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Nonlinear systems subjected to Gaussian inputs are studied based on the Wiener-like stochastic functional Fourier series . Analytic and nonanalytic cascade, bilinear, and feedback nonlinear structures are considered and their th-order Fourier-Hermite kernels are calculated analytically. The characteristic kernel features thus revealed are discussed as a guide to interpret data from multidimensional cross-correlation experiments for nonparametric nonlinear system identification. The results are shown to be useful also for the mean-square-signal analysis of nonlinear systems whose structures and parameters are known a priori. For the feedback case, a certain approximation is employed for finding the th-order closed-loop kernel. This is a generalization of the describing function technique, and using examples, the algorithm is compared to existing procedures for random-input nonlinear servosynthesis.