Generalized Morse Wavelets as a Superfamily of Analytic Wavelets

TLDR

The generalized Morse wavelets form a superfamily that subsumes eight commonly used analytic wavelets, including the Airy wavelets that emulate the Morlet wavelet while avoiding its deficiencies. This framework enables systematic investigation of wavelet suitability for diverse applications. The wavelets are parameterized by a time‑domain duration or Fourier‑domain bandwidth, with a second shape parameter γ that adjusts the wavelet form at fixed bandwidth. For integer γ, the member with γ = 3 is the most symmetric, nearly Gaussian, and time‑frequency concentrated, and these Airy wavelets are recommended as an ideal starting point for general‑purpose use.

Abstract

The generalized Morse wavelets are shown to constitute a superfamily that essentially encompasses all other commonly used analytic wavelets, subsuming eight apparently distinct types of analysis filters into a single common form. This superfamily of analytic wavelets provides a framework for systematically investigating wavelet suitability for various applications. In addition to a parameter controlling the time-domain duration or Fourier-domain bandwidth, the wavelet shape with fixed bandwidth may be modified by varying a second parameter, called γ. For integer values of γ, the most symmetric, most nearly Gaussian, and generally most time-frequency concentrated member of the superfamily is found to occur for γ = 3. These wavelets, known as "Airy wavelets," capture the essential idea of popular Morlet wavelet, while avoiding its deficiencies. They may be recommended as an ideal starting point for general purpose use.