Circuits, Systems, and Signal Processing

, Volume 37, Issue 8, pp 3313–3329 | Cite as

Eigenvalue Decomposition of Hankel Matrix-Based Time-Frequency Representation for Complex Signals

  • Rishi Raj SharmaEmail author
  • Ram Bilas Pachori


The analysis of non-stationary signals using time-frequency representation (TFR) presents simultaneous information in time and frequency domain. Most of TFR methods are developed for real-valued signals. In several fields of science and technology, the study of unique information presented in the complex form of signals is required. Therefore, an eigenvalue decomposition of Hankel matrix-based TFR method, which is a data-driven technique, has been extended for the analysis of complex-valued signals. In this method, the positive and negative frequency components of complex signals are separately decomposed using recently developed eigenvalue decomposition of Hankel matrix-based method. Further, the Hilbert transform is applied on decomposed components to obtain TFR for both positive and negative frequency ranges. The proposed method for obtaining TFR is compared with the existing methods. Results for synthetic and natural complex signals provide support to the proposed method to perform better than compared methods.


Complex signal analysis Eigenvalue decomposition Time-frequency analysis Non-stationary signals Data-driven methods 


Compliance with Ethical Standards

Conflict of interest

All authors declare that they have no conflict of interest.


  1. 1.
    M.U.B. Altaf, T. Gautama, T. Tanaka, D.P. Mandic, Rotation invariant complex empirical mode decomposition. in IEEE International Conference on Acoustics, Speech and Signal Processing, 2007 (ICASSP 2007), vol. 3, pp. III–1009 (2007)Google Scholar
  2. 2.
    V. Bajaj, R.B. Pachori, Classification of seizure and nonseizure EEG signals using empirical mode decomposition. IEEE Trans. Inf Technol. Biomed. 16, 1135–1142 (2012)CrossRefGoogle Scholar
  3. 3.
    D. Bhati, R.B. Pachori, V.M. Gadre, A novel approach for time-frequency localization of scaling functions and design of three-band biorthogonal linear phase wavelet filter banks. Digit. Signal Proc. 69, 309–322 (2017)CrossRefGoogle Scholar
  4. 4.
    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
  5. 5.
    A. Bhattacharyya, L. Singh, R.B. Pachori, Fourier–Bessel series expansion based empirical wavelet transform for analysis of non-stationary signals. Digit. Signal Proc. 78, 185–196 (2018)CrossRefGoogle Scholar
  6. 6.
    B. Bjelica, M. Dakovic, L. Stankovic, T. Thayaparan, Complex empirical decomposition method in radar signal processing, in Proceedings of 2012 Mediterranean Conference on Embedded Computing (MECO) (2012). pp. 88–91Google Scholar
  7. 7.
    L. Cohen, Time-Frequency Analysis, vol. 778 (Prentice Hall PTR, Englewood Cliffs, NJ, 1995)Google Scholar
  8. 8.
    I. Daubechies, Ten Lectures on Wavelets (SIAM, Philadelphia, 1992)CrossRefzbMATHGoogle Scholar
  9. 9.
    I. Daubechies, J. Lu, H.T. Wu, Synchrosqueezed wavelet transforms: an empirical mode decomposition-like tool. Appl. Comput. Harmonic Anal. 30, 243–261 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    K. Dragomiretskiy, D. Zosso, Variational mode decomposition. IEEE Trans. Signal Process. 62, 531–544 (2014)MathSciNetCrossRefGoogle Scholar
  11. 11.
    M.G. Frei, I. Osorio, Intrinsic time-scale decomposition: time-frequency-energy analysis and real-time filtering of non-stationary signals, in Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences vol. 463 (The Royal Society, London, 2007). pp. 321–342Google Scholar
  12. 12.
    Y. Gao, E. Sang, Z. Shen, Comparison of EMD and complex EMD in signal processing, in Congress on Image and Signal Processing, 2008 vol. 1 (IEEE, New York, 2008). pp. 141–145Google Scholar
  13. 13.
    J. Gilles, Empirical wavelet transform. IEEE Trans. Signal Process. 61, 3999–4010 (2013)MathSciNetCrossRefGoogle Scholar
  14. 14.
    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, 1971 (The Royal Society, London, 1998). pp. 903–995Google Scholar
  15. 15.
    IEM-AWOS data, Iowa State University, USA (2017)
  16. 16.
    P. Jain, R.B. Pachori, GCI identification from voiced speech using the eigen value decomposition of Hankel matrix, in 8th International Symposium on Image and Signal Processing and Analysis (2013). pp. 371–376Google Scholar
  17. 17.
    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 Language Process. 22, 1467–1482 (2014)CrossRefGoogle Scholar
  18. 18.
    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. Franklin Inst. 352, 4017–4044 (2015)CrossRefGoogle Scholar
  19. 19.
    D. Looney, D.P. Mandic, Fusion of visual and thermal images using complex extension of EMD, in Second ACM/IEEE International Conference on Distributed Smart Cameras, 2008. ICDSC 2008 (2008). pp. 1–8Google Scholar
  20. 20.
    D. Looney, D.P. Mandic, Multiscale image fusion using complex extensions of EMD. IEEE Trans. Signal Process. 57, 1626–1630 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    S. Mallat, A Wavelet Tour of Signal Processing (Academic Press, Cambridge, 1999)zbMATHGoogle Scholar
  22. 22.
    R.B. Pachori, A. Nishad, Cross-terms reduction in the Wigner-Ville distribution using tunable-Q wavelet transform. Sig. Process. 120, 288–304 (2016)CrossRefGoogle Scholar
  23. 23.
    R.B. Pachori, P. Sircar, A new technique to reduce cross terms in the wigner distribution. Digit. Signal Proc. 17, 466–474 (2007)CrossRefGoogle Scholar
  24. 24.
    R.B. Pachori, P. Sircar, Analysis of multicomponent AM-FM signals using FB-DESA method. Digit. Signal Proc. 20, 42–62 (2010)CrossRefGoogle Scholar
  25. 25.
    C. Park, D. Looney, M.M. Van Hulle, D.P. Mandic, The complex local mean decomposition. Neurocomputing 74, 867–875 (2011)CrossRefGoogle Scholar
  26. 26.
    N. Rehman, D.P. Mandic, Multivariate empirical mode decomposition, in Proceedings of The Royal Society of London A: Mathematical, Physical and Engineering Sciences (The Royal Society, London, 2009). p. rspa.2009.0502Google Scholar
  27. 27.
    P. Richardson, J. Price, D. Walsh, L. Armi, M. Schröder, Tracking three meddies with sofar floats. J. Phys. Oceanogr. 19, 371–383 (1989)CrossRefGoogle Scholar
  28. 28.
    G. Rilling, P. Flandrin, P. Gonçalves, J.M. Lilly, Bivariate empirical mode decomposition. IEEE Signal Process. Lett. 14, 936–939 (2007)CrossRefGoogle Scholar
  29. 29.
    R.R. Sharma, P. Chandra, R.B. Pachori, Electromyogram signal analysis using eigenvalue decomposition of the Hankel matrix, in Proceedings of the International Conference on Machine Intelligence and Signal Processing (Springer, Berlin, 2017)Google Scholar
  30. 30.
    R.R. Sharma, M. Kumar, R.B. Pachori, Automated CAD identification system using time-frequency representation based on eigenvalue decomposition of ECG signals, in Proceedings of the International Conference on Machine Intelligence and Signal Processing (Springer, Berlin, 2017)Google Scholar
  31. 31.
    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
  32. 32.
    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 (SPIN 2017) Noida India (2017). pp. 484–488Google Scholar
  33. 33.
    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)MathSciNetCrossRefGoogle Scholar
  34. 34.
    L. Stankovic, M. Dakovic, T. Thayaparan, Time-Frequency Signal Analysis with Applications (Artech House, Cambridge, 2014)zbMATHGoogle Scholar
  35. 35.
    L. Stanković, D. Mandić, M. Daković, M. Brajović, Time-frequency decomposition of multivariate multicomponent signals. Sig. Process. 142, 468–479 (2018)CrossRefGoogle Scholar
  36. 36.
    Subsurface float data: National Oceanic and Atmospheric Administration, USA (2017)
  37. 37.
    T. Tanaka, D.P. Mandic, Complex empirical mode decomposition. IEEE Signal Process. Lett. 14(2), 101–104 (2007)CrossRefGoogle Scholar
  38. 38.
    R. Vautard, M. Ghil, Singular spectrum analysis in nonlinear dynamics, with applications to paleoclimatic time series. Physica D 35, 395–424 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    R. Vautard, P. Yiou, M. Ghil, Singular-spectrum analysis: a toolkit for short, noisy chaotic signals. Physica D 58, 95–126 (1992)CrossRefGoogle Scholar
  40. 40.
    Y. Wang, F. Liu, Z. Jiang, S. He, Q. Mo, Complex variational mode decomposition for signal processing applications. Mech. Syst. Signal Process. 86, 75–85 (2017)CrossRefGoogle Scholar
  41. 41.
    M.H. Yeh, The complex bidimensional empirical mode decomposition. Sig. Process. 92, 523–541 (2012)CrossRefGoogle Scholar
  42. 42.
    B. Yuan, Z. Chen, S. Xu, Micro-Doppler analysis and separation based on complex local mean decomposition for aircraft with fast-rotating parts in ISAR imaging. IEEE Trans. Geosci. Remote Sens. 52, 1285–1298 (2014)CrossRefGoogle Scholar
  43. 43.
    X. Zhang, Comparison of EMD based image fusion methods. Int. Conf. Comput. Autom. Eng. 2009, 302–305 (2009)Google Scholar

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Authors and Affiliations

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

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