Abstract
This paper presents an efficient methodology based on empirical wavelet transform (EWT) to remove cross-terms from the Wigner–Ville distribution (WVD). An EWT-based filter bank method is suggested to remove the cross-terms that occur due to nonlinearity in modulation. The mean-square error-based filter bank bandwidth selection is done which has been applied for the boundaries selection in EWT. In this way, a signal-dependent adaptive boundary selection is performed. Thereafter, energy-based segmentation is applied in time domain to eliminate inter-cross-terms generated between components. Moreover, the WVD of all the components is added together to produce a complete cross-terms-free time–frequency distribution. The proposed method is compared with other existing methods, and normalized Rényi entropy measure is also computed for validating the performance.
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Andria, G., Savino, M.: Interpolated smoothed pseudo Wigner–Ville distribution for accurate spectrum analysis. IEEE Trans. Instrum. Meas. 45, 818–823 (1996)
Boashash, B.: Time–Frequency Signal Analysis and Processing: A Comprehensive Reference. Elsevier, Amsterdam (2003)
Baydar, N., Ball, A.: 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)
Bhattacharyya, A., Pachori, R.B.: A multivariate approach for patient-specific EEG seizure detection using empirical wavelet transform. IEEE Trans. Biomed. Eng. 64, 2003–2015 (2017)
Bhattacharyya, A., Singh, L., Pachori, R.B.: Fourier–Bessel series expansion based empirical wavelet transform for analysis of non-stationary signals. Digit. Signal Process. 78, 185–196 (2018)
Chen, J., Pan, J., Li, Z., Zi, Y., Chen, X.: Generator bearing fault diagnosis for wind turbine via empirical wavelet transform using measured vibration signals. Renew. Energy 89, 80–92 (2016)
Chen, V.C., Ling, H.: Time–Frequency Transforms for Radar Imaging and Signal Analysis. Artech House, Norwood (2002)
Choi, H.I., Williams, W.J.: Improved time–frequency representation of multicomponent signals using exponential kernels. IEEE Trans. Acoust. Speech Signal Process. 37, 862–871 (1989)
Claasen, T.A.C.M., Mecklenbrauker, W.F.G.: The Wigner distribution—a tool for time–frequency signal analysis. Part I: continuous-time signals. Philips J. Res 35(3), 217–250 (1980)
Climente-Alarcon, V., Antonino-Daviu, J.A., Riera-Guasp, M., Vlcek, M.: Induction motor diagnosis by advanced notch FIR filters and the Wigner–Ville distribution. IEEE Trans. Ind. Electron. 61, 4217–4227 (2014)
Daubechies, I.: Ten Lectures on Wavelets, vol. 61. SIAM, Philadelphia (1992)
Dragomiretskiy, K., Zosso, D.: Variational mode decomposition. IEEE Trans. Signal Process. 62, 531–544 (2014)
Flandrin, P., Escudié, B.: An interpretation of the pseudo-Wigner–Ville distribution. Signal Process. 6, 27–36 (1984)
Gaikwad, C.J., Sircar, P.: Reduced interference Wigner–Ville time frequency representations using signal support information. In: 2016 IEEE Annual India Conference, pp. 1–5. IEEE (2016)
Gaikwad, C.J., Sircar, P.: Bispectrum-based technique to remove cross-terms in quadratic systems and Wigner–Ville distribution. Signal Image Video Process. 12(4), 703–710 (2018)
Gavrovska, A., Bogdanović, V., Reljin, I., Reljin, B.: 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)
Gilles, J.: Empirical wavelet transform. IEEE Trans. Signal Process. 61, 3999–4010 (2013)
Hu, X., Peng, S., Hwang, W.L.: EMD revisited: a new understanding of the envelope and resolving the mode-mixing problem in AM–FM signals. IEEE Trans. Signal Process. 60, 1075–1086 (2012)
Huang, N.E., Shen, Z., Long, S.R., Wu, M.C., Shih, H.H., Zheng, Q., Yen, N.C., Tung, C.C., Liu, H.H.: 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, pp. 903–995. The Royal Society (1998)
Jain, P., Pachori, R.B.: Time-order representation based method for epoch detection from speech signals. J. Intell. Syst. 21, 79–95 (2012)
Jain, P., Pachori, R.B.: 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)
Kadambe, S., Boudreaux-Bartels, G.F.: 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)
Kazemi, K., Amirian, M., Dehghani, M.J.: The S-transform using a new window to improve frequency and time resolutions. Signal Image Video Process. 8(3), 533–541 (2014)
Stankovic, L., Daković, M., Thayaparan, T.: Time–Frequency Signal Analysis with Applications. Artech House, Norwood (2013)
Liu, W., Cao, S., Chen, Y.: Seismic time–frequency analysis via empirical wavelet transform. IEEE Geosci. Remote Sens. Lett. 13, 28–32 (2016)
Meyer, Y.: Wavelets and Operators, vol. 1. Cambridge University Press, Cambridge (1992)
Pachori, R.B., Nishad, A.: Cross-terms reduction in the Wigner–Ville distribution using tunable-Q wavelet transform. Signal Process. 120, 288–304 (2016)
Pachori, R.B., Sircar, P.: A new technique to reduce cross terms in the Wigner distribution. Digit. Signal Process. 17, 466–474 (2007)
Pachori, R.B., Sircar, P.: Time–frequency analysis using time-order representation and Wigner distribution. In: 2008 IEEE Region 10 Conference, pp. 1–6 (2008)
Ping, D., Zhao, P., Deng, B.: Cross-terms suppression in Wigner–Ville distribution based on image processing. In: 2010 IEEE International Conference on Information and Automation, pp. 2168–2171. IEEE (2010)
Ren, H., Ren, A., Li, Z.: A new strategy for the suppression of cross-terms in pseudo Wigner–Ville distribution. Signal Image Video Process. 10, 139–144 (2016)
Sharma, R.R., Pachori, R.B.: 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, pp. 484–488. IEEE (2017)
Sharma, R.R., Pachori, R.B.: Eigenvalue decomposition of Hankel matrix based time–frequency representation for complex signals. Circuits Syst. Signal Process. 37(8), 3313–3329 (2018)
Sharma, R.R., Pachori, R.B.: Improved eigenvalue decomposition based approach for reducing cross-terms in Wigner–Ville distribution. Circuits Syst. Signal Process. 37(8), 3330–3350 (2018)
Sharma, R.R., Pachori, R.B.: Time–frequency representation using IEVDHM–HT with application to classification of epileptic EEG signals. IET Sci. Meas. Technol. 12, 72–82 (2018)
Stanković, L.: A measure of some time–frequency distributions concentration. Signal Process. 81, 621–631 (2001)
Upadhyay, A., Pachori, R.B.: Instantaneous voiced/non-voiced detection in speech signals based on variational mode decomposition. J. Frankl. Inst. 352, 2679–2707 (2015)
Wang, Z., Bovik, A.C.: Mean squared error: love it or leave it? A new look at signal fidelity measures. IEEE Signal Process. Mag. 26(1), 98–117 (2009)
Wu, Y., Li, X.: Elimination of cross-terms in the Wigner–Ville distribution of multi-component LFM signals. IET Signal Process. 11(6), 657–662 (2017)
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Sharma, R.R., Kalyani, A. & Pachori, R.B. An empirical wavelet transform-based approach for cross-terms-free Wigner–Ville distribution. SIViP 14, 249–256 (2020). https://doi.org/10.1007/s11760-019-01549-7
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DOI: https://doi.org/10.1007/s11760-019-01549-7