Defining the Wavelet Bispectrum

Newman, Julian and Pidde, Aleksandra and Stefanovska, Aneta (2021) Defining the Wavelet Bispectrum. Applied and Computational Harmonic Analysis, 51. pp. 171-224. ISSN 1063-5203

Text (Bispectra 02062020)
Bispectra_02062020.pdf - Accepted Version
Available under License Creative Commons Attribution-NonCommercial.
Download (3MB)

Archive (Supplementary codes)
bispectrum.zip - Accepted Version
Available under License Creative Commons Attribution-ShareAlike.
Download (5kB)

Abstract

Bispectral analysis is an eective signal processing tool for analysing 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 quantication of the bispectral content of an area of scale-scale space. This makes the capacity for quantitative rather than merely qualitative interpretation of wavelet bispectrum computations very limited. In this paper, we 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 denition of the wavelet bispectrum matches the traditional bispectrum of sums of sinusoids, in the limit as the frequency resolution tends to innity. We illustrate the improved quantitative power of our denition with numerical and experimental data.