Perfect periodic sequences for nonlinear Wiener filters

Abstract

A periodic sequence is defined as a perfect periodic sequence for a certain nonlinear filter if the cross-correlation between any two of the filter basis functions, estimated over a period, is zero. Using a perfect periodic sequence as input signal, an unknown nonlinear system can be efficiently identified with the cross-correlation method. Moreover, the basis functions that guarantee the most compact representation according to some information criterion can also be easily estimated. Perfect periodic sequences have already been developed for even mirror Fourier, Legendre and Chebyshev nonlinear filters. In this paper, we show they can be developed also for nonlinear Wiener filters. Their development is non-trivial and differs from that of the other nonlinear filters, since Wiener filters have orthogonal basis functions for white Gaussian input signals. Experimental results highlight the usefulness of the proposed perfect periodic sequences in comparison with the Gaussian input signals commonly used for Wiener filter identification

A study about Chebyshev nonlinear filters

“…In [32,[35][36][37], it has been proved that perfect periodic sequences (PPSs) can be developed for the identification of EMFN and LN filters. A periodic sequence is called perfect for a modeling filter if all cross-correlations between two of its basis functions, estimated over a period, are zero.…”

Multivariance nonlinear system identification using Wiener basis functions and perfect sequences

“…PPS are periodic sequences that guarantee the perfect orthogonality of the basis functions over a period and thus can accurately estimate the coefficients of the filter with the cross-correlation approach replacing the expectations in (4) with time averages over a period. A PPS x p (n) of period L suitable for the identification of the Wiener filters up to an order P and memory N and with Gaussian variance σ 2 x can be developed following the approach of [8]. The PPS x p (n) can be derived by imposing that all joint moments of the input signal, estimated over a period, involved in the identification of the WN filter, are equal to those of a white Gaussian signal x(n) ∈ N (0, σ 2 x ).…”

“…Therefore, an unknown system can be efficiently identified with the cross-correlation method using a PPS as input signal. PPSs suitable for the identification of WN filters have been recently proposed in [8] to reduce the accuracy problems in estimating the kernels diagonal points thanks to the perfect orthogonality of the Wiener basis functions for PPSs.…”

“…The Wiener basis functions [8] are a set of polynomial basis functions, which are orthogonal for white Gaussian noise inputs. They can arbitrarily well approximate any discrete time, time-invariant, finite memory, continuous, nonlinear system.…”

“…Perfect periodic sequences (PPSs) [9], [10] have been proposed for linear [11], [12] and nonlinear [8], [13]- [17] system identification as an alternative to stochastic inputs. A PPS ensures that the cross-correlation between any two different basis functions of the system, estimated over a period, is zero.…”