Skip to main content
SpringerLink
Log in

Time-Extracting Wavelet Transform for Characterizing Impulsive-Like Signals and Theoretical Analysis

  • Published:
Circuits, Systems, and Signal Processing Aims and scope Submit manuscript

Abstract

In this paper, a high-resolution time–frequency (TF) analysis method, called time-extracting wavelet transform (TEWT), is introduced to analyze impulsive-like signals whose TF ridge curves are nearly perpendicular to the time axis. For impulsive-like signals, the instantaneous frequency with almost infinite rate of change is difficult to estimate, but the group delay (GD) with nearly zero rate of change is easier to estimate. Since the GD is the key feature of frequency-domain signals, it indicates that one can try to understand impulsive-like signals from the perspective of frequency-domain signals. In this regard, for an impulsive signal and its Fourier transform (i.e., the frequency-domain harmonic signal), we propose the TEWT that achieves highly concentrated TF representations while allowing signal reconstruction, only by retaining the TF coefficients closely related to TF features of signals, while removing weakly related TF information. The two contributions of this paper are the proposal of TEWT and the theoretical analysis of TEWT for frequency-domain signals. On the other hand, we provide a rigorous theoretical analysis of TEWT under a mathematical framework for frequency-domain signals. Specifically, we define a function class as a set of all superposition of well-separated frequency-domain harmonic-like functions, where each function can be locally regarded as a sum of a finite number of harmonic signals in the frequency domain, and establish error bounds for WT approximate expression, GD estimation, and component recovery. Finally, we verify the effectiveness of TEWT in terms of the energy concentration, robustness, and invertibility through numerical experiments with synthetic and real signals.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14

Similar content being viewed by others

Data Availability

The datasets analyzed during the current study are available from the corresponding author on reasonable request.

References

  1. F. Auger, P. Flandrin, Improving the readability of time-frequency and time-scale representations by the reassignment method. IEEE Trans. Signal Process. 43(5), 1068–1089 (1995)

    Article  Google Scholar 

  2. F. Auger, P. Flandrin, Y. Lin, S. McLaughlin, S. Meignen, T. Oberlin, H.T. Wu, Time-frequency reassignment and synchrosqueezing: an overview. IEEE Signal Process. Mag. 30(6), 32–41 (2013)

    Article  Google Scholar 

  3. S. Aviyente, W.J. Williams, Minimum entropy time-frequency distributions. IEEE Signal Process. Lett. 12(1), 37–40 (2005)

    Article  Google Scholar 

  4. R.G. Baraniuk, P. Flandrin, A.J.E.M. Janssen, O.J.J. Michel, Measuring time-frequency information content using the Rényi entropies. IEEE Trans. Inf. Theory. 47, 1391–1409 (2001)

  5. R. Behera, S. Meignen, T. Oberlin, Theoretical analysis of the second order synchrosqueezing transform. Appl. Comput. Harmon. Anal. 45(2), 379–404 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  6. E. Chassande-Mottin, F. Auger, P. Flandrin, Time frequency/time scale reassignment. Wavelets and signal processing. 233–268 (2003)

  7. L. Cohen, Time-Frequency Analysis (Prentice-Hall, Englewood Cliffs, 1995)

    Google Scholar 

  8. I. Daubechies, J.F. Lu, H.T. Wu, Synchrosqueezed wavelet transforms: an empirical mode decomposition-like tool. Appl. Comput. Harmon. Anal. 30, 243–261 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. I. Daubechies, S. Maes, A nonlinear squeezing of the continuous wavelet transform. Wavelets in Medecine and Biology. 527–546 (1996)

  10. P. Flandrin, F. Auger, E. Chassande-Mottin, Time-frequency reassignment: from principles to algorithms, Applications in Time-Frequency Signal Processing (CRC, Arizona). 179–203 (2003)

  11. D. Fourer, F. Auger, P. Flandrin, Recursive versions of the Levenberg–Marquardt reassigned spectrogram and of the synchrosqueezed STFT, in Proc. IEEE ICASSP. 4880–4884 (2016)

  12. D. Fourer, F. Auger, K. Czarnecki, S. Meignen, P. Flandrin, Chirp rate and instantaneous frequency estimation: application to recursive vertical synchrosqueezing. IEEE Signal Process. Lett. 24(11), 1724–1728 (2017)

    Article  Google Scholar 

  13. B. Han, Y. Zhou, G. Yu, Second-order synchroextracting wavelet transform for nonstationary signal analysis of rotating machinery. Signal Process. 186, 108123 (2021)

    Article  Google Scholar 

  14. D. He, H. Cao, S. Wang, X. Chen, Time-reassigned synchrosqueezing transform: the algorithm and its applications in mechanical signal processing. Mech. Syst. Signal Process. 117, 255–279 (2019)

    Article  Google Scholar 

  15. Y. Hu, X. Tu, F. Li, High-order synchrosqueezing wavelet transform and application to planetary gearbox fault diagnosis. Mech. Syst. Signal Process. 131, 126–151 (2019)

    Article  Google Scholar 

  16. N.E. Huang, Z. Shen, S.R. Long, M.L. Wu, H.H. Shih, Q. Zheng, N.C. Yen, C.C. Tung, H.H. Liu, The empirical mode decomposition and Hilbert spectrum for nonlinear and non-stationary time series analysis. Proc. R. Soc. Lond. A. 454, 903–995 (1998)

  17. Z. Huang, J. Zhang, Z. Zou, Synchrosqueezing S-transform and its application in seismic spectral decomposition. IEEE Trans. Geosci. Remote Sens. 54(2), 817–825 (2016)

    Article  Google Scholar 

  18. K. Kodera, R. Gendrin, C. de Villedary, Analysis of time-varying signals with small BT values. IEEE Trans. Acoust. Speech Signal Process. ASSP. 26(1), 64–76 (1978)

  19. K. Kodera, C. de Villedary, R. Gendrin, A new method for the numerical analysis of nonstationary signals. Phys. Earth. Planet. Int. 12, 142–150 (1976)

    Article  Google Scholar 

  20. C. Li, M. Liang, A generalized synchrosqueezing transform for enhancing signal time–frequency representation. Signal Process. 92(9), 2264–2274 (2012)

    Article  Google Scholar 

  21. W. Li, F. Auger, Z. Zhang, X. Zhu, Self-matched extracting wavelet transform and signal reconstruction. Digital Signal Process. 128, 103602 (2022)

    Article  Google Scholar 

  22. W. Li, Z. Zhang, F. Auger, X. Zhu, Theoretical analysis of time-reassigned synchrosqueezing wavelet transform. Appl. Math. Lett. 132, 108141 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  23. Z. Li, J. Gao, H. Li et al., Synchroextracting transform: the theory analysis and comparisons with the synchrosqueezing transform. Signal Process. 166, 107243 (2020)

    Article  Google Scholar 

  24. J.M. Lilly, S.C. Olhede, On the analytic wavelet transform. IEEE Trans. Inf. Theor. 56(8), 4135–4156 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. N. Liu, J. Gao, X. Jiang, Z. Zhang, Q. Wang, Seismic time–frequency analysis via STFT-based concentration of frequency and time. IEEE Geosci. Remote Sens. Lett. 14(1), 127–131 (2017)

    Article  Google Scholar 

  26. S. Mallat, A Wavelet Tour of Signal Processing: The Sparse Way, 3rd edn. (Academic Press, Burlington, 2009)

    MATH  Google Scholar 

  27. Z.H. Michalopoulou, Underwater transient signal processing: Marine mammal identification, localization, and source signal deconvolution, in ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing-Proceedings. (1), 503–506 (1997)

  28. Z.K. Peng, G. Meng, F.L. Chu, Z.Q. Lang, W.M. Zhang, Y. Yang, Polynomial chirplet transform with application to instantaneous frequency estimation. IEEE Trans. Instrum. Meas. 60(9), 3222–3229 (2011)

    Article  Google Scholar 

  29. D.H. Pham, S. Meignen, High-order synchrosqueezing transform for multicomponent signals analysis-with an application to gravitational-wave signal. IEEE Trans. Signal Process. 65(12), 3168–3178 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  30. J. Pons-Llinares, J.A. Antonino-Daviu, M. Riera-Guasp, S. Bin Lee, T.J. Kang, C. Yang, Advanced induction motor rotor fault diagnosis via continuous and discrete time–frequency tools. IEEE Trans. Ind. Electron. 62(3), 1791–1802 (2015)

    Article  Google Scholar 

  31. J. Shi, M. Liang, D.S. Necsulescu, Y. Guan, Generalized stepwise demodulation transform and synchrosqueezing for time-frequency analysis and bearing fault diagnosis. J. Sound Vib. 368, 202–222 (2016)

    Article  Google Scholar 

  32. G. Thakur, H.T. Wu, Synchrosqueezing-based recovery of instantaneous frequency from nonuniform samples. SIAM J. Math. Anal. 43(5), 2078–2095 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  33. X. Tu, Y. Hu, F. Li, S. Abbas, Z. Liu, W. Bao, Demodulated high-order synchrosqueezing transform with application to machine fault diagnosis. IEEE Trans. Ind. Electron. 66(4), 3071–3081 (2018)

    Article  Google Scholar 

  34. S. Wang, X. Chen, G. Cai, B. Chen, X. Li, Z. He, Matching demodulation transform and synchrosqueezing in time–frequency analysis. IEEE Trans. Signal Process. 62(1), 69–84 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  35. S. Wang, X. Chen, I.W. Selesnick, Y. Guo, C. Tong, X. Zhang, Matching synchrosqueezing transform: a useful tool for characterizing signals with fast varying instantaneous frequency and application to machine fault diagnosis. Mech. Syst. Signal Process. 100, 242–288 (2018)

    Article  Google Scholar 

  36. G. Yu, A concentrated time–frequency analysis tool for bearing fault diagnosis. IEEE Trans. Instrum. Meas. 69(2), 371–381 (2019)

    Article  Google Scholar 

  37. K. Yu, H. Ma, H. Han, Second order multi-synchrosqueezing transform for rub-impact detection of rotor systems. Mech. Mach. Theory. 140, 321–349 (2019)

  38. G. Yu, Z. Wang, P. Zhao, Multi-synchrosqueezing transform. IEEE Trans. Ind. Electron. 66(7), 5441–5455 (2019)

    Article  Google Scholar 

  39. G. Yu, M. Yu, C. Xu, Synchroextracting transform. IEEE Trans. Ind. Electron. 64(10), 8042–8054 (2017)

    Article  Google Scholar 

  40. X. Zhu, Z. Zhang, J. Gao, Three-dimension extracting transform. Signal Process. 179, 107830 (2021)

    Article  Google Scholar 

  41. X. Zhu, Z. Zhang, J. Gao, B. Li, Z. Li, X. Huang, G. Wen, Synchroextracting chirplet transform for accurate IF estimate and perfect signal reconstruction. Dig. Signal Process. 93, 172–186 (2019)

    Article  Google Scholar 

  42. X. Zhu, Z. Zhang, Z. Li, J. Gao, X. Huang, G. Wen, Multiple squeezes from adaptive chirplet transform. Signal Process. 163, 26–40 (2019)

    Article  Google Scholar 

Download references

Acknowledgements

This work was supported by National Natural Science Foundation of China under Grant U20B2075, the ANR ASCETE project with Grant Number ANR-19-CE48-0001-01, and the Fundamental Research Funds for the Central Universities under Grant G2021KY05103. The authors thank the editor and the anonymous reviewers for their valuable comments and suggestions that helped improve the quality of this paper.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Xi’an Jiaotong University, 710049, Xi’an, China

    Wenting Li & Zhuosheng Zhang

  2. Institut de Recherche en Énergie Électrique de Nantes Atlantique (IREENA, UR 4642), Nantes Université, Saint-Nazaire, France

    François Auger

  3. School of Mathematics and Statistics, Northwestern Polytechnical University, 710049, Xi’an, China

    Xiangxiang Zhu

Authors
  1. Wenting Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhuosheng Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. François Auger
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Xiangxiang Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhuosheng Zhang.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, W., Zhang, Z., Auger, F. et al. Time-Extracting Wavelet Transform for Characterizing Impulsive-Like Signals and Theoretical Analysis. Circuits Syst Signal Process 42, 3873–3901 (2023). https://doi.org/10.1007/s00034-022-02253-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-022-02253-7

Keywords

Search

Navigation