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The paper deals with the identification of a class of nonlinear filters recently introduced in the literature, the even mirror Fourier nonlinear filters, and shows that perfect periodic sequences can be developed for these filters. A periodic sequence is perfect for a nonlinear filter if all cross-correlations between two different basis functions, estimated over a period, are zero. By applying perfect periodic sequences as input signals to even mirror Fourier nonlinear filters, it is possible to model unknown nonlinear systems using the cross-correlation method. Moreover, the most relevant basis functions, i.e., those that guarantee the most compact representation of the nonlinear system according to some information criterion, can be easily estimated. Experimental results on the identification of real nonlinear systems illustrate the main advantage of the proposed method which is the remarkable reduction in the computational complexity.

This paper introduces computationally efficient NLMS and RLS adaptive algorithms for identifying non-recursive, linear-in-the-parameters (LIP) nonlinear systems using periodic input sequences. The algorithms presented in the paper are exact and require a real-time computational effort of a single multiplication, an addition and a subtraction per input sample. The transient, steady state, and tracking behavior of the algorithms as well as the effect of model mismatch is studied in the paper. The low computational complexity, good performance and broad applicability make the approach of this paper a valuable alternative to the current techniques for nonlinear system identification. Highlights? Adaptive algorithms for linear-in-the-parameters (LIP) nonlinear filters are discussed. ? NLMS and RLS algorithms for LIP filters using periodic sequences are derived. ? Computational cost is only three arithmetic operations per sample time. ? Transient, steady state, and tracking behavior of the algorithms is analyzed. ? Effects of a model mismatch between unknown and adaptive system are also analyzed.