Advertisement

Generalized Ridge Reconstruction Approaches Toward more Accurate Signal Estimate

  • Xiangxiang Zhu
  • Zhuosheng ZhangEmail author
  • Hanqiu Zhang
  • Jinghuai Gao
  • Bei Li
Article
  • 31 Downloads

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.

Keywords

Multicomponent signal decomposition Ridge reconstruction Time–frequency representation Demodulation FM signals 

Notes

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.

References

  1. 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)CrossRefGoogle Scholar
  2. 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)CrossRefGoogle Scholar
  3. 3.
    R. Behera, S. Meignen, T. Oberlin, Theoretical analysis of the second order synchrosqueezing transform. Appl. Comput. Harmon. Anal. 45(2), 379–404 (2018)MathSciNetCrossRefGoogle Scholar
  4. 4.
    E.T. Bell, Exponential polynomials. Ann. Math. 35(2), 258–277 (1934)MathSciNetCrossRefGoogle Scholar
  5. 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)CrossRefGoogle Scholar
  6. 6.
    R.A. Carmona, W.L. Hwang, B. Torrésani, Multiridge detection and time–frequency reconstruction. IEEE Trans. Signal Process. 47, 480–492 (1999)CrossRefGoogle Scholar
  7. 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)CrossRefGoogle Scholar
  8. 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)CrossRefGoogle Scholar
  9. 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)CrossRefGoogle Scholar
  10. 10.
    L. Cohen, Time–Frequency Analysis (Prentice-Hall, Englewood Cliffs, 1995)Google Scholar
  11. 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)MathSciNetCrossRefGoogle Scholar
  12. 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)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Discovery of Sound in the Sea [Online]. Available: http://www.dosits.org/
  14. 14.
    I. Djurović, L. Stanković, Quasi-maximum-likelihood estimator of polynomial phase signals. IET Signal Process. 8(4), 347–359 (2013)CrossRefGoogle Scholar
  15. 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)CrossRefGoogle Scholar
  16. 16.
    D. Groth, S. Hartmann, S. Klie, J. Selbig, Principal components analysis. Comput. Toxicol. 2, 527–547 (2013)CrossRefGoogle Scholar
  17. 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)CrossRefGoogle Scholar
  18. 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)CrossRefGoogle Scholar
  19. 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)MathSciNetCrossRefGoogle Scholar
  20. 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)CrossRefGoogle Scholar
  21. 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)MathSciNetCrossRefGoogle Scholar
  22. 22.
    C. Li, M. Liang, A generalized synchrosqueezing transform for enhancing signal time–frequency representation. Signal Process. 92, 2264–2274 (2012)CrossRefGoogle Scholar
  23. 23.
    J.M. Lilly, S.C. Olhede, On the analytic wavelet transform. IEEE Trans. Inf. Theor. 56(8), 4135–4156 (2010)MathSciNetCrossRefGoogle Scholar
  24. 24.
    J. Lin, Ridges reconstruction based on inverse wavelet transform. J. Sound Vib. 294, 916–926 (2006)CrossRefGoogle Scholar
  25. 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. 98570NGoogle Scholar
  26. 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. C0073Google Scholar
  27. 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. 799–803Google Scholar
  28. 28.
    S. Mallat, A Wavelet Tour of Signal Processing: The Sparse Way, 3rd edn. (Academic Press, Burlington, 2009)zbMATHGoogle Scholar
  29. 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)MathSciNetCrossRefGoogle Scholar
  30. 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)CrossRefGoogle Scholar
  31. 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)MathSciNetCrossRefGoogle Scholar
  32. 32.
    D. Nelson, Instantaneous higher order phase derivatives. Dig. Signal Process. 12, 416–428 (2002)CrossRefGoogle Scholar
  33. 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)MathSciNetCrossRefGoogle Scholar
  34. 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)MathSciNetCrossRefGoogle Scholar
  35. 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)MathSciNetCrossRefGoogle Scholar
  36. 36.
    S. Pei, S. Huang, STFT with adaptive window width based on the chirp rate. IEEE Trans. Signal Process. 60, 4065–4080 (2012)MathSciNetCrossRefGoogle Scholar
  37. 37.
    S. Qian, D. Chen, Joint time–frequency analysis. IEEE Signal Process. Mag. 26(2), 52–67 (1999)CrossRefGoogle Scholar
  38. 38.
    Y. Qin, B. Tang, Y. Mao, Adaptive signal decomposition based on wavelet ridge and its application. Signal Process. 120, 480–494 (2016)CrossRefGoogle Scholar
  39. 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)CrossRefGoogle Scholar
  40. 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)CrossRefGoogle Scholar
  41. 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)CrossRefGoogle Scholar
  42. 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)CrossRefGoogle Scholar
  43. 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)MathSciNetCrossRefGoogle Scholar
  44. 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)CrossRefGoogle Scholar
  45. 45.
    G. Yu, M. Yu, C. Xu, Synchroextracting transform. IEEE Trans. Ind. Electron. 64(10), 8042–8054 (2017)CrossRefGoogle Scholar
  46. 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)CrossRefGoogle Scholar
  47. 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)CrossRefGoogle Scholar
  48. 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)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Xiangxiang Zhu
    • 1
  • Zhuosheng Zhang
    • 1
    Email author
  • Hanqiu Zhang
    • 2
  • Jinghuai Gao
    • 3
  • Bei Li
    • 1
  1. 1.School of Mathematics and StatisticsXi’an Jiaotong UniversityXi’anChina
  2. 2.Melbourne School of EngineeringThe University of MelbourneVictoriaAustralia
  3. 3.National Engineering Laboratory for Offshore Oil ExplorationXi’anChina

Personalised recommendations