One of the first requirements in implementing white noise analysis of a nonlinear system is that the system must be stationary; i.e., its parameters (gains, rate constants, natural frequencies, etc.) must not change in time, at least over the time required to sample the data (x(t) and y(t)) required to compute the kernels h0, hj, h2, etc.
White noise is an engineering idealization (similar to an ideal voltage source, or an ideal op amp), made for computational expedience. In practice, noise is called "white" when its power density spectrum is flat and overlaps the low- and high-frequency limits of the NSUS bandpass by at least two octaves. Because kernel estimations are carried out digitally, attention must be given to problems caused by aliasing, quantization noise, and to data window functions. To avoid aliasing, the sampling rate for the noise x(t) must be well above the frequency where the amplitude of the autopower spectrum, ®xx(f), of the noise is down to 10% of its peak value. The noise autopower spectrum must also be sharply attenuated above its cutoff frequency to avoid aliasing.
Maramarelis and Naka (1974) have shown that aside from the bandwidth requirements imposed on ®xx(f) by the NSUS and by alias-free sampling, an excessive input noise spectral bandwidth will contribute to large statistical errors in computing the estimate of the linear kernel, h1 (and presumably also to the higher-order kernels as well). They show that, for a strictly linear system where kernels of order S 2 are zero, when the input noise has a two-sided, rectangular spectrum with cutoff frequency, rao r/s, then the variance of the Lee-Schetzen estimate of h1(T) is the variance of 9xy(T), and is given by
VAR[hj(T)] = (co0/4n3)| |HX(j®)| dffl + (l/4rc2)| Ht((®)exp((®T)d®
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