# An Improved Signal Processing Approach Based on Analysis Mode Decomposition and Empirical Mode Decomposition

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Basic Principles of EMD

_{i}refers to the ith IMF component and r

_{n}is the residual function.

_{n}becomes a monotonic function, but in the real decomposition process, when r

_{n}meets the conditions of monotonic function, the cycle number is usually large, and the number of IMF components will be too large. Furthermore, a mass of EMD tests have revealed that most of these final components obtained from EMD are false IMF components, without substantive physical significance. Thus, a good sifting stop criterion can not only improve the decomposition efficiency, but also increase the decomposition accuracy.

_{d}[25]. In particular, S

_{d}is defined as:

## 3. The Sifting Stop Criteria Based on the Valid Data Segment

_{k}(t) = 0, the iteration threshold will become an uncertain value. It is evidently inappropriate if the iteration threshold is used to control sifting number at the time. In addition, the sifting stop criteria ignored the influence of end effect. Consequently, the decomposition result may produce errors if the distorted endpoint data are adopted. In most cases, the end effect of EMD is obvious, though some effective measures will been taken to restrain it, the end effect still exists.

_{k}(t) is an IMF, it satisfies the following inequality:

_{1}(t) and x

_{2}(t) (as shown in Figure 1, Figure 2 and Figure 3) are the decomposition results by using EMD based on the proposed sifting stop criterion and the traditional one, respectively. It is worth noting that the EMD can efficiently decompose the original signal by using the two sifting stop criteria, but it is clear from the two edges of these two figures that the decomposition result c

_{1}and c

_{2}in Figure 2 are more accurate than imf

_{1}and imf

_{2}in Figure 3, which show an evident end effect.

_{1}contains massive fault information of the gear. Note from its time domain graph (see Figure 5) that the waveform of imf

_{1}presents an obvious modulation feature, and its cycle of modulation wave (T, approximately 0.074 s) accordingly had a frequency of about 13.6 Hz, which is exactly the rotating frequency of the faulty gear. Thus, it can be concluded that the fault information has been exacted from the vibration signal of the practical gears by using the proposed sifting stop criterion.

## 4. AEMD Method

#### 4.1. Analytical Mode Decomposition (AMD) Method

_{b}, one is the fast-changing signal x

_{f}(t) and another is slowly-changing signals x

_{s}(t):

_{f}(ω), X

_{s}(ω) have no overlap in the frequency band. The Hilbert transforms of $\mathrm{cos}(2\mathsf{\pi}{f}_{b}t)\cdot {x}_{0}(t)$ and $\mathrm{sin}(2\mathsf{\pi}{f}_{b}t)\cdot {x}_{0}(t)$ are as follows:

_{s}(t) and x

_{f}(t) can be derived from the above formulas:

_{s}(t) and x

_{f}(t) by using the analysis mode decomposition.

#### 4.2. Steps of the AEMD Method

_{1}(t) with mode mixing is performed by AMD. The boundary frequency f

_{b1}can be determined according to the spectrum. The average frequency of the two confused modes is used as the boundary frequency in this analysis. Then IMF

_{1}(t) is separated into the two signals ${c}_{1}(t)$ and ${c}_{1}^{^}(t)$ by using AMD. ${c}_{1}(t)$ is the correction component of IMF

_{1}(t) and ${c}_{1}^{^}(t)$ is the residual component of ${c}_{1}(t)$. Thus, IMF

_{1}(t) can be expressed as follows:

_{2}(t) and get the renewed IMF

_{2}(t), denoted as IMF*

_{2}(t). The boundary frequency f

_{b}

_{2}can be obtained according to the amplitude spectrum of IMF

^{*}

_{2}(t). Then IMF

^{*}

_{2}(t) is decomposed into the two signals ${c}_{2}(t)$ and ${c}_{2}^{^}(t)$ by using AMD. ${c}_{2}(t)$ is the correction component of IMF*

_{2}(t) and ${c}_{2}^{^}(t)$ is the residual component of ${c}_{2}(t)$. Similarly, IMF*

_{2}(t) is expressed as follows:

^{^}

_{k}(t) is added to the residual error r and obtain the final residual error, denote as v

_{k}(t).

#### 4.3. The Comparison of Simulation Signal Analysis by Different Methods

_{1}(t) is confused with the sinusoidal signal at 30 Hz, for the second mode c

_{2}(t), the sinusoidal signal at 30 Hz with the sinusoidal signal at 14 Hz. The third and fourth modes are false IMF components. The decomposition result of EEMD is imf

_{1}, imf

_{2}, imf

_{3}, imf

_{4,}as shown in Figure 8, with ensemble average number N = 30, and the amplitude of noise is set as 0.01 standard deviation of the original signal. The decomposition result of AEMD is shown in Figure 9. A successful decomposition result should distinctly obtain the several components of the original signal. In order to compare the results of decomposition results by these different methods, the green lines in Figure 7, Figure 8 and Figure 9 represent the several components of the original signal; The blue lines represent the composition results by these different methods. Table 1 lists the run time by these three methods using the same computer.

#### 4.4. Case Study

_{2}diagram, the characteristic frequency of the rotor rubbing fault is confused in multiple frequencies, caused by the slight rubbing fault of the rotor, which affects the identification for the fault. In the imf

_{3}diagram, the triple frequency of the characteristic frequency of the rotor rubbing fault—150 Hz is nearly invisible, which is mixed in the fundamental frequency of the rotor rubbing fault—50 Hz.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Chen, Z.; Cao, S.; Mao, Z. Remaining useful life estimation of aircraft engines using a modified similarity and supporting vector machine (SVM) approach. Energies
**2017**, 11, 28. [Google Scholar] [CrossRef] - Han, J.; Ji, G.Y. Gear Fault Diagnosis Based on Improved EMD Method and the Energy Operator Demodulation Approach. J. Changsha Univ. Sci. Technol.
**2015**, 12, 66–71. [Google Scholar] - Yang, Y.F.; Wu, Y.F. Application of Empirical Mode Decomposition in the Analysis of Vibration; National Defence of Industry Press: Beijing, China, 2013; pp. 17–45. [Google Scholar]
- Huang, D.S. The Method of False Modal Component Elimination in Empirical Mode Decomposition. Vibration. Meas. Diagn.
**2011**, 31, 381–384. [Google Scholar] - Wang, L. Fault Diagnosis of Rotor System Based on EMD-Fuzzy Entropy and SVM. Noise Vib. Control
**2012**, 6, 172–173. [Google Scholar] - Ben Ali, J.; Fnaiech, N.; Saidi, L.; Chebel-Morello, B.; Fnaiech, F. Application of empirical mode decomposition and artificial neural network for automatic bearing fault diagnosis based on vibration signals. Appl. Acoust.
**2015**, 89, 16–27. [Google Scholar] [CrossRef] - Xue, X.; Zhou, J.; Xu, Y.; Zhu, W.; Li, C. An Adaptively Fast Ensemble Empirical Mode Decomposition Method and Its Applications to Rolling Element Bearing Fault Diagnosis. Mech. Syst. Sig. Process.
**2015**, 62–63, 444–459. [Google Scholar] [CrossRef] - Yu, D.J.; Cheng, J.S. The Hilbert-Huang Transform Method in Mechanical Fault Diagnosis; Science Press: Beijing, China, 2006; pp. 179–190. [Google Scholar]
- Cheng, J.; Xu, Y.L. Application of HHT Method in Structural Modal Parameter Identification. J. Vib. Eng.
**2003**, 16, 383–387. [Google Scholar] - Cheng, J.; Yang, Y. The application of EMD Method in Local Touch Friction Fault Diagnosis of Rotor. Vibration. Meas. Diagn.
**2006**, 26, 24–27. [Google Scholar] - Rilling, G.; Flandrin, P. One or Two Frequencies? The Empirical Mode Decomposition Answers. IEEE Trans. Signal Process.
**2008**, 56, 85–95. [Google Scholar] [CrossRef] - Dou, D.Y.; Zhao, Y.K. Application of Ensemble Empirical Mode Decomposition in Failure Analysis of Rotating Machinery. Trans. Chin. Soc. Agric. Eng.
**2010**, 26, 190–196. [Google Scholar] - Pustelnik, N.; Borgnat, P.; Flandrin, P. Empirical mode decomposition revisited by multicomponent non-smooth convex optimization. Signal Process.
**2014**, 102, 313–331. [Google Scholar] [CrossRef] - Lei, Y.G.; He, Z.J.; Zi, Y.Y. Application of the EEMD Method to Rotor Fault Diagnosis of Rotating Machinery. Mech. Syst. Signal Process.
**2009**, 23, 1327–1338. [Google Scholar] [CrossRef] - Zhao, J.P. Study on the Effects of Abnormal Events to Empirical Mode Decomposition Method and the Removal Method for Abnormal Signal. J. Ocean Univ. Qingdao
**2001**, 31, 805–814. [Google Scholar] - Deering, R.; Kaiser, J.F. The Use of a Masking Signal to Improve Empirical Mode Decomposition. Acoust. Speech Signal Process. ICASSP
**2005**, 4, 485–488. [Google Scholar] - Hu, A.J. Research on the Application of Hilbert-Huang Transform in Vibration Signal Analysis of Rotating Machinery; North China Electric Power University: Bao Ding, Hebei, China, 2008. [Google Scholar]
- Wu, Z.; Huang, N.E. Ensemble Empirical Mode Decomposition: A Noise-Assisted Data Analysis Method. Adv. Adapt. Data Anal.
**2009**, 1, 1–41. [Google Scholar] [CrossRef] - Chen, G.D.; Wang, Z.C. A Signal Decomposition Theorem with Hilbert Transform and Its Application to Narrow Band Time Series with Closely Spaced Frequency Components. Mech. Syst. Signal Process.
**2012**, 28, 258–279. [Google Scholar] [CrossRef] - Lei, Y.G.; Kong, D.T. Adaptive Ensemble Empirical Mode Decomposition and Application to Fault Detection of Planetary Gear Boxes. J. Mech. Eng.
**2014**, 50, 64–70. [Google Scholar] [CrossRef] - Zheng, J.D.; Cheng, J.S.; Yang, Y. Partly Ensemble Empirical Mode Decomposition: An Improved Noise-Assisted Method for Eliminating Mode Mixing. Signal Process.
**2014**, 96, 362–374. [Google Scholar] [CrossRef] - Hu, J.S.; Yang, S.X. Energy-Based Stop Condition of Empirical Mode Decomposition of Vibration Signal. J. Vib. Meas. Diagn.
**2009**, 29, 19–22. [Google Scholar] - Mohammad, S.H.; Siamak, E.K.; Mohammad, S.S. Quantitative diagnosis for bearing faults by improving ensemble empirical mode decomposition. ISA Trans.
**2018**, 83, 261–275. [Google Scholar] - Yang, Y.; Yu, D.J.; Cheng, J.S. A Roller Bearing Fault Diagnosis Method Based on EMD Energy Entropy and ANN. J. Sound Vib.
**2006**, 294, 269–277. [Google Scholar] - 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. Proc. R. Soc. Lond. A
**1998**, 454, 903–995. [Google Scholar] [CrossRef]

**Figure 2.**The result of empirical mode decomposition (EMD) method using the sifting criterion based on the valid data segment.

**Figure 8.**The decomposition result of the simulation signal by ensemble empirical mode decomposition (EEMD).

**Figure 14.**(

**a**) The spectrum of intrinsic mode functions (IMFs) by EMD. (

**b**) The spectrum of IMFs by AEMD.

Method | First | Second | Third | Fourth | Fifth | Sixth |
---|---|---|---|---|---|---|

EMD | 1.3988 | 1.3860 | 1.3930 | 1.3846 | 1.3820 | 1.3940 |

EEMD | 41.4875 | 44.8383 | 44.1800 | 42.8480 | 43.2321 | 44.024 |

AEMD | 1.4027 | 1.3832 | 1.3843 | 1.3934 | 1.3880 | 1.3848 |

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**MDPI and ACS Style**

Chen, Z.; Liu, B.; Yan, X.; Yang, H.
An Improved Signal Processing Approach Based on Analysis Mode Decomposition and Empirical Mode Decomposition. *Energies* **2019**, *12*, 3077.
https://doi.org/10.3390/en12163077

**AMA Style**

Chen Z, Liu B, Yan X, Yang H.
An Improved Signal Processing Approach Based on Analysis Mode Decomposition and Empirical Mode Decomposition. *Energies*. 2019; 12(16):3077.
https://doi.org/10.3390/en12163077

**Chicago/Turabian Style**

Chen, Zhongzhe, Baqiao Liu, Xiaogang Yan, and Hongquan Yang.
2019. "An Improved Signal Processing Approach Based on Analysis Mode Decomposition and Empirical Mode Decomposition" *Energies* 12, no. 16: 3077.
https://doi.org/10.3390/en12163077