On the Design of Optimal Analytic Wavelets

Properties determining optimal behavior of continuous analytic wavelet functions are investigated, and wavelet functions nearly obtaining such behavior are identified. This is accomplished through detailed investigation of the generalized Morse wavelets, a highly flexible family of exactly analytic continuous wavelets. The degree of time/frequency localization, the existence of a unique interpretation of scale as frequency, and the extent of bias involved in estimating properties of modulated oscillatory signals, are proposed as important considerations. Optimal wavelet behavior is then found to be achieved by minimizing third central moments in both the frequency and the time domains. A particular subset of the generalized Morse wavelets, recognized as deriving from an inhomogeneous Airy function, are shown to be nearly optimal. These ``Airy wavelets'' are expected to substantially outperform the only approximately analytic Morlet wavelets for high time localization. Further investigation of the generalized Morse wavelets reveals a remarkably broad range of behavior, and suggests that these wavelets can be considered as a generic analytic wavelet family appropriate for a variety of applications.