The terms, y 9xx(t1 - t2) and 9xxyi(Ti, t2) are zero for t1 | t2. The functions, h2(Ti, t2) and 9xxy(Ti, t2) can be visualized as three-dimensional surfaces over a t1, t2 plane. Also, the h2(Ti, t2) kernel is symmetrical around the line t1 = t2.
According to Marmarelis and Marmarelis (1978): "The value of h2(Ti, t2) gives a quantitative measure of the nonlinear deviation from superposition due to interaction between portions of the stimulus signal that are t1 and t2 sec in the past, in affecting the system's response in the present. Or if t1 = t2, it denotes the amplitude-dependent nonlinearities."
Because of the computational complexity of finding k S 3 cross-correlation functions and kernels, and problems in visualizing S four-dimensional functionals, they are seldom used in white noise analysis of nonlinear systems. This means that if the system is sharply nonlinear, its description only in terms of h0, hi, and h2 will be inaccurate. The number of terms required to accurately describe a nonlinear system can be estimated by examining the response of the system to a steady-state sinusoidal input. The number and relative magnitudes of the harmonics in the system output can predict how many G terms will be required, because theoretically, Gn can produce at most an nth order harmonic in y(t).
Maramarelis and Naka (1974) pointed out that computation of the system kernels by using the fast Foutier transform (FFT) to find the cross-power spectrums, and then inverting these functions by FFT, provides a considerable economy in time and computer effort. The catfish retina studies reported by Marmarelis and Naka (1973a,b; 1974) use no higher than h2 terms in the Wiener functional series models of retinal behavior. They justify the truncation of the series by arguing that the biosystem is weakly nonlinear and does not generate a significant amount of third-and higher-order harmonics when driven sinusoidally. They also observed very relevantly:
The fact that we cannot easily [in terms of computational effort] estimate cross-correlations of higher (than third) degree limits the applicability of the method to systems whose nonlinearities allow an acceptable representation in terms of the first few terms of the series. Thus systems with "sharp" nonlinearities (thresholds, sharp limiters, etc.) cannot be described accurately. However, if the series is truncated after the nth order term, the resulting approximation is the best nth order characterization in the MSE sense. This derives directly from the fact that the terms of the series are orthogonal.
Thus, it is apparent that because of difficulties in interpretation of high-order kernels, and computational difficulties, it is not practical to use kernels of order three and higher in implementing the Lee-Schetzen approach to white noise analysis. These are important, practical restrictions.
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