Practical nonlinear system analysis by Wiener kernel estimation in the frequency domain

Nonlinear systems which have finite memories and are time invariant can be completely described by the Wiener functional expansion, in which a series of multidimensional kernels provide a polynomial approximation to the nonlinear behaviour. The kernels give a best fitting estimation to the total system behaviour in the least mean square sense and can therefore be used to describe systems in which the nonlinearities include discontinuous functions. A modification of the Wiener method described by Lee and Schetzen, which uses kernels defined in terms of cross correlation functions, has been used in most practical attempts to analyse nonlinear systems, but we have previously described how the cross correlations may be replaced with complex multiplications in the frequency domain. The speed of domain translation offered by the fast Fourier transform makes this method more efficient than time domain estimation. In this paper the practical implementation of the technique on a medium sized digital computer is described for nonlinear systems whose outputs are continuous or pulsatile signals. This description should be adequate to allow others to implement the analysis scheme. The technique is well suited to the analysis of nonlinear biological systems, particularly those encountered in neurophysiology, because of its generality, ability to deal with hard nonlinearities and ease of use with systems having pulsatile outputs.