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On the Analytic Wavelet Transform

Jonathan M. Lilly, Sofia C. Olhede

arXiv:0711.3834v3math.STmath.FAstat.ME

TL;DR

Wavelet-ridge analysis estimates modulated oscillations but becomes biased when amplitude and frequency modulation is non-negligible. This paper derives an exact analytic wavelet-transform representation using local modulation functions, then identifies how wavelet properties govern the resulting time-varying bias. The framework supports wavelet selection matched to signal variability and bias reduction.

  • Problem

    Time-dependent errors from moderate or strong modulation, and the wavelet conditions governing those errors, had not been examined despite their importance for ridge-based amplitude and frequency estimates.

  • Method

    The paper locally demodulates the analytic signal at its instantaneous frequency, expands it into instantaneous modulation functions, and derives the resulting analytic wavelet transform.

  • Results

    The analytic wavelet transform is expressed as interactions between increasingly higher-order signal modulation functions and frequency-domain derivatives of the wavelet.

  • Takeaways & Limitations

    The deviation terms provide conditions for matching wavelet properties to signal variability and quantifying time-varying ridge-estimation bias.

Abstract

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An exact and general expression for the analytic wavelet transform of a real-valued signal is constructed, resolving the time-dependent effects of non-negligible amplitude and frequency modulation. The analytic signal is first locally represented as a modulated oscillation, demodulated by its own instantaneous frequency, and then Taylor-expanded at each point in time. The terms in this expansion, called the instantaneous modulation functions, are time-varying functions which quantify, at increasingly higher orders, the local departures of the signal from a uniform sinusoidal oscillation. Closed-form expressions for these functions are found in terms of Bell polynomials and derivatives of the signal's instantaneous frequency and bandwidth. The analytic wavelet transform is shown to depend upon the interaction between the signal's instantaneous modulation functions and frequency-domain derivatives of the wavelet, inducing a hierarchy of departures of the transform away from a perfect representation of the signal. The form of these deviation terms suggests a set of conditions for matching the wavelet properties to suit the variability of the signal, in which case our expressions simplify considerably. One may then quantify the time-varying bias associated with signal estimation via wavelet ridge analysis, and choose wavelets to minimize this bias.

I. INTRODUCTION

The paper develops the analytic wavelet transform for modulated oscillations, motivated by the need to estimate time-varying signal properties without imposing a parametric model. It introduces analytic wavelets and wavelet-ridge estimation while framing modulation-dependent bias as the central problem.

  • Motivation and analytic signals: Modulated oscillations are represented through the analytic signal, whose amplitude, phase, instantaneous bandwidth, and instantaneous frequency describe time-varying signal properties.The analytic signal provides canonical amplitude and phase, while their rates of change define instantaneous bandwidth and frequency.
  • Motivation and analytic signals: Direct Hilbert-transform construction can fail for contaminated multicomponent signals because amplitude and phase reflect aggregate components rather than the signal of interest.The paper therefore seeks a method that isolates the desired component while preserving analyticity.
  • Wavelet-ridge estimation: Wavelet ridge analysis uses localized complex wavelet transforms to estimate properties of an underlying analytic signal, but localization introduces bias when modulation is substantial.Earlier negligible-modulation conditions are described as a strong constraint for many real-world signals.
  • Wavelet-ridge estimation: The paper targets time-dependent errors from moderate or strong modulation and uses them to guide wavelet choices that minimize estimation bias.This extends earlier weak-modulation error bounds and focuses on interpreting ridge-based amplitude and frequency estimates.
  • Analytic wavelet transform: Analytic wavelets vanish at negative frequencies, and the analytic wavelet transform projects signals onto rescaled and translated wavelets indexed by time and scale.The wavelet is normalized so that its peak-frequency transform magnitude equals the sinusoid amplitude for a pure cosine.
  • Analytic wavelet transform: Wavelet duration and frequency-domain derivatives provide dimensionless properties used to characterize wavelet behavior and its interaction with modulated signals.The duration measure is related to the number of peak-frequency oscillations within the wavelet’s central time window.

C. A General Family of Analytic Wavelets

Generalized Morse wavelets form a broad, exactly analytic two-parameter family whose duration and higher-order frequency-domain properties can be varied separately. Wavelet-ridge analysis evaluates the transform along amplitude or phase ridge curves, but modulation creates time-varying estimation bias.

  • C. A General Family of Analytic Wavelets: Generalized Morse wavelets are exactly analytic and controlled by positive parameters β and γ, which generate a broad range of wavelet characteristics.The family includes the Cauchy or Klauder wavelets when γ = 1 and analytic Derivative of Gaussian wavelets when γ = 2.
  • C. A General Family of Analytic Wavelets: The parameters β and γ independently vary the dimensionless duration Pβ,γ and the third-order frequency-domain derivative eΨ3;β,γ(ωβ,γ).This separates second-order duration properties from third-order asymmetry-related properties.
  • C. A General Family of Analytic Wavelets: Increasing β with γ = 3 makes the wavelet more oscillatory in time and more tightly peaked in frequency as Pβ,γ increases.The third-order derivative vanishes in this row of examples.
  • C. A General Family of Analytic Wavelets: With Pβ,γ fixed, changing γ and β alters long-time behavior and shifts frequency-domain enhancement from the right to the left of the peak as γ increases.The examples illustrate separate control of second- and third-order wavelet properties.
  • Wavelet ridge analysis: Amplitude ridge points select local maxima of transform magnitude at each time, whereas phase ridge points match transform phase rate to the frequency associated with scale.Ridge points are grouped into continuous ridge curves, and transform values along a ridge form the signal estimate.
  • Wavelet ridge analysis: For non-negligible modulation, the time-varying error and wavelet-selection conditions remain unresolved by earlier results that considered only vanishing modulation strength.The paper frames these unresolved effects as bias in ridge-based estimation.

E. Application to Oceanographic Data

The paper applies wavelet ridge analysis to an oceanographic oscillation and develops a local modulation expansion to interpret signal variability. Three wavelets produce reasonable estimates, but their residual variability and smoothness differ with wavelet duration.

  • E. Application to Oceanographic Data: The example uses eastward velocity from a freely drifting subsurface oceanographic float, whose oscillatory structure reflects an oceanic vortex.The record has been used to infer properties of the modulated oscillation.
  • E. Application to Oceanographic Data: Three wavelet transforms based on the wavelets from Fig. 1a–c yield ridge-based signal estimates and residuals for the differentiated position record.Time derivatives are displayed to emphasize oscillatory structure rather than lower-frequency meandering.
  • E. Application to Oceanographic Data: 74/21 = 9.2 cycles are used for Fig. 2a, b, and c, and each transform contains one ridge extending nearly throughout the record.The ridge structure is consistent across all three cases.
  • E. Application to Oceanographic Data: All three estimates appear reasonable, but residual variability and estimated-signal smoothness increase with Pβ,γ.A major low-frequency fluctuation near t = 0 is missed by the smoothest estimate.
  • A. A Local Representation: The local modulation expansion represents an analytic signal as increasingly higher-order departures from a constant-amplitude, constant-frequency sinusoid.Instantaneous modulation functions are defined by demodulating with the local instantaneous frequency and taking dimensionless local-time derivatives.

B. The Instantaneous Modulation Functions

The paper defines instantaneous modulation functions by locally demodulating the analytic signal and expanding its departures from a uniform oscillation. Bell-polynomial expressions combine bandwidth and derivatives of complex instantaneous frequency, while examples show how modulation variability affects higher-order terms.

  • The complex instantaneous frequency combines instantaneous bandwidth and frequency through the time derivative of ln x+(t).
  • Complete Bell polynomials provide closed-form expressions for instantaneous modulation functions in terms of bandwidth and the first n − 1 derivatives of complex instantaneous frequency.
  • The nth modulation function combines powers of bandwidth with derivatives of complex instantaneous frequency to measure nth-order departure from uniform oscillation.
  • Instantaneous modulation functions are dimensionless local derivatives of the demodulated analytic signal and quantify departures from a pure oscillation at successive orders.
  • Rapid fluctuations in instantaneous frequency or bandwidth can prevent higher-order modulation functions from becoming negligible.
  • In the examples, longer-duration wavelets produce smoother estimates, while amplitude modulation remains generally small relative to instantaneous frequency.
  • Because bandwidth is small relative to instantaneous frequency, the second-order modulation contribution from eρ1(t) is minor, while bandwidth and frequency derivatives contribute comparably.

E. Signal Variability

The paper uses instantaneous modulation functions to quantify local signal stability and to match wavelet properties to signal variability. The resulting suitability criteria constrain wavelet duration and frequency-domain derivatives, with generalized Morse wavelets favoring a specific parameter range.

  • The local stability level δNT quantifies a signal’s departure from a uniform oscillation through order NT over a time interval.
  • When δNT ≪1, the analytic wavelet transform admits a greatly simplified representation and supports closed-form modulation effects.
  • Wavelet suitability criteria match frequency-domain derivatives of the wavelet to the signal’s local stability level and constrain appropriate wavelet choices.
  • At n = 2, the criterion implies Pψ = |eΨ2|1/2 ≤ 2/δNT, limiting wavelet duration relative to signal variability or limiting frequency localization.
  • The criteria constrain odd wavelet moments more tightly than even moments, while the first derivative vanishes at the peak frequency and the third is the lowest relevant odd derivative.
  • For generalized Morse wavelets, normalized derivatives decay rapidly and remain below unity for 1 ≤ γ ≤6, unlike many cases outside that range.
  • Choosing Pβ,γ = 2/δNT and β > 1 with 1 ≤γ ≤6 satisfies the stated suitability criteria for the corresponding local stability level.

IV. ANALYSIS OF MODULATED OSCILLATIONS

The paper derives an exact analytic wavelet transform representation for highly variable modulated signals by combining local signal expansions with wavelet frequency-domain derivatives. The theorem exposes a hierarchy of signal–wavelet interactions and supports interpretation of ridge-based estimation and localized analyticity.

  • The AWT representation makes explicit how the analytic signal interacts with the analyzing wavelet for potentially highly variable signals.
  • The derivation expands the locally demodulated analytic signal, transforms time-domain moments into frequency-domain derivatives, and handles finite truncation with wavelet decay conditions.
  • The theorem shows that each higher-order wavelet frequency derivative interacts with a corresponding time-varying instantaneous modulation function.
  • Unlike earlier weak-modulation treatments, the representation resolves nonlinear modulation terms for a broader range of local signal behavior.
  • Evaluating the AWT along the instantaneous frequency curve defines a localized analytic signal that reflects joint signal and wavelet properties.
  • The localized analytic signal is a non-uniform, nonlinear filtering whose wavelet scale changes with the local instantaneous period.
  • Localization can compromise exact analyticity because interactions between modulation functions and wavelet terms may distribute energy to negative frequencies.

E. Bias of the Localized Analytic Signal

The localized analytic signal differs from the true analytic signal through a hierarchy of modulation–wavelet interactions. Matching wavelet duration and derivative properties to signal stability controls these bias terms.

  • Wavelet suitability: Wavelet suitability keeps higher-order modulation interactions small, so localized-signal deviations diminish with increasing order.The expansion resolves interactions between instantaneous modulation functions and frequency-domain wavelet derivatives.
  • Assumptions: The analysis assumes a signal stability level over a time interval and wavelets satisfying the stated suitability criteria.The truncation level is constrained by the signal’s differentiability and affects the resulting stability level.
  • Leading bias term: The leading deviation is governed by wavelet duration and the signal’s second-order instantaneous modulation function, rather than its first-order function.The lowest-order contribution is associated with eρ2(t) because the first wavelet derivative vanishes at the peak frequency.
  • Amplitude and phase bias: Amplitude and phase deviations scale with the curvature of the corresponding quantity over the wavelet’s time support.Both are underestimated at local maxima and overestimated at local minima.
  • Wavelet design: Choosing a wavelet requires simultaneously limiting the duration-related term and higher-order derivative terms, including frequency-domain asymmetry.For sufficiently small δNT, higher-order contributions can be neglected relative to the leading duration term.
  • Transform representation: The AWT representation theorem separates the transform near an instantaneous-frequency curve into scale-independent and scale-dependent perturbations.The scale-independent term contains the first-order contribution, while the scale-dependent term begins at second order in δNT.

B. Expressions for the Ridge-Based Signal Estimates

Amplitude and phase ridge curves provide second-order approximations to the instantaneous-frequency curve, while their signal estimate recovers the localized analytic signal to the leading order. In practice, amplitude ridges are favored because phase ridges can break under strong modulation.

  • Ridge curves: Both amplitude and phase ridge conditions have unique solutions within the second-order instantaneous-frequency neighborhood.Their perturbation terms begin at second order in δNT, with no first-order terms under the suitability criteria.
  • Ridge curves: Amplitude and phase ridge curves differ except for special wavelet choices, although both deviate from the instantaneous-frequency curve only at higher order.Their deviations from each other and from the instantaneous-frequency curve are higher-order than the localized-signal bias.
  • Signal estimate: The ridge-based signal estimate is identical to the localized analytic signal apart from a residual term.Thus its leading error comes from the localized analytic signal’s scale-independent departure from the true analytic signal.
  • Practical caveat: Amplitude ridges are preferred in practice because phase ridges more often break at isolated points under particularly strong modulation.The authors note that this performance difference may depend on their numerical implementation.
  • Signal estimate: Estimated amplitude and phase match those of the localized analytic signal up to second order in δNT.The ridge estimate is formed by evaluating the transform along the estimated ridge curves.

C. Instantaneous Frequency and Bandwidth Estimation

The ridge framework estimates instantaneous frequency and bandwidth through transform derivatives evaluated before ridge lookup. Instantaneous frequency is recovered more faithfully than bandwidth at leading perturbation order, while frequency estimation includes a joint modulation effect.

  • Practical limitations: Direct estimates formed from discrete scale levels are unsatisfactory, and numerical differentiation of estimated amplitude and phase tends to be noisy.The transform-based derivative quantities avoid reversing the differentiation and ridge-lookup operations.
  • Estimation method: Transform instantaneous frequency and bandwidth are obtained by differentiating before lookup along the ridge curve.This ordering is presented as a better estimation procedure than differentiating quantities after ridge lookup.
  • Estimation accuracy: The instantaneous-frequency estimate is perturbed at second order in δNT, whereas the bandwidth estimate is perturbed at first order.The bandwidth’s first-order perturbation follows because υ(t)/ω(t) is itself first order.
  • Estimation accuracy: The ridge-based instantaneous-frequency estimate is identical at leading order for amplitude and phase ridges.This differs from direct instantaneous-frequency estimates, which differ at order δNT^2.
  • Frequency-modulation effect: The additional term in the estimated instantaneous frequency reflects the joint effect of contemporaneous amplitude and frequency modulation.It arises from motion of the instantaneous-frequency curve across scales at fixed time.
  • Modulation-function estimation: The second-order instantaneous modulation function can be estimated along a ridge with relative error beginning at O(δNT).The paper also illustrates these estimates using three different wavelets.

D. Application

The application evaluates how local stability and wavelet suitability govern ridge-based estimates, then uses these results to guide wavelet choice for modulated signals.

  • Application: Ridge estimates are close to true signal properties when local stability δNT is small and the wavelet is suitably chosen.The estimates include the analytic signal, amplitude, phase, instantaneous frequency, bandwidth, and second-order modulation function.
  • Application: The mean ratios |eρ2(t)|/(Pβ,γ/4) are 0.54, 0.74, and 2.10 for the three estimates, while the medians are 0.47, 0.57, and 1.74.Because suitability requires this ratio to be below unity, the third-column wavelets are too long and the estimate is expected to be poor.
  • Application: 0.040, 0.036, and 0.057 are the mean iterated relative deviations for the three estimates, with corresponding medians of 0.024, 0.014, and 0.022.The calculation treats each estimated signal in turn as the true signal.
  • Implications for Choice of Wavelet: For generalized Morse wavelets, 1 ≤ γ ≤ 6 satisfies the suitability conditions, while γ = 3 removes the third-order wavelet-derivative contribution.The next contribution for γ = 3 involves fourth-order signal variability, making second-order expansions particularly accurate.
  • Implications for Choice of Wavelet: The leading-order error is minimized by making the wavelet as short as possible, but noise requires sufficient duration for stabilization and bandpass behavior.The paper therefore identifies competing constraints on Pβ,γ.
  • Implications for Choice of Wavelet: Highly variable signals with δNT of order unity are problematic because the shortest suitable generalized Morse wavelet still spans one full cycle.The paper notes that such signals are not aptly described as modulated oscillations.
  • Discussion: The exact AWT representation uses instantaneous modulation functions to quantify departures from constant-amplitude, constant-frequency oscillations.The results connect these signal functions with wavelet derivatives for analyzing non-negligible modulation.
  • Discussion: Leading-order ridge-estimation error comes from wavelet smoothing along the instantaneous frequency curve, rather than ridge-type differences.Amplitude, phase, bandwidth, and frequency can be estimated faithfully when modulation is not too strong and the wavelet is appropriate.

APPENDIX I A FREELY DISTRIBUTED SOFTWARE PACKAGE

This appendix documents the software package accompanying the paper and develops the residual bounds used for the AWT representation theorem.

  • Software Package: The associated Jlab Matlab toolbox includes routines for instantaneous frequency, bandwidth, second-order modulation, generalized Morse wavelets, and figure generation.Jsignal provides wavelet ridge-analysis routines designed for large data sets.
  • Residual Construction: The AWT expansion is formed by inserting the local modulation expansion into the wavelet transform and separating the finite summation from the residual.The residual is defined as the difference between the transform of the signal and the transform of the truncated expansion.
  • Residual Construction: The truncation level must satisfy N ≤ rψ − 2 so the transformed polynomial terms remain square integrable under the wavelet’s long-time decay.The representation theorem follows by combining the expansion and associated definitions.
  • Residual Bounds: Residual bounds split the transform into inner and outer time ranges using an energy level α and wavelet half-width Lψ(α).The outer contribution is negligible when α is sufficiently close to unity, while the terms are antagonistic.
  • Residual Bounds: The three residual components are bounded using triangle and Cauchy–Schwarz inequalities, together with positive constants controlling the wavelet terms.These bounds combine to bound the full residual.

APPENDIX III PROOF OF THE AWT SCALE DEVIATION EXPANSION

This appendix derives the AWT scale-deviation expansion by expanding normalized wavelet derivatives and powers of the scale-frequency deviation.

  • Scale-Deviation Expansion: The derivation inserts the Taylor series of normalized wavelet derivatives into the AWT representation theorem.The expansion is centered at the wavelet peak frequency.
  • Scale-Deviation Expansion: Expanding powers of sω(t)/ωψ with the binomial theorem produces the triple summation for the AWT scale deviation.Terms through m = 2 and n = 2 are written explicitly, while higher-order terms are collected in εψ,3(t,s) and the ellipses.
  • Scale-Deviation Expansion: The first wavelet derivative vanishes at the peak frequency, and Ψ(ωψ) is normalized to 2.These peak-frequency properties simplify the displayed expansion.

APPENDIX IV BOUNDING THE DIFFERENTIATED RESIDUAL

This appendix bounds time and scale derivatives of the transform residual by representing them with differentiated wavelets and applying the residual-bound construction.

  • Differentiated Residual: Time and scale derivatives of the residual are analyzed through differentiated transform terms denoted Uψ(t,s) and Vψ(t,s).The appendix introduces bounds for their numerators before constructing the modified-wavelet representation.
  • Differentiated Residual: Throughout the analysis, sω(t)/ωψ is restricted to remain near unity, and the orders of the derivative residual terms are then evaluated.The time- and scale-derivative calculations use the same residual-bounding strategy as the undifferentiated transform.
  • Differentiated Residual: The modified wavelets have Fourier transforms Θ(ω) = −i(ω/ωψ)Ψ(ω) and Φ(ω) = ω dΨ(ω)/dω.This incorporates differentiation into the wavelet while retaining the original signal in the transform integrands.
  • Differentiated Residual: The modified wavelets require finite energy and admissibility conditions inherited from the original wavelet’s Fourier transform.The appendix also assumes truncation N ≤ rθ − 2 and N ≤ rϕ − 2 based on their long-time decay.
  • Differentiated Residual: The local modulation expansion is inserted into the differentiated transforms, making the resulting summation terms derivatives of the original AWT expansion.Differentiated residuals are then defined from these corresponding expansions.

APPENDIX V PROOFS OF THE FORMS OF THE RIDGE CURVES

The appendix derives amplitude- and phase-ridge curves by expanding the analytic wavelet transform near an instantaneous-frequency curve. The derivation controls residual terms and establishes that ridge solutions lie within a specified neighborhood.

  • The derivation assumes the ridge deviation order before obtaining expressions for the ridge curves.
  • The ridge curve lies in the 2-neighborhood of the instantaneous frequency curve, and the ridge equations have solutions within that neighborhood.
  • Amplitude ridges: Amplitude ridges are derived by expanding the logarithm of the analytic wavelet transform, retaining the squared term and truncating at NT = 3.
  • Phase ridges: The phase-ridge derivation defines a complex-valued transform quantity analogous to the signal’s complex instantaneous frequency and differentiates the wavelet transform.
  • Phase ridges: With truncation level NT = 2, scale differentiation and residual expansions yield terms whose bracketed remainder is second order in δNT.
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