Defining the wavelet bispectrum
Julian M. I. Newman, Aleksandra Pidde, Aneta Stefanovska · Applied and Computational Harmonic Analysis · 2020
Bispectral analysis is an effective signal processing tool for investigating interactions between oscillations, and has been adapted to the continuous wavelet transform for time-evolving analysis of open systems. However, one unaddressed question for the wavelet bispectrum is quantification of the bispectral content of an area of scale-scale space. Without this, interpretation of wavelet bispectrum computations is merely qualitative. We now overcome this limitation by providing suitable normalisations of the wavelet bispectrum formula that enable it to be treated as a density to be integrated. These are roughly analogous to the normalisation for second-order wavelet spectral densities. We prove that our definition of the wavelet bispectrum matches the traditional bispectrum of sums of sinusoids, in the limit as the frequency resolution tends to infinity. We illustrate the improved quantitative power of our definition with numerical and experimental data. We also discuss notions of bicoherence and its practical implementation.