where P = O^f) is the power density spectrum of the input Gaussian white noise in mean-squared units/Hz. Beyond order k = 3, the physical significance of the kernels and functionals strains the imagination; they can be considered hypersurfaces.
Marmarelis (1972) noted that each kernel hk is a symmetrical function of its arguments, and that the nonlinear kernels (k S 2) give a quantitative description of the nonlinear "cross-talk" between different portions of the input as it affects the present system response. In a second-order nonlinear system, the kernel h2(t, t - to) describes the deviation from linear superposition between an impulse at t = 0 and another at time to.
For example, let the input to a "pure" second-order NL system be x(t) = S(t) + S(t - to). Assume h0 = 0 and hk = 0 for k S 3. Hence, the Wiener formulation becomes y(t) = Gi[hi,x(t)] + G2 [h2,x(t)]= Phi (x)[8(t-x) + 8(t -10-Ti )dt
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