An Improved EMD Method Based on Utilizing Certain Inflection Points in the Construction of Envelope Curves
Abstract
The empirical mode decomposition (EMD) algorithm is widely used as an adaptive time-frequency analysis method to decompose nonlinear and non-stationary signals into sets of intrinsic mode functions (IMFs). In the traditional EMD, the lower and upper envelopes should interpolate the minimum and maximum points of the signal, respectively. In this paper, an improved EMD method is proposed based on the new interpolation points, which are special inflection points (SIPn) of the signal. These points are identified in the signal and its first (n − 1) derivatives and are considered as auxiliary interpolation points in addition to the extrema. Therefore, the upper and lower envelopes should not only pass through the extrema but also these SIPn sets of points. By adding each set of SIPi (i = 1, 2, n) to the interpolation points, the frequency resolution of EMD is improved to a certain extent. The effectiveness of the proposed SIPn-EMD is validated by the decomposition of synthetic and experimental bearing vibration signals.Keywords:
empirical mode decomposition (EMD), interpolation points, envelope curve, inflection points, rolling element bearing fault diagnosisReferences
1. Bouchikhi A., Boudraa A.-O. (2012), Multicomponent AM–FM signals analysis based on EMD-B-splines ESA, Signal Processing, 92(9): 2214–2228, https://doi.org/10.1016/j.sigpro.2012.02.014
2. Case Western Reserve University (n.d.), Bearing Data Center, https://engineering.case.edu/bearingdatacenter/download-data-file (access date: 21.02.2023).
3. Chen Q., Huang N., Riemenschneider S., Xu Y. (2006), A B-spline approach for empirical mode decompositions, Advances in Computational Mathematics, 24(1): 171–195, https://doi.org/10.1007/s10444-004-7614-3
4. Chu P.C., Fan C., Huang N. (2012), Compact empirical mode decomposition: An algorithm to reduce mode mixing, end effect, and detrend uncertainty, Advances in Adaptive Data Analysis, 4(3): 1250017, https://doi.org/10.1142/S1793536912500173
5. Deering R., Kaiser J.F. (2005), The use of a masking signal to improve empirical mode decomposition, [in:] Proceedings (ICASSP’05). IEEE International Conference on Acoustics, Speech, and Signal Processing, 4: 485–488, https://doi.org/10.1109/ICASSP.2005.1416051
6. Egambaram A., Badruddin N., Asirvadam V.S., Begum T. (2016), Comparison of envelope interpolation techniques in empirical mode decomposition (EMD) for eyeblink artifact removal from EEG, [in:] 2016 IEEE EMBS Conference on Biomedical Engineering and Sciences (IECBES), pp. 590–595, https://doi.org/10.1109/IECBES.2016.7843518
7. Guo T., Deng Z. (2017), An improved EMD method based on the multi-objective optimization and its application to fault feature extraction of rolling bearing, Applied Acoustics, 127: 46–62, https://doi.org/10.1016/j.apacoust.2017.05.018
8. Huang N.E et al. (1998), The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 454(1971): 903–995, https://doi.org/10.1098/rspa.1998.0193
9. Huang N.E. et al. (2003), A confidence limit for the empirical mode decomposition and Hilbert spectral analysis, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 459(2037): 2317–2345, https://doi.org/10.1098/rspa.2003.1123
10. Kopsinis Y., McLaughlin S. (2007), Investigation and performance enhancement of the empirical mode decomposition method based on a heuristic search optimization approach, [in:] IEEE Transactions on Signal Processing, 56(1): 1–13, https://doi.org/10.1109/TSP.2007.901155
11. Kopsinis Y., McLaughlin S. (2008), Improved EMD using doubly-iterative sifting and high order spline interpolation, EURASIP Journal on Advances in Signal Processing, 2008(1): 128293, https://doi.org/10.1155/2008/128293
12. Lei Y., Lin J., He Z., Zuo M.J. (2013), A review on empirical mode decomposition in fault diagnosis of rotating machinery, Mechanical Systems and Signal Processing, 35(1–2): 108–126, https://doi.org/10.1016/j.ymssp.2012.09.015
13. Li H., Qin X., Zhao D., Chen J.,Wang P. (2018), An improved empirical mode decomposition method based on the cubic trigonometric B-spline interpolation algorithm, Applied Mathematics and Computation, 332: 406–419, https://doi.org/10.1016/j.amc.2018.02.039
14. Li H., Wang C., Zhao D. (2015a), An improved EMD and its applications to find the basis functions of EMI signals, Mathematical Problems in Engineering, 2015: 150127, https://doi.org/10.1155/2015/150127
15. Li Y., Xu M., Wei Y., Huang W. (2015b), An improvement EMD method based on the optimized rational Hermite interpolation approach and its application to gear fault diagnosis, Measurement, 63: 330–345, https://doi.org/10.1016/j.measurement.2014.12.021
16. Pegram G.G.S., Peel M.C., McMahon T.A. (2008), Empirical mode decomposition using rational splines: An application to rainfall time series, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 464(2094): 1483–1501, https://doi.org/10.1098/rspa.2007.0311
17. Qin S.R., Zhong Y.M. (2006), A new envelope algorithm of Hilbert–Huang Transform, Mechanical Systems and Signal Processing, 20(8): 1941–1952, https://doi.org/10.1016/j.ymssp.2005.07.002
18. Rilling G., Flandrin P. (2007), One or two frequencies? The empirical mode decomposition answers, [in:] IEEE Transactions on Signal Processing, 56(1): 85–95, https://doi.org/10.1109/TSP.2007.906771
19. Rilling G., Flandrin P., Gonçalvès P. (2003), On empirical mode decomposition and its algorithms, [in:] IEEE-EURASIP workshop on nonlinear signal and image processing, 3(3): 8–11, Grado.
20. Shu L., Deng H., Liu X., Pan Z. (2022), A Comprehensive working condition identification scheme for rolling bearings based on modified CEEMDAN as well as modified hierarchical amplitude-aware permutation entropy, Measurement Science and Technology, 33(7): 075111, https://doi.org/10.1088/1361-6501/ac5b2c
21. Singh P., Joshi S.D., Patney R.K., Saha K. (2014), Some studies on nonpolynomial interpolation and error analysis, Applied Mathematics and Computation, 244: 809–821, https://doi.org/10.1016/j.amc.2014.07.049
22. SKF (n.d.), Bearing Select, https://www.skfbearingselect.com/#/bearing-selection-start (access date: 21.02.2023).
23. Smith W.A., Randall R.B. (2015), Rolling element bearing diagnostics using the Case Western Reserve University data: A benchmark study, Mechanical Systems and Signal Processing, 64–65: 100–131, https://doi.org/10.1016/j.ymssp.2015.04.021
24. Sun Y., Li S., Wang X. (2021), Bearing fault diagnosis based on EMD and improved Chebyshev distance in SDP image, Measurement, 176: 109100, https://doi.org/10.1016/j.measurement.2021.109100
25. Torres M.E., Colominas M.A., Schlotthauer G., Flandrin P. (2011), A complete ensemble empirical mode decomposition with adaptive noise, [in:] 2011 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 4144–4147, https://doi.org/10.1109/ICASSP.2011.5947265
26. Wang J., Du G., Zhu Z., Shen C., He Q. (2020), Fault diagnosis of rotating machines based on the EMD manifold, Mechanical Systems and Signal Processing, 135: 106443, https://doi.org/10.1016/j.ymssp.2019.106443
27. Wang J.-L., Li Z.-J. (2013), Extreme-point symmetric mode decomposition method for data analysis, Advances in Adaptive Data Analysis, 5(3): 1350015, https://doi.org/10.1142/S1793536913500155
28. Wang Z., Yang J., Guo Y. (2022), Unknown fault feature extraction of rolling bearings under variable speed conditions based on statistical complexity measures, Mechanical Systems and Signal Processing, 172: 108964, https://doi.org/10.1016/j.ymssp.2022.108964
29. Wu Z., Huang N.E. (2009), Ensemble empirical mode decomposition: A noise-assisted data analysis method, Advances in Adaptive Data Analysis, 1(1): 1–41, https://doi.org/10.1142/S1793536909000047
30. Xu Z., Huang B., Li K. (2010), An alternative envelope approach for empirical mode decomposition, Digital Signal Processing, 20(1): 77–84, https://doi.org/10.1016/j.dsp.2009.06.009
31. Yang L., Yang Z., Zhou F., Yang L. (2014), A novel envelope model based on convex constrained optimization, Digital Signal Processing, 29(1): 138–146, https://doi.org/10.1016/j.dsp.2014.02.017
32. Yeh J.-R., Shieh J.-S., Huang N.E. (2010), Complementary ensemble empirical mode decomposition: A novel noise enhanced data analysis method, Advances in Adaptive Data Analysis, 2(2): 135–156, https://doi.org/10.1142/S1793536910000422
33. Yuan J., He Z., Ni J., Brzezinski A.J., Zi Y. (2013), Ensemble noise-reconstructed empirical mode decomposition for mechanical fault detection, Journal of Vibration and Acoustics, 135(2): 021011, https://doi.org/10.1115/1.4023138
34. Yuan J., Xu C., Zhao Q., Jiang H., Weng Y. (2022), High-fidelity noise-reconstructed empirical mode decomposition for mechanical multiple and weak fault extractions, ISA Transaction, 129(Part B): 380–397, https://doi.org/10.1016/j.isatra.2022.02.017
35. Zhao D., Huang Z., Li H., Chen J., Wang P. (2017), An improved EEMD method based on the adjustable cubic trigonometric cardinal spline interpolation, Digital Signal Processing, 64: 41–48, https://doi.org/10.1016/j.dsp.2016.12.007
36. Zheng J., Cao S., Pan H., Ni Q. (2022), Spectral envelope-based adaptive empirical Fourier decomposition method and its application to rolling bearing fault diagnosis, ISA Transactions, 129(Part B): 476–492, https://doi.org/10.1016/j.isatra.2022.02.049
2. Case Western Reserve University (n.d.), Bearing Data Center, https://engineering.case.edu/bearingdatacenter/download-data-file (access date: 21.02.2023).
3. Chen Q., Huang N., Riemenschneider S., Xu Y. (2006), A B-spline approach for empirical mode decompositions, Advances in Computational Mathematics, 24(1): 171–195, https://doi.org/10.1007/s10444-004-7614-3
4. Chu P.C., Fan C., Huang N. (2012), Compact empirical mode decomposition: An algorithm to reduce mode mixing, end effect, and detrend uncertainty, Advances in Adaptive Data Analysis, 4(3): 1250017, https://doi.org/10.1142/S1793536912500173
5. Deering R., Kaiser J.F. (2005), The use of a masking signal to improve empirical mode decomposition, [in:] Proceedings (ICASSP’05). IEEE International Conference on Acoustics, Speech, and Signal Processing, 4: 485–488, https://doi.org/10.1109/ICASSP.2005.1416051
6. Egambaram A., Badruddin N., Asirvadam V.S., Begum T. (2016), Comparison of envelope interpolation techniques in empirical mode decomposition (EMD) for eyeblink artifact removal from EEG, [in:] 2016 IEEE EMBS Conference on Biomedical Engineering and Sciences (IECBES), pp. 590–595, https://doi.org/10.1109/IECBES.2016.7843518
7. Guo T., Deng Z. (2017), An improved EMD method based on the multi-objective optimization and its application to fault feature extraction of rolling bearing, Applied Acoustics, 127: 46–62, https://doi.org/10.1016/j.apacoust.2017.05.018
8. Huang N.E et al. (1998), The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 454(1971): 903–995, https://doi.org/10.1098/rspa.1998.0193
9. Huang N.E. et al. (2003), A confidence limit for the empirical mode decomposition and Hilbert spectral analysis, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 459(2037): 2317–2345, https://doi.org/10.1098/rspa.2003.1123
10. Kopsinis Y., McLaughlin S. (2007), Investigation and performance enhancement of the empirical mode decomposition method based on a heuristic search optimization approach, [in:] IEEE Transactions on Signal Processing, 56(1): 1–13, https://doi.org/10.1109/TSP.2007.901155
11. Kopsinis Y., McLaughlin S. (2008), Improved EMD using doubly-iterative sifting and high order spline interpolation, EURASIP Journal on Advances in Signal Processing, 2008(1): 128293, https://doi.org/10.1155/2008/128293
12. Lei Y., Lin J., He Z., Zuo M.J. (2013), A review on empirical mode decomposition in fault diagnosis of rotating machinery, Mechanical Systems and Signal Processing, 35(1–2): 108–126, https://doi.org/10.1016/j.ymssp.2012.09.015
13. Li H., Qin X., Zhao D., Chen J.,Wang P. (2018), An improved empirical mode decomposition method based on the cubic trigonometric B-spline interpolation algorithm, Applied Mathematics and Computation, 332: 406–419, https://doi.org/10.1016/j.amc.2018.02.039
14. Li H., Wang C., Zhao D. (2015a), An improved EMD and its applications to find the basis functions of EMI signals, Mathematical Problems in Engineering, 2015: 150127, https://doi.org/10.1155/2015/150127
15. Li Y., Xu M., Wei Y., Huang W. (2015b), An improvement EMD method based on the optimized rational Hermite interpolation approach and its application to gear fault diagnosis, Measurement, 63: 330–345, https://doi.org/10.1016/j.measurement.2014.12.021
16. Pegram G.G.S., Peel M.C., McMahon T.A. (2008), Empirical mode decomposition using rational splines: An application to rainfall time series, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 464(2094): 1483–1501, https://doi.org/10.1098/rspa.2007.0311
17. Qin S.R., Zhong Y.M. (2006), A new envelope algorithm of Hilbert–Huang Transform, Mechanical Systems and Signal Processing, 20(8): 1941–1952, https://doi.org/10.1016/j.ymssp.2005.07.002
18. Rilling G., Flandrin P. (2007), One or two frequencies? The empirical mode decomposition answers, [in:] IEEE Transactions on Signal Processing, 56(1): 85–95, https://doi.org/10.1109/TSP.2007.906771
19. Rilling G., Flandrin P., Gonçalvès P. (2003), On empirical mode decomposition and its algorithms, [in:] IEEE-EURASIP workshop on nonlinear signal and image processing, 3(3): 8–11, Grado.
20. Shu L., Deng H., Liu X., Pan Z. (2022), A Comprehensive working condition identification scheme for rolling bearings based on modified CEEMDAN as well as modified hierarchical amplitude-aware permutation entropy, Measurement Science and Technology, 33(7): 075111, https://doi.org/10.1088/1361-6501/ac5b2c
21. Singh P., Joshi S.D., Patney R.K., Saha K. (2014), Some studies on nonpolynomial interpolation and error analysis, Applied Mathematics and Computation, 244: 809–821, https://doi.org/10.1016/j.amc.2014.07.049
22. SKF (n.d.), Bearing Select, https://www.skfbearingselect.com/#/bearing-selection-start (access date: 21.02.2023).
23. Smith W.A., Randall R.B. (2015), Rolling element bearing diagnostics using the Case Western Reserve University data: A benchmark study, Mechanical Systems and Signal Processing, 64–65: 100–131, https://doi.org/10.1016/j.ymssp.2015.04.021
24. Sun Y., Li S., Wang X. (2021), Bearing fault diagnosis based on EMD and improved Chebyshev distance in SDP image, Measurement, 176: 109100, https://doi.org/10.1016/j.measurement.2021.109100
25. Torres M.E., Colominas M.A., Schlotthauer G., Flandrin P. (2011), A complete ensemble empirical mode decomposition with adaptive noise, [in:] 2011 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 4144–4147, https://doi.org/10.1109/ICASSP.2011.5947265
26. Wang J., Du G., Zhu Z., Shen C., He Q. (2020), Fault diagnosis of rotating machines based on the EMD manifold, Mechanical Systems and Signal Processing, 135: 106443, https://doi.org/10.1016/j.ymssp.2019.106443
27. Wang J.-L., Li Z.-J. (2013), Extreme-point symmetric mode decomposition method for data analysis, Advances in Adaptive Data Analysis, 5(3): 1350015, https://doi.org/10.1142/S1793536913500155
28. Wang Z., Yang J., Guo Y. (2022), Unknown fault feature extraction of rolling bearings under variable speed conditions based on statistical complexity measures, Mechanical Systems and Signal Processing, 172: 108964, https://doi.org/10.1016/j.ymssp.2022.108964
29. Wu Z., Huang N.E. (2009), Ensemble empirical mode decomposition: A noise-assisted data analysis method, Advances in Adaptive Data Analysis, 1(1): 1–41, https://doi.org/10.1142/S1793536909000047
30. Xu Z., Huang B., Li K. (2010), An alternative envelope approach for empirical mode decomposition, Digital Signal Processing, 20(1): 77–84, https://doi.org/10.1016/j.dsp.2009.06.009
31. Yang L., Yang Z., Zhou F., Yang L. (2014), A novel envelope model based on convex constrained optimization, Digital Signal Processing, 29(1): 138–146, https://doi.org/10.1016/j.dsp.2014.02.017
32. Yeh J.-R., Shieh J.-S., Huang N.E. (2010), Complementary ensemble empirical mode decomposition: A novel noise enhanced data analysis method, Advances in Adaptive Data Analysis, 2(2): 135–156, https://doi.org/10.1142/S1793536910000422
33. Yuan J., He Z., Ni J., Brzezinski A.J., Zi Y. (2013), Ensemble noise-reconstructed empirical mode decomposition for mechanical fault detection, Journal of Vibration and Acoustics, 135(2): 021011, https://doi.org/10.1115/1.4023138
34. Yuan J., Xu C., Zhao Q., Jiang H., Weng Y. (2022), High-fidelity noise-reconstructed empirical mode decomposition for mechanical multiple and weak fault extractions, ISA Transaction, 129(Part B): 380–397, https://doi.org/10.1016/j.isatra.2022.02.017
35. Zhao D., Huang Z., Li H., Chen J., Wang P. (2017), An improved EEMD method based on the adjustable cubic trigonometric cardinal spline interpolation, Digital Signal Processing, 64: 41–48, https://doi.org/10.1016/j.dsp.2016.12.007
36. Zheng J., Cao S., Pan H., Ni Q. (2022), Spectral envelope-based adaptive empirical Fourier decomposition method and its application to rolling bearing fault diagnosis, ISA Transactions, 129(Part B): 476–492, https://doi.org/10.1016/j.isatra.2022.02.049
