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Generalized Morse Wavelets as a Superfamily of Analytic Wavelets
Jonathan M. Lilly, Sofia C. Olhede
TL;DR
The paper addresses the difficulty of choosing among many apparently distinct analytic wavelets. It analyzes generalized Morse wavelets as a two-parameter superfamily, showing that they encompass common analytic wavelets and that γ = 3 provides the most favorable integer-γ balance of concentration, Gaussianity, and symmetry.
Problem
The profusion of analytic wavelet types lacks a unifying framework, making wavelet selection for particular applications appear arbitrary.
Method
The paper studies generalized Morse wavelets by varying duration-related parameter β and shape parameter γ across their parameter space.
Results
For fixed duration, the γ = 3 Airy wavelet family has the highest time/frequency concentration among integer-γ families except at durations below about Pβ,γ/π ≈ 1.
Takeaways & Limitations
Airy wavelets at γ = 3 are recommended as a natural default analytic wavelet, particularly for time-localized applications where Morlet wavelet concentration degrades more rapidly.
Abstract
from arXiv · showhide
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 {\em 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.
I. INTRODUCTION
The paper introduces analytic wavelets and frames the choice among many apparently distinct wavelet types as a practical problem. It presents generalized Morse wavelets as a unified family whose parameters support systematic comparison of wavelet behavior.
- Analytic wavelets are complex-valued filters localized in time and frequency with vanishing support on negative frequencies.
- The proliferation of analytic wavelet types can make selecting a wavelet for a particular application appear arbitrary.
- Generalized Morse wavelets unify the commonly mentioned wavelet types, analytic filters, and, in a limiting sense, complex exponentials within one broad family.
- The wavelet form is controlled by parameters β and γ, while scale s dilates or compresses a given wavelet.
- The family includes an additional order k for constructing increasingly oscillatory wavelets orthogonal to a chosen wavelet, but this degree of freedom is not pursued here.
II. GENERALIZED MORSE WAVELET PARAMETER SPACE
Generalized Morse wavelets span a broad parameter space in which β and γ control distinct aspects of wavelet form, duration, decay, and concentration. This parameterization recovers several established analytic wavelet families and identifies limiting forms.
- II. GENERALIZED MORSE WAVELET PARAMETER SPACE: Varying β and γ produces a wide variety of wavelet forms beyond the dilation or compression provided by scale s.
- II. GENERALIZED MORSE WAVELET PARAMETER SPACE: The duration Pβ,γ tracks the number of peak-frequency oscillations in the central time-domain window and remains constant along diagonal parameter-space lines.For Pβ,γ = 3, the central window contains approximately Pβ,γ/π ≈ 1 oscillation.
- II. GENERALIZED MORSE WAVELET PARAMETER SPACE: β controls behavior near zero frequency and time-domain decay, whereas γ controls high-frequency decay; fixed-duration changes exchange these decay properties.The time-domain decay is |ψβ,γ(t)/ψβ,γ(0)| ∼ 1/t^(β+1).
- II. GENERALIZED MORSE WAVELET PARAMETER SPACE: The family contains Morse or Cauchy-Klauder-Paul wavelets at γ = 1 and analytic Derivative of Gaussian wavelets at γ = 2, while other limits approach analytic filters and complex exponentials.
III. CONCENTRATION AND GAUSSIANITY
The paper compares generalized Morse and Morlet wavelets by time-frequency concentration and Gaussian similarity, finding that integer γ=3 provides the most symmetric and nearly Gaussian family at fixed duration. The Morlet wavelet becomes substantially less concentrated in highly time-localized settings, while all generalized Morse families have unbounded Heisenberg area at β=1.
- Concentration: For small wavelet durations, Morlet concentration degrades more rapidly than that of the γ=2, 3, and 4 generalized Morse families.The difference is especially pronounced for highly time-localized analyses of rapidly varying signals.
- Concentration: The Heisenberg areas of all generalized Morse wavelet families become unbounded at β=1.The paper also notes that the temporal standard deviation and Heisenberg area are unbounded in the corresponding limiting case.
- Parameters: The wavelet duration Pβ,γ measures the number of peak-frequency oscillations within the central time-domain window, while γ controls high-frequency decay and wavelet shape.Pβ,γ/π gives the approximate number of oscillations in the central window; 1/Pβ,γ measures frequency-domain bandwidth.
- Gaussianity: γ=3 gives the most symmetric and nearly Gaussian generalized Morse wavelets at fixed duration, because the cubic deviation vanishes and the quartic deviation is smallest.The cubic term controls local asymmetry, while the quartic term controls local narrowing or broadening relative to a Gaussian.
IV. SPECIAL FORMS OF GENERALIZED MORSE WAVELETS
Generalized Morse wavelets approach several established analytic wavelets in limiting regimes, including lognormal, Shannon, and Bessel forms. These limits reveal distinct trade-offs in symmetry, decay, and localization.
- Lognormal limit: As γ approaches zero with fixed Pβ,γ, generalized Morse wavelets become highly asymmetric lognormal wavelets in the frequency domain.Here 1/Pβ,γ acts as the standard deviation around the maximum at ω = 1.
- Shannon limit: As γ approaches infinity and β approaches zero, the wavelets become constant-amplitude bandpass filters on 0 < ω ≤ 1, matching the Shannon wavelet.The corresponding time-domain form is a sinc function.
- Shannon limit: The Shannon limit has slow 1/t time decay, an undesirable property that limits its usefulness.
- Limiting behavior: Shannon and lognormal wavelets delimit extreme properties attainable for fixed Pβ,γ, but their limits differ in Heisenberg area and remaining family structure.The Shannon limit has unbounded Heisenberg area, whereas the lognormal limit retains a one-parameter family with fixed asymptotic area for each Pβ,γ.
- Bessel correspondence: The Bessel wavelet is the only exactly analytic wavelet in common use not formally included in the generalized Morse family.A numerical search nevertheless finds a maximum magnitude-squared inner product of 0.9995 at (β, γ) = (22, 1/10), with visually indistinguishable time-domain wavelets.
V. DISCUSSION
The generalized Morse family unifies commonly used analytic wavelets and adds a shape parameter alongside duration or inverse bandwidth. Among integer γ values, γ = 3 yields the most Gaussian, concentrated, and symmetric family, while other choices remain useful for specific signals and frequency settings.
- Discussion: Commonly used continuous analytic wavelets and analysis filters can be treated as special cases of one generalized Morse wavelet family.
- Discussion: Pβγ sets wavelet duration or inverse bandwidth and the number of oscillations, while γ shifts shape between qualitatively different families at fixed duration.
- Discussion: For integer γ, Airy wavelets at γ = 3 are the most Gaussian, time/frequency concentrated, and symmetric generalized Morse wavelets.The paper presents them as a natural default analytic wavelet for general-purpose applications and as preferable to Morlet for time-localized settings.
- Discussion: Other generalized Morse wavelets may suit signals resembling Gaussian or Cauchy families, or oscillatory variability near the Nyquist frequency.Near the Nyquist frequency, greater high-frequency decay may be favored over symmetry.
APPENDIX
Software associated with the paper is distributed through the freely available MATLAB toolbox JLAB, which implements the generalized Morse wavelets and related analysis functions.
- APPENDIX: JLAB distributes the paper’s software, including morsewave for generalized Morse wavelets and wavetrans, which uses them by default.Additional functions compute wavelet properties and generate the paper’s figures.