High-order symmetrical hyperbolic wavelets

This paper studies high-order wavelets of the first-order hyperbolic, Choi–Williams (CW) and nth-order hyperbolic kernels for analyses of digital time series, by using their second- and higher-order derivatives. For time-domain investigations, normalisation constants of the second-, fourth-, sixth-, eighth- and tenth-order hyperbolic and CW wavelets are numerically given. For frequency-domain investigations, wavelet parameters including band-peak frequencies, minimum numbers of sampling points, scale limits, scale resolutions and total number of scales are explicitly given and numerically estimated for the fourth-order hyperbolic and CW wavelets. Parameter comparisons among the Morlet wavelet, hyperbolic and CW second- and fourth-order wavelets are also given. Detection of periodicity and chaos in the Duffing oscillator is discussed.