The Lee Schetzen Approach to White Noise Analysis

Lee and Schetzen (1965) showed how the terms of the Wiener functional series may be estimated through cross-correlation (or cross-power spectrum) techniques to arrive at a useful, general, quantitative model for a nonlinear dynamic system. Marmarelis and Naka (1973a) extended the Lee-Schetzen method of white noise analysis to nonlinear systems with two inputs and one output, and have applied this approach to describing the dynamics of signal processing in the catfish retina. They also analyzed the statistical errors associated with finding the kernel estimates, restricting their analysis to second-order nonlinear systems having all {hk} = 0 for k S 3.

Lee and Schetzen (1965) showed that the Wiener kernels are, in general, given by hk(Tj, T2, ... Tk) = 1(k! Pk) E[y(t) x(t - Tj) x(t - T2) ... x(t - Tk)] 8.3-10

which is valid for Tm | Tn. This expectation is a k-dimensional cross-correlation. A more general form for hk is valid for Tm = Tn:

To clarify the meaning of Equation 8.3-11, the first three kernels are written out:

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