Statistics of the estimates of tricoherence are obtained analytically for nonlinear harmonic random processes with known true tricoherence. Expressions are presented for the bias, variance, and probability distributions of estimates of tricoherence as functions of the true tricoherence and the number of realizations averaged in the estimates. The expressions are applicable to arbitrary higherorder coherence and arbitrary degree of interaction between modes. Theoretical results are compared with those obtained from numerical simulations of nonlinear harmonic random processes. Estimation of true values of tricoherence given observed values is also discussed.
Polynomial models are shown to simulate accurately the quadratic and cubic nonlinear interactions (e.g. higher-order spectra) of time series of voltages measured in Chua's circuit. For circuit parameters resulting in a spiral attractor, bispectra and trispectra of the polynomial model are similar to those from the measured time series, suggesting that the individual interactions between triads and quartets of Fourier components that govern the process dynamics are modeled accurately. For parameters that produce the double-scroll attractor, both measured and modeled time series have small bispectra, but nonzero trispectra, consistent with higher-than-second order nonlinearities dominating the chaos.
A technique to generate realizations of quadratically nonlinear non-Gaussian time series with a desired ("target") power spectrum and bispectrum is presented. Specifically, by generating a Gaussian time series (using amplitude information from the target power spectrum and random phases) and passing it through a quadratic filter (that uses phase information from the target bispectrum), a realization of a quadratically nonlinear random process with a specified power spectrum and bispectrum can be produced. Second- and third-order statistics from many realizations of simulated nonlinear time series compare well to those from the original time series providing the target power spectrum and bispectrum, with deviations consistent with theory. The simulation technique is shown to simulate accurately ocean waves in shallow water, which are well known to be quadratically nonlinear.
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