The synchrosqueezing transform (SST) is a time-frequency technique that sharpens the continuous wavelet transform by reallocating its energy along estimated instantaneous frequencies, and it is widely used as an alternative to empirical mode decomposition (EMD) for extracting oscillatory modes from nonstationary signals. Wu, Flandrin and Daubechies (2011) analyzed when two components with close frequencies are recognized and represented as two (rather than fused into one, or corrupted by beating) for an idealized two-harmonic signal and a Gaussian-like wavelet, explicitly framed as an oversimplified model. The generalization problem left open there is to establish a predictive, quantitative theory of mode separation by SST for realistic signals — components with time-varying amplitudes and nonlinear instantaneous frequencies, general wavelet profiles, additive noise, and finite samples — including sharp resolution criteria, error quantification of the extracted modes, principled choice of the wavelet support width, and criterion-based comparison with EMD-type decompositions.

Background

Many physical and engineering signals are nonstationary superpositions of oscillatory components — vibration responses of structures, physiological waves, seismic traces, audio. For such signals the classical Fourier frequency is inadequate, and one instead wants the instantaneous frequency (IF), i.e. the time-varying frequency of each component, together with its amplitude. The empirical mode decomposition (EMD) of Huang, Shen, Long et al. (1998) addresses this by iteratively sifting a signal into a small number of intrinsic mode functions, oscillatory components to which the Hilbert transform then assigns an IF; the resulting time-frequency-amplitude description is the Hilbert spectrum. EMD is fully data-adaptive but has no precise definition of what it computes, so its behavior on borderline inputs must be studied empirically.

A canonical probe is the two-tone test: given f(t)=A1cos⁡(ω1t)+A2cos⁡(ω2t)f(t)=A_1\cos(\omega_1 t)+A_2\cos(\omega_2 t) with ω1\omega_1 and ω2\omega_2 close, does the method output two components or one? Rilling and Flandrin (2008) answered this for EMD, showing that EMD separates the two tones only when their frequency gap exceeds a method-dependent critical threshold, and that below it the decomposition merges them or produces beating-like amplitude fluctuations (beating: the slow periodic swelling and fading of the summed amplitude that two close tones inevitably produce, since their phase difference drifts through cycles of constructive and destructive interference). The question 'one or two frequencies?' thus became a benchmark for decomposition methods.

The synchrosqueezing transform was introduced by Daubechies, Lu and Wu (2011) as an EMD-like tool with a firmer footing. The continuous wavelet transform (CWT) represents a signal at each time across scales, using a translated and dilated mother wavelet (in practice often a Morlet wavelet, a windowed complex exponential whose effective time support width is controlled by a parameter, here denoted Δ\Delta). Synchrosqueezing is a reassignment-type nonlinear post-processing: from the CWT phase it estimates the local instantaneous frequency and squeezes the diffuse time-frequency energy onto that frequency curve, producing a concentrated, invertible representation from which individual components and their IFs can be read off. Daubechies, Lu and Wu proved that for well-separated modes with slowly varying amplitudes and IFs, SST reconstructs the components and their IFs accurately.

Wu, Flandrin and Daubechies (2011) then ran the two-tone experiment for synchrosqueezing, in deliberate parallel to the EMD study of Rilling and Flandrin. Working with an idealized two-harmonic model and a Gaussian-like wavelet profile, they characterized when two close frequencies are recognized and represented as two — quantifying, in the borderline regime, the transition to beating-type artifacts and the attenuation of the recovered components, and measuring the reconstruction fidelity by an error metric of the form e(Sf,Sf1)e(S_f,S_{f_1}), comparing the synchrosqueezed representation SfS_f of the full two-component signal with the representation Sf1S_{f_1} of the isolated first component. The mother wavelet's support width Δ\Delta enters as the knob trading frequency resolution against temporal localization and robustness.

The analysis, however, is tied to its idealized setting — fixed amplitudes, fixed harmonic frequencies, and a simple Gaussian-like wavelet — a simplification the authors themselves acknowledge. Real signals have components with time-varying amplitudes and curving (nonlinear) instantaneous frequencies, are observed in noise and only at finite sampling rates, and are analyzed with non-Gaussian wavelets; for these, the conditions under which SST genuinely separates close components, and how large the resulting reconstruction error is, are not covered by the two-harmonic theory. The open problem is precisely this generalization.

Problem Statement

Establish a quantitative, predictive separation theory for the wavelet-based synchrosqueezing transform on realistic multicomponent signals, generalizing the idealized two-harmonic analysis of Wu, Flandrin and Daubechies (2011). The common objective is to determine, by explicit criteria rather than case-by-case simulation, when synchrosqueezing reports two components as two and when it merges them or corrupts them with beating-type artifacts. Concretely: (a) derive resolution criteria — ideally necessary and sufficient, at minimum sharp quantitative sufficient conditions — for the separation of two or more components with time-varying amplitudes and nonlinear instantaneous frequencies, for general (non-Gaussian) mother wavelet profiles, together with bounds of the e(Sf,Sf1)e(S_f,S_{f_1})-type error between the synchrosqueezed representation of the full signal and that of the isolated component; (b) characterize how additive noise and finite-sample discretization shift these criteria and the achievable reconstruction error; (c) give a principled (fixed or adaptive) rule for selecting the mother wavelet support width Δ\Delta that balances frequency resolution against robustness, tied to the criteria in (a) and (b); and (d) use these criteria to delimit the signal regimes in which synchrosqueezing's separation behavior compares favorably with empirical mode decomposition and related decomposition methods. The four components (a)-(d) describe one predictive separation criterion: each supplies a part (threshold, degradation, parameter choice, comparative scope) of it, and none of them is currently available in the published literature beyond the idealized two-tone / parallel-chirp cases under a Gaussian-windowed Gabor transform.

References

  1. Wu, Hau-Tieng; Flandrin, Patrick; Daubechies, Ingrid (2011). One or two frequencies? The synchrosqueezing answers. Advances in Adaptive Data Analysis 3(1-2), 15-39. DOI: 10.1142/S179353691100074X. https://doi.org/10.1142/S179353691100074X
  2. Rilling, G.; Flandrin, P. (2008). One or two frequencies? The empirical mode decomposition answers. IEEE Transactions on Signal Processing 56(1), 85-95. DOI: 10.1109/TSP.2007.906771. https://doi.org/10.1109/TSP.2007.906771
  3. Huang, Norden E.; Shen, Zheng; Long, Steven R.; Wu, Manli C.; Shih, Hsing H.; Zheng, Quanan; Yen, Nai-Chyuan; Tung, Chi Chao; Liu, Henry H. (1998). The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454(1971), 903-995. DOI: 10.1098/rspa.1998.0193. https://doi.org/10.1098/rspa.1998.0193
  4. Daubechies, Ingrid; Lu, Jianfeng; Wu, Hau-Tieng (2011). Synchrosqueezed wavelet transforms: An empirical mode decomposition-like tool. Applied and Computational Harmonic Analysis 29(3), 275-293. DOI: 10.1016/j.acha.2010.08.002. https://doi.org/10.1016/j.acha.2010.08.002
  5. Meignen, Sylvain; Laurent, Nils; Oberlin, Thomas (2022). One or two ridges? An exact mode separation condition for the Gabor transform. IEEE Signal Processing Letters 29, 1816-1820. DOI: 10.1109/LSP.2022.3226948. https://doi.org/10.1109/LSP.2022.3226948
  6. Oberlin, Thomas; Meignen, Sylvain; Perrier, Valerie (2015). Second-order synchrosqueezing transform or invertible reassignment? Towards ideal time-frequency representations. IEEE Transactions on Signal Processing 63(3), 879-891. DOI: 10.1109/TSP.2015.2391077. https://doi.org/10.1109/TSP.2015.2391077
  7. Behera, Ratikanta; Meignen, Sylvain; Oberlin, Thomas (2018). Theoretical analysis of the second-order synchrosqueezing transform. Applied and Computational Harmonic Analysis 45(2), 379-404. DOI: 10.1016/j.acha.2016.11.001. https://doi.org/10.1016/j.acha.2016.11.001
  8. Yang, Haizhao (2018). Statistical analysis of synchrosqueezed transforms. Applied and Computational Harmonic Analysis 45(2), 326-352. DOI: 10.1016/J.ACHA.2017.01.001. https://doi.org/10.1016/J.ACHA.2017.01.001
  9. Li, Lin; Cai, Haiyan; Jiang, Qingtang (2020). Adaptive synchrosqueezing transform with a time-varying parameter for non-stationary signal separation. Applied and Computational Harmonic Analysis 49(2), 692-714. DOI: 10.1016/J.ACHA.2019.06.002. https://doi.org/10.1016/J.ACHA.2019.06.002
  10. Daubechies, Ingrid; Wang, Yi (Grace); Wu, Hau-Tieng (2016). ConceFT: concentration of frequency and time via a multitapered synchrosqueezed transform. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 374(2065), 20150193. DOI: 10.1098/rsta.2015.0193. https://doi.org/10.1098/rsta.2015.0193
  11. Yang, Haizhao (2015). Synchrosqueezed wave packet transforms and diffeomorphism based spectral analysis for 1D general mode decompositions. Applied and Computational Harmonic Analysis 39(3), 533-570. DOI: 10.1016/J.ACHA.2014.08.004. https://doi.org/10.1016/J.ACHA.2014.08.004
  12. Wu, Hau-Tieng (2013). Instantaneous frequency and wave shape functions (I). Applied and Computational Harmonic Analysis 35(2), 141-174. DOI: 10.1016/J.ACHA.2012.08.008. https://doi.org/10.1016/J.ACHA.2012.08.008
  13. Iatsenko, Dmytro; McClintock, Peter V. E.; Stefanovska, Aneta (2015). Linear and synchrosqueezed time–frequency representations revisited: Overview, standards of use, resolution, reconstruction, concentration, and algorithms. Digital Signal Processing 42, 1-24. DOI: 10.1016/j.dsp.2015.03.004. https://doi.org/10.1016/j.dsp.2015.03.004
  14. Cicone, Antonio; Serra-Capizzano, Stefano; Zhou, Haomin (2024). One or two frequencies? The iterative filtering answers. Applied Mathematics and Computation 460, 128322. DOI: 10.1016/j.amc.2023.128322. https://doi.org/10.1016/j.amc.2023.128322
  15. Lostanlen, Vincent; Cohen-Hadria, Alice; Bello, Juan Pablo (2021). One or two frequencies? The scattering transform answers. Proceedings of the 28th European Signal Processing Conference (EUSIPCO 2020), 1943-1947. DOI: 10.23919/eusipco47968.2020.9287216. https://doi.org/10.23919/eusipco47968.2020.9287216
  16. Chand, Shrikant; Nolen, James; Wu, Hau-Tieng (2026). On spectral interference of the short-time Fourier transform and its nonlinear variations. arXiv:2601.10910 [math.CA, math.NA]. DOI: 10.48550/arXiv.2601.10910. https://arxiv.org/abs/2601.10910