Integrating Equation 8.3-6 yields y(t) = hj(t) + hj(t - to) + h2(t, t) + h2(t, t - to)

Subtracting the output of the second-order, nonlinear system caused by each input acting alone, from Equation 8.3-7 yields

Since h2(tj, t2) is symmetrical around the line tj = t2, Equation 8.3-8 can be written:

Equation 8.3-9 gives a measure of the cross-talk between the two inputs. It can be shown that if the system consists of a no-memory nonlinearity, f(x), followed by a linear dynamics, then h2(xj, t2) = 0 for Tj | t2, and the system obeys "time superposition"; i.e., in this case the responses of the system to the sum of two or more impulses is equal to the sum of the responses of the system to each impulse separately. The values of h2(TJ, t2) for tj = t2 are a continuous series of impulses of varying areas. Thus, the magnitudes of the kernels gives an indication of the nonlinear cross-talk between different (in past time) portions of the input (Marmarelis and Marmarelis, 1978).

Marmarelis (1972) clearly showed the awesome computational complexity required to find estimates of a finite set of Wiener kernels, {hk}, required to characterize the nonlinear system. Wiener's approach, although possessing great mathematical sophistication, is so very unwieldy in an experimental situation that it cannot satisfactorily cope even with a low-order linear system.

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