Skip to main content
SpringerLink
Log in

Generalized Ridge Reconstruction Approaches Toward more Accurate Signal Estimate

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

Abstract

Ridge reconstruction (RR) method is one of the most commonly used ways for multicomponent signal reconstruction from time–frequency representations. However, this method leads to large reconstruction error when dealing with strongly amplitude-modulated and frequency-modulated (AM–FM) signals. In this paper, we first give the error analysis of RR method based on short-time Fourier transform when the amplitude and frequency modulations are not negligible. Then, two generalized ridge reconstruction approaches are proposed to overcome the limitations existing in the standard RR method. The first approach relies on a second-order local expansion of phase function, and the chirp rate is employed to improve the reconstruction. The second one is supported by the fact that RR is exact for pure sinusoidal signals; thus, demodulation operator is performed to facilitate the ridge reconstruction. A simple theoretical analysis of the proposed two approaches is provided. Numerical experiments on simulated and real signals demonstrate that the proposed approaches can obtain a more accurate signal estimate for strongly FM signals, being stable for the selection of window length and keeping a good noise robustness.

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

Similar content being viewed by others

References

  1. V.S. Amin, Y.D. Zhang, B. Himed, Sparsity-based time–frequency representation of FM signals with burst missing samples. Signal Process 155, 25–43 (2019)

    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)

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  4. E.T. Bell, Exponential polynomials. Ann. Math. 35(2), 258–277 (1934)

    MathSciNet  MATH  Google Scholar 

  5. R.A. Carmona, W.L. Hwang, B. Torrésani, Characterization of signals by the ridges of their wavelet transforms. IEEE Trans. Signal Process. 45, 2586–2590 (1997)

    Google Scholar 

  6. R.A. Carmona, W.L. Hwang, B. Torrésani, Multiridge detection and time–frequency reconstruction. IEEE Trans. Signal Process. 47, 480–492 (1999)

    Google Scholar 

  7. S. Chen, X. Dong, G. Xing, Z. Peng, W. Zhang, G. Meng, Separation of overlapped non-stationary signals by ridge path regrouping and intrinsic chirp component decomposition. IEEE Sens. J. 17(18), 5994–6005 (2017)

    Google Scholar 

  8. S. Chen, Y. Yang, K. Wei, X. Dong, Z. Peng, W. Zhang, Time-varying frequency-modulated component extraction based on parameterized demodulation and singular value decomposition. IEEE Trans. Instrum. Meas. 65(2), 276–285 (2016)

    Google Scholar 

  9. S. Chen, Z. Peng, Y. Yang, X. Dong, W. Zhang, Intrinsic chirp component decomposition by using Fourier series representation. Signal Process 137, 319–327 (2017)

    Google Scholar 

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

    Google Scholar 

  11. 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)

    MathSciNet  MATH  Google Scholar 

  12. N. Delprat, B. Escudie, P. Guillemain, R. Kronland-Martinet, P. Tchamitchian, B. Torrésani, Asymptotic wavelet and Gabor analysis: extraction of instantaneous frequencies. IEEE Trans. Inf. Theor. 38(2), 644–664 (1992)

    MathSciNet  MATH  Google Scholar 

  13. Discovery of Sound in the Sea [Online]. Available: http://www.dosits.org/

  14. I. Djurović, L. Stanković, Quasi-maximum-likelihood estimator of polynomial phase signals. IET Signal Process. 8(4), 347–359 (2013)

    Google Scholar 

  15. 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)

    Google Scholar 

  16. D. Groth, S. Hartmann, S. Klie, J. Selbig, Principal components analysis. Comput. Toxicol. 2, 527–547 (2013)

    Google Scholar 

  17. D. Iatsenko, P.V.E. McClintock, A. Stefanovska, Linear and synchrosqueezed time–frequency representations revisited: overview, standards of use, reconstruction, resolution, concentration, and algorithms. Dig. Signal Process. 42, 1–26 (2015)

    Google Scholar 

  18. D. Iatsenko, P.V.E. McClintock, A. Stefanovska, Extraction of instantaneous frequencies from ridges in time–frequency representations of signals. Signal Process. 125, 290–303 (2016)

    Google Scholar 

  19. C. Ioana, A. Jarrot, C. Gervaise, Y. Stephan, A. Quinquis, Localization in underwater dispersive channels using the time–frequency-phase continuity of signals. IEEE Trans. Signal Process. 58(8), 4093–4107 (2010)

    MathSciNet  MATH  Google Scholar 

  20. F. Jing, C. Zhang, W. Si, Y. Wang, S. Jiao, Polynomial phase estimation based on adaptive short-time Fourier transform. Sens. J. 18, 568–580 (2018)

    Google Scholar 

  21. N.A. Khan, M. Mohammadi, Reconstruction of non-stationary signals with missing samples using time–frequency filtering. Circuits Syst. Signal Process. 37(8), 3175–3190 (2018)

    MathSciNet  Google Scholar 

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

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  24. J. Lin, Ridges reconstruction based on inverse wavelet transform. J. Sound Vib. 294, 916–926 (2006)

    Google Scholar 

  25. S. Liu, Y.D. Zhang, T. Shan, S. Qin, M.G. Amin, Structure-aware Bayesian compressive sensing for frequency-hopping spectrum estimation, in Proceedings of SPIE 9857, Compressive Sensing V: From Diverse Modalities to Big Data Analytics (Baltimore, MD, 2016), p. 98570N

  26. S. Liu, Y. Ma, T. Shan, Segmented discrete polynomial-phase transform with coprime sampling, in Proceedings of 2018 IET International Radar Conference (Nanjing, China, 2018), p. C0073

  27. S. Liu, Z. Zeng, Y.D. Zhang, T. Fan, T. Shan, R. Tao, Automatic human fall detection in fractional Fourier domain for assisted living, in Proceedings of 2016 IEEE International Conference on Acoustics, Speech and Signal Processing (Shanghai, China, 2016), p. 799–803

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

    MATH  Google Scholar 

  29. S. Meignen, D.H. Pham, S. McLaughlin, On demodulation, ridge detection and synchrosqueezing for multicomponent signals. IEEE Trans. Signal Process. 65(8), 2093–2103 (2017)

    MathSciNet  MATH  Google Scholar 

  30. S. Meignen, T. Oberlin, P. Depalle, P. Flandrin, S. McLaughlin, Adaptive multimode signal reconstruction from time–frequency representations. Philos. Trans. R. Soc. A 374(2065), 20150205 (2016)

    Google Scholar 

  31. M. Mohammadi, A.A. Pouyan, N.A. Khan, V. Abolghasemi, Locally optimized adaptive directional time–frequency distributions. Circuits Syst. Signal Process. 37(3), 1223–1242 (2018)

    MathSciNet  Google Scholar 

  32. D. Nelson, Instantaneous higher order phase derivatives. Dig. Signal Process. 12, 416–428 (2002)

    Google Scholar 

  33. T. Oberlin, S. Meignen, V. Perrier, Second-order synchrosqueezing transform or invertible reassignment? Towards ideal time-frequency representations. IEEE Trans. Signal Process. 63(5), 1335–1344 (2015)

    MathSciNet  MATH  Google Scholar 

  34. S. Olhede, A.T. Walden, A generalized demodulation approach to time–frequency projections for multicomponent signals. Proc. R. Soc. A Math. Phys. Eng. Sci. 461, 2159–2179 (2005)

    MathSciNet  MATH  Google Scholar 

  35. 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)

    MathSciNet  MATH  Google Scholar 

  36. S. Pei, S. Huang, STFT with adaptive window width based on the chirp rate. IEEE Trans. Signal Process. 60, 4065–4080 (2012)

    MathSciNet  MATH  Google Scholar 

  37. S. Qian, D. Chen, Joint time–frequency analysis. IEEE Signal Process. Mag. 26(2), 52–67 (1999)

    Google Scholar 

  38. Y. Qin, B. Tang, Y. Mao, Adaptive signal decomposition based on wavelet ridge and its application. Signal Process. 120, 480–494 (2016)

    Google Scholar 

  39. 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)

    Google Scholar 

  40. L. Stanković, M. Brajović, Analysis of the reconstruction of sparse signals in the DCT domain applied to audio signals. IEEE/ACM Trans. Audio Speech Lang. Process. 26(7), 1220–1235 (2018)

    Google Scholar 

  41. S. Wang, G. Cai, Z. Zhu, W. Huang, X. Zhang, Transient signal analysis based on Levenberg–Marquardt method for fault feature extraction of rotating machines. Mech. Syst. Signal Process. 54, 16–40 (2015)

    Google Scholar 

  42. 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)

    Google Scholar 

  43. H. Xie, J. Lin, Y. Lei, Y. Liao, Fast-varying AM–FM components extraction based on an adaptive STFT. Dig. Signal Process. 22, 664–670 (2012)

    MathSciNet  Google Scholar 

  44. Y. Yang, X. Dong, Z. Peng, W. Zhang, G. Meng, Component extraction for non-stationary multi-component signal using parameterized de-chirping and band-pass filter. IEEE Signal Process. Lett. 22(9), 1373–1377 (2015)

    Google Scholar 

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

    Google Scholar 

  46. G. Yu, Z. Wang, P. Zhao, Z. Li, Local maximum synchrosqueezing transform: an energy-concentrated time–frequency analysis tool. Mech. Syst. Signal Process. 117, 537–552 (2019)

    Google Scholar 

  47. X. Zhu, Z. Zhang, J. Gao, W. Li, Two robust approaches to multicomponent signal reconstruction from STFT ridges. Mech. Syst. Signal Process. 115, 720–735 (2019)

    Google Scholar 

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

    Google Scholar 

Download references

Acknowledgements

This work is supported by the Major Research Plan of the National Natural Science Foundation of China (Grant No. 91730306) and National Key R and D Program of the Ministry of Science and Technology of China with the Project Integration Platform Construction for Joint Inversion and Interpretation of Integrated Geophysics (Grant No. 2018YFC0603501). X. Zhu thanks the China Scholarship Council for their support. The authors thank the anonymous reviewers for their valuable comments and suggestions that improve the quality of this paper.

Author information

Authors and Affiliations

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

    Xiangxiang Zhu, Zhuosheng Zhang & Bei Li

  2. Melbourne School of Engineering, The University of Melbourne, Victoria, 3010, Australia

    Hanqiu Zhang

  3. National Engineering Laboratory for Offshore Oil Exploration, Xi’an, 710049, China

    Jinghuai Gao

Authors
  1. Xiangxiang Zhu
    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. Hanqiu Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jinghuai Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Bei Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhuosheng Zhang.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhu, X., Zhang, Z., Zhang, H. et al. Generalized Ridge Reconstruction Approaches Toward more Accurate Signal Estimate. Circuits Syst Signal Process 39, 2574–2599 (2020). https://doi.org/10.1007/s00034-019-01278-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-019-01278-9

Keywords

Search

Navigation