinear filters: the Chebyshev nonlinear (CN) filters.
• CN filters are linear-in-the-parameters and have polynomial basis functions.
• They are universal approximators according to the Stone–Weierstrass theorem.
• They admit perfect periodic sequences (PPSs) simply related to those of EMFN filters.
• Using PPS inputs, CN filters can be identified with the cross-correlation method.

The paper studies a novel family of nonlinear filters based on Chebyshev polynomials of the first kind, the Chebyshev nonlinear filters. This family shares many of the characteristics of the recently introduced Legendre and even mirror Fourier nonlinear filters, but has also peculiar properties. Chebyshev nonlinear filters belong to the class of linear-in-the-parameters nonlinear filters. Their basis functions are polynomials, specifically, products of Chebyshev polynomial expansions of the input signal samples. According to the Stone–Weierstrass theorem, they are universal approximators for causal, time-invariant, finite-memory, continuous, nonlinear systems. Their basis functions are mutually orthogonal for white input signals with a particular nonuniform distribution. They admit perfect periodic sequences, i.e., periodic input sequences that guarantee the mutual orthogonality of the basis functions on a finite period. Using perfect periodic input signals, an unknown nonlinear system and its most relevant basis functions can be identified with the cross-correlation method. It is shown in the paper that the perfect periodic sequences of Chebyshev nonlinear filters are simply related to those of even mirror Fourier nonlinear systems. Experimental results involving a real nonlinear system illustrate the potentialities of these filters.