Abstract
The empirical mode decomposition (EMD) is a useful method for processing nonlinear and nonstationary signals. However, it usually suffers from the mode mixing problem due to the existence of intermittence and interferences of noises in signals. In this paper, a preconditioning framework for the EMD method is proposed in order to alleviate the mode mixing problem. The key points of the preconditioning before implementing the EMD method lie in two aspects: On the one hand, the interferences of noises in the original signal are reduced by filtering; on the other hand, the assisted signals are added to the denoised signal to improve properties of the signal data. Under this framework, the preconditioned forms of the complementary ensemble empirical mode decomposition method and the masking signal-assisted empirical mode decomposition method are presented, respectively. The effectiveness of the proposed methods is illustrated by numerical simulations and applications to real-world signals.
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References
A. Bouchikhi, A. Boudraa, Multicomponent AM–FM signals analysis based on EMD-B-splines ESA. Signal Process. 92, 2214–2228 (2012)
M. Colominas, G. Schlotthauer, M. Torres, Improved complete ensemble EMD: a suitable tool for biomedical signal processing. Biomed. Signal Process. Control 14, 19–29 (2014)
R. Deering, J. Kaiser, The use of a masking signal to improve empirical mode decomposition. In Proceedings of IEEE International Conference on Acoustics Speech and Signal Processing (ICASSP05), vol. 4 (Philadelphia, Pennsylvania, USA, 2005), pp. 485–488
I. Daubechies, Ten Lectures on Wavelets. CBMS-NSF Regional Conference Series in Applied Mathematics (SIAM, Philadelphia, PA, USA, 1992)
N.E. Huang et al., The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis. Proc. R. Soc. Lond. A 454, 903–995 (1998)
N.E. Huang, Z. Shen, A new view of nonlinear water waves: the Hilbert spectrum. Fluid Mech. 31, 417–457 (1999)
H. Hong, X. Wang, Z. Tao, Local integral mean-based sifting for empirical mode decomposition. IEEE Signal Process. Lett. 16, 841–844 (2016)
X. Jin, Preconditioning Techniques for Toeplitz systems (Higher Education Press, Beijing, 2010)
H. Li, C. Wang, D. Zhao, An improved EMD and its applications to find the basis functions of EMI Signals. Math. Probl. Eng. (2015). https://doi.org/10.1155/2015/150127
L. Lin, H. Ji, Signal feature extraction based on an improved EMD method. Measurement 42, 796–803 (2008)
M. Li, X. Wu, X. Liu, An improved EMD method for time–frequency feature extraction of telemetry vibration signal based on multi-scale median filtering. Circuits Syst. Signal Process. 34, 815–830 (2015)
H. Li et al., Reconstruction and basis function construction of electromagnetic interference source signals based on Toeplitz-based singular value decomposition. IET Signal Process. 11(1), 59–65 (2016). https://doi.org/10.1049/iet-spr.2016.0307
S. Mallat, A Wavelet Tour of Signal Processing: The Sparse Way, 3rd edn. (Elsevier, New York, 2009)
B. Ning, D. Zhao, H. Li, Improved Schur complement preconditioners for block-Toeplitz systems with small size blocks. J. Comput. Appl. Math. 311, 655–663 (2016)
J. Proakis, D. Manolakis, Digital Signal Processing, 4th edn. (Pearson Education Limited, New York, 2006)
N. Rehman et al., EMD via MEMD: multivariate noise-aided computation of standard EMD. Adv. Adapt. Data Anal. 5, 1350007 (2013)
N. Rehman, D. Mandic, Filter bank property of multivariate empirical mode decomposition. IEEE Trans. Signal Process. 59, 2421–2426 (2011)
L. Reichel, F. Sgallari, Q. Ye, Tikhonov regularization based on generalized Krylov subspace methods. Appl. Numer. Methods 62, 1215–1228 (2012)
D. Singh, Q. Zhao, Pseudo-fault signal assisted EMD for fault detection and isolation in rotating machines. Mech. Syst. Signal Proc. 81, 202–218 (2016)
C. Wang, H. Li, D. Zhao, A new signal classification method based on EEMD and FCM and its application in bearing fault diagnosis. Appl. Mech. Mater. 602–605, 1803–1806 (2014)
Z. Wu, N.E. Huang, Ensemble empirical mode decomposition: a noise assisted data analysis method. Adv. Adapt. Data Anal. 1, 1–41 (2009)
C. Wang, Q. Kemao, F. Da, Regenerated phase-shifted sinusoid-assisted empirical mode decomposition. IEEE Signal Process. Lett. 23, 556–560 (2016)
C. Wang, H. Li, D. Zhao, Preconditioning Toeplitz-plus-diagonal linear systems using the Sherman–Morrison–Woodbury formula. J. Comput. Appl. Math. 309, 312–319 (2017)
C. Wang, H. Li, D. Zhao, An explicit formula for the inverse of a pentadiagonal Toeplitz matrix. J. Comput. Appl. Math. 278, 12–18 (2015)
X. Xie, Illumination preprocessing for face images based on empirical mode decomposition. Signal Process. 103, 250–257 (2014)
J. Yeh, J. Shieh, N.E. Huang, Complementary ensemble empirical mode decomposition: a novel noise enhanced data analysis method. Adv. Adapt. Data Anal. 2, 135–156 (2010)
Y. Yang, J. Deng, D. Kang, An improved empirical mode decomposition by using dyadic masking signals. Signal Image Video Process. 9, 1–5 (2015)
D. Zhao, Z. Huang, H. Li, An improved EEMD method based on the adjustable cubic trigonometric cardinal spline interpolation. Digit. Signal Prog. 64, 41–48 (2017)
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This work was supported by the National Natural Science Foundation of China (Grant 61771001), the scholarship of China Scholarship Council (No. 201606020068).
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Wang, C., Li, H. & Zhao, D. A Preconditioning Framework for the Empirical Mode Decomposition Method. Circuits Syst Signal Process 37, 5417–5440 (2018). https://doi.org/10.1007/s00034-018-0821-9
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DOI: https://doi.org/10.1007/s00034-018-0821-9