A Fault Feature Extraction Method for Rolling Bearing Based on Intrinsic Time-Scale Decomposition and AR Minimum Entropy Deconvolution

It is very diﬃcult to extract the feature frequency of the vibration signal of the rolling bearing early weak fault and in order to extract its feature frequency quickly and accurately. A method of extracting early weak fault vibration signal feature frequency of the rolling bearing by intrinsic time-scale decomposition (ITD) and autoregression (AR) minimum entropy deconvolution (MED) is proposed in this paper. Firstly, the original early weak fault vibration signal of the rolling bearing is decomposed by the ITD algorithm to proper rotations (PRs) with fault feature frequency. Then, the sample entropy value of each PR is calculated to ﬁnd the largest PRs of the sample entropy. Finally, the AR-MED ﬁltering algorithm is utilized to ﬁlter and reduce the noise of the largest PRs of the sample entropy value, and the early weak fault vibration signal feature frequency of the rolling bearing is accurately extracted. The results show that the ITD-AR-MED method can extract the early weak fault vibration signal feature frequency of the rolling bearing accurately.


Introduction
Rolling bearings play a pivotal role in modern industry and their applications are very extensive, such as aircraft engines [1,2], gas turbines [3], landing gear [4], flexible mechanism [5], and wind turbines [6][7][8][9][10][11]. However, the rolling bearing is very easy to cause damage in a strong vibration environment, and the rolling bearing is very prone to failure in most rotating machinery systems. e failure of the rolling bearing will even cause a catastrophic accident to the rotating machinery system, which will indirectly cause huge economic losses. erefore, it is significant to accurately extract the early weak fault vibration signal feature frequency of the rolling bearing [12].
At present, vibration signal [13][14][15] is mostly used for rolling bearing fault diagnosis analysis. Because the rolling bearing usually works in complex environmental noise, and the collected fault vibration signal feature frequency of the rolling bearing is usually weak [16,17]. Many researchers use the signal decomposition algorithms such as empirical mode decomposition (EMD) [18,19], wavelet packet decomposition (WPT) [20][21][22], and local mode decomposition (LMD) [23][24][25] to reduce the noise of the original signal. However, the wavelet basis function (WBF) is fixed before the WPT decomposition and the decomposition effect according to the selection of the WBF. erefore, WPT has obvious defects in vibration signal decomposition. e EMD and LMD are prone to adverse factors such as endpoint effects and mode mixing [26][27][28][29]. erefore, Frei and Osorio [30] proposed an intrinsic time-scale decomposition (ITD) adaptive signal decomposition algorithm. e ITD algorithm can effectively solve the unfavorable factors between EMD and LMD. Xing et al. [31] developed the ITD method combined with singular value decomposition (SVD) and support vector machine (SVM) method. e original vibration signal is decomposed by the ITD algorithm, and the proper rotations (PRs) with the greatest correlation was selected for subsequent analysis. An [32] utilized the ITD algorithm in fault diagnosis under four fault states collected. e first few PRs were selected for spectral analysis to find out their feature frequency. Feng et al. [33] utilized the ITD algorithm in fault diagnosis and feature extraction of the planetary gearbox. e fault vibration signal of the rolling bearing usually presents the characteristics of the impact waveform [34] when the rolling bearing fails. Wiggins [35] proposed a minimum entropy deconvolution (MED) filtering method to enhance the impact component in the fault vibration signal. is method can enhance the impact component and the fault feature in the signal. Sawalhi et al. [36] and Endo and Randall [37] utilized the MED to enhance the impact component of the rolling bearing and gear vibration signal, respectively, and the experiment results proved that this method could effectively enhance the feature frequency. Jiang et al. [38] proposed a condition monitoring method of the rolling bearing based on the MED and envelope spectrum. After the rolling bearing fault vibration signal was filtered by the MED, its fault feature frequency was enhanced, and then an envelope spectrum analysis was performed on it; thus, the rolling bearing fault feature frequency was separated accurately. Li et al. [39] developed an autoregression (AR) MED and variational mode decomposition (VMD) fault diagnosis method. Among them, the AR-MED method can eliminate the influence of background noise effectively, which can enhance the feature frequency in the fault signal.
In summary, this paper studied the problem of extracting early weak fault vibration signal feature frequency of the rolling bearing under complex background noise. is paper proposes a method for extracting early weak fault vibration signal feature frequency of the rolling bearing based on ITD and AR-MED.

ITD Signal Decomposition Method
Suppose that the original signal X t to be decomposed is a real value discrete signal, ξ is defined as the baseline extraction factor of X t , H t � X t − ξX t is defined as the proper rotations (PRs), where ξX t is the mean curve of the signal, denoted as L t � ξX t ; then, the primary decomposition of the original signal can be expressed as en, the steps of the ITD algorithm are as follows: (1) Extract the extreme point X k of X t and the corresponding time point τ k . Among them, k � 1, 2, . . . , N and definition τ 0 � 0, so definition L k+1 is where α ∈ [0, 1], α � 0.5, and k � 1, 2, . . . , N − 2.
(2) e piecewise linear baseline extraction factor of signal X t is defined as (3) Repeat steps (1)- (2) with baseline signal L t as the original signal; then, the original signal X t is decomposed into where Hξ k X t is the k + 1 component of proper rotations (PRs) and ξ p X t is the monotone trend signal. According to the above equation, the first PR component can be obtained. en, L 1 t is decomposed until the new signal is generated, L p t is monotone function or constant function, and then the decomposition is stopped. e time domain and frequency domain waveform of fault signals of the rolling bearing is shown in Figure 1.
It easy can be seen from the above ITD decomposition process that the baseline signal L t is obtained through the linear transformation of the signal, so the calculation process of ITD can be greatly simplified and the decomposition speed greatly improved.

AR-MED Filter
3.1. AR Filter. AR filtering is a parametric modeling method based on the rational transfer function. Because the system parameters of the AR filter are very sensitive to the state change of the system, it is of great value to apply it to signal analysis.
When the data sequence slides along the signal sequence, the adaptive AR spectrum of the signal is formed. e parameter estimation of signal sequence AR model is to select appropriate parameters so that the residual of the AR model ε k is white noise. e commonly used methods are time series theory method and optimization theory estimation method. If there is a time series x n , there is a positive integer p, which makes where Γ is the Wold decomposition operator; p is analysis order; a p is AR model parameter. e AR model of signal sequence x n can be expressed as where ε k is a white noise process, and its variance represents the estimation error of the AR filter. e process of linear predictive filtering using the AR filter is shown in Figure 2. After the original signal passes through p-order AR linear filter for predictive filtering, a new signal Y n is obtained. Due to the convolution operation    Shock and Vibration in the filtering process, Y n is also intercepted to obtain a signal Y n−p .

MED Method.
e purpose of MED is to find the source of impact in the signal. e principle is to select an appropriate filter coefficient to construct an inverse filter so that the output signal is restored to the input after the inverse filter. e minimum entropy of the signal is used as the determination condition, which can significantly improve the signalto-noise ratio (SNR) of pulse impact signals, which is very suitable for bearing crack fault diagnosis. e fault vibration signal generated by the bearing in the event of failure usually has impact characteristics. e MED can be applied to the noise reduction treatment of impact fault of the rolling bearing to better highlight the impact pulse. It is assumed that when impact failure occurs to the rolling bearing, the collected discrete signals can be expressed as follows: where g t is the signal of rolling bearing fault impact source after AR filtering; n t is noise interference; * is the convolution operator; and h t is the influence of the transfer path. e purpose of deconvolution is to find the filter coefficient f of the inverse filter so that the output signal y t is restored to the input signal g t ; that is, where e t is the output signal of the inverse filter and the input signal g t can be recovered. L is the filter length. In this paper, the objective function method is selected to realize the MED, and the optimal inverse filter coefficient is found so that the objective function is maximized. e objective function can be defined as Take the derivative of the objective function and set it equal to 0 to obtain Since Equation (11) can be written as matrix where b is the cross-correlation matrix of input signal g t and output signal e t of the inverse filter; A is Toeplitz autocorrelation matrix of inverse filter input signal g t ; and f is the filtering coefficient of the inverse filter. e solution process of MED is as follows: (1) Toeplitz autocorrelation matrix A is calculated, and the filtering coefficient f (0) of the inverse filter is initialized, which is usually set as the delayed pulse. (2) According to the initial filter coefficient f (0) and the inverse filter input signal g (0) , the output signal e (0) is calculated. (1) is calculated based on the output signal e (0) and the input signal g (0) . (4) According to equation (12), When the expected value of the iteration error is greater than the set error threshold, that is, (2), repeat the calculation until E(δ) ≤ τ or E(δ) diverges, stop the calculation, and get the MED FIR filter parameters.

Early Weak Fault Feature Frequency Enhance Method of Rolling Bearing Based on ITD-AR-MED
In order to enhance the feature frequency in the complex background noise, a fault feature extraction method of the rolling bearing based on the ITD-AR-MED is proposed in this paper. e flow chart of fault feature frequency enhance method of rolling bearing based on ITD-AR-MED algorithm is shown in Figure 3. e steps of the ITD-AR-MED algorithm are as follows: (1) Firstly, the early weak fault vibration signal of rolling bearing, which decomposed by the ITD algorithm into several PRs. (2) en, the sample entropy value of each PR is solved, and the PRs with the maximum entropy value is selected for subsequent processing. e ITD-AR-MED algorithm decomposition advantages are compared by the fast kurtogram algorithm. e simulation signals are decomposed by the ITD-AR-MED algorithm and fast kurtogram algorithm, respectively. Its simulation signal is where

Fault Diagnosis of the Rolling Bearing Experiment.
e standard rolling bearing data set selected from Case Western Reserve University (CWRU) was tested by the ITD-AR-MED method for fault diagnosis of bearing outer and inner rings [40]. e test rig is shown in Figure 7. e test rig equipment consists of motor, power meter, torque sensor, acceleration sensor, and electronic control equipment. Among them, the fault bearing should be as close as possible to the motor. e single point of bearing fault is processed by Electric Discharge Machining (EDM) technology, with a diameter of 0.117 mm and a depth of 0.011 mm. e fault vibration signal generated by the test rig is collected by the acceleration sensor, which is installed directly above the motor drive end shell. e motor frequency is 30 Hz, and the 16-channel DAT data acquisition system is adopted to collect the vibration signal of the rolling bearing fault. e signal sampling frequency is 12 kHz, the spindle speed is 1797 r/min, and the rolling bearing model is 6205-2rs JEM SKF.
According to the empirical equations (15) and (16), the fault feature frequency is 107.3 Hz and 162.2 Hz of the inner race (IR) and the outer race (OR): where D is the section bearing diameter; N is the spindle speed; d is the diameter of the roll; and z is the number of the ball.
It is necessary to conduct noise reduction decomposition base on Figure 1's results. e decomposition results of the ITD algorithm for the IR fault signal and OR fault signal of the rolling bearing are shown in Figures 8 and 9, respectively.
It can be seen from Figures 8 and 9 that the IR fault signal of rolling bearing has the widest frequency range of PR1 after ITD decomposition, so it covers the most characteristic information. e OR fault signal of rolling bearing has the widest frequency range of PR2 after ITD decomposition, so it covers the most characteristic information.
e sample entropy value of each PR is obtained. Table 1 shows the sample entropy values of each PR of the rolling bearing.
According to the obtained sample entropy value of each PR component, the fault vibration signal of the IR is decomposed by the ITD, and the PR1 component is selected for subsequent analysis. After the fault vibration signal of the OR is decomposed by ITD, the PR2 component is selected for subsequent analysis.
Firstly, the PR1 of the IR fault is filtered by AR, and Figure 10 is obtained. Figure 10 shows the spectrum of the PR1 component filtered by ITD-AR. After AR filtering, the     Shock and Vibration signal is processed by MED. Figure 11 shows the spectrum diagram of the PR1 component by the ITD-AR-MED algorithm. Figures 10 and 11 show that the feature frequency of the IR fault vibration signal is 162.2 Hz. e envelope spectrum of PR1 by the ITD-AR-MED algorithm is shown in Figure 12.
As shown in Figure 12, the fault frequency of the PR1 can be accurately obtained, which is close to the theoretical value of 161.9 Hz. In addition, the envelope spectrum also    processed by MED. Figure 14 shows the spectrum diagram of PR2 component filtered by ITD-AR. As shown in Figures 13 and 14, the spectrum diagram shows that the fault feature frequency is not clearly displayed at 107 Hz, which covers a large amount of background noise frequency. erefore, the spectrum diagram obtained by the MED is used for envelope spectrum analysis. e envelope spectrum of PR2 by the ITD-AR-MED algorithm is shown in Figure 15.
As shown in Figure 15, the fault frequency of the PR2 component can be accurately obtained, which is similar to the theoretical value 107.7 Hz. In addition, the envelope  spectrum also accurately extracted the fault feature frequency, both 215.3 Hz and 323 Hz, which are two-time frequency and three-time frequency, as well as the motor rotation frequency 30 Hz. is proves the effectiveness of the proposed fault diagnosis method.
In order to highlight the advantages of the ITD-AR-MED algorithm, the fast kurtogram algorithm is substituted for the ITD algorithm to obtain the envelope spectrum. e fault vibration signal is filtered and enhanced by the fast kurtogram algorithm. Figure 16 shows the fast kurtosis  spectrum of rolling bearing IR fault vibration signal. It can be seen from Figure 16 that the IR bearing fault signal is obtained by the fast kurtogram algorithm and the center frequency and bandwidth of the bandpass filter are 2250 Hz and 1500 Hz, respectively. Figure 17 shows the envelope spectrum of the rolling bearing IR fault vibration signal after the fast kurtogram algorithm. By comparing Figure 12 with Figure 17, the fast kurtogram algorithm can extract IR fault feature frequency, but its accuracy is slightly lower than that of the ITD-AR-MED algorithm. Figure 18 shows the fast kurtosis spectrum of rolling bearing OR fault vibration signal. It can be seen from Figure 18 that the OR bearing fault signal is obtained by the fast kurtogram algorithm and the center frequency and bandwidth of the bandpass filter are 3000 Hz and 2000 Hz, respectively. Figure 19 shows the envelope spectrum of the rolling bearing OR fault vibration signal after the fast kurtogram algorithm. By comparing Figure 15 with Figure 19, the fast kurtogram algorithm can extract OR fault feature frequency, but its accuracy is slightly lower than that of the ITD-AR-MED algorithm.

Fault Diagnosis of the Desulfurization Fan Rolling Bearing Experiment.
e desulfurization fan rolling bearing test rig is shown in Figure 20. e test rig equipment consists of motor, bearing, hydraulic coupler, acceleration sensor, and fan. e signal sampling frequency is 25.6 kHz. e OR fault feature frequency is 60.31 Hz of the desulfurization fan rolling bearing. e decomposition results of the ITD algorithm for the OR fault signal of the desulfurization fan rolling bearing are shown in Figure 21.
It can be seen from Figure 21 that the OR fault signal of the desulfurization fan rolling bearing has the widest frequency range of PR2 after ITD decomposition, so it covers the most characteristic information.
e sample entropy value of each PR is obtained. Table 2 shows the sample entropy values of each PR of the desulfurization fan rolling bearing.
According to the obtained sample entropy value of each PR component, the fault vibration signal of the OR is decomposed by the ITD, and PR2 component is selected for subsequent analysis.

Shock and Vibration 13
Firstly, the PR2 of the IR fault is filtered by AR, and Figure 22 is obtained. Figure 22 shows the spectrum of the PR2 component filtered by ITD-AR. After AR filtering, the signal is processed by MED. Figure 23 shows the spectrum diagram of the PR2 component by the ITD-AR-MED algorithm. Figures 22 and 23 show that the feature frequency of the OR fault vibration signal is 50 Hz. e envelope spectrum of PR1 by the ITD-AR-MED algorithm is shown in Figure 24.
As shown in Figure 24, the fault frequency of the PR2 can be accurately obtained, which is close to the theoretical value of 60.31 Hz. Figure 25 shows the fast kurtosis spectrum of the desulfurization fan rolling bearing OR fault vibration signal. It can be seen from Figure 25 that the OR bearing fault signal is obtained by the fast kurtogram algorithm and the center frequency and bandwidth of the bandpass filter are 3000 Hz and 2000 Hz, respectively. Figure 26 shows the envelope spectrum of the rolling bearing OR fault vibration signal after the fast kurtogram     16 Shock and Vibration  algorithm. By comparing Figure 24 with Figure 26, the fast kurtogram algorithm can extract OR fault feature frequency, but its accuracy is slightly lower than that of the ITD-AR-MED algorithm.

Conclusions
Aiming at the difficulty of extracting the early weak fault feature frequency, the characteristics of the shock waveform of the IR and OR of rolling bearing are combined. In this paper, the ITD-AR-MED method is used for weak fault feature extraction, and the following conclusions are as follows.
(1) e ITD algorithm can decompose the early weak fault vibration signal effectively, retaining the fault impact characteristics of the rolling bearing. (2) e MED algorithm can be used to separate the noise from the original signal. e envelope spectrum can extract the early weak fault feature frequency of the rolling bearing, which verifies the effectiveness and accuracy of the method.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.