Application of the bispectrum for detection of small nonlinearities excited sinusoidally

The nonlinear behaviour of damaged systems excited by vibration or ultrasound offers potential as a technique for damage detection in machine condition monitoring and non-destructive testing applications. The bispectrum, a third-order spectrum, has properties that lend themselves to the measurement of nonlinearities in systems. The properties of interest are insensitivity to Gaussian noise and ability to detect quadratic phase coupling. However, thus far analysis of the statistics of bispectrum estimation has been mainly aimed at stochastic systems. Many applications to vibration and ultrasound involve primarily deterministic, periodic excitations in the presence of stochastic noise. This paper considers the properties of a bispectrum estimate when applied to a system with weak quadratic nonlinearity excited by the superposition of two sinusoids in the presence of additive Gaussian noise. This is compared, using signal-to-noise ratios, to the powerspectrum, with the results validated using numerical data. Also addressed is the effect of quadratic phase coupling on such a system (in the absence of noise).