Legendre nonlinear filters

A study about Chebyshev nonlinear filters

“…Let us assume that the nonlinear system is identified with a PPS of order K and memory N. As discussed in [32,36], when the nonlinear system in (8) is a linear combination of basis functions with memory N and maximum order greater than K, the identification is affected by an error that influences mainly the coefficients of the higher-order basis functions, while, in general, has only a marginal effect on the coefficients of the lower-order basis functions. When the system to be identified is a linear combination of basis functions with order K but memory greater than N, the identification is also affected by an error, which influences mainly the coefficients of basis functions associated with the most recent samples x(n), xðn À1Þ; …, while, in general, the coefficients of basis functions associated with less recent samples xðn À N þ 1Þ, xðn ÀN þ 2Þ; … are only marginally affected.…”

“…Recently, the finite-memory LIP class has been enriched with novel sub-classes of nonlinear filters that guarantee the orthogonality of the basis functions for white uniform input signals in the range ½À1; þ1: the Fourier nonlinear (FN) filters [28,29], the even mirror Fourier nonlinear (EMFN) filters [29,30], and the Legendre nonlinear (LN) filters [31,32]. FN and EMFN filters are based on trigonometric function expansions of the input signal samples, and do not include a linear term among the basis functions.…”

“…While [29] suggested that EMFN filters should be preferred to FN filters, because they often provide a much more compact representation of nonlinear systems, it was found in [34] that also FN filters can be very useful for modeling nonlinear systems. It has been shown that EMFN filters can also be better models than LN and Volterra filters in the presence of strong nonlinearities, while Volterra and LN filters provide better results for weak or medium nonlinearities [29,32].…”

“…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.…”

A reduced-rank approach for implementing higher-order Volterra filters

“…To meet this challenge, one important filter characteristic that needs to be considered is the trade-off between implementation complexity and approximation capability. The well-known Volterra filter [1] represents one extreme of this trade-off, since its universal approximation capability [2][3][4] comes at the cost of a high computational complexity (which is due to the large number of coefficients required for the implementation) [1,[5][6][7][8][9]. In this context, one topic that has drawn attention from researchers in the last decades is the development of Volterra implementations having an enhanced tradeoff between computational complexity and approximation capability.…”