The Bispectrum and Bicoherence for Quadratically Nonlinear Systems Subject to Non-Gaussian Inputs

• Published in: IEEE transactions on signal processing Vol. 57; no. 10; pp. 3879 - 3890
• Main Authors: Nichols, J.M., Olson, C.C., Michalowicz, J.V., Bucholtz, F.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.10.2009; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Acoustic signal detection; Applied sciences; Autocorrelation; Bicoherence; bispectrum; Data analysis; Density; Detection, estimation, filtering, equalization, prediction; Dynamical systems; Exact sciences and technology; Exact solutions; Fourier transforms; Information, signal and communications theory; Laboratories; Mathematical analysis; Miscellaneous; Non-Gaussian; Nonlinear dynamics; Nonlinear systems; Optimization; polyspectra; Probability density function; Probability distribution; Random processes; Signal and communications theory; Signal processing; Signal, noise; Telecommunications and information theory; Underwater tracking; Autocorrelation function; polyspectra; Second order; Probabilistic approach; Optimal filter; Non linear filter; Gaussian distribution; Probability distribution; Bicoherence; Joint distribution; Non linear system; Non linear phenomenon; Statistical method; Probability density; Analytical method; Probability density function; Signal processing; Bispectrum; Non linear effect
• ISSN: 1053-587X; 1941-0476
• DOI: 10.1109/TSP.2009.2024267

In the analysis of data from nonlinear systems both the bispectrum and the bicoherence have emerged as useful tools. Both are frequently used to detect the influence of a nonlinear system on the joint probability distribution of the system input. Previous work has provided an analytical expression for the bispectrum of a quadratically nonlinear system output if the input is stationary, jointly Gaussian distributed. This work significantly generalizes the previous analysis by providing an analytical expression for the bispectrum of the response of quadratically nonlinear systems subject to stationary, jointly non-Gaussian inputs possessing arbitrary auto-correlation function. The expression is then used to determine the optimal input probability density function for detecting a quadratic nonlinearity in a second-order system. It is also shown how the expression can be used to design an optimal nonlinear filter for detecting deviations from normality in the probability density of a signal.