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High-Order Synchrosqueezing Transform for Multicomponent Signals Analysis -- With an Application to Gravitational-Wave Signal
Duong-Hung Pham, Sylvain Meignen
TL;DR
Existing time-frequency methods and synchrosqueezing techniques face localization or scope limitations for multicomponent signals, especially under strong modulation. The paper generalizes STFT-based synchrosqueezing with higher-order amplitude and phase approximations for instantaneous-frequency estimation. Numerical experiments on simulated and real signals, including gravitational waves, show more concentrated representations and accurate mode reconstruction across a wider variety of AM-FM modes.
Problem
Existing time-frequency representations have limited simultaneous localization, while synchrosqueezing applicability is hindered by weak-modulation requirements and restrictive signal classes.
Method
The paper generalizes STFT-based synchrosqueezing by using higher-order approximations of amplitude and phase to estimate instantaneous frequencies.
Results
Numerical experiments on simulated and real signals show more concentrated time-frequency representations and successful handling of a wider variety of multicomponent signals.
Takeaways & Limitations
The approach supports concentrated analysis and mode reconstruction for AM-FM multicomponent signals, including gravitational-wave signals.
Abstract
from arXiv · showhide
This study puts forward a generalization of the short-time Fourier-based Synchrosqueezing Transform using a new local estimate of instantaneous frequency. Such a technique enables not only to achieve a highly concentrated time-frequency representation for a wide variety of AM-FM multicomponent signals but also to reconstruct their modes with a high accuracy. Numerical investigation on synthetic and gravitational-wave signals shows the efficiency of this new approach.
I. INTRODUCTION
The paper addresses limitations of time-frequency analysis and existing synchrosqueezing methods for strongly modulated multicomponent signals. It proposes a higher-order STFT-based synchrosqueezing approach intended to improve concentration and mode reconstruction across a wider range of AM-FM modes.
- The uncertainty principle limits simultaneous time and frequency localization in standard time-frequency representations.
- Reassignment sharpens time-frequency representations, but its transformed representation is no longer invertible and cannot reconstruct modes.
- SynchroSqueezing Transform improves concentration while retaining mode retrieval, but standard SST requires weak frequency modulation.
- Strongly modulated AM-FM modes occur in chirps from radar, speech processing, and gravitational-wave signals.
- The proposed method estimates instantaneous frequencies using higher-order amplitude and phase approximations.
- The resulting representation enables perfect concentration and reconstruction for a wider variety of AM-FM modes than earlier synchrosqueezing techniques.
B. STFT-based SST (FSST)
FSST sharpens STFT coefficients by reassigning them along the frequency axis using a local instantaneous-frequency estimate, while preserving approximate mode reconstruction. Its ridge-based recovery procedure requires assumptions about the modes and can involve difficult non-convex optimization.
- FSST sharpens the blurred STFT representation by reassigning coefficients to estimated instantaneous frequencies.
- The instantaneous-frequency estimate uses the argument and real part of complex quantities and a threshold for reassignment.
- Because reassignment occurs along the frequency axis, FSST preserves causality and supports approximate reconstruction of individual modes.
- Mode retrieval commonly relies on ridge extraction, assuming the transform and number of modes are known.
- The ridge energy functional balances phase smoothness and energy through regularization parameters, but its non-convexity makes implementation difficult.
- FSST theory assumes weak frequency modulation and well-separated component frequencies.
- Under its assumptions, FSST concentrates energy in narrow time-frequency bands and reconstructs modes with reasonably high accuracy.
C. Second Order STFT-based SST (FSST2)
FSST2 replaces the first-order frequency estimate with a second-order estimate derived from local modulation operators. It is exact for Gaussian-modulated linear chirps and can be computed from a small set of STFTs.
- FSST2 defines a second-order local modulation operator and uses it to construct an improved instantaneous-frequency estimate.The operator is formed from derivatives with respect to time of the reassignment operators.
- For a Gaussian-modulated linear chirp, ℜ{˜qt,f(t, η)} = φ′′(t).This establishes exact recovery of the phase curvature for the specified signal class.
- The operators ˜ωf, ˜τf, and ˜qt,f can be computed using only five STFTs.The required STFTs use windows derived from g and its derivatives or time-weighted variants.
- The corresponding frequency-domain operator ˜qη,f(t, η) has the same properties as the time-derivative-based operator.It is defined using partial derivatives with respect to η and supports the improved IF estimate.
- The improved estimate recovers φ′(t) for the Gaussian-modulated linear-chirp setting.The construction combines the reassignment frequency, modulation operator, and reassigned time coordinate.
III. HIGHER ORDER SYNCHROSQUEEZING TRANSFORM
The higher-order transform addresses the limited signal class for which FSST2 had been demonstrated. It extends the construction to AM-FM modes with higher-order amplitude and phase variation.
- FSST2 had only been demonstrated to work well on perturbed linear chirps with Gaussian-modulated amplitudes.This defines the scope boundary motivating the higher-order extension.
- The unresolved setting includes more general AM-FM modes with non-negligible higher-order phase derivatives.The limitation concerns modes beyond the previously studied chirp and amplitude assumptions.
- The proposed transform introduces synchrosqueezing operators based on approximation orders higher than three for both amplitude and phase.This is the central generalization presented in the section.
A. Nth-order IF Estimate
The Nth-order IF estimate uses high-order Taylor expansions of mode amplitude and phase, with recursively constructed modulation operators. Under the stated model, it recovers the instantaneous frequency.
- The new IF estimate is based on high-order Taylor expansions of the amplitude and phase.The mode model bounds the expansion order by L ≤ N.
- The first-order estimate ℜ{˜ωf(t, η)} does not equal φ′(t) when the higher-order terms have a nonzero real part.The higher-order modulation construction is introduced to account for these terms.
- The construction derives N − 1 local modulation operators recursively from STFT-based quantities differentiated with respect to frequency.Frequency differentiation is chosen because it leads to simpler expressions than time differentiation.
- The Nth-order local complex IF estimate is defined from these recursively obtained modulation operators.Its construction requires nonzero partial derivatives of the relevant intermediate quantities.
- For modes satisfying Definition III.1 with L ≤ N, Proposition III.2 gives φ′(t) from the Nth-order estimate.The result follows from canceling the higher-order amplitude and phase terms represented in the Taylor expansion.
B. Efficient Computation of Modulation Operators
The modulation operators are computed analytically from multiple STFTs rather than by discrete differentiation. This avoids numerical instability, especially in noisy signals, while giving an explicit STFT cost.
- Discrete differentiation should not be used to approximate the modulation operators because it can generate numerical instability, especially in noise.The paper instead recommends analytic expressions in terms of different STFTs.
- For N = 4, Proposition III.3 gives analytic formulas for the modulation operators with k = 2, 3, 4.The formulas are expressed through STFTs computed with time-weighted windows and related derivatives.
- For general N, the modulation operators can be computed using 3N − 1 STFTs.The paper derives this count by generalizing the explicit computation procedure.
C. Nth-order STFT-based SST (FSSTN)
The Nth-order FSST generalizes STFT-based synchrosqueezing by replacing the instantaneous-frequency estimate and supports mode reconstruction. Numerical experiments compare it with reassignment and existing FSST variants using concentration and Earth mover’s distance.
- The Nth-order FSST replaces the instantaneous-frequency estimate with an Nth-order estimate and defines the corresponding FSSTN operator.
- The method reconstructs multicomponent-signal modes by modifying the synchrosqueezed representation around the estimated instantaneous frequencies.
- The simulations use a two-component AM-FM signal sampled at 1024 Hz on [0, 1], with a polynomial chirp f1 and a strongly nonlinear damped-sine component f2.
- The Gaussian STFT window requires a time–frequency localization trade-off because an inappropriate length can impair ridge extraction and mode retrieval.
- FSST2 is sharp for f1, whereas FSST3 and FSST4 sharpen reassignment for f2; FSST4 is exact for f1 under Definition III.1 and therefore perfectly reassigns its STFT.
A. Evaluation of TF Concentration
The evaluation compares TF concentration using normalized energy and Earth mover’s distance across RM, FSST2, FSST3, and FSST4. FSST4 gives the strongest concentration for f2, while FSST3 and FSST4 are more accurate under low noise for f1 and remain beneficial at high noise.
- Normalized-energy evaluation: Normalized energy increases toward 1 faster when a TF representation is more concentrated.For the single-mode signal, the first M coefficients correspond to abscissa 1 because M is the sampling rate.
- Normalized-energy evaluation: FSST4 perfectly localizes f1’s energy because f1 obeys Definition III.1, but its advantage over the other methods is otherwise difficult to determine.The normalized-energy study alone does not clearly distinguish FSST3 and FSST4 from RM and FSST2 for f1.
- Normalized-energy evaluation: FSST4 concentrates f2’s normalized energy more than the other methods, while FSST3 also outperforms FSST2 and RM.
- Noisy-signal evaluation: The global EMD averages per-time 1D EMD between each resultant TF representation and the ideal one; smaller EMD indicates better concentration and fewer noise fluctuations.
- Noisy-signal evaluation: At low noise for f1, FSST3 and FSST4 are more accurate than the other methods, and FSST3 or FSST4 remains advantageous for f2 even at high noise.This EMD analysis reveals differences that normalized energy does not, including the continued importance of the proposed methods at high noise levels.
B. Evaluation of Mode Reconstruction Performance
The study evaluates mode reconstruction by extracting ridges from FSST representations and measuring output SNR. Using FSST3 and FSST4 improves reconstruction performance over lower-order alternatives.
- Reconstruction procedure: Mode reconstruction retrieves each mode from an FSST representation using ridge detection and information restricted to the detected ridge.The integer parameter d compensates for frequency-resolution inaccuracy, with d = 0 corresponding to ridge-only reconstruction.
- Reconstruction procedure: Output SNR is used to measure reconstruction accuracy for modes f1, f2, and f using FSST2, FSST3, or FSST4.
- Results: The improvement from FSST3 and FSST4 is clear and consistent with the preceding evaluation of TF-representation accuracy.
PERFORMANCE OF MODE RECONSTRUCTION IN THE NOISE-FREE CASE
For GW150914, the sharpened FSST2 and FSST4 representations make the signal’s single upward-sweeping mode easier to interpret. FSST4 improves ridge detection during the ringdown and produces a reconstruction much closer to an independently computed numerical-relativity waveform.
- Signal representation: The gravitational-wave signal is a transient generated by the coalescence of two stellar-mass black holes and contains one mode sweeping sharply upward.
- Signal representation: FSST2 and FSST4 sharpen the TF representation, making the gravitational-wave TF information easier to interpret.The improvement of high-order synchrosqueezing is not obvious from the initial TF display alone.
- Ridge detection: FSST4 detects the three collision stages better than FSST2, especially the ringdown, where the mode’s instantaneous-frequency curvature changes suddenly.The higher-order transform better accounts for this sudden curvature variation.
- Mode reconstruction: FSST4 reconstructs a waveform much closer to the numerical-relativity waveform than FSST2, with the comparison evaluated through residual l2-norm errors.The numerical-relativity waveform comes from an independent calculation.
- Conclusion: The experiments on simulated and real signals show that the proposed method produces more concentrated TF pictures than other synchrosqueezing or reassignment methods while improving invertibility.
- Conclusion: The paper identifies theoretical analysis under noisy signals, including non-Gaussian noise, as future work.
APPENDIX A
The appendix derives the modulation operators by rewriting the defining equations as an upper-triangular linear system. Back-substitution then recovers the operators and establishes their relation to phase derivatives.
- Matrix formulation: The derivation rewrites the relevant expression in matrix form and defines row vectors and the associated system variables.
- Back-substitution: The resulting matrix is upper triangular with nonzero diagonal coefficients, so back-substitution obtains r_k for k = 1, . . . , N.
- Operator formulation: The proof concludes by introducing notation G_k and G_j,k to rewrite the derived expressions for the new synchrosqueezing operators.