*2.2. Information Entropy*

The concept of entropy originates from thermodynamics and is used to describe the disorder degree of the system. Based on this idea, scholar Shannnon proposed the concept of information entropy [28]. Information entropy is a concept used to measure the amount of information in information theory. When a system is more orderly, the value of information entropy will be smaller. On the contrary, when the disorder degree of the system becomes higher, the value of information entropy will become larger. The calculation formula of information entropy is as follows:

$$\mathbf{G}(\mathbf{x}) = -\sum\_{i=1}^{n} P(\mathbf{x}\_{i}) \log P(\mathbf{x}\_{i}) \tag{8}$$

where *<sup>P</sup>*(*xi*) represents the probability of occurrence of an event *xi* in the system, and *n* represents the number of samples to be analyzed.

#### *2.3. Robust Independent Component Analysis*

RobustICA algorithm is proposed by Zarzoso et al. [29,30]. Because the observation data used in it does not need whitening preprocessing, it can only meet the condition that its mean value is zero, so the problem of introducing error is avoided. The algorithm realizes the kurtosis optimization by using linear search and algebraic calculation of the global optimal step size. The frame of independent component analysis is shown in Figure 1 [31]. Assuming that the mixed data containing noise is *X* and the output signal is *Y* = *WX*, the kurtosis formula can be expressed as follows:

$$K(\mathcal{W}) = \frac{E\left\{|\;\mathcal{Y}|^4\right\} - 2E^2\{|\;\mathcal{Y}|^2\} - |E\left\{\mathcal{Y}^2\right\}|^2}{E^2\{|\!|\mathcal{Y}|^2\}}\tag{9}$$

where *E* {·} represents mathematical expectation, *W* represents separation matrix.

**Figure 1.** Frame of independent component analysis.

An exact linear search is performed by using the absolute value of kurtosis as the objective function:

$$
\mu\_{opt} = \text{argmax}\_{\mu} |\mathcal{K}(\mathcal{W} + \mu \mathcal{g})| \tag{10}
$$

 The search direction *g* is usually gradient, namely. *g* = ∇*w<sup>K</sup>*(*W*). It is expressed as follows:

$$g = \nabla\_w K(\mathcal{W}) = \frac{4}{E^2 \{ |Y|^2 \}} \left\{ E \left\{ \left| Y^2 \right| \right\} YX - E \{ YX \} E \left\{ Y^2 \right\} - \frac{(E \left\{ |Y|^4 \right\} - |E \{ Y^2 \}|^2) E \{ YX \}}{E \{ |Y|^2 \}} \right. \tag{11}$$

where *K* indicates kurtosis. *E* {·} represents mathematical expectation. ∇*w* indicates gradient.

In the process of each iterative operation, the operation steps of RobustICA is as follows:

(1) Find the coefficients of the optimal step polynomial:

$$P(\boldsymbol{\mu}) = \sum\_{k=0}^{4} a\_k \boldsymbol{\mu}^k \tag{12}$$

(2) Extract the root of the optimal polynomial (12).

(3) In the search direction, the root value that maximizes the kurtosis is selected as the optimal step

$$
\mu\_{opt} = \text{argmax}\_{\boldsymbol{\mu}} |K(\mathcal{W} + \boldsymbol{\mu}\mathcal{g})| \tag{13}
$$

(4) Update the separation vector: *W*<sup>+</sup> : *W*<sup>+</sup> = *W* + *μoptg*;

(5) Normalized the separation vector: *W*<sup>+</sup> : *W*<sup>+</sup> ← *W*<sup>+</sup> *W*+;

(6) If there is no convergence, return to Step (1), otherwise the solution of the separation vector is completed.

#### **3. Fault Feature Extraction Based on VMD Optimized with Information Entropy and RobustICA**

#### *3.1. Parameter Optimization of VMD*

During the working process of rolling bearing, because the inner ring will rotate with the shaft, the pressure of therolling bearing changes periodically. When the rolling bearing fails, the damaged part's surface will contact other parts and collide. Therefore, it will produce periodic pulse impact. According to this bearing characteristic, the information entropy theory can be used to measure the above changes. When the periodic impact is more uniform, the signal is more orderly, so the value of information entropy will be smaller. Therefore, if the IMF component obtained by VMD decomposition contains more fault information, its performance will be more orderly, and the value of information entropy will be smaller. In this research, envelope entropy is introduced to measure signal sparsity. The envelope entropy of IMF*i*(*j*) component of signal decomposed by VMD algorithm can be expressed as follows:

$$E\_i = -\sum\_{j=1}^{n} P\_{i,j} \log P\_{i,j} \tag{14}$$

$$P\_{\bar{1}, \bar{j}} = a\_{\bar{i}}(\bar{j}) / \sum\_{j=1}^{n} a\_{\bar{i}}(\bar{j}) \tag{15}$$

where, *i* represents the sequence number of IMF obtained by the decomposition of signal *x*(*j*) (*i* = 1, 2, 3, ··· ). *Pi*,*<sup>j</sup>* represents the normalized result of *ai*(*j*). *ai*(*j*) is the envelope signal of signal component IMF*i*(*j*) after Hilbert envelope demodulation. Based on this, this paper adopts the principle of minimum envelope entropy to determine the initialization parameters of VMD.

In parameter optimization, the value of α can be given first, and the optimal value of mode *k* can be determined based on the principle of minimum envelope entropy. After obtaining the value of *k*, the optimal value of α is further determined based on the value of *k*. Its optimization objective can be expressed as shown in Formula (16). Where *k*min and *k*max represent the minimum and maximum values of the interval range of modal *k* search, respectively. *α*min and *α*max represent the minimum and maximum values of the interval range searched by the penalty factor α, respectively. The specific optimization process is shown in Figure 2.

$$\begin{cases} \min\{E\} \\ \text{s.t. } k\_{\text{min}} \le k\_z \le k\_{\text{max}} \\ \text{s.t. } a\_{\text{min}} \le a\_z \le a\_{\text{max}} \end{cases} \tag{16}$$

The specific steps are as follows:

⎨


**Figure 2.** Parameter optimization process of VMD.

#### *3.2. Screening Criteria for IMF Components*

As a dimensionless parameter, kurtosis is often used for the distribution characteristics of vibration signals. For the IMF signal components obtained by VMD decomposition, when the signal component's kurtosis value is more prominent, it contains more impact components. At the same time, the cross-correlation coefficient represents the correlation between signals. The greater the correlationcoefficient for the IMF signals components, the more sensitive information it contains. Conversely, the more interference components it contains. Based on this, in the fault signal processing of rolling bearings, this research will comprehensively use the above two criteria to determine which signal components are used to construct the observation signal channel of the RobustICA, and then achieve the purpose of noise reduction. In selecting signal components, the principle adopted is that the correlation number is more significant than 0.3 and the kurtosis value is greater than 3. In this way, it is possible to avoid the problem of using a single index to select only the signal component with the most significantkurtosis value, which leads to the loss of sensitive information of part of the fault signal. The formula for calculating kurtosis and cross-correlation coefficient is as follows:

$$\text{Kurtosis}\_{i} = \frac{1}{\mu} \sum\_{i=1}^{n} \left( \frac{\theta\_{i} - \overline{\chi}}{\hat{\updownarrow}} \right)^{4} \tag{17}$$

In the original vibration signal, *θi* and *x* are its actual value and average value. is the standard deviation. *μ* is the number of samples.

$$Correlation = \frac{\sum\_{i=0}^{n} \left(\theta\_i - \overline{\mathfrak{x}}\right) \left(\delta\_i - \overline{\mathfrak{y}}\right)}{\sqrt{\sum\_{i=0}^{n} \left(\theta\_i - \overline{\mathfrak{x}}\right)^2 \left(\delta\_i - \overline{\mathfrak{y}}\right)^2}}\tag{18}$$

In the original vibration signal, *θi* and *x* its specific value and average value. Meanwhile, *δi* and *y* are the specific and average values of signal *ϑ*.

#### *3.3. Algorithm Steps and Flow*

The specific steps of the fault feature extraction method based on information entropy optimization VMD and RobustICA are as follows. The fault feature extraction process is shown in Figure 3.


**Figure 3.** Fault feature extraction method based on VMD optimized with information entropy and RobustICA.

#### **4. Simulations and Comparative Analysis**

Bearing is one of the most commonly used general parts in all kinds of rotating machinery. It plays a role in bearing and transmitting load in mechanical equipment, and it is very prone to failure. Therefore, to verify the performance of the algorithm proposed

in this paper, a typical model was used to simulate the periodic impact signal caused by bearing fault [32]. Firstly, a set of periodic pulse signals was simulated to simulate the fault impact signal, and on this basis, Gaussian white noise was added to the fault impact signal to simulate the bearing fault vibration signal polluted by environmental noise. In this study, Matlab (version R2009a) was selected as the vibration signal modeling and simulation software to build the signal model. The expression of the simulated signal is as follows:

$$s(t) = y\_0 e^{-2\pi f\_n \overline{\xi} t} \sin(2\pi f\_n \sqrt{1 - \overline{\xi}^2} t) \tag{19}$$

In the above formula: carrier frequency *fn* = 3000 Hz, displacement constant *y*0 = 5, damping coefficient *ξ* = 0.1, period *T* = 0.01 s, sampling frequency *fs* = 20 KHz, and number of sampling points *n* = 4096, where *t* represents the sampling time. Through calculation, it can be seen that the fault frequency *f* 0 = 100 Hz. In order to simulate the noise interference of rolling bearing during operation, SNR = −5 dB white Gaussian noise was added to the original signal *s*(*t*).

The time–domain waveform of the original signal is shown in Figure 4. After adding noise, the time–domain waveform and frequency–domain waveform of the mixedsignal after adding noise is shown in Figures 5 and 6. Analyzing the above diagrams showsthat most of the impact signal features were covered up under the interference of background noise, which brings some difficulty to the fault feature extraction.

**Figure 4.** Time–domain waveform of the simulated signal.

**Figure 5.** Time–domain waveform of the mixed signal.

**Figure 6.** Frequency–domain waveform of the mixed signal.

Since the mixed signal was severely interfered with by noise, next, the signal will be decomposed by VMD. Before VMD decomposition, information entropy must be used to optimize VMD to determine the parameters *k* and *α* in VMD. Firstly, our experiment used the default value a, which was 2500. Meanwhile, we initialized *k* = 3, and the search range of *k* was set to [3,15]. The value of optimal mode *k* was searched according to the principle of minimum envelope spectral entropy. The relationship between *k* and envelope spectral entropy is shown in Figure 7. From the transformation trend of the value of *k* in the figure, it can be seen that with the increasing value of *k*, the corresponding envelope spectral entropy was also increasing. Therefore, *k* = 3 was taken as the optimal value.

**Figure 7.** Curve of fitness varying with *k* value.

After selecting the value of the optimal mode *K*, the value of *α* was then initialized. The search range was set to [100, 2500]. Similarly, the value of the optimal mode *α* was searched according to the principle of minimum envelope spectral entropy. The relationship with envelope spectral entropy is shown in Figure 8. From the transformation trend of the value of *α* in the figure, it can be seen that with the continuous increase of *α*, the value of the corresponding envelope spectral entropy was decreasing and gradually tends to be flat. When the value of *α* was 2500, the optimal value can be obtained. Therefore, after parameter optimization, the selected optimal parameter combination *K* and *α* was [3, 2500].

**Figure 8.** Curve of fitness varying with *α* value.

Next, we performed VMD decomposition on the mixed signal. After VMD decomposition optimized by information entropy, three IMF components were obtained. The decomposition results are shown in Figure 9. In order to compare the effects of different methods, the traditional LMD decomposition method, EMD decomposition method, and EEMD decomposition method were used for the time–frequency analysis of mixed signals. The signal decomposition results obtained based on the above three methods are shown in Figure 10a,b. Figures 10 and 11 showed that there were more signal components decomposed by LMD, EMD and EEMD methods, and the signal components obtained by EMD and EEMD had certain modal aliasing and endpoint effect. At the same time, faultycomponents were generated.

**Figure 9.** VMD decomposition result.

To select the appropriate signal component from the decomposition results obtained by the above methods, this experiment will combine kurtosis and cross-correlation coefficient to select. Firstly, all signal components' correlation coefficient *C*(*t*) and kurtosis value *Q*(*t*) were calculated. The calculated results are shown in Tables 1–4. It can be seen from Table 1 that the IMF1 component and IMF2 component obtained by VMD decomposition meet the conditions that the correlation value was more significant than 0.3 and the kurtosis value was greater than 3. This shows that the correlation between the above two signal components and the original signal was high, and the signal contained more impact components. Therefore, the IMF1 and IMF2 components were selected to reconstruct the observation signal channel. Secondly, it can be seen from Table 2 that the PF1 component and PF2 component obtained by LMD decomposition met the conditions that the correlation value is more significantthan 0.3 and the kurtosis value is greater than 3. Therefore, PF1 component and PF2 component were selected to reconstruct the observation signal channel.

Meanwhile, it can be seen from Table 3 that the IMF1 component and IMF2 component obtained by EMD decomposition met the conditions that the correlation value is more significantthan 0.3 and the kurtosis value is greater than 3. Therefore, IMF1 and IMF2 components were selected to reconstruct the observation signal channel. It can be seen from Table 4 that the IMF1 component, IMF2 component and IMF3 component obtained by EEMD decomposition meet the conditions that the correlation value is more significant than 0.3 and the kurtosis value is greater than 3. Therefore, the above three signal components were selected to reconstruct the observation signal channel, and the remaining signal components were used to reconstruct the noise signal channel. Finally, on this basis, the RobustICA algorithm was used to separate signal and noise. The noise-reduction results obtained by using the method proposed in this paper and LMD–RobustICA, EMD– RobustICA, and EEMD–RobustICA are shown in Figures 12–15.

**Figure 10.** LMD (**a**) andEMD (**b**) decomposition result.


**Figure 11.** EEMD decomposition result.

**Table 1.** Index values of different IMF signal components (VMD).


**Table 2.** Index values of different PF signal components (LMD).


**Table 3.** Index values of different IMF signal components (EMD).



**Table 4.** Index values of different IMF signal components (EEMD).

**Figure 12.** Noise-reduction results by the method proposed in this paper.

**Figure 13.** LMD–RobustICA noise-reduction results.

**Figure 14.** EMD–RobustICAnoise-reduction results.

**Figure 15.** EEMD–RobustICAnoise-reduction results.

By analyzing the noise-reduction results in Figures 12–15, it can be seen that after the noise-reduction method proposed in this article, the impact components in the signal have been revealed. In contrast, the results obtained by the LMD–RobustICA, EMD– RobustICA, and EEMD–RobustICA were not significant. To further analyze the effect of noise reduction, the experiment selected four indicators of kurtosis value, cross-correlation coefficient, root mean square error (RMSE), and mean absolute error (MAE) as evaluation indicators. After calculation, the results obtained are shown in Table 5. In Table 5, by using the method proposed in this paper, the correlation value and kurtosis value obtained after noise reduction were the largest, while the RMSE and the MAE were the smallest. However, the four groups of evaluation index values obtained after noise reduction using LMD– RobustICA, EMD–RobustICA, and EEMD–RobustICA were relatively poor. Therefore, after quantitative analysis, it can be known that the signal obtained after noise reduction using the method proposed in this paper contained a higher impact component, a greater degree of correlation with the original signal, and a relatively higher waveform similarity.


**Table 5.** Comparison of noise-reduction effect index values.

Next, Hilbert envelope demodulation was performed on the results obtained based on the above three methods, and then the corresponding fault features were extracted. The envelope spectrum is shown in Figures 16–19. It can be seen from Figure 16 that the envelope spectrum obtained by the method proposed in this paper can clearly show multiple peaks with higher amplitudes, and the above peaks correspond to the frequency of one to eight times of the fault frequency, and so on. It shows that the useful signal and noise have been separated well, and the fault feature can be successfully extracted. At the same time, it can be seen from Figures 17–19 that the envelope spectrum obtained by the LMD– RobustICA, EMD–RobustICA, and EEMD–RobustICA can extracted components such as two to eight times of the fault frequency. However, it is difficult to clearly distinguish the fundamental frequency of fault frequency from each peak. In addition, compared with Figure 16, the amplitude of the fault frequency in Figures 17–19 was relatively low. It can be seen that the effect of using the method proposed in this paper to extract fault features was more significant.

**Figure 16.** Analysis of signal envelope spectrum after noise reduction based on the proposed method.

**Figure 17.** Analysis of signal envelope spectrum after noise reduction based on LMD–RobustICA.

**Figure 18.** Analysis of signal envelope spectrum after noise reduction based on EMD–RobustICA.

**Figure 19.** Analysis of signal envelope spectrum after noise reduction based on EEMD–RobustICA.

## **5. Case Analysis**

To test whether the method proposed in this paper can effectively extract the fault characteristics of rolling bearing, an example was given to verify it. The experimental data used in the experiment were from Case Western Reserve University [33]. The structure diagram of the rolling bearing test platform and rolling bearing is shown in Figure 20 [31]. The experimental platform mainly consists of the drive motor, torque speed sensor, and power meter. Among them, the model of rolling bearing at the driving end was 6205- 2RSJEMSKF. Its specific parameters are shown in Table 6. During the experiment, the set sampling frequency was 12 KHz. The acceleration sensor collected the vibration signals of the inner and outer rings with a damage diameter of 0.007 inches for experimental analysis. By substituting the relevant parameters in Table 6 into the calculation formula of fault characteristic fundamental frequency of inner ring and outer ring of rolling bearing, it can be calculated that the fundamental frequency of inner ring fault was 162.19 Hz and that of outer ring fault was 107.36 Hz.

**Figure 20.** Illustration of the bearing experimental platform and rolling bearing. (**a**)Illustration of the bearing experimental platform. (**b**) Rolling bearing.

**Table 6.** The bearing structure factor.


#### *5.1. Inner Ring Signal Analysis*

By processing the collected signal, the time domain waveform and frequency domain waveform of the fault signal of the inner ring of the rolling bearing are shown in Figures 21 and 22 can be obtained. It can be seen from that the noise interference in the signal was relatively small, and the fault impact characteristics were prominent. To test the method's effectiveness, SNR = −2 dB white Gaussian noise was added to the inner ring fault signal in this experiment. The time–domain waveform and frequency–domain waveform of the finally mixed signal is shown in Figures 23 and 24, respectively.

**Figure 21.** Time–domain analysis of the inner ring fault signal.

**Figure 22.** Frequency–domain analysis of the inner ring fault signal.

**Figure 23.** Time domain analysis of the mixed signal.

**Figure 24.** Frequency domain analysis of the mixed signal.

Next, the fault signal of the inner ring of the rolling bearing after adding noise will be processed. Firstly, the VMD was optimized by information entropy to select the values of parameters *α* and *k* to be initialized in VMD. The default value of *α* was 2500. Then, we initialized *k* = 3, and the search range of *k* was set to [3,15]. The value of optimal mode *k* was searched according to the principle of minimum envelope spectral entropy. The relationship between *k* and envelope spectral entropy as shown in Figure 25 can be obtained through calculation. As can be seen from Figure 25, with the continuous increase of *k* value, the value of the corresponding envelope spectral entropy was first decreased, then gradually increased, and, finally, decreased. Therefore, *k* = 4 was taken as the optimal value.

**Figure 25.** Curve of fitness varying with *K* value.

Next, the value of *α* was determined by experiment. The search range of *α* was set as [100, 2500], and then the value of the optimal mode was searched according to the principle of minimum envelope spectral entropy. The relationship between *α* and envelope spectral entropy as shown in Figure 26 can be obtained through calculation. As the value of *α* increases, the corresponding envelope spectral entropy decreases and gradually tends to be flat. When *α* = 2050, the envelope spectrum entropy was the smallest. Therefore, the value of *α* was 2050. After the above parameter optimization, it can be obtained that the value of the optimal parameter combination *k* and *α* was [4, 2050].

**Figure 26.** Curve of fitness varying with *α* value.

Next, this experiment substituted the selected initialization parameters into VMD, and then decomposed the signal. As is shown in Figure 27, after VMD decomposition optimized by information entropy, four IMF components were obtained. Because VMD has excellent advantages in suppressing mode aliasing and endpoint effect, it has better signal characterization ability. Next, all signal components *C*(*t*) and *Q*(*t*) were calculated. The calculated results are shown in Table 7. It can be seen that the IMF1 component and IMF1 component obtained by VMD decomposition met the conditions for screening the optimal signal component. That is, *C*(*t*) was greater than 0.3, and *Q*(*t*) was more significant than 3. Therefore, the above two IMF components were selected to reconstruct the observation signal channel. Finally, on this basis, RobustICA algorithm was used to separate signal and noise. The signal noise-reduction results obtained by using the proposed method are shown in Figure 28. Some impact characteristics of the signal can already be shown in Figure 28.

**Figure 27.** VMD decomposition result.

**Table 7.** The correlation coefficient and kurtosis between IMFand original signal (VMD).


**Figure 28.** Noise-reduction results by the proposed method.

Finally, the denoised signal was demodulated by the Hilbert envelope, and then the corresponding fault features were extracted. To compare the experimental effects of different methods, LMD–RobustICA, EMD–RobustICA, and EEMD–RobustICA were used to denoise the signal. Then Hilbert envelope spectrum was generated.The envelope spectra obtained based on the above three methods are shown in Figures 29–32, respectively. After analyzing the above four graphs, it can be seen that from the envelope spectrum of the above methods, the one-time frequency of the fault frequency, as well as the two-time frequency and five-time frequency of the fault frequency could be extracted. However, the amplitude of the fault frequency in the envelope spectrum obtained based on the method proposed in this paper was relatively high, especially the fundamental frequency amplitude. Therefore, the effect of using the proposed method to extract fault features was more significant.

**Figure 29.** Analysis of signal envelope spectrum after noise reduction based on the proposed method.

**Figure 30.** Analysis of signal envelope spectrum after noise reduction based on LMD–RobustICA.

**Figure 31.** Analysis of signal envelope spectrum after noise reduction based on EMD–RobustICA.

**Figure 32.** Analysis of signal envelope spectrum after noise reduction based on EEMD–RobustICA.

#### *5.2. Outer Ring Signal Analysis*

By extracting the rolling bearing outer ring fault signal data, the time domain waveform and frequency domain waveform of the outer ring fault signal is shown in Figures 33 and 34. Similarly, SNR = −2 dB white Gaussian noise was added to the inner ring fault signal in this experiment to test the method's effectiveness.The time domain waveform and frequency domain waveform of the finally obtained mixed signal are shown in Figures 35 and 36. Due to noise interference, it wasnot easy to distinguish the impact features from the diagram. Next, the signal was further processed.

**Figure 33.** Time domain analysis of the outer ring fault signal.

**Figure 34.** Frequency domain analysis of the outer ring fault signal.

**Figure 35.** Time domain analysis of the mixed signal.

Next, the VMD was optimized by information entropy to select the parameters *α* and *k* was initialized. Similarly, the value of *α* was the default value of 2500. First, *k* = 3 was initialized, and the search range of k was set to [3,15]. The value of optimal mode kwas searched according to the principle of minimum envelope spectral entropy. Through the search, the results shown in Figure 37 can be obtained. It is shown that the corresponding envelope spectral entropy's value increased with the continuous increase of the *k* value. Therefore, *k* = 3 was taken as the optimal value.

**Figure 36.** Frequency domain analysis of the mixed signal.

**Figure 37.** Curve of fitness varying with *k* value.

Based on the above calculation results, the value of *α* was searched. The search range of α was set as [100, 2500], and then the value of the optimal mode α was searchedaccording to the principle of minimum envelope spectral entropy. Through calculation, the relationship between α and envelope spectral entropy as shown in Figure 38 can be obtained. As shown in Figure 38, with the increasing value of *α*, the corresponding envelope spectral entropy was decreasing and gradually tends to be flat. When *α* = 1900, the envelope spectral entropy was the smallest. Therefore, the value of *α* was 2050. After the above parameter optimization, it can be obtained that the value of the optimal parameter combination k and *α* was [3, 1900].

**Figure 38.** Curve of fitness varying with a value.

Next, the optimal parameters obtained by the search were substituted into the VMD, and then the signal was decomposed. As is shown in Figure 39, after VMD decomposition optimized by information entropy, three IMFs were obtained. Based on the above decomposition results, the *C*(*t*) and *Q*(*t*) of IMF1, IMF2, and IMF3 are further calculated. The calculated results are shown in Table 8. It can be seen that the IMF2 component and IMF3 component obtained by VMD decomposition met the conditions for screening the optimal signal component. That is, *C*(*t*) was greater than 0.3 and *Q*(*t*) was more significant than 3. Therefore, the above two signal components were selected and used to reconstruct the observation signal channel. Then, RobustICA was used for signal noise reduction. The final signal noise-reduction result is shown in Figure 40. It can be seen that the periodic impact had a high similarity with the waveform of the original signal.

**Figure 39.** VMD decomposition result.

**Table 8.** The correlation coefficient and kurtosis between IMFand original signal (VMD).


**Figure 40.** Noise-reduction results by the proposed method.

Next, the outer ring fault signal after signal noise reduction was demodulated by the Hilbert envelope to extract the fault feature. To furtheranalyze the fault feature extraction effect of the proposed method more intuitively, LMD–RobustICA, EMD–RobustICA, and EMD–RobustICA methods were used to denoise the signal. Then Hilbert envelope spectrum was generated. The envelope spectra obtained based on the above methods are shown in Figures 41–44, respectively. Through the analysis of Figures 41–44, it can be seen that the frequency doubling component of the outer ring fault characteristic frequency could be extracted by using the above methods for fault feature extraction. Meanwhile,

the peak value of the components of the fault frequency from one to six times frequency is much higher than that obtained by the other three methods. The surrounding interference could not affect the identification of frequency doubling. However, the peak value of fault characteristic frequency in the envelope spectrum based on the other three methods was relatively low, and the interference components near these frequencies were close to the fault characteristic frequency. This brought some interference to fault feature extraction.

**Figure 41.** Analysis of signal envelope spectrum after noise reduction based on the proposed method.

**Figure 42.** Analysis of signal envelope spectrum after noise reduction based on LMD–RobustICA.

**Figure 43.** Analysis of signal envelope spectrum after noise reduction based on EMD–RobustICA.

**Figure 44.** Analysis of signal envelope spectrum after noise reduction based on EEMD–RobustICA.
