Higher-order spectra: The bispectrum and trispectrum

This paper is concerned with the development of techniques to detect and analyse non-linearities. The methods developed are based on the concepts of higher-order spectra (HOS), in particular the bispectrum and trispectrum. The study of HOS has been dominated by work on the bispectrum. The bispectrum can be viewed as a decomposition of the third moment (skewness) of a signal over frequency and proves useful for analysing systems with asymmetric non-linearities. In studying symmetric non-linearities, the trispectrum is a more powerful tool, as it represents a decomposition of kurtosis over frequency. Techniques are presented that enable the estimation and display of bispectra and trispectra. HOS are studied in detail with particular attention being paid to normalisation methods. Two traditional methods, the bicoherence and skewness function, are studied and these are extended to their fourth-order equivalents, the tricoherence and kurtosis functions. Under certain conditions, notably narrowband signals, the above normalisation methods are shown to fail, and so a new technique based on pre-whitening the signal in the time domain is developed. Examples of these functions are given both for an amplitude modulated process and the Duffing oscillator. (C) 1998 Academic Press Limited.