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Circuits, Systems, and Signal Processing

, Volume 37, Issue 8, pp 3330–3350 | Cite as

Improved Eigenvalue Decomposition-Based Approach for Reducing Cross-Terms in Wigner–Ville Distribution

  • Rishi Raj Sharma
  • Ram Bilas Pachori
Article

Abstract

In this work, a novel data-driven methodology is proposed to reduce cross-terms in the Wigner–Ville distribution (WVD) using improved eigenvalue decomposition of the Hankel matrix (IEVDHM). The IEVDHM method decomposes a multi-component non-stationary (NS) signal into mono-component NS signals. After that, amplitude-based segmentation is applied to obtain the components which are separated in time domain. Further, frequency modulation (FM) rate of components is observed to achieve an adaptive window. The adaptive window successfully removes intra-cross-terms which are generated due to nonlinearity present in FM. Finally, the sum of WVD of all the components is considered the WVD of the multi-component NS signal. The simulation study has been carried out on synthetic and natural signals to show the effectiveness of the proposed method. Performance of the proposed method is compared with the existing methods. We have also evaluated the performance of the proposed method in additive white Gaussian noise environment. The normalized Renyi entropy measure is computed to show the efficacy of the proposed method and all the compared methods for obtaining time–frequency representation.

Keywords

Eigenvalue decomposition Hankel matrix Wigner–Ville distribution Cross-terms Time–frequency representation 

Notes

Compliance with Ethical Standards

Conflict of interest

All authors declare that they have no conflict of interest.

References

  1. 1.
    S.S. Abeysekera, B. Boashash, Methods of signal classification using the images produced by the Wigner-Ville distribution. Pattern Recognit. Lett. 12, 717–729 (1991)CrossRefGoogle Scholar
  2. 2.
    J.P. Amezquita-Sanchez, H. Adeli, A new music-empirical wavelet transform methodology for time-frequency analysis of noisy nonlinear and non-stationary signals. Digit. Signal Process. 45, 55–68 (2015)CrossRefGoogle Scholar
  3. 3.
    M.G. Amin, D. Borio, Y.D. Zhang, L. Galleani, Time-frequency analysis for GNSSs: from interference mitigation to system monitoring. IEEE Signal Process. Mag. 34, 85–95 (2017)CrossRefGoogle Scholar
  4. 4.
    G. Andria, M. Savino, Interpolated smoothed pseudo Wigner-Ville distribution for accurate spectrum analysis. IEEE Trans. Instrum. Meas. 45, 818–823 (1996)CrossRefGoogle Scholar
  5. 5.
    F. Auger P. Flandrin, P. Gonçalvès, O. Lemoine, Time-frequency toolbox. CNRS France-Rice University 46 (1996)Google Scholar
  6. 6.
    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)CrossRefGoogle Scholar
  7. 7.
    A. Bhattacharyya, R.B. Pachori, A multivariate approach for patient-specific EEG seizure detection using empirical wavelet transform. IEEE Trans. Biomed. Eng. 64, 2003–2015 (2017)CrossRefGoogle Scholar
  8. 8.
    A. Bhattacharyya, L. Singh, R.B. Pachori, Fourier-Bessel series expansion based empirical wavelet transform for analysis of non-stationary signals. Digit. Signal Process. 78, 185–196 (2018)CrossRefGoogle Scholar
  9. 9.
    B. Boashash, Time-Frequency Signal Analysis and Processing: a Comprehensive Reference (Elsevier, New York, 2003)Google Scholar
  10. 10.
    B. Boashash, P. Black, An efficient real-time implementation of the Wigner-Ville distribution. IEEE Trans. Acoust. Speech Signal Process. 35, 1611–1618 (1987)CrossRefGoogle Scholar
  11. 11.
    J. Brynolfsson, M. Sandsten, Classification of one-dimensional non-stationary signals using the Wigner-Ville distribution in convolutional neural networks. in 2017 25th European Signal Processing Conference (IEEE, 2017), pp. 326–330Google Scholar
  12. 12.
    J. Burriel-Valencia, R. Puche-Panadero, J. Martinez-Roman, A. Sapena-Bano, M. Pineda-Sanchez, Short-frequency Fourier transform for fault diagnosis of induction machines working in transient regime. IEEE Trans. Instrum. Meas. 66, 432–440 (2017)CrossRefGoogle Scholar
  13. 13.
    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. 99020WGoogle Scholar
  14. 14.
    V.C. Chen, H. Ling, Time-Frequency Transforms for Radar Imaging and Signal Analysis (Artech House, Norwood, 2002)zbMATHGoogle Scholar
  15. 15.
    H.I. Choi, W.J. Williams, Improved time-frequency representation of multicomponent signals using exponential kernels. IEEE Trans. Acoust. Speech Signal Process. 37, 862–871 (1989)CrossRefGoogle Scholar
  16. 16.
    T.A.C.M. Claasen, W.F.G. Mecklenbrauker, The Wigner distribution- A tool for time-frequency signal analysis. Part I: continuous-time signals. Philips J. Res. 35(3), 217–250 (1980)MathSciNetzbMATHGoogle Scholar
  17. 17.
    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, 4217–4227 (2014)CrossRefGoogle Scholar
  18. 18.
    L. Cohen, Time-frequency distributions-a review. Proc. IEEE 77, 941–981 (1989)CrossRefGoogle Scholar
  19. 19.
    L. Faping, W. Hongxing, L. Xiao, L. Chuanhui, K. Jiafang, M. Xingji, Time-frequency characteristics of PSWF with Wigner–Ville Distributions. in IEEE International Conference on Signal and Image Processing (IEEE, 2016), pp. 568–572Google Scholar
  20. 20.
    P. Flandrin, B. Escudié, An interpretation of the pseudo-Wigner-Ville distribution. Signal Process. 6, 27–36 (1984)CrossRefGoogle Scholar
  21. 21.
    P. Flandrin, O. Rioul, Affine smoothing of the Wigner–Ville distribution. in 1990 International Conference on Acoustics, Speech, and Signal Processing (IEEE, 1990), pp. 2455–2458Google Scholar
  22. 22.
    C.J. Gaikwad, P. Sircar, Reduced interference Wigner–Ville time frequency representations using signal support information. in 2016 IEEE Annual India Conference( IEEE, 2016), pp. 1–5Google Scholar
  23. 23.
    C.J. Gaikwad, P. Sircar, Bispectrum-based technique to remove cross-terms in quadratic systems and Wigner-Ville distribution. Signal Image Video Process. 12, 703–710 (2018)CrossRefGoogle Scholar
  24. 24.
    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)CrossRefGoogle Scholar
  25. 25.
    J. Gilles, Empirical wavelet transform. IEEE Trans. Signal Process. 61, 3999–4010 (2013)MathSciNetCrossRefGoogle Scholar
  26. 26.
    P. Gonçalves, R.G. Baraniuk, Pseudo affine Wigner distributions: definition and kernel formulation. IEEE Trans. Signal Process. 46, 1505–1516 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    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. 903–995Google Scholar
  28. 28.
    N.E. Huang, Z. Wu, A review on Hilbert-Huang transform: method and its applications to geophysical studies. Rev. Geophys. 46(2) (2008)Google Scholar
  29. 29.
    P. Jain, R.B. Pachori, Time-order representation based method for epoch detection from speech signals. J. Intell. Syst. 21, 79–95 (2012)Google Scholar
  30. 30.
    P. Jain, R.B. Pachori, GCI identification from voiced speech using the eigen value decomposition of Hankel matrix. in 2013 8th International Symposium on Image and Signal Processing and Analysis (2013), pp. 371–376Google Scholar
  31. 31.
    P. Jain, R.B. Pachori, Marginal energy density over the low frequency range as a feature for voiced/non-voiced detection in noisy speech signals. J. Frankl. Inst. 350, 698–716 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    P. Jain, R.B. Pachori, Event-based method for instantaneous fundamental frequency estimation from voiced speech based on eigenvalue decomposition of the Hankel matrix. IEEE/ACM Trans. Audio Speech Lang. Process. 22, 1467–1482 (2014)CrossRefGoogle Scholar
  33. 33.
    P. Jain, R.B. Pachori, An iterative approach for decomposition of multi-component non-stationary signals based on eigenvalue decomposition of the Hankel matrix. J. Frankl. Inst. 352, 4017–4044 (2015)CrossRefGoogle Scholar
  34. 34.
    Q. Jiang, B.W. Suter, Instantaneous frequency estimation based on synchrosqueezing wavelet transform. Signal Process. 138, 167–181 (2017)CrossRefGoogle Scholar
  35. 35.
    S. Kadambe, G.F. Boudreaux-Bartels, A comparison of the existence of ‘cross terms’ in the Wigner distribution and the squared magnitude of the wavelet transform and the short-time Fourier transform. IEEE Trans. Signal Process. 40, 2498–2517 (1992)CrossRefzbMATHGoogle Scholar
  36. 36.
    N.A. Khan, M. Sandsten, Time-frequency image enhancement based on interference suppression in Wigner-Ville distribution. Signal Process. 127, 80–85 (2016)CrossRefGoogle Scholar
  37. 37.
    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, 590–599 (2011)CrossRefzbMATHGoogle Scholar
  38. 38.
    C. Li, M. Liang, A generalized synchrosqueezing transform for enhancing signal time-frequency representation. Signal Process. 92, 2264–2274 (2012)CrossRefGoogle Scholar
  39. 39.
    S. Liu, G. Tang, X. Wang, Y. He, Time-frequency analysis based on improved variational mode decomposition and Teager energy operator for rotor system fault diagnosis. Math. Probl. Eng. 2016, (2016)Google Scholar
  40. 40.
    W. Liu, Auto term window method and its parameter selection. Measurement 46, 3113–3118 (2013)CrossRefGoogle Scholar
  41. 41.
    W. Liu, S. Cao, Y. Chen, Seismic time-frequency analysis via empirical wavelet transform. IEEE Geosci. Remote Sens. Lett. 13, 28–32 (2016)CrossRefGoogle Scholar
  42. 42.
    X. Liu, Y. Jia, Z. He, J. Zhou, Application of EMD-WVD and particle filter for gearbox fault feature extraction and remaining useful life prediction. J. Vibroeng. 19(3) (2017)Google Scholar
  43. 43.
    X. Lv, M. Xing, S. Zhang, Z. Bao, Keystone transformation of the Wigner-Ville distribution for analysis of multicomponent LFM signals. Signal Process. 89, 791–806 (2009)CrossRefzbMATHGoogle Scholar
  44. 44.
    Y. Meyer, Wavelets and Operators, vol. 1 (Cambridge University Press, Cambridge, 1995)zbMATHGoogle Scholar
  45. 45.
    S.V. Narasimhan, A.R. Haripriya, B.K.S. Kumar, Improved Wigner-Ville distribution performance based on DCT/DFT harmonic wavelet transform and modified magnitude group delay. Signal Process. 88, 1–18 (2008)CrossRefzbMATHGoogle Scholar
  46. 46.
    S.V. Narasimhan, M.B. Nayak, Improved Wigner-Ville distribution performance by signal decomposition and modified group delay. Signal Process. 83, 2523–2538 (2003)CrossRefzbMATHGoogle Scholar
  47. 47.
    M.B. Nayak, S. Narasimhan, Autoregressive modeling of the Wigner-Ville distribution based on signal decomposition and modified group delay. Signal Process. 84, 407–420 (2004)CrossRefzbMATHGoogle Scholar
  48. 48.
    J.M. OToole, Choi-williams method code. http://otoolej.github.io/code/fast-tfds/ (Accessed 16 Apr 2018)
  49. 49.
    R.B. Pachori, A. Nishad, Cross-terms reduction in the Wigner-Ville distribution using tunable-Q wavelet transform. Signal Process. 120, 288–304 (2016)CrossRefGoogle Scholar
  50. 50.
    R.B. Pachori, P. Sircar, A new technique to reduce cross terms in the Wigner distribution. Digit. Signal Process. 17, 466–474 (2007)CrossRefGoogle Scholar
  51. 51.
    R.B. Pachori, P. Sircar, Time-frequency analysis using time-order representation and Wigner distribution. in 2008 IEEE Region 10 Conference (2008), pp. 1–6Google Scholar
  52. 52.
    R.B. Pachori, P. Sircar, Analysis of multicomponent AM-FM signals using FB-DESA method. Digit. Signal Process. 20, 42–62 (2010)CrossRefGoogle Scholar
  53. 53.
    D. Ping, P. Zhao, B. Deng, Cross-terms suppression in Wigner-Ville distribution based on image processing. in 2010 IEEE International Conference on Information and Automation (IEEE, 2010), pp. 2168–2171Google Scholar
  54. 54.
    H. Ren, A. Ren, Z. Li, A new strategy for the suppression of cross-terms in pseudo Wigner-Ville distribution. Signal Image Video Process. 10, 139–144 (2016)CrossRefGoogle Scholar
  55. 55.
    F. Sattar, G. Salomonsson, The use of a filter bank and the Wigner-Ville distribution for time-frequency representation. IEEE Trans. Signal Process. 47, 1776–1783 (1999)CrossRefzbMATHGoogle Scholar
  56. 56.
    E. Sejdić, I. Djurović, J. Jiang, Time-frequency feature representation using energy concentration: an overview of recent advances. Digit. Signal Process. 19, 153–183 (2009)CrossRefGoogle Scholar
  57. 57.
    R.R. Sharma, P. Chandra, R.B. Pachori, Electromyogram signal analysis using eigenvalue decomposition of the Hankel matrix. in International Conference on Machine Intelligence and Signal Processing, 22–24 Dec 2017Google Scholar
  58. 58.
    R.R. Sharma, M. Kumar, R.B. Pachori, Automated CAD identification system using time-frequency representation based on eigenvalue decomposition of ECG signals. in International Conference on Machine Intelligence and Signal Processing, 22–24 Dec, 2017Google Scholar
  59. 59.
    R.R. Sharma, R.B. Pachori, A new method for non-stationary signal analysis using eigenvalue decomposition of the Hankel matrix and Hilbert transform. in Fourth International Conference on Signal Processing and Integrated Networks (IEEE, 2017), pp. 484–488Google Scholar
  60. 60.
    R.R. Sharma, R.B. Pachori, Time-frequency representation using IEVDHM-HT with application to classification of epileptic EEG signals. IET Sci. Meas. Technol. 12, 72–82 (2017)CrossRefGoogle Scholar
  61. 61.
    R.R. Sharma, R.B. Pachori, Eigenvalue decomposition of Hankel matrix based time-frequency representation for complex signals (Syst. Signal Process, Circuits, 2018).  https://doi.org/10.1007/s00034-018-0834-4 Google Scholar
  62. 62.
    P. Shi, W. Yang, M. Sheng, M. Wang, An enhanced empirical wavelet transform for features extraction from wind turbine condition monitoring signals. Energies 10, 972 (2017)CrossRefGoogle Scholar
  63. 63.
    Y.S. Shin, J.J. Jeon, Pseudo Wigner-Ville time-frequency distribution and its application to machinery condition monitoring. Shock Vib. 1, 65–76 (1993)CrossRefGoogle Scholar
  64. 64.
    P. Singh, S.D. Joshi, R.K. Patney, K. Saha, The Fourier decomposition method for nonlinear and non-stationary time series analysis. Proc. R. Soc. A 473, 20160,871 (2017)Google Scholar
  65. 65.
    L. Stankovic, A method for time-frequency analysis. IEEE Trans. Signal Process. 42, 225–229 (1994)CrossRefGoogle Scholar
  66. 66.
    L. Stanković, A measure of some time-frequency distributions concentration. Signal Process. 81, 621–631 (2001)CrossRefzbMATHGoogle Scholar
  67. 67.
    R.G. Stockwell, L. Mansinha, R.P. Lowe, Localization of the complex spectrum: the S transform. IEEE Trans. Signal Process. 44, 998–1001 (1996)CrossRefGoogle Scholar
  68. 68.
    B.P. Tang, F. Li, W.Y. Liu, Using ASTFT spectrum to suppress cross terms in WVD and its application in fault diagnosis, Materials Science Forum, vol. 626 (Trans Tech Publ, Zurich, 2009), pp. 535–540Google Scholar
  69. 69.
    A. Upadhyay, R.B. Pachori, Instantaneous voiced/non-voiced detection in speech signals based on variational mode decomposition. J. Frankl. Inst. 352, 2679–2707 (2015)CrossRefGoogle Scholar
  70. 70.
    A. Upadhyay, R.B. Pachori, Speech enhancement based on mEMD-VMD method. Electron. Lett. 53, 502–504 (2017)CrossRefGoogle Scholar
  71. 71.
    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)CrossRefGoogle Scholar
  72. 72.
    S. Wang, X. Chen, Y. Wang, G. Cai, B. Ding, X. Zhang, Nonlinear squeezing time-frequency transform for weak signal detection. Signal Process. 113, 195–210 (2015)CrossRefGoogle Scholar
  73. 73.
    Y. Wang, K.C. Veluvolu, Time-frequency analysis of non-stationary biological signals with sparse linear regression based Fourier linear combiner. Sensors 17, 1386 (2017)CrossRefGoogle Scholar
  74. 74.
    Y. Wu, X. Li, Elimination of cross-terms in the Wigner-Ville distribution of multi-component LFM signals. IET Signal Process. 11, 657–662 (2017)CrossRefGoogle Scholar
  75. 75.
    M. Xing, R. Wu, Y. Li, Z. Bao, New ISAR imaging algorithm based on modified Wigner-Ville distribution. IET Radar Sonar Navig. 3, 70–80 (2009)CrossRefGoogle Scholar
  76. 76.
    Y.S. Yan, C.C. Poon, Zhang Yt, Reduction of motion artifact in pulse oximetry by smoothed pseudo Wigner-Ville distribution. J. NeuroEng. Rehabil. 2, 3 (2005)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Discipline of Electrical EngineeringIndian Institute of Technology IndoreIndoreIndia

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