Skip to main content

Sliding Mode Singular Spectrum Analysis for the Elimination of Cross-Terms in Wigner–Ville Distribution

Abstract

The Wigner–Ville distribution (WVD) is a signal processing approach to evaluate a high-resolution time–frequency representation (TFR) of a multi-component signal. The WVD of a multi-component signal produces unwanted cross-terms in the TFR. The elimination of these cross-terms using various signal processing techniques is a challenging research problem. In this paper, a data-driven signal decomposition technique is investigated for the elimination of these cross-terms in the WVD-based time–frequency representation of a multi-component signal. The approach is based on the decomposition of a multicomponent signal into its mono-components using sliding mode singular spectrum analysis (SM-SSA). The WVD of each mono-component is evaluated, and the sum of WVDs of all mono-components represents the cross-term free WVD representation of multi-component signals. Renyi entropy (RE) is used to quantify the performance of the proposed approach. Simulations are carried out using synthetic and real signals to verify the effectiveness of the proposed approach for the removal of cross-terms in WVD. The results demonstrated that SM-SSA has better performance with the lowest RE value as compared to other data-driven signal decomposition approaches such as automated SSA (AutoSSA) and swarm decomposition for the elimination of cross-terms from the WVD-based TFR of multi-component signals.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Data Availability Statement

The signal files and the codes of the proposed work are available upon request to the authors.

Notes

  1. 1.

    https://physionet.org/content/ptbdb/1.0.0/.

References

  1. 1.

    K.T. Andersen, M. Moonen, Adaptive time-frequency analysis for noise reduction in an audio filter bank with low delay. IEEE/ACM Trans. Audio Speech Lang. Process. 24(4), 784–795 (2016)

    Google Scholar 

  2. 2.

    G.K. Apostolidis, L.J. Hadjileontiadis, Swarm decomposition: a novel signal analysis using swarm intelligence. Signal Process. 132, 40–50 (2017)

    Google Scholar 

  3. 3.

    B. Boashash, Time-Frequency Signal Analysis and Processing: A Comprehensive Reference (Elsevier, Amsterdam, 2003)

    Google Scholar 

  4. 4.

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

    MATH  Google Scholar 

  5. 5.

    N. Baydar, A. Ball, A comparative study of acoustic and vibration signals in detection of gear failures using Wigner–Ville distribution. Mech. Syst. Signal Process. 15, 1091–1107 (2001)

    Google Scholar 

  6. 6.

    M. Bayram, R.G. Baraniuk, Multiple window time-frequency analysis, in Proceedings of Third International Symposium on Time-Frequency and Time-Scale Analysis (TFTS-96) (IEEE, 1996), pp. 173–176

  7. 7.

    B. Boashash, Time-Frequency Signal Analysis and Processing: A Comprehensive Reference (Academic Press, Cambridge, 2015)

    Google Scholar 

  8. 8.

    R. Bousseljot, D. Kreiseler, A. Schnabel, Nutzung der ekg-signaldatenbank cardiodat der ptb über das internet. Biomedizinische Technik/Biomed. Eng. 40(s1), 317–318 (1995)

    Google Scholar 

  9. 9.

    Y. Chai, X. Zhang, EMD-WVD time-frequency distribution for analysis of multi-component signals, in Fourth International Conference on Wireless and Optical Communications, vol. 9902 (International Society for Optics and Photonics, 2016), p. 99020W

  10. 10.

    V.C. Chen, H. Ling, Time-Frequency Transforms for Radar Imaging and Signal Analysis (Artech House, Boston, 2001)

    Google Scholar 

  11. 11.

    S.H. Cho, G. Jang, S.H. Kwon, Time-frequency analysis of power-quality disturbances via the Gabor–Wigner transform. IEEE Trans. Power Deliv. 25(1), 494–499 (2009)

    Google Scholar 

  12. 12.

    H.I. Choi, W.J. Williams, Improved time-frequency representation of multicomponent signals using exponential kernels. IEEE Trans. Acoust. Speech Signal Process. 37(6), 862–871 (1989)

    Google Scholar 

  13. 13.

    V. Cizek, Discrete Hilbert transform. IEEE Trans. Audio Electroacoust. 18(4), 340–343 (1970)

    Google Scholar 

  14. 14.

    T. Claasen, W. Mecklenbrauker, The Wigner distribution—A tool for time-frequency signal analysis. Philips J. Res. 35(3), 217–250 (1980)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    V. Climente-Alarcon, J.A. Antonino-Daviu, M. Riera-Guasp, M. Vlcek, Induction motor diagnosis by advanced notch FIR filters and the Wigner–Ville distribution. IEEE Trans. Ind. Electron. 61(8), 4217–4227 (2013)

    Google Scholar 

  16. 16.

    P. Dash, B. Panigrahi, G. Panda, Power quality analysis using s-transform. IEEE Trans. Power Deliv. 18(2), 406–411 (2003)

    Google Scholar 

  17. 17.

    Y. Ding, W. He, B. Chen, Y. Zi, I.W. Selesnick, Detection of faults in rotating machinery using periodic time-frequency sparsity. J. Sound Vib. 382, 357–378 (2016)

    Google Scholar 

  18. 18.

    P. Flandrin, O. Rioul, Affine smoothing of the Wigner–Ville distribution, in International Conference on Acoustics, Speech, and Signal Processing (IEEE, 1990), pp. 2455–2458

  19. 19.

    A. Gavrovska, V. Bogdanović, I. Reljin, B. Reljin, Automatic heart sound detection in pediatric patients without electrocardiogram reference via pseudo-affine Wigner–Ville distribution and Haar wavelet lifting. Comput. Methods Programs Biomed. 113, 515–528 (2014)

    Google Scholar 

  20. 20.

    A.L. Goldberger, L.A. Amaral, L. Glass, J.M. Hausdorff, P.C. Ivanov, R.G. Mark, J.E. Mietus, G.B. Moody, C.K. Peng, H.E. Stanley, Physiobank, physiotoolkit, and physionet: components of a new research resource for complex physiologic signals. Circulation 101(23), e215–e220 (2000)

    Google Scholar 

  21. 21.

    N. Golyandina, A. Zhigljavsky, Singular Spectrum Analysis for Time Series (Springer, Berlin, 2013)

    MATH  Google Scholar 

  22. 22.

    J. Han, M. van der Baan, Empirical mode decomposition for seismic time-frequency analysis. Geophysics 78(2), O9–O19 (2013)

    Google Scholar 

  23. 23.

    G. Hao, F. Tan, X. Hu, Y. Bai, Y. Lv, A matching pursuit-based method for cross-term suppression in WVD and its application to the ENPEMF. IEEE Geosci. Remote Sens. Lett. 16, 1304–1308 (2019)

    Google Scholar 

  24. 24.

    J. Harmouche, D. Fourer, F. Auger, P. Borgnat, P. Flandrin, The sliding singular spectrum analysis: a data-driven nonstationary signal decomposition tool. IEEE Trans. Signal Process. 66(1), 131–136 (2018). https://doi.org/10.1109/TSP.2017.2752720

    MathSciNet  Article  MATH  Google Scholar 

  25. 25.

    N.E. Huang, Z. Shen, S.R. Long, M.C. Wu, H.H. Shih, Q. Zheng, N.C. Yen, C.C. Tung, H.H. Liu, The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis, in Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 454 (The Royal Society, 1998), pp. 903–995

  26. 26.

    S. Jain, R. Panda, R.K. Tripathy, Multivariate sliding mode singular spectrum analysis for the decomposition of multisensor timeserie. IEEE Sens. Lett. 4(6), 1–4 (2020)

    Google Scholar 

  27. 27.

    A. Kareem, T. Kijewski, Time-frequency analysis of wind effects on structures. J. Wind Eng. Ind. Aerodyn. 90(12–15), 1435–1452 (2002)

    MATH  Google Scholar 

  28. 28.

    N.A. Khan, M. Sandsten, Time-frequency image enhancement based on interference suppression in Wigner–Ville distribution. Signal Process. 127, 80–85 (2016)

    Google Scholar 

  29. 29.

    N.A. Khan, I.A. Taj, M.N. Jaffri, S. Ijaz, Cross-term elimination in Wigner distribution based on 2D signal processing techniques. Signal Process. 91(3), 590–599 (2011)

    MATH  Google Scholar 

  30. 30.

    F. Li, R. Li, L. Tian, L. Chen, J. Liu, Data-driven time-frequency analysis method based on variational mode decomposition and its application to gear fault diagnosis in variable working conditions. Mech. Syst. Signal Process. 116, 462–479 (2019)

    Google Scholar 

  31. 31.

    Y. Li, Q. Liu, S.R. Tan, R.H. Chan, High-resolution time-frequency analysis of eeg signals using multiscale radial basis functions. Neurocomputing 195, 96–103 (2016)

    Google Scholar 

  32. 32.

    Y. Li, M.L. Luo, K. Li, A multiwavelet-based time-varying model identification approach for time-frequency analysis of EEG signals. Neurocomputing 193, 106–114 (2016)

    Google Scholar 

  33. 33.

    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 (2016)

    Google Scholar 

  34. 34.

    D.P. Mandic, N. ur Rehman, Z. Wu, N.E. Huang, Empirical mode decomposition-based time-frequency analysis of multivariate signals: the power of adaptive data analysis. IEEE Signal Process. Mag. 30(6), 74–86 (2013)

    Google Scholar 

  35. 35.

    Y. Meyer, Wavelets and Operators, vol. 1 (Cambridge University Press, Cambridge, 1992)

    MATH  Google Scholar 

  36. 36.

    R.B. Pachori, A. Nishad, Cross-terms reduction in the Wigner–Ville distribution using tunable-q wavelet transform. Signal Process. 120, 288–304 (2016)

    Google Scholar 

  37. 37.

    R.B. Pachori, P. Sircar, A new technique to reduce cross terms in the Wigner distribution. Digit. Signal Process. 17(2), 466–474 (2007)

    Google Scholar 

  38. 38.

    C.R. Pinnegar, L. Mansinha, The s-transform with windows of arbitrary and varying shape. Geophysics 68(1), 381–385 (2003)

    MATH  Google Scholar 

  39. 39.

    H. Ren, A. Ren, Z. Li, A new strategy for the suppression of cross-terms in pseudo Wigner–Ville distribution. SIViP 10(1), 139–144 (2016)

    Google Scholar 

  40. 40.

    S. Sanei, T.K. Lee, V. Abolghasemi, A new adaptive line enhancer based on singular spectrum analysis. IEEE Trans. Biomed. Eng. 59(2), 428–434 (2011)

    Google Scholar 

  41. 41.

    D.H. Schoellhamer, Singular spectrum analysis for time series with missing data. Geophys. Res. Lett. 28(16), 3187–3190 (2001)

    Google Scholar 

  42. 42.

    E. Sejdic, L. Stankovic, M. Dakovic, J. Jiang, Instantaneous frequency estimation using the s-transform. IEEE Signal Process. Lett. 15, 309–312 (2008)

    Google Scholar 

  43. 43.

    R.R. Sharma, A. Kalyani, R.B. Pachori, An empirical wavelet transform-based approach for cross-terms-free Wigner–Ville distribution. Signal Image Video Process. 14, 249–256 (2019). https://doi.org/10.1007/s11760-019-01549-7

    Article  Google Scholar 

  44. 44.

    R.R. Sharma, P. Meena, R.B. Pachori, Enhanced time-frequency representation based on variational mode decomposition and Wigner–Ville distribution, in Recent Trends in Image and Signal Processing in Computer Vision (Springer, 2020), pp. 265–284

  45. 45.

    R.R. Sharma, R. Pachori, Improved eigenvalue decomposition-based approach for reducing cross-terms in Wigner–Ville distribution. Circuits Syst. Signal Process. 37, 3330–3350 (2018). https://doi.org/10.1007/s00034-018-0846-0

    MathSciNet  Article  MATH  Google Scholar 

  46. 46.

    H. Singh, R.K. Tripathy, R.B. Pachori, Detection of sleep apnea from heart beat interval and ECG derived respiration signals using sliding mode singular spectrum analysis. Digit. Signal Process. 104, 102796 (2020)

    Google Scholar 

  47. 47.

    P. Sircar, S. Sharma, Complex FM signal model for non-stationary signals. Signal Process. 57(3), 283–304 (1997)

    MATH  Google Scholar 

  48. 48.

    L. Stanković, A measure of some time-frequency distributions concentration. Signal Process. 81(3), 621–631 (2001)

    MATH  Google Scholar 

  49. 49.

    L. Stankovic, M. Daković, T. Thayaparan, Time-Frequency Signal Analysis with Applications (Artech House, Boston, 2013)

    MATH  Google Scholar 

  50. 50.

    R.G. Stockwell, L. Mansinha, R. Lowe, Localization of the complex spectrum: the s transform. IEEE Trans. Signal Process. 44(4), 998–1001 (1996)

    Google Scholar 

  51. 51.

    R. Tripathy, M.R.A. Paternina, J.G. Arrieta, P. Pattanaik, Automated detection of atrial fibrillation ECG signals using two stage VMD and atrial fibrillation diagnosis index. J. Mech. Med. Biol. 17(07), 1740044 (2017)

    Google Scholar 

  52. 52.

    R. Tripathy, L. Sharma, S. Dandapat, Detection of shockable ventricular arrhythmia using variational mode decomposition. J. Med. Syst. 40(4), 79 (2016)

    Google Scholar 

  53. 53.

    R.K. Tripathy, M.R. Paternina, J.G. Arrieta, A. Zamora-Méndez, G.R. Naik, Automated detection of congestive heart failure from electrocardiogram signal using stockwell transform and hybrid classification scheme. Comput. Methods Programs Biomed. 173, 53–65 (2019)

    Google Scholar 

  54. 54.

    D. Waldo, P.R. Chitrapu, On the Wigner Ville distribution of finite duration signals. Signal Process. 24(2), 231–237 (1991)

    Google Scholar 

  55. 55.

    S. Wan, B. Peng, An integrated approach based on swarm decomposition, morphology envelope dispersion entropy, and random forest for multi-fault recognition of rolling bearing. Entropy 21(4), 354 (2019)

    MathSciNet  Google Scholar 

  56. 56.

    L. Wang, Z. Liu, Q. Miao, X. Zhang, Time-frequency analysis based on ensemble local mean decomposition and fast kurtogram for rotating machinery fault diagnosis. Mech. Syst. Signal Process. 103, 60–75 (2018)

    Google Scholar 

  57. 57.

    W. Yang, Z. Peng, K. Wei, P. Shi, W. Tian, Superiorities of variational mode decomposition over empirical mode decomposition particularly in time-frequency feature extraction and wind turbine condition monitoring. IET Renew. Power Gener. 11(4), 443–452 (2016)

    Google Scholar 

Download references

Funding

Funding was provided by Birla Institute of Technology and Science, Pilani (Grant No. FR/SCM/150618/EEE).

Author information

Affiliations

Authors

Corresponding author

Correspondence to R. K. Tripathy.

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

Verify currency and authenticity via CrossMark

Cite this article

Panda, R., Jain, S., Tripathy, R.K. et al. Sliding Mode Singular Spectrum Analysis for the Elimination of Cross-Terms in Wigner–Ville Distribution. Circuits Syst Signal Process 40, 1207–1232 (2021). https://doi.org/10.1007/s00034-020-01537-0

Download citation

Keywords

  • Wigner–Ville distribution
  • Cross-terms
  • Sliding mode singular spectrum analysis
  • Time–frequency representation
  • Renyi entropy