# Enhanced Time–Frequency Representation Based on Variational Mode Decomposition and Wigner–Ville Distribution

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## Abstract

The Wigner–Ville distribution (WVD) gives a very high-resolution time–frequency distribution but diminishes due to the existence of cross-terms. The cross-terms suppression in WVD is crucial to get the actual energy distribution in time–frequency (TF) plane. This chapter proposes a method to remove both inter and intra cross-terms from TF distribution obtained using WVD. The variational mode decomposition is applied to decompose a multicomponent signal into corresponding mono-components and inter cross-terms are suppressed due to the separation of mono-components. Thereafter, segmentation is applied in time domain to remove intra cross-terms present due to nonlinearity in frequency modulation. The obtained components are processed to get WVD of each component. Finally, all the collected WVDs are added to get complete time–frequency representation. Efficacy of the proposed method is checked using Renyi entropy measure over one synthetic and two natural signals (bat echo sound and speech signal) in clean and noisy environment. The method presented works well and gives better results in comparison to the WVD and pseudo WVD techniques.

## Notes

### Acknowledgements

The authors wish to thank Curtis Condon, Ken White, and Al Feng of the Beckman Institute of the University of Illinois for the bat data and for permission to use it in this chapter.

## References

- 1.B. Boashash,
*Time-Frequency Signal Analysis and Processing: A Comprehensive Reference*(Elsevier, Amsterdam, 2003)Google Scholar - 2.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 (1998), pp. 903–995MathSciNetCrossRefGoogle Scholar - 3.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*(2017), pp. 484–488Google Scholar - 4.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 - 5.L. Stankovic, M. Dakovic, T. Thayaparan,
*Time-Frequency Signal Analysis with Applications*(Artech House, Norwood, 2013)Google Scholar - 6.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 Processcess.
**40**, 2498–2517 (1992)CrossRefGoogle Scholar - 7.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 - 8.Y. Meyer,
*Wavelets and Operators*, vol. 1 (Cambridge University Press, Cambridge, 1995)Google Scholar - 9.R.R. Sharma, R.B. Pachori, Time-frequency representation using IEVDHM-HT with application to classification of epileptic EEG signals. IET Sci. Measur. Technol.
**12**(1), 72–82 (2018)CrossRefGoogle Scholar - 10.R.R. Sharma, R.B. Pachori, Eigenvalue decomposition of Hankel matrix-based time-frequency representation for complex signals. Circuits, Syst., Signal Process.
**37**(8), 3313–3329 (2018)MathSciNetCrossRefGoogle Scholar - 11.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 - 12.L. Cohen, Time-frequency distributions-a review. Proc. IEEE
**77**, 941–981 (1989)CrossRefGoogle Scholar - 13.R.R. Sharma, R.B. Pachori, Improved eigenvalue decomposition-based approach for reducing cross-terms in Wigner-Ville distribution. Circuits, Syst., Signal Process.
**37**(08), 3330–3350 (2018)MathSciNetCrossRefGoogle Scholar - 14.W.J. Staszewski, K. Worden, G.R. Tomlinson, Time-frequency analysis in gearbox fault detection using the Wigner-Ville distribution and pattern recognition. Mech. Syst. Signal Process.
**11**(5), 673–692 (1997)CrossRefGoogle Scholar - 15.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*(2017), pp. 326–330Google Scholar - 16.Y.S. Yan, C.C. Poon, Y.T. Zhang, Reduction of motion artifact in pulse oximetry by smoothed pseudo Wigner-Ville distribution. J. Neuro Eng. Rehabil.
**2**(1), 3 (2005)CrossRefGoogle Scholar - 17.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)MathSciNetCrossRefGoogle Scholar - 18.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*(2017), pp. 597–608Google Scholar - 19.R.R. Sharma, M. Kumar, R.B. Pachori, Joint time-frequency domain-based CAD disease sensing system using ECG signals. IEEE Sens. J.
**19**(10), 3912–3920 (2019)CrossRefGoogle Scholar - 20.R.R. Sharma, P. Chandra, R.B. Pachori, Electromyogram signal analysis using eigenvalue decomposition of the Hankel matrix, in
*Machine Intelligence and Signal Analysis*(Springer, Singapore, 2019), pp. 671–682Google Scholar - 21.R.R. Sharma, M. Kumar, R.B. Pachori, Classification of EMG signals using eigenvalue decomposition-based time-frequency representation, in
*Biomedical and Clinical Engineering for Healthcare Advancement*(IGI Global, 2020), pp. 96–118Google Scholar - 22.C. Xude, X. Bing, X. Xuedong, Z. Yuan, W. Hongli, Suppression of cross-terms in Wigner-Ville distribution based on short-term fourier transform, in
*2015 12th IEEE International Conference on Electronic Measurement and Instruments (ICEMI)*(2015), pp. 472–475Google Scholar - 23.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. 1–8 (2019). https://doi.org/10.1007/s11760-019-01549-7CrossRefGoogle Scholar
- 24.R.B. Pachori, P. Sircar, A novel technique to reduce cross terms in the squared magnitude of the wavelet transform and the short time Fourier transform, in
*IEEE International Workshop on Intelligent Signal Processing*(Faro, Portugal, 2005), pp. 217–222Google Scholar - 25.P. Flandrin, B. EscudiÃl’, An interpretation of the pseudo-Wigner-Ville distribution. Signal Process.
**6**, 27–36 (1984)Google Scholar - 26.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*(2010), pp. 2168–2171Google Scholar - 27.P. Meena, R.R. Sharma, R.B. Pachori, Cross-term suppression in the Wigner-Ville distribution using variational mode decomposition, in
*5th International Conference on Signal Processing, Computing, and Control (ISPCC-2k19)*(Waknaghat, India, 2019)Google Scholar - 28.R.B. Pachori, P. Sircar, A new technique to reduce cross terms in the Wigner distribution. Digital Signal Process.
**17**, 466–474 (2007)CrossRefGoogle Scholar - 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**, 590–599 (2011)CrossRefGoogle Scholar - 30.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 - 31.R.B. Pachori, P. Sircar, Analysis of multicomponent nonstationary signals using Fourier-Bessel transform and Wigner distribution, in
*14th European Signal Processing Conference*(2006)Google Scholar - 32.R.B. Pachori, P. Sircar, Time-frequency analysis using time-order representation and Wigner distribution, in
*IEEE Tencon Conference*, Article no. 4766782 (2008)Google Scholar - 33.K. Dragomiretskiy, D. Zosso, Variational mode decomposition. IEEE Trans. Signal Process.
**62**(3) 531–544 (2014)MathSciNetCrossRefGoogle Scholar - 34.S. Mohanty, K.K. Gupta, Bearing fault analysis using variational mode decomposition. J. Instrum. Technol. Innov.
**4**, 20–27 (2014)Google Scholar - 35.A. Upadhyay, M. Sharma, R.B. Pachori, Determination of instantaneous fundamental frequency of speech signals using variational mode decomposition. Comput. Electr. Eng.
**62**, 630–647 (2017)CrossRefGoogle Scholar - 36.A. Upadhyay, R.B. Pachori, Instantaneous voiced/non-voiced detection in speech signals based on variational mode decomposition. J. Frankl. Inst.
**352**(7), 2679–2707 (2015)CrossRefGoogle Scholar - 37.A. Upadhyay, R.B. Pachori, Speech enhancement based on mEMD-VMD method. Electron. Lett.
**53**(07), 502–504 (2017)CrossRefGoogle Scholar - 38.
- 39.F. Auger, P. Flandrin, P. Goncalves, O. Lemoine,
*Time-Frequency Toolbox*, vol. 46 (CNRS France-Rice University, 1996)Google Scholar - 40.L. Stankovic, A measure of some time-frequency distributions concentration. Signal Process.
**81**, 621–631 (2001)CrossRefGoogle Scholar - 41.R. Baraniuk, Bat Echolocation Chirp, http://dsp.rice.edu/software/TFA/RGK/BAT/batsig.bin.Z/, (2009)
- 42.R.B. Pachori, P. Sircar, Analysis of multicomponent AM-FM signals using FB-DESA method. Digital Signal Process.
**20**, 42–62 (2010)CrossRefGoogle Scholar - 43.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