Application of bispectral analysis in nonlinear ultrasonic structural health monitoring

Bispectral analysis has excited some interest in the field of non-linear nondestructive testing (NDT), machine condition monitoring and structural health monitoring (SHM). This is due to its apparent insensitivity to Gaussian noise and ability to detect coupled signals at different frequencies such as those produced by non-linearities. It has been demonstrated that damage in many engineering components (both metal and carbon-fibre-reinforced plastic) displays non-linear behaviour and that the bispectrum can be used to analyse the resulting response to ultrasonic excitation. Much of the analysis underpinning the use of the bispectrum relates to stochastic systems, whereas non-linear SHM frequently involves the excitation of the system with a deterministic signal and analysing the response in the presence of noise. This paper aims to bridge the gap between the practical application of the bispectrum as a tool for analysing non-linear responses to deterministic excitations and the existing analytical understanding for stochastic systems. Appropriate signal-to-noise ratios are defined and evaluated analytically for weakly non-linear systems excited with sinusoidal signals with Gaussian noise added to the response. Comparison is made of analysis using the power spectrum and the bispectrum to analyse the data and signal-to-noise ratios expressed as a function of the number of averages and the number of data points per average. Numerical simulations confirm, and illustrate, the behaviours of the signal-to-noise ratios. The other issue of practical interest regarding the bispectrum is its ability to detect signals which are quadratically phase coupled. As non-linear systems result in signals that are quadratically phase coupled to the excitation signals quadratic phase coupling offers the possibility of ignoring contributions to the signal that are not due to non-linearity. The estimation method and related excitation scheme required for this to be exploited when the excitations are deterministic is considered analytically and numerically. Excitation and estimation strategies are outlined in the light of these calculations in order to ensure that the advantageous properties of the bispectrum are used.