The hyperbolic wavelet function

A survey of known wavelet groups is listed and properties of the symmetrical first-order hyperbolic wavelet function are studied This new wavelet is the negative second derivative function of the hyperbolic kernel function, [sech(beta theta)](n) where n = 1, 3, 5,... and n = 1 corresponds to the first-order hyperbolic kernel, which was recently proposed by the authors as a useful kernel for studying time-frequency power spectrum Members of the "crude" wavelet group, which includes the hyperbolic, Mexican hat (Choi-Williams) and Morlet wavelets, are compared in terms of band-peak frequency, aliasing effects, scale limit, scale resolution and the total number of computed scales. The hyperbolic wavelet appears to have the finest scale resolution for well-chosen values of beta less than or equal to 0.5 and the Morlet wavelet seems to have the largest total number of scales.