PaperlayerView exact revision on arXiv

Source-linked AI summary

Higher-Order Properties of Analytic Wavelets

J. M. Lilly, S. C. Olhede

arXiv:0802.2377v2stat.MEmath.ST

TL;DR

The paper asks how higher-order properties affect analytic wavelet transforms and investigates these effects through the two-parameter generalized Morse family. It finds that localization, scale–frequency mapping, and oscillatory-signal bias depend strongly on time- and frequency-domain asymmetry, with Airy wavelets offering particularly desirable properties.

  • Problem

    The paper examines which analytic wavelet properties govern localization, scale–frequency interpretation, and bias when analyzing modulated oscillatory signals.

  • Method

    The authors analyze generalized Morse wavelets, a two-parameter family of exactly analytic continuous wavelets, using higher-order derivatives and moments.

  • Results

    Time- and frequency-domain asymmetry, quantified through third-order measures, strongly affects transform behavior and oscillatory-signal analysis.

  • Takeaways & Limitations

    Airy wavelets, the γ = 3 generalized Morse subfamily derived from an inhomogeneous Airy function, combine exact analyticity with desirable localization and symmetry properties.

Abstract

from arXiv · show

The influence of higher-order wavelet properties on the analytic wavelet transform behavior is investigated, and wavelet functions offering advantageous performance are identified. This is accomplished through detailed investigation of the generalized Morse wavelets, a two-parameter family of exactly analytic continuous wavelets. The degree of time/frequency localization, the existence of a mapping between scale and frequency, and the bias involved in estimating properties of modulated oscillatory signals, are proposed as important considerations. Wavelet behavior is found to be strongly impacted by the degree of asymmetry of the wavelet in both the frequency and the time domain, as quantified by the third central moments. A particular subset of the generalized Morse wavelets, recognized as deriving from an inhomogeneous Airy function, emerge as having particularly desirable properties. These "Airy wavelets" substantially outperform the only approximately analytic Morlet wavelets for high time localization. Special cases of the generalized Morse wavelets are examined, revealing a broad range of behaviors which can be matched to the characteristics of a signal.

I. INTRODUCTION

The paper examines how higher-order properties shape analytic wavelet transforms, focusing on generalized Morse wavelets as an exactly analytic, adjustable family. It evaluates localization, scale–frequency interpretation, and oscillatory-signal bias to guide wavelet selection.

  • Motivation: Analytic wavelets isolate variability in time and scale while estimating instantaneous amplitude and phase near each time/scale location.The analytic transform also underlies wavelet-ridge estimates of time-varying signal properties.
  • Wavelet family: Generalized Morse wavelets provide two adjustable parameters and a broad range of forms, enabling their properties to be matched to signal characteristics.The family includes Cauchy wavelets and analytic versions of derivative-of-Gaussian wavelets as special cases.
  • Goals: Third-order frequency- and time-domain measures of wavelet asymmetry emerge as key quantities governing transform behavior.The paper expects comparable behavior for discrete analytic wavelets at long time scales.
  • Goals: The study evaluates three criteria: time/frequency concentration, unique scale-to-frequency correspondence, and minimized bias for modulated oscillatory signals.The latter two criteria are linked to higher-order asymmetry measures.
  • Analyticity: The paper focuses on exactly analytic wavelets because negative-frequency leakage in approximately analytic wavelets can produce spurious transform-phase variation.Morlet wavelets are only approximately analytic for sufficiently large carrier frequency.

C. Importance of Analyticity

Exact analyticity and higher-order wavelet structure determine transform behavior, especially for short time windows. The paper shows that analyticity prevents leakage artifacts, while symmetry supports consistent frequency interpretations.

  • Short-time behavior: Narrowing both wavelet time windows by a factor of 10 causes Morlet negative-frequency leakage and substantial central instantaneous-frequency fluctuations.The generalized Morse wavelet remains nearly constant in instantaneous frequency and its Wigner-Ville distribution remains entirely concentrated.
  • Short-time behavior: Negative-frequency leakage in the Morlet transform of a Gaussian-enveloped chirp produces interference and an irregular transform structure.The Morse transform’s maxima-line follows the instantaneous frequency of the total signal rather than the derivative of the chirp phase.
  • Analyticity: Departures from analyticity also deteriorate statistical transform properties: Gaussian-process inputs no longer yield Gaussian proper transform coefficients.The paper distinguishes these statistical effects from the deterministic interference illustrated by the chirp example.
  • Mapping scale to frequency: Three scale-to-frequency mappings use the peak frequency, energy frequency, and central instantaneous frequency, each valid for different signal or feature types.The mappings correspond respectively to sinusoid-transform maxima, energy-mean scale, and phase propagation at an infinitesimally narrow impulse.
  • Mapping scale to frequency: For γ = 3 generalized Morse wavelets, the three frequency measures are nearly equal while exact analyticity and good time localization are retained.A unique scale interpretation requires symmetry about the peak frequency and equality of the energy mean and mode.

E. Energy Localization

The paper uses Heisenberg area as a conventional measure of time/frequency energy localization while noting that other localization notions also exist. Generalized Morse wavelets, especially γ = 3, remain highly concentrated under exact analyticity.

  • Heisenberg area: Heisenberg area quantifies time/frequency energy localization from nondimensional time- and frequency-domain standard deviations.A Gaussian envelope reaches the theoretical minimum area of one-half, but is not a zero-mean wavelet.
  • Localization criteria: Generalized Morse wavelets are optimally localized under a joint time-frequency localization operator, while Heisenberg area enables conventional comparisons among functions.The paper treats these as distinct notions of localization.
  • Localization criteria: γ = 3 generalized Morse wavelets approach the theoretical minimum Heisenberg area while remaining exactly analytic, even for narrow time-domain settings.Their concentration is comparable to or greater than that of the Morlet wavelet.

F. Minimally Biased Signal Inference

The paper analyzes how wavelet frequency-domain derivatives bias analytic-signal estimates for modulated oscillatory signals, identifying third-order derivatives as especially important. It argues that minimizing the third derivative at fixed duration provides a practical route to reducing the dominant higher-order bias.

  • Signal inference: Wavelet ridge analysis estimates analytic-signal properties from an analytic wavelet transform, including amplitude, phase, and frequency.The transform produces a smoothed quantity that can differ from the analytic signal because of interactions between signal derivatives and wavelet frequency derivatives.
  • Signal inference: Bias in the wavelet-based analytic-signal estimate arises from interactions between time-domain derivatives of the signal and frequency-domain derivatives of the wavelet.The expansion includes a bounded residual from truncating the integration range.
  • Bias hierarchy: For locally oscillatory signals, the term involving eΨ3(ωψ) is smaller than the eΨ2(ωψ) term, while the eΨ4(ωψ) contribution is smaller still.Under the stated smoothness assumption, this ordering makes the third derivative the most important higher-order target after the second derivative.
  • Bias minimization: A wavelet with vanishing eΨ3(ωψ) at fixed eΨ2(ωψ) removes the next-highest-order bias term for a given wavelet duration.The generalized Morse wavelets achieve this when γ = 3; Shannon wavelets also have the property but are disfavored for poor time localization.
  • Design criteria: The paper frames high concentration, a unique scale–frequency relationship, and minimized oscillatory-signal bias as the three useful properties guiding analytic-wavelet selection.It then derives the relevant quantities explicitly for generalized Morse wavelets.

B. Frequency-Domain Derivatives / Time-Domain Moments

The paper characterizes generalized Morse wavelets using duration and demodulate skewness, linking second- and third-order moments to wavelet shape and transform behavior. The special choice γ = 3 eliminates demodulate skewness while maximizing demodulate kurtosis at fixed duration.

  • Time-domain moments: Demodulate moments are central moments of the wavelet after demodulation by its peak frequency.The paper normalizes higher-order moments by the second moment to obtain dimensionless shape measures.
  • Time-domain moments: Demodulate skewness α3;ψ and demodulate kurtosis α4;ψ formally resemble probability-distribution skewness and kurtosis, although the demodulated wavelet is generally complex-valued.Because it is not a nonnegative probability density, these are formal analogues rather than ordinary distribution moments.
  • Duration: Pψ/π measures the number of peak-frequency oscillations within the central wavelet window, using the standard deviation of the demodulated wavelet.Increasing Pψ narrows the wavelet in frequency and broadens it in time.
  • Generalized Morse parameters: γ = 3 makes demodulate skewness and eΨ3(ωψ) vanish while maximizing the magnitude of demodulate kurtosis at fixed Pβ,γ.For locally oscillatory signals, the paper emphasizes skewness because it is expected to contribute more substantially to analytic-signal bias than kurtosis.
  • Generalized Morse parameters: For generalized Morse wavelets, Pβ,γ and ℑ{α3;β,γ} uniquely determine β and γ and provide a two-parameter description of duration and time-domain asymmetry.At fixed duration, varying the imaginary demodulate skewness produces different wavelet shapes, while kurtosis is implicitly determined.
  • Parameter mapping: The generalized Morse parameters β and γ increase or decrease systematically with Pβ,γ and ℑ{α3;β,γ}, with γ = 3 corresponding to zero imaginary demodulate skewness.The admissible parameter region has a lower bound on skewness for γ = 0 but no upper bound.

C. Wavelet Frequency Measures

The paper examines how generalized Morse wavelet frequency measures vary with duration and demodulate skewness, emphasizing γ = 3 as a near-ideal compromise. At sufficient duration, this family aligns peak, energy, and central instantaneous frequencies while eliminating instantaneous-frequency curvature.

  • Instantaneous-frequency curvature: Frequency-domain skewness corresponds to the curvature of instantaneous frequency, with the curvature having the opposite sign of the frequency-domain skewness coefficient.This connects a third-order frequency-domain moment to the local shape of instantaneous frequency.
  • Frequency measures: Frequency measures are evaluated as functions of Pβ,γ/π and ℑ{α3;β,γ}, including energy frequency, central instantaneous frequency, and instantaneous-frequency curvature.Their behavior changes character at γ = 3.
  • Scale–frequency mapping: For Pβ,γ/π > 1, γ = 3 wavelets make energy, central instantaneous, and peak frequencies nearly identical.This provides an approximately unambiguous interpretation of scale as frequency while retaining exact analyticity.
  • Instantaneous-frequency curvature: Negative curvature produces concave wavelets whose central instantaneous frequency is maximal, whereas positive curvature produces convex wavelets whose central instantaneous frequency is minimal.For Pβ,γ/π > 1, γ = 3 divides these two behaviors.
  • Time–frequency concentration: Wavelets with small time-domain skewness have small Heisenberg area, which approaches the theoretical limit of one-half as duration increases.The minimum is not attained except in the long-duration limit, apparently because analyticity induces asymmetry.
  • Section synthesis: Near γ = 3, generalized Morse wavelets attain minimum Heisenberg area; at sufficiently large duration, their three frequency measures become indistinguishable and curvature vanishes.These combined properties identify γ = 3 as a special family within the generalized Morse wavelets.

IV. SPECIAL CASES OF GENERALIZED MORSE WAVELETS

The paper treats generalized Morse wavelets as a broad analytic family whose parameters generate different time- and frequency-domain behaviors. Their construction proceeds through analyticity, frequency warping, and time-domain differentiation, while admissibility excludes only boundary cases.

  • Family scope: The generalized Morse family is examined as a generic collection of analytic wavelets suitable for signals with differing characteristics.The discussion explores special cases, family boundaries, and relationships among members.
  • Parameter roles: β and γ have distinct interpretations related to relationships among family members and to time- and frequency-domain decay.The alternate construction is designed to make these parameter roles more transparent.
  • Filters and wavelets: Across γ ≥ 0 and β ≥ 0, the inverse Fourier transform defines a valid filtering function, while only a subset satisfies the stricter wavelet conditions.For β > 0 the filter is zero-mean.
  • Admissibility: For γ > 0 and β > 0, admissibility and finite energy are satisfied; β = 0 and γ = 0 are the only invalid wavelet boundary cases.The full nonnegative parameter range therefore includes filters beyond the valid-wavelet subset.
  • Warping: Increasing γ is implemented by warping frequency content, with γ = 1 giving an identity delta kernel in the time-domain transformation.The frequency-domain substitution maps the β = 0, γ = 1 power distribution onto different Fourier components.
  • Differentiation: For integer β ≥ 1, differentiation in time generates higher-β filters from the β = 0 filter at fixed γ.Thus the family can be generated from the analytic ψ0,1(t) through analyticity, frequency warping, and differentiation.
  • Decay: The parameter γ controls high-frequency decay, whereas β controls time-domain decay.This separates the principal decay roles of the two generalized Morse parameters.

2) Frequency and Time Decay:

Generalized Morse wavelets separate time-domain decay, central-window width, and symmetry through their β and γ parameters. Their higher-order behavior determines how wavelets localize and respond to signals.

  • Frequency and Time Decay: O(t^−(β+1)) time-domain decay follows from the singularity of the (β + 1)st frequency-domain derivative at ω = 0.The smallest inverse-time power dominates at large times.
  • Frequency and Time Decay: Increasing β broadens the central filter while making long-time decay more rapid.This change reflects β’s role as a decay or compactness parameter.
  • Frequency and Time Decay: Increasing γ broadens the central filter by reducing envelope curvature without changing the long-time decay.Adjusting β and γ together controls inner window width independently from long-time decay.
  • Symmetry Versus Compactness: β controls decay or compactness, whereas γ controls time-domain symmetry through the demodulate skewness parameter.The skewness is α3;β,γ = i(γ − 3)/Pβ,γ.
  • Symmetry Versus Compactness: Time-domain symmetry and compactness are antagonistic for γ < 3 but covary for γ > 3.At γ = 3, the wavelet is most symmetric when its time decay is strongest.

B. Domain Boundaries

At parameter boundaries, generalized Morse filters connect analytic filtering, differentiation, sinusoidal behavior, and several named wavelet families. These limits show the breadth of the two-parameter construction.

  • Domain Boundaries: At β = γ = 0, the generalized Morse filter becomes the analytic filter rather than a wavelet.Applying it to x(t) recovers the analytic version of the signal independently of scale.
  • Domain Boundaries: For γ = 0, applying ψβ,0 is essentially equivalent to taking the βth derivative of the analytic signal.This follows by commuting differentiation with the analytic filter.
  • Domain Boundaries: As β approaches infinity, normalized generalized Morse wavelets approach a complex sinusoid in their fixed-order moments.Matching the wavelet and sinusoid over finite times requires care with terms whose order scales as O(β).
  • Domain Boundaries: The γ = 1 family corresponds to Cauchy wavelets, generated from the analytic Cauchy filter and related to the Witch of Agnesi curve.Integer-order filters arise through repeated differentiation, and the representation extends to all β > 0.
  • Domain Boundaries: All generalized Morse filters with integer β and γ ≥ 1 can be generated from the Witch of Agnesi through analytization, warping, and differentiation.The source function is not itself a wavelet but forms the basis for the generalized Morse family.
  • Domain Boundaries: The γ = 2 family gives analytic Derivative of Gaussian wavelets, whose instantaneous-frequency curve is concave and less suitable for oscillation analysis than γ = 3 wavelets.These wavelets were proposed for singularity analysis.

3) The Airy Wavelets :

The γ = 3 generalized Morse family consists of Airy wavelets derived from an inhomogeneous Airy function. This family combines exact analyticity with near-constant instantaneous frequency and favorable higher-order properties.

  • The Airy Wavelets: Airy wavelets are obtained by differentiating the analytic Airy filter ψ0,3(t) β times.The resulting expression defines ψβ,3(t) for integer β ≥ 1.
  • The Airy Wavelets: The β = 1 Airy wavelet lies outside the localization regime β > (γ − 1)/2.Within the family, instantaneous frequency remains nearly constant across the wavelet width.
  • The Airy Wavelets: Unique scale–frequency mapping requires frequency-domain symmetry, equality of mean and mode, and third-order symmetry conditions.These conditions support reliable analysis of oscillatory signals.
  • The Airy Wavelets: The γ = 3 wavelet has zero demodulated time-domain asymmetry and remains nearly frequency-symmetric with near-optimal Heisenberg area at high time concentration.It preserves exact analyticity while retaining properties associated with the Morlet wavelet.
  • The Airy Wavelets: The generalized Morse family includes Cauchy and analytic Derivative of Gaussian wavelets, while Airy wavelets mark the approximate boundary between concave and convex behavior.The parameters β and γ separately govern decay, central-window width, and related filter properties.

APPENDIX A THE MORLET WAVELET

The Morlet appendix derives its peak frequency and higher-order frequency-domain properties from the carrier-frequency parameter. It also relates wavelet moments and cumulants through Bell-polynomial expansions.

  • Appendix A: The Morlet peak frequency ων differs from the carrier frequency ν and is found by setting the first frequency-domain derivative to zero.The resulting relation is solved numerically after introducing eν ≡ ν/ων.
  • Appendix A: The normalization function aν is chosen so that Ψν(ων) = 2.The parametric peak-frequency relation also determines aν as a function of carrier frequency.
  • Appendix A: Higher-order derivatives evaluated at the peak frequency determine additional Morlet wavelet properties, including duration.The appendix gives the normalized second derivative and wavelet duration.
  • Appendix A: As ν becomes large, the Morlet peak frequency satisfies ων ∼ ν.Thus peak and carrier frequencies become asymptotically similar in this limit.
  • Appendix A: Moments and cumulants are related using complete Bell polynomials, with M0;ψ = exp(K0;ψ) because the wavelet is not normalized as a probability density.A recursion relation provides the general conversion between moments and cumulants.

APPENDIX C CONVERGENCE OF MORSE MOMENT EXPANSION

The appendix establishes convergence properties for the generalized Morse wavelet moment expansion and derives higher-order frequency-domain derivative relationships. It also explains the special γ = 3 frequency behavior through a gamma-function ratio.

  • Convergence: The moment expansion converges for |t| < r, where r is the positive radius of convergence determined using the ratio test.For fixed (β, γ), the radius is finite when γ = 1 and infinite when γ > 1.
  • Convergence: For generalized Morse wavelets, the moment expansion has radius r = 1 when γ = 1 and infinite radius when γ > 1.
  • Frequency-domain derivatives: Taylor expansion and Bell-polynomial identities relate normalized wavelet derivatives to derivatives of the logarithm of the frequency-domain wavelet.The complete Bell polynomial Bn(c1, c2, . . . cn) is defined implicitly by the stated relationship.
  • Frequency-domain derivatives: At the peak frequency, the first normalized derivative is zero, while the second and third are −βγ and −βγ(γ − 3), respectively.The fourth normalized derivative is given as 3(βγ)^2 − βγ in the supplied expression.
  • Energy and peak frequencies: For γ = 3, the gamma-function ratio governing energy and peak frequencies remains close to unity for x ≥ 1, explaining their near-indistinguishability.The ratio rapidly approaches its asymptotic value as x increases, and its minimum departure from unity occurs near r = 1/3.
Loading 0802.2377v2…