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Circuits, Systems, and Signal Processing

, Volume 38, Issue 1, pp 191–217 | Cite as

Separation of Single Frequency Component Using Singular Value Decomposition

  • Xuezhi ZhaoEmail author
  • Bangyan Ye
Article

Abstract

It is proved that for an arbitrary frequency component, no matter how much the frequency and its amplitude are, this frequency component always generates only two nonzero singular values. Based on this relationship, an approach based on singular value decomposition (SVD) is proposed to separate the single frequency, and the condition for SVD to separate the single frequency is that the amplitude of each frequency is not equal to each other. To separate the frequencies with the same amplitude, it is proposed to add the white noise to the original signal, thus the amplitudes of frequencies become different, and the influence of noise on the singular values is studied. Three principles for selecting singular values are proposed. The quantitative relation among the singular values and frequency parameters is obtained, so the place of the nonzero singular values of each frequency in the singular value sequence can be located. Under these conditions, as long as a frequency can be distinguished in the amplitude spectrum of original signal, it can always be separated from the original signal by SVD. Simulation and practical signal separation examples verify the effectiveness of this approach, and compared with the existing methods, SVD has higher frequency separation accuracy.

Keywords

Singular value decomposition Single frequency component Signal separation Separation condition Noise 

Notes

Acknowledgements

This work is supported by the National Natural Science Foundation of China ((NSFC, Grant No. 51375178) and Natural Science Foundation of Guangdong province (Grant No. S2012010008789). We are grateful to the editor and the anonymous reviewers for their helpful suggestions to improve the quality of the paper.

References

  1. 1.
    M.B. Ashtiani, S.M. Shahrtash, Partial discharge de-noising employing adaptive singular value decomposition. IEEE Trans. Dielectr. Electr. Insul. 21(2), 775–782 (2014)CrossRefGoogle Scholar
  2. 2.
    N.L. Bihan, J. Mars, Singular value decomposition of quaternion matrices: a new tool for vector-sensor signal processing. Sig. Process. 84(7), 1177–1199 (2004)CrossRefzbMATHGoogle Scholar
  3. 3.
    M. Bydder, J. Du, Noise reduction in multiple-echo data sets using singular value decomposition. Magn. Reson. Imaging 24(7), 849–856 (2006)CrossRefGoogle Scholar
  4. 4.
    L. Cattani, D. Maillet, F. Bozzoli et al., Estimation of the local convective heat transfer coefficient in pipe flow using a 2D thermal quadrupole model and truncated singular value decomposition. Int. J. Heat Mass Transf. 91, 1034–1045 (2015)CrossRefGoogle Scholar
  5. 5.
    S. Chen, Y. Yang, K. Wei et al., Time-varying frequency-modulated component extraction based on parameterized demodulation and singular value decomposition. IEEE Trans. Instrum. Meas. 65(2), 276–285 (2016)CrossRefGoogle Scholar
  6. 6.
    A. Daneshmand, F. Facchinei, V. Kungurtsev, G. Scutari, Hybrid random/deterministic parallel algorithms for convex and nonconvex big data optimization. IEEE Trans. Signal Process. 63(15), 3914–3929 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    I. Dassios, K. Fountoulakis, J. Gondzio, A second-order method for compressed sensing problems with coherent and redundant dictionaries. arXiv preprint arXiv:1405.4146, 2014
  8. 8.
    I. Dassios, K. Fountoulakis, J. Gondzio, A preconditioner for a primal-dual newton conjugate gradient method for compressed sensing problems. SIAM J. Sci. Comput. 37(6), A2783–A2812 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    A. Ghaffari, E. Fatemizadeh, RISM: Single-modal image registration via rank-induced similarity measure. IEEE Trans. Image Process. 24(12), 5567–5580 (2015)MathSciNetCrossRefGoogle Scholar
  10. 10.
    R. Golafshan, K.Y. Sanliturk, SVD and Hankel matrix based de-noising approach for ball bearing fault detection and its assessment using artificial faults. Mech. Syst. Signal Process. 70–71(3), 36–50 (2016)CrossRefGoogle Scholar
  11. 11.
    G.H. Golub, C.F. Van Loan, Matrix Computations, 3rd edn. (John Hopkins University Press, Baltimore, 1996)zbMATHGoogle Scholar
  12. 12.
    B. Hu, R.G. Gosine, A new eigenstructure method for sinusoidal signal retrieval in white noise: estimation and pattern recognition. IEEE Trans. Signal Process. 45(12), 3073–3083 (1997)CrossRefGoogle Scholar
  13. 13.
    P.P. Kanjilal, S. Palit, On multiple pattern extraction using singular value decomposition. IEEE Trans. Signal Process. 43(6), 1536–1540 (1995)CrossRefGoogle Scholar
  14. 14.
    P.P. Kanjilal, G. Saha, Fetal ECG extraction from single channel maternal ECG using SVD and SVR spectrum, in Proceedings of the 17th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (Montreal, Canada, 1997), Vol. 1, pp. 187–188Google Scholar
  15. 15.
    J.B. Maj, L. Royackers, M. Moonen et al., SVD-based optimal filtering for noise reduction in dual microphone hearing aids: a real time implementation and perceptual evaluation. IEEE Trans. Biomed. Eng. 52(9), 1563–1573 (2005)CrossRefGoogle Scholar
  16. 16.
    A.K. Marzook, A. Ismail, B.M. Ali et al., Singular value decomposition rank-deficient-based estimators in TD-SCDMA systems. Circuits Syst. Signal Process. 32(1), 205–221 (2013)MathSciNetCrossRefGoogle Scholar
  17. 17.
    H.M. Nguyen, X. Peng, M.N. Do et al., Denoising MR spectroscopic imaging data with low-rank approximations. IEEE Trans. Biomed. Eng. 60(1), 78–89 (2013)CrossRefGoogle Scholar
  18. 18.
    J.G. Proakis, D.G. Manolakis, Digital Signal Processing: Principles, Algorithms, and Applications, 4th edn. (Pearson Prentice Hall, Upper Saddle River, 2007)Google Scholar
  19. 19.
    R.B. Randall, J. Antoni, Rolling element bearing diagnostics—a tutorial. Mech. Syst. Signal Process. 25(2), 485–520 (2011)CrossRefGoogle Scholar
  20. 20.
    P.A. Reninger, G. Martelet, J. Deparis et al., Singular value decomposition as a denoising tool for airborne time domain electromagnetic data. J. Appl. Geophys. 75(2), 264–276 (2011)CrossRefGoogle Scholar
  21. 21.
    H.R. Seresht, S.M. Ahadi, S. Seyedin, Spectro-temporal power spectrum features for noise robust ASR. Circuits Syst Signal Process. 36(8), 3222–3242 (2017)CrossRefGoogle Scholar
  22. 22.
    O.H. Simon, Adaptive Filter Theory, 5th edn. (McMaster University, Hamilton, 2013)Google Scholar
  23. 23.
    N. Tie, J. Zhai, A matrix-free smoothing algorithm for large-scale support vector machines. Inf. Sci. 358–359(9), 29–43 (2016)Google Scholar
  24. 24.
    C.Y. Wang, T.S. Qiu, J.C. Li et al., A novel spectrum-sensing technique in cognitive radio based on singular value decomposition. Sig. Process. 27(5), 727–731 (2011)Google Scholar
  25. 25.
    S. Wang, D. Cui, B. Wang et al., A perceptual image quality assessment metric using singular value decomposition. Circuits Syst. Signal Process. 34(1), 209–229 (2015)CrossRefGoogle Scholar
  26. 26.
    Z. Wu, N.E. Huang, Ensemble empirical mode decomposition: a noise-assisted data analysis method. Adv. Adapt. Data Anal. 1(1), 1–41 (2009)CrossRefGoogle Scholar
  27. 27.
    X. Xiang, J. Zhou, X. An et al., Fault diagnosis based on Walsh transform and support vector machine. Mech. Syst. Signal Process. 22(7), 1685–1693 (2008)CrossRefGoogle Scholar
  28. 28.
    W.X. Yang, P.W. Tse, Development of an advanced noise reduction method for vibration analysis based on singular value decomposition. NDT&E Int. 36(6), 419–432 (2003)CrossRefGoogle Scholar
  29. 29.
    X. Zhao, B. Ye, Similarity of signal processing effect between Hankel matrix-based SVD and wavelet transform and its mechanism analysis. Mech. Syst. Signal Process. 23(4), 1062–1075 (2009)CrossRefGoogle Scholar
  30. 30.
    X. Zhao, B. Ye, Convolution wavelet packet transform and its application to signal processing. Digit. Signal Process. 20(5), 1352–1364 (2010)MathSciNetCrossRefGoogle Scholar
  31. 31.
    X. Zhao, B. Ye, Selection of effective singular values using difference spectrum and its application to fault diagnosis of headstock. Mech. Syst. Signal Process. 25(5), 1617–1631 (2011)MathSciNetCrossRefGoogle Scholar
  32. 32.
    X. Zhao, B. Ye, The influence of formation manner of component on signal processing effect of singular value decomposition. J. Shanghai Jiaotong Univ. 45(3), 368–374 (2011)MathSciNetGoogle Scholar
  33. 33.
    X. Zhao, B. Ye, Singular value decomposition packet and its application to extraction of weak fault feature. Mech. Syst. Signal Process. 70–71(3), 73–86 (2016)CrossRefGoogle Scholar
  34. 34.
    X. Zhao, B. Ye, T. Chen, Selection of effective singular values based on curvature spectrum of singular values. J. South China Univ. Technol. (Nat. Sci. Edn.) 38(6), 11–18 (2010)Google Scholar
  35. 35.
    C. Zheng, R. Peng, J. Li et al., A constrained MMSE LP residual estimator for speech dereverberation in noisy environments. IEEE Signal Process. Lett. 21(12), 1462–1466 (2014)CrossRefGoogle Scholar
  36. 36.
    B. Zhu, J. Ma, S. Li, A novel L2 − L filtering strategy for two kinds of network-based linear time-invariant systems. Circuits Syst. Signal Process. 36(8), 3098–3113 (2017)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mechanical and Automotive EngineeringSouth China University of TechnologyGuangzhouPeople’s Republic of China

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