# A Comparative Study of Empirical Mode Decomposition-Based Filtering for Impact Signal

^{*}

## Abstract

**:**

_{in}). In particular, this approach is successful in filtering impact signal. The results of the filtering are evaluated by a de-trended fluctuation analysis (DFA) algorithm, revised mean square error (RMSE), and revised signal-to-noise ratio (RSNR), respectively. The filtering results of simulated and impact signal show that the filtering method based on CEEMDAN and fuzzy entropy outperforms other signal filtering methods.

## 1. Introduction

## 2. Related Works

#### 2.1. Improved Empirical Mode Decomposition

- (1)
- Decompose signal $x\left(t\right)+{w}_{0}{\epsilon}^{i}\left(t\right)$ to obtain the first mode by using EMD algorithm.$${c}_{1}\left(t\right)=\frac{1}{N}{\displaystyle \sum _{i=1}^{N}{c}_{1}^{i}\left(t\right)}i\in \left\{1,\dots ,N\right\}$$
- (2)
- Compute the difference signal$${r}_{1}\left(t\right)=x\left(t\right)-{c}_{1}\left(t\right)$$
- (3)
- Decompose ${r}_{1}\left(t\right)+{w}_{1}{E}_{1}\left({\epsilon}^{i}\left(t\right)\right)$ to obtain the first mode and define the second mode by$${c}_{2}\left(t\right)=\frac{1}{N}{\displaystyle \sum}_{i=1}^{N}{E}_{1}\left({r}_{1}\left(t\right)+{w}_{1}{E}_{1}\left({\epsilon}^{i}\left(t\right)\right)\right)$$
- (4)
- For k = 2, …, K, calculate the k-th residue and obtain the first mode. Define the (k + 1)-th mode as follows:$${c}_{k+1}\left(t\right)=\frac{1}{N}{\displaystyle \sum}_{i=1}^{N}{E}_{1}\left({r}_{k}\left(t\right)+{w}_{k}{E}_{k}\left({\epsilon}^{i}\left(t\right)\right)\right)$$
- (5)
- Repeat step (4) until the residue contains no more than two extremes. The residual mode is then defined as:$$\mathrm{R}\left(t\right)=x\left(t\right)-{\displaystyle \sum}_{k=1}^{K}{c}_{k}\left(t\right)$$$$x\left(t\right)={\displaystyle \sum}_{k=1}^{K}{c}_{k}\left(t\right)+\mathrm{R}\left(t\right)$$

#### 2.2. Fuzzy Entropy

- (1)
- Given a time sequence $\left\{x\left(i\right):1\le i\le N\right\}$ ($N$ is the sample size) to form a sequence segment by choosing $m$ consecutive values in time sequence $x\left(i\right)$$${X}_{i}^{m}=\left\{x\left(i\right),x\left(i+1\right),\dots ,x\left(i+m-1\right)\right\}-{x}_{0}\left(i\right)\text{}(i=1,2,\dots ,N-m)$$The ${x}_{0}\left(i\right)$ expression is shown as follows:$${x}_{0}\left(i\right)=\frac{1}{m}{\displaystyle \sum _{j=0}^{m-1}x\left(i+j\right)}\text{}(i=1,2,\dots ,N-m)$$
- (2)
- The distance $d\left[{X}_{i}^{m},{X}_{j}^{m}\right]$ between the vector ${X}_{i}^{m}$ and ${X}_{j}^{m}$ is defined$$\begin{array}{cc}\hfill {d}_{ij}^{m}& =d\left[{X}_{i}^{m},{X}_{j}^{m}\right]\hfill \\ & =\underset{k\in \left(0,m-1\right)}{\mathrm{max}}\left\{\left|\left(x\left(i+k\right)-{x}_{0}\left(i\right)\right)-\left(x\left(j+k\right)-{x}_{0}\left(j\right)\right)\right|\right\}\hfill \end{array}$$
- (3)
- The similarity degree ${D}_{ij}^{m}$ between ${X}_{i}^{m}$ and ${X}_{j}^{m}$ is expressed by the fuzzy function $\mu \left({d}_{ij}^{m},n,r\right)$. The expression of similarity ${D}_{ij}^{m}$ is as follows:$${D}_{ij}^{m}=\mu \left({d}_{ij}^{m},n,r\right)={e}^{-{\left({d}_{ij}^{m}/r\right)}^{n}}$$
- (4)
- Definition function$${\varphi}^{m}\left(n,r\right)=\frac{1}{N-m}{\displaystyle \sum _{i=1}^{N-m}\left(\frac{1}{N-m}{\displaystyle \sum _{\begin{array}{l}j=1\\ j\ne i\end{array}}^{N-m}{D}_{ij}^{m}}\right)}$$
- (5)
- For $m+1$ dimension function, repeat steps (1) to (4), and obtain ${\varphi}^{m+1}\left(n,r\right)$$${\varphi}^{m+1}\left(n,r\right)=\frac{1}{N-m}{\displaystyle \sum _{i=1}^{N-m}\left(\frac{1}{N-m}{\displaystyle \sum _{\begin{array}{l}j=1\\ j\ne i\end{array}}^{N-m}{D}_{ij}^{m+1}}\right)}$$
- (6)
- The fuzzy entropy is defined by:$$\mathrm{Fuzzy}En\left(m,n,r\right)=\mathrm{ln}{\varphi}^{m}\left(n,r\right)-\mathrm{ln}{\varphi}^{m+1}\left(n,r\right)$$

## 3. Filtering Method

#### 3.1. Identifying the Relevant Mode

- (1)
- The noisy signal x(t) is decomposed to obtain IMF
_{i}(i = 1, …, N) by EMD or the improved version. - (2)
- Each IMF is sorted in ascending order, and the energy of the sorted data is calculated and then normalized to obtain a new waveform (the normalized new waveform is defined as NNW).
- (3)
- The complexity of each NNW
_{i}(except residue) is calculated by fuzzy entropy, and the value of fuzzy entropy is marked as E_{i}(i = 1, …, M − 1). Here, M is the number of modes by EMD or the improved version. - (4)
- The difference of adjacent E
_{i}is obtained, and the absolute value of the difference is calculated.$${D}_{j}=\left|{E}_{j}-{E}_{j+1}\right|\left(j=1,2,\dots ,M-2\right)$$ - (5)
- The relevant mode is identified.$$\mathrm{r}=\mathrm{argmax}\left({D}_{j}\right)+1$$
- (6)
- The filtered signal is obtained.$$\tilde{x}\left(t\right)={\displaystyle \sum}_{m=r}^{M}IM{F}_{m}\left(t\right)$$

#### 3.2. Application

_{in}) is 3 dB, and the length of data is 1024. The signal $x\left(t\right)$ is decomposed into nine modes by CEEMDAN (the ratio of the standard deviation of added white noise is 0.15, and ensemble number is 80). If the signal $x\left(t\right)$ is known, the seventh and eighth modes are the useful signal mode, and the other are the noisy modes (except residue). The filtered signal is the sum of last three modes (the seventh, eighth, and ninth modes).

## 4. Results and Discussion

#### 4.1. Simulated Signal Filtering

_{in}). The SNR

_{in}ranges from 1 dB to 11 dB with a fixed step of 2 dB. To quantize the filtering result, the output signal-to-noise ratio (SNR

_{out}) and mean square error (MSE) are performed to compare.

_{in}, the different values of ensemble number are set to obtain corresponding SNR

_{out}. The statistical results can be obtained by averaging the SNR

_{out}. Here, about a hundred values for ensemble number for each SNR

_{in}have been tested: that is, values 10–500 with steps of 5. Figure 7 and Figure 8 are the statistical SNR

_{out}and MSE for four filtering methods (according to references [31,32] and experimentally, the three parameters $\left(m,n,r\right)$ for the fuzzy entropy are (1,0.2,1) and for sample entropy, the parameters $\left(m,r\right)$ are (2,0.15)).

_{out}of the filtering method based on CEEMDAN and fuzzy entropy is larger than the others, and the MSE of the filtering method based on CEEMDAN and fuzzy entropy is smaller than the others. This shows that the proposed filtering method outperforms the other filtering methods.

#### 4.2. Impact Signal Filtering

_{out}), it is found that the original signal $\mathrm{y}\left(\mathrm{n}\right)$ is unknown in Equations (19) and (20), which is often used to evaluate the filtering effectiveness. Now, the noisy signal and filtered signal are known for the measured signal. To evaluate the filtering effectiveness, the original signal $y\left(n\right)$ in Equations (19) and (20) is replaced with noisy signal, these equations being re-defined as RMSE and RSNR

_{out}, respectively. It is obvious that the evaluated result is opposite to MSE and SNR

_{out}. That means that the larger (smaller) the RMSE (RSNR

_{out}) is, the better the filtering result [27]. Here, the RSNR

_{out}and RMSE are introduced to evaluate the filtering performance. Table 3 shows the values of RSNR

_{out}and RMSE. To be easily observed, the fractal scaling index (α) is also in Table 3. From Table 3, it is found that the RSNR, RMSE, and the fractal scaling index (α) of the filtering method based on CEEMDAN and fuzzy entropy are smallest, largest, and largest respectively. The result shows that the filtering method based on CEEMDAN and fuzzy entropy had the best performance in removing noise.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**The intrinsic mode functions (IMFs) obtained by the CEEMDAN (complete Ensemble Empirical Mode Decomposition (EEMD) with adaptive noise) algorithm (except residue).

**Figure 5.**The noisy signal and filtered signal. (

**a**) The noisy signal; (

**b**) The original signal and filtered signal.

**Figure 6.**The five kinds of simulated signal. (

**a**) Blocks; (

**b**) Bumps; (

**c**) Heavysine; (

**d**) $y=\mathrm{cos}\left(2\pi t\right)+\mathrm{sin}\left(5\pi t\right)$; (

**e**) $y=\mathrm{cos}\left(3\pi t\right)+\mathrm{sin}\left(11\pi t\right)+\mathrm{sin}\left(2.5\pi t\right)$.

**Figure 7.**The statistical output signal-to-noise ratio (SNR

_{out}) for different simulated signals with various input SNR (SNR

_{in}). (

**a**) Blocks; (

**b**) Bumps; (

**c**) Heavysine; (

**d**) $y=\mathrm{cos}\left(2\pi t\right)+\mathrm{sin}\left(5\pi t\right)$; (

**e**) $y=\mathrm{cos}\left(3\pi t\right)+\mathrm{sin}\left(11\pi t\right)+\mathrm{sin}\left(2.5\pi t\right)$.

**Figure 8.**The statistical mean square error (MSE) for different simulated signals with various SNR

_{in}. (

**a**) Blocks; (

**b**) Bumps; (

**c**) Heavysine; (

**d**) $y=\mathrm{cos}\left(2\pi t\right)+\mathrm{sin}\left(5\pi t\right)$; (

**e**) $y=\mathrm{cos}\left(3\pi t\right)+\mathrm{sin}\left(11\pi t\right)+\mathrm{sin}\left(2.5\pi t\right)$.

**Figure 11.**The fractal scaling index of the four filtering methods. (

**a**) CEEMDAN + fuzzy entropy filter method; (

**b**) Wavelet filter method; (

**c**) Median filter method; (

**d**) Moving averging filter method.

Changed Waveform | NNW1 | NNW2 | NNW3 | NNW4 | NNW5 | NNW6 | NNW7 | NNW8 |

Fuzzy Entropy | 0.0438 | 0.0446 | 0.0458 | 0.0469 | 0.0455 | 0.0406 | 0.0246 | 0.0212 |

Order | Signal | Length of Data |
---|---|---|

a | Blocks | 2048 |

b | Bumps | 2048 |

c | Heavysine | 2048 |

d | $y=\mathrm{cos}\left(2\pi t\right)+\mathrm{sin}\left(5\pi t\right)$ | 2048 |

e | $y=\mathrm{cos}\left(3\pi t\right)+\mathrm{sin}\left(11\pi t\right)+\mathrm{sin}\left(2.5\pi t\right)$ | 2048 |

**Table 3.**Values of revised mean squared error (RMSE), revised output signal-to-noise ratio (RSNR

_{out}), and fractal scaling index for different filtering methods.

Filter Method | RSNR_{out} | RMSE | Fractal Scaling Index (α) |
---|---|---|---|

CEEMDAN | 7.5352 | 0.0086 | 0.2578 |

Wavelet | 7.9124 | 0.0082 | 0.2473 |

Median | 8.5065 | 0.0075 | 0.2453 |

Moving Averaging | 8.2266 | 0.0080 | 0.2461 |

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Zhan, L.; Li, C.
A Comparative Study of Empirical Mode Decomposition-Based Filtering for Impact Signal. *Entropy* **2017**, *19*, 13.
https://doi.org/10.3390/e19010013

**AMA Style**

Zhan L, Li C.
A Comparative Study of Empirical Mode Decomposition-Based Filtering for Impact Signal. *Entropy*. 2017; 19(1):13.
https://doi.org/10.3390/e19010013

**Chicago/Turabian Style**

Zhan, Liwei, and Chengwei Li.
2017. "A Comparative Study of Empirical Mode Decomposition-Based Filtering for Impact Signal" *Entropy* 19, no. 1: 13.
https://doi.org/10.3390/e19010013