# Theoretical Analysis of Empirical Mode Decomposition

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Principle of Oscillation Signal Decomposition

#### 2.1. Theoretical Principle

_{k}and t

_{k}

_{+ 2}should be equal to zero. Therefore, from Equation (1), one has

_{o}(t) is the original function of r(t), that is, r

_{o}′(t) = r(t). Obviously, r

_{o}(t) is a real-valued function defined on interval [t

_{k}, t

_{k}

_{+ 2}], and differentiable at every point on the interval, with continuity and a finite value of its derivative. According to the Lagrange differential theorem of mean, it is known that there is at least one point t

_{l}between t

_{k}and t

_{k}

_{+ 2}to make

_{m}between t

_{k}

_{+1}and t

_{k}

_{+3}to make

_{k}to t

_{k}

_{+ 1}, as shown in Figure 2, can derive new relations from Equations (4) and (5), respectively, given by

_{k}

_{+ 1}− t

_{k})/2.

_{k}, t

_{k}

_{+ 1}, t

_{k}

_{+ 2}, and t

_{k}

_{+ 3}, one can use the following approximate relations

_{k}and ∆tx

_{k}

_{+ 3}are approximate to two integrals of this equation, as shown in Figure 2. There are two shadow areas caused by this approximation. If they are almost equal to each other, such approximation is very good due to their difference considered here. Similarly, Equation (9) is also a good approximation when the other two areas in the integral calculation almost keep the same value. In Equation (10), the relevant two integrals have the same initial point, except for different end points. Both of them represent the integral means over their individual intervals with slight differences, and thus this equation should be a good approximation. Additionally, t

_{m}− t

_{l}approximated by 2 ∆t can be derived from the same step of movement for these two time points as that of their corresponding intervals.

_{k}

_{+ 3}− t

_{k}

_{+ 1}) and Equation (6) divided by (t

_{k}

_{+ 2}− t

_{k}), after simplifying, yields

_{max}(t) between two local maxima and that of the curve x

_{min}(t) between two local minima. With due regard to the behavior of r(t), x

_{max}(t) and x

_{min}(t) uniformly varying with time over the interval [t

_{k}, t

_{k}

_{+ 3}], one has

_{max}(t) and x

_{min}(t) generated by local maxima of the signal and local minima, respectively. Through Equation (1), the oscillation component c(t) can be computed so as to realize the decomposition of oscillation signal.

#### 2.2. Basic Steps

- (1)
- Consider an oscillation signal x(t) varying with time t and take the piece of this signal as a signal analyzed. Assuming that this signal is composed of a pure oscillation component c(t) of proper rotation, with mean zero and a residual term (the trend or the baseline signal) r(t), Equation (1) follows.
- (2)
- Since c(t) integral in the interval of two local maxima (or minima) points t
_{k}and t_{k}_{+ 2}should be equal to zero. Equation (1) can be transformed into Equation (2). - (3)
- According to the Lagrange differential theorem of mean, it is known that there is at least one point t
_{l}between t_{k}and t_{k}_{+ 2}to make Equation (3). Thus, Equation (2) can be rewritten as Equation (4).It can be derived that there is a point t_{m}between t_{k}_{+ 1}and t_{k}_{+ 3}to make Equation (5). - (4)
- Choosing a point C located at the central point from t
_{k}to t_{k}_{+ 1}, it can derive new relations from Equations (4) and (5). Equations (6) and (7) follow. - (5)
- Where ∆t = (t
_{k}_{+ 1}− t_{k})/2. - (6)
- With consideration of these approximate relations from Equations (8) to (11), after simplifying, this yields Equation (12).
- (7)
- In Equation (12), the term on the left-hand side is approximate to the differential of r(t) with respect to time t, and the first and the second terms on the right-hand side. To the differential of the curve x
_{max}(t) between two local maxima and that of the curve x_{min}(t) between two local minima. With due regard to the behavior of r(t), x_{max}(t), and x_{min}(t) uniformly varying with time over the interval [t_{k}, t_{k}_{+ 3}], one has Equation (13). - (8)
- Finally, the component as in Equation (14) is obtained.

## 3. Characteristics of EMD Algorithm

- (1)
- The successive extrema of x(t) are firstly identified, then the local maxima are connected by a cubic spline as the upper envelope, and the local minima are similarly connected as the lower envelope.
- (2)
- These two envelopes are used to calculate the mean as a function of time designated as m
_{1}(t). - (3)
- The difference h
_{1}(t) between the signal x(t) and the mean m_{1}(t) is calculated by the relation h_{1}(t) = x(t) − m_{1}(t), which can be regarded as the primary description of the first IMF. - (4)
- To determine the first IMF more accurately, h
_{1}(t) is treated as a new signal, its upper and lower envelopes, and their new mean m_{2}(t) are calculated, and a new difference h_{2}(t) = h_{1}(t) − m_{2}(t) is determined. This h_{2}(t) is again treated as a new signal, and the process, referred to as iteration, is repeated many times designated by k until a stopping criterion satisfies. h_{k}(t) is the first IMF, designated by H_{1}(t). - (5)
- The first residue d
_{1}(t) = x(t) − H_{1}(t) is analyzed by the same steps (1)–(4) to obtain the second IMF H_{2}(t). This sifting process continues until the last residue shows no apparent variation.

#### 3.1. Primary Description of the First IMF

_{1}(t) correspond essentially to x

_{max}(t) and x

_{min}(t) in Equation (14), and the mean m

_{1}(t) is equal to the function r(t). Therefore, the theoretical principle of oscillation signal decomposition suggested above provides a theoretical basis available for the analysis of EMD.

_{max}(t) and x

_{min}(t) uniformly and continuously varying with time over the interval [t

_{k}, t

_{k}

_{+ 4}], there are no other special requirements introduced. Obviously, these links can exclude the choice of straight lines due to the smooth property of r(t). Consequently, all nonlinear expressions should be appropriate for the links between the nearest maxima or between the nearest minima only if they meet the requirement of varying with time uniformly and smoothly, which can explain this phenomenon that it is very difficult to agree in the best implementation for so many studies.

#### 3.2. Iteration and Sifting Process

^{(0)}(t). Assuming that the extrema positions of c

^{(0)}(t) represent the real extrema positions of the oscillation component c(t), and taking one of them as the point analyzed at which the extremum is M

_{a}and the value of another curve linking the maxima or minima is M

_{b}, may yield, after the first iteration, that the corresponding mean (or trend) r

^{(1)}can be calculated by means of Equation (14) and expressed as

^{(1)}is calculated through Equation (1), given by

^{(2)}and the oscillation value c

^{(2)}are given by

^{(2)}and the oscillation value c

^{(2)}are given by

^{(j)}→0. This result can be derived from any one of the extrema positions of c

^{(0)}(t). Since the function r(t) varies monotonically on most of the intervals between two neighbor extrema, it can be seen that r

^{(j)}→0 at all extrema means that the approximate function r(t) calculated by Equation (14), after many iterations, tends to zero at every time point, that is

## 4. Analysis and Discussion

#### 4.1. Interpolation

_{max}(t) and x

_{min}(t) had best to monotonically vary with time on the intervals between two nearest maxima or minima [23].

#### 4.2. Frequency Resolving Ability

_{l}and t

_{m}. This interval is not sure to cover the periodic time of the oscillation component, an interval with three successive local extrema of the signal. Thus, it is necessary to add the piece of signal up to next extremum into the signal analyzed. That is to say, decomposing the oscillation component described by three extrema requires the piece of signal, at least, with five extrema. It shows that the theoretical limit of frequency resolving ability of EMD is just the frequency ratio larger than 1/0.6 and less than 0.6, while the frequency ratio more than 2 or less than 0.5 should be its optimum frequency resolving ability. To break through this theoretical limit, a number of approaches have been made so far, all showing their advantage in reducing the so-call mode mixing [3,18,20,21]. Unfortunately, evidence provided by the suggested theoretical principle has led to fruitless efforts, essentially, that the EMD is modified under the present framework.

_{k}

_{+4}, a relation similar to Equations (4) or (5) on the interval [x

_{k}

_{+ 2}, x

_{k}

_{+ 4}] can be added as

_{n}is a point in the interval (x

_{k}

_{+2}, x

_{k}

_{+4}). Generally, one has, t

_{l}< t

_{m}< t

_{n}. It can be found that the data set {(t

_{l}, r(t

_{l})); (t

_{m}, r(t

_{m})); (t

_{n}, r(t

_{n})} provides interpolation points for the solution of r(t) by means of the spline implementation. Since only five local extrema points are used, the interpolation solution makes the frequency resolving ability increase to the range of the frequency ratio larger than 5/3 and less than 3/5, which just corresponds to the actual limitation of the EMD. However, t

_{l}, t

_{m}and t

_{n}are unknown to operate this decomposition. The method of EMD makes full use of the information associated with extrema of the signal and coincidently avoids the difficulty to determine parameters t

_{l}, t

_{m}and t

_{n}at the cost of lowering its frequency resolving ability.

_{l}, t

_{m}, and t

_{n}can roughly be taken as the centre of its corresponding time interval. Figure 3 presents the decomposition results of two digital simulation signals with two components. One simulation signal has frequency ratio 3, and another 1.4. Comparing with the results obtained by the EMD, one can see that results from the interpolation solution are not better than those from the EMD for lack of optimal interpolation time estimates, but the former can obviously increase the frequency-resolving ability of oscillation signal decomposition, which should indicate a developing direction for reducing the mode mixing of EMD.

#### 4.3. Assumption and Approximation in the Theoretical Principle

_{l}on the interval (x

_{k}, x

_{k}

_{+ 2}), t

_{m}on (x

_{k}

_{+ 1}, x

_{k}

_{+ 3}), or t

_{n}on (x

_{k}

_{+ 2}, x

_{k}

_{+ 4}) (impossible for all of them to occur simultaneously), Equations (12) and (13) simultaneously turn the exact relations into approximate expressions, which means that the central point of several t

_{l}, t

_{n}, or t

_{m}, or any one of them can also work well. Since the interval considered moves by a step of a half wave length, this relationship t

_{l}< t

_{m}< t

_{n}is always following. Examples of interpolation solution decomposition, as shown in Figure 3, illustrate that t

_{l}, t

_{m}, and t

_{n}should be located near the centre of their corresponding intervals. Affirmatively, more than one t

_{l}, t

_{m}, or t

_{n}will have influence on the effect of interpolation solution decomposition, but have little on the EMD.

_{max}(t), x

_{min}(t), and r(t) are very close to straight lines over the interval (t

_{l}, t

_{m}), can the approximation involved in this equation be correct. The upper envelope generated by local maxima of a signal and the lower one by local minima can be controlled properly to describe the link between two nearest maxima or minima close to a straight line. Naturally, r(t) close to a straight line can be regarded. One cannot adopt the assumption that the segments of two envelopes between two nearest maxima and minima are straight lines, because straight lines cannot cover the signal well and also cannot portrait its property smoothly varying with time. In view of this, Equation (13) is an inevitable inference drawn from Equation (12).

#### 4.4. Bearing Fault Data Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Huang, N.E.; Shen, Z.; Long, S.R.; Wu, M.L.; Shih, H.H.; Zheng, Q.; Yen, N.C.; Tung, C.C.; Liu, H.H. The empirical mode decomposition and Hilbert spectrum for nonlinear and non-stationary time series analysis. Proc. R. Soc. Lond. A
**1998**, 454, 903–995. [Google Scholar] [CrossRef] - Rilling, G.; Flandrin, P.; Goncalves, P. On empirical mode decomposition and its algorithms. In Proceedings of the IEEE-EURASIP Workshop on Nonlinear Signal and Image Processing NSIP-03, Grado, Italy, 8–11 June 2003. [Google Scholar]
- Lee, Y.S.; Tsakirtzis, S.; Vakakis, A.F.; Bergman, L.A.; McFarlan, D.M. Physics-based foundation for empirical mode decomposition. AIAA J.
**2009**, 47, 2938–2963. [Google Scholar] [CrossRef] - Feldman, M. Hilbert transform in vibration analysis. Mech. Syst. Signal Process.
**2011**, 25, 735–802. [Google Scholar] [CrossRef] - Yang, J.N.; Lei, Y.; Pan, S.; Huang, N.H. System identification of linear structures based on Hilbert-Huang spectral analysis, part 1: Normal modes. Earthq. Eng. Struct. Dyn.
**2003**, 32, 1443–1467. [Google Scholar] [CrossRef] - Yang, J.N.; Lei, Y.; Pan, S.; Huang, N.H. System identification of linear structures based on Hilbert-Huang spectral analysis, part 2: Complex modes. Earthq. Eng. Struct. Dyn.
**2003**, 32, 1533–1554. [Google Scholar] [CrossRef] - Pai, P.F. Time–frequency characterization of nonlinear normal modes and challenges in nonlinearity identification of dynamical systems. Mech. Syst. Signal Process.
**2011**, 25, 2358–2374. [Google Scholar] [CrossRef] - Lee, Y.S.; Vakakis, A.F.; McFarland, D.M.; Bergman, L.A. A global-local approach to nonlinear system identification: A review. Struct. Control Health Monit.
**2010**, 17, 742–760. [Google Scholar] [CrossRef] - Li, H.; Deng, X.; Dai, H. Structural damage detection using the combination method of EMD and wavelet analysis. Mech. Syst. Signal Process.
**2007**, 21, 298–306. [Google Scholar] [CrossRef] - Liu, B.; Riemenschneider, S.; Xu, Y. Gearbox fault diagnosis using empirical mode decomposition and Hilbert spectrum. Mech. Syst. Signal Process.
**2006**, 20, 718–734. [Google Scholar] [CrossRef] - Gao, Q.; Duan, C.; Fan, H.; Meng, Q. Rotating machine fault diagnosis using empirical mode decomposition. Mech. Syst. Signal Process.
**2008**, 22, 1072–1081. [Google Scholar] [CrossRef] - Ricci, R.; Pennacchi, P. Diagnostics of gear faults based on EMD and automatic selection of intrinsic mode functions. Mech. Syst. Signal Process.
**2011**, 25, 821–838. [Google Scholar] [CrossRef] [Green Version] - Kopsinis, Y.; McLaughlin, S. Investigation and performance enhancement of the empirical mode decomposition method based on a heuristic search optimization approach. IEEE Trans. Signal Process.
**2008**, 56, 1–13. [Google Scholar] [CrossRef] - Deléchelle, E.; Lemoine, J.; Niang, O. Empirical mode decomposition: An analytical approach for sifting process. IEEE Signal Process. Lett.
**2005**, 12, 764–767. [Google Scholar] [CrossRef] - Niang, O.; Deléchelle, E.; Lemoine, J. A spectral approach for sifting process in empirical mode decomposition. IEEE Trans. Signal Process.
**2010**, 58, 5612–5623. [Google Scholar] [CrossRef] - Rilling, G.; Flandrin, P. One or two frequencies? The empirical mode decomposition answers. IEEE Trans. Signal Process.
**2008**, 56, 85–95. [Google Scholar] [CrossRef] - Feldman, M. Analytical basics of the EMD: Two harmonics decomposition. Mech. Syst. Signal Process.
**2009**, 23, 2059–2071. [Google Scholar] [CrossRef] - Huang, N.E.; Shen, Z.; Long, S.R. A new review of nonlinear water waves: The Hilbert spectrum. Ann. Rev. Fluid Mech.
**1999**, 31, 417–457. [Google Scholar] [CrossRef] - Smith, J.S. The local mean decomposition and its application to EEG perception data. J. R. Soc. Interface
**2005**, 2, 443–454. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Hong, H.; Wang, X.; Tao, Z. Local integral mean-based sifting for empirical mode decomposition. IEEE Signal Process. Lett.
**2009**, 16, 841–844. [Google Scholar] [CrossRef] - Li, C.; Wang, X.; Tao, Z.; Wang, Q.; Du, S. Extraction of time varying information from noisy signals: An approach based on the empirical mode decomposition. Mech. Syst. Signal Process.
**2011**, 25, 812–820. [Google Scholar] [CrossRef] - Hawley, S.D.; Atlas, L.E.; Chizeck, H.J. Some properties of an empirical mode type signal decomposition algorithm. IEEE Signal Process. Lett.
**2010**, 17, 24–27. [Google Scholar] [CrossRef] - Xu, Z.; Huang, B.; Li, K. An alternative envelope approach for empirical mode decomposition. Digit. Signal Process.
**2010**, 20, 77–84. [Google Scholar] [CrossRef] - Datig, M.; Schlurmann, T. Performance and limitations of the Hilbert-Huang transformation (HHT) with an application to irregular water waves. Ocean Eng.
**2004**, 31, 1783–1834. [Google Scholar] [CrossRef] - Case Western Reserve University Data Center. Available online: https://csegroups.case.edu/bearingdatacenter/home (accessed on 9 November 2018).
- Dang, Z.; Lv, Y.; Li, Y.; Wei, G. Improved Dynamic Mode Decomposition and Its Application to Fault Diagnosis of Rolling Bearing. Sensors
**2018**, 18, 1972. [Google Scholar] [CrossRef] [PubMed] - Pang, B.; Tang, G.; Tian, T.; Zhou, C. Rolling Bearing Fault Diagnosis Based on an Improved HTT Transform. Sensors
**2018**, 18, 1203. [Google Scholar] [CrossRef] [PubMed] - Wan, S.; Zhang, X. Teager Energy Entropy Ratio of Wavelet Packet Transform and Its Application in Bearing Fault Diagnosis. Entropy
**2018**, 20, 388. [Google Scholar] [CrossRef]

**Figure 3.**Comparison of decomposition results between interpolation solution and empirical mode decomposition (EMD).

**Figure 5.**Spectrum diagram of Figure 4.

**Figure 8.**Instantaneous frequency distribution of component parts obtained by EMD method decomposition.

**Figure 9.**Instantaneous frequency distribution of component parts obtained by wavelet method decomposition.

**Figure 10.**Partial modal component spectrum obtained by wavelet decomposition. (

**a**) IMF5 component spectrum, (

**b**) IMF6 component spectrum.

Method Type | SNR | RMSE |
---|---|---|

Wavelet method | 18.69 | 0.19 |

EMD method | 21.12 | 0.13 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ge, H.; Chen, G.; Yu, H.; Chen, H.; An, F.
Theoretical Analysis of Empirical Mode Decomposition. *Symmetry* **2018**, *10*, 623.
https://doi.org/10.3390/sym10110623

**AMA Style**

Ge H, Chen G, Yu H, Chen H, An F.
Theoretical Analysis of Empirical Mode Decomposition. *Symmetry*. 2018; 10(11):623.
https://doi.org/10.3390/sym10110623

**Chicago/Turabian Style**

Ge, Hengqing, Guibin Chen, Haichun Yu, Huabao Chen, and Fengping An.
2018. "Theoretical Analysis of Empirical Mode Decomposition" *Symmetry* 10, no. 11: 623.
https://doi.org/10.3390/sym10110623