Generalized Morse wavelets

This paper examines the class of generalized Morse wavelets, which are eigenfunction wavelets suitable for use in time-varying spectrum estimation via averaging of time-scale eigenscalograms. Generalized Morse wavelets of order k (the corresponding eigenvalue order) depend on a doublet of parameters (β, γ); we extend results derived for the special case β = γ = 1 and include a proof of "the resolution of identity." The wavelets are easy to compute using the discrete Fourier transform (DFT) and, for (β, γ) = (2m, 2), can be computed exactly. A correction of a previously published eigenvalue formula is given. This shows that for γ > 1, generalized Morse wavelets can outperform the Hermites in energy concentration, contrary to a conclusion based on the γ = 1 case. For complex signals, scalogram analyses must be carried out using both the analytic and anti-analytic complex wavelets or odd and even real wavelets, whereas for real signals, the analytic complex wavelet is sufficient.