A Fault Feature Extraction Method Based on LMD and Wavelet Packet Denoising

Aiming at the problem of fault feature extraction of a diaphragm pump check valve, a fault feature extraction method based on local mean decomposition (LMD) and wavelet packet transform is proposed. Firstly, the collected vibration signal was decomposed by LMD. After several amplitude modulation (AM) and frequency modulation (FM) components were obtained, the effective components were selected according to the Kullback-Leible (K-L) divergence of all component signals for reconstruction. Then, wavelet packet transform was used to denoise the reconstructed signal. Finally, the characteristics of the fault signal were extracted by Hilbert envelope spectrum analysis. Through experimental analysis, the results show that compared with other traditional methods, the proposed method can effectively overcome the phenomenon of mode aliasing and extract the fault characteristics of a check valve more effectively. Experiments show that this method is feasible in the fault diagnosis of check valve.


Introduction
With the vigorous development of mineral pipeline transportation technology, highpressure diaphragm pump operation, maintenance, and fault monitoring have become a concern. The high-pressure diaphragm pump provides the power required in the process of mineral transportation [1], and most of its failures are caused by one of the core components of the pump check valve. The check valve is a directional element used to control the feeding and discharging, so that the conveying medium flows in one direction and cannot flow back. Generally, the average price of each diaphragm pump exceeds CNY20 million, and the daily delivery of pulp exceeds 30,000 tons. If the stroke coefficient of the high-pressure diaphragm pump is 50 R/min, the feed and discharge check valves in a normal operating day need to act in a reciprocating manner 72,000 times. Therefore, the check valve is the most frequent and fault-prone component in a high-pressure diaphragm pump. Local faults and unplanned shutdowns in its operation state easily cause equipment pressure and flow fluctuations, resulting in equipment vibration and damage to the mineral transmission pipeline. To sum up, monitoring the operation state of the check valve is very important for the safe, stable, and efficient operation of mineral pipeline transportation. By detecting the vibration signal of the check valve and extracting the characteristics that can characterize the operation state for diagnosis and identification, the engineers and technicians can learn the operation state of the equipment in time and significantly improve the ability of operation and maintenance management. However, in the actual detection process, due to the complex field background noise, there is often strong impulse interference and random noise, making it difficult to distinguish the useful signal and noise interference, affecting the effective extraction of fault features.
Traditional signal processing methods are mostly based on Fourier transform, such as short-time Fourier transform, Wigner-Ville, power spectrum analysis, etc. [2][3][4]. Although 3 of 16 of non-stationary signals. Kuanfang he et al. [15] proposed a wavelet packet denoising method for welding acoustic emission signals. The experimental results show that the proposed method can effectively process the acoustic emission signals of welding cracks. Wang X et al. [16] proposed a method based on optimized variational mode decomposition (VMD) and wavelet packet threshold denoising and applied it to remove strong white noise signals. The results show that this method retains the practical components of the signal well. Sun W et al. [17] used a wavelet packet to remove the noise in the collected bearing signal and then combined it with the LMD method to extract the fault feature. The results show that this method can effectively extract fault features.
Based on the above analysis, a fault feature extraction method based on local mean decomposition (LMD) and wavelet packet denoising is proposed in this paper. Firstly, the vibration signal of the check valve was decomposed by LMD, and the K-L divergence of each AM-FM component was obtained. The practical component signal was selected for reconstruction through the comparative analysis with the set threshold. Then, wavelet packet transform was used to denoise the reconstructed signal to reduce the noise interference. Finally, Hilbert envelope spectrum analysis asper formed on the denoised signal to extract the fault signal features. The effectiveness of the proposed method was verified by analyzing the actual fault data of the check valve.
The rest of the paper is organized as follows: In Section 2, we introduce the basic principle of the LMD algorithm and K-L divergence and describe the process of PF component selection. In Sections 3 and 4, we introduce the basic principles of Hilbert envelope demodulation and wavelet packet denoising. In Section 5, we describe the modeling process of check valve fault feature extraction based on LMD and wavelet packet denoising. Section 6 is experimental analysis. The effectiveness of the method proposed in this paper was verified by the collected actual fault data of the check valve. Section 7 concludes the paper.

LMD Algorithm
LMD is a time-frequency analysis method proposed by Smith. Its principle is to decompose the signal into product function components of different frequencies and a margin, and each component is obtained by multiplying a pure FM signal and envelope signal. If the original signal is x(t), the decomposition step can be described in the following form.
(1) Calculate all local extreme points of the original signal x(t), and then calculate the average value m i of adjacent local extreme points n i and n i+1 : The moving average method is used to smooth the line composed of m i , and the local mean function m 11 (t) can be obtained.
(2) The envelope estimation value a i is calculated according to the adjacent extreme points n i and n i+1 : Similarly, the envelope estimation function a 11 (t) is obtained by the moving average method.
(3) The local mean function m 11 (t) is separated from the original signal x(t).
(4) Divide h 11 (t) by a 11 (t) to obtain FM signal s 11 (t) When a 12 (t) = 1, s 11 (t) is a standard pure FM signal, and when a 12 (t) is not equal to 1, s 11 (t) is taken as the original data, and the above processes repeated. Until a 1(n+1) (t) = 1 is met, the pure FM signal s 1n (t) is calculated, as shown in the following formula: The conditions for iteration termination are as follows: (5) Calculate the envelope signal: The product function can be obtained by multiplying the pure FM signal s 1n (t) and the envelope signal a 1 (t) (7) Separate PF 1 (t) from x(t) and find the signal u 1 (t). Then, repeat the above steps k times. When u k (t) becomes a monotone function, the operation is terminated.
The following formula can be obtained by combining all PF components and recombining.
From the above description, it can be seen that the LMD is an adaptive decomposition method based on the local extreme points of the signal itself. The decomposition process is a multi-cycle iterative process.

PF Component Selection Based on K-L Divergence
Several PF components obtained by LMD have a different correlation with the original signal. To effectively screen component signals, appropriate screening methods must be adopted. K-L divergence is called directional divergence, describing the difference between the two probability distributions [18]. Let the two probability distributions be p 1 (x) and p 2 (x), then the K-L distance is defined as follows: Coatings 2022, 12, 156 5 of 16 The K-L divergence values of p 1 (x) and p 2 (x) are: The vibration signals collected in the equipment characterize the vibration of the equipment. Therefore, assuming that the probabilities between the two signals are p 1 (x) and p 2 (x), the closer the vibration between them, the closer p 1 (x) and p 2 (x) will be, and the smaller the K-L divergence value will be.
Since several PF components are obtained through LMD, the solution of K-L divergence for the original signal X = {x 1 , x 2 . . . x n } and component Y = {y 1 , y 2 . . . y n } decomposed by the LMD algorithm is as follows: (1) Solve the probability distribution of the above two signals and let the function p 1 (x) be the kernel density estimation of signal X: where k(·) is the kernel function, which usually uses the Gaussian kernel function, which is k(u) = 1 √ 2π e −u 2 /2 , and h is a given positive number. Similarly, p 2 (x) can be obtained in the above way.
(4) Normalize all calculated K-L divergence values. (5) Filter according to the preset threshold. If the component is less than the threshold, it will be regarded as the component containing an obvious fault signal. According to the data obtained in this study, the set threshold is 0.03.

Hilbert Envelope Demodulation
The Hilbert envelope demodulation method mainly converts the actual signal into an analytical signal by Hilbert transform and then takes the modulus to obtain its envelope. Assuming that the signal x(t) has amplitude modulation envelope A(t) and phase modulation function ϕ(t), its expression is as follows: The Hilbert transform of signal x(t) can be an approximately 90 • phase shift of x(t): Construct the analytical signal Z(t) so that: By calculating the modulus of the above formula, the following formula can be obtained: In the above formula, |Z(t)| is the envelope of the signal.

Principle of Wavelet Packet Denoising
As one of the more detailed signal analysis methods, the advantage of wavelet packet transform is that it can adaptively select the corresponding frequency band and divide the frequency band at multiple levels [19,20]. For a one-dimensional signal containing noise, it can be described as the following formula: where f (i) is the actual signal, S(i) is the signal containing noise, and β(i) is noise. The process of wavelet packet denoising is as follows: (1) Determine the corresponding wavelet base and decomposition level, and then start wavelet packet decomposition; (2) According to the established entropy standard, the optimal tree is obtained, and the optimal wavelet basis is determined; (3) The appropriate threshold is selected, and the high-frequency coefficients with different decomposition scales are quantified at the same time; (4) The signal reconstruction operation is carried out according to the decomposition coefficients and quantization coefficients of the N-th layer wavelet packet.
The decomposition structure of wavelet packet is shown in Figure 1 [21].
As one of the more detailed signal analysis methods, the advantage of wavelet packet transform is that it can adaptively select the corresponding frequency band and divide the frequency band at multiple levels [19,20]. For a one-dimensional signal containing noise, it can be described as the following formula: where ( ) f i is the actual signal, ( ) S i is the signal containing noise, and ( ) i b is noise. The process of wavelet packet denoising is as follows: (1) Determine the corresponding wavelet base and decomposition level, and then start wavelet packet decomposition; (2) According to the established entropy standard, the optimal tree is obtained, and the optimal wavelet basis is determined; (3) The appropriate threshold is selected, and the high-frequency coefficients with different decomposition scales are quantified at the same time; (4) The signal reconstruction operation is carried out according to the decomposition coefficients and quantization coefficients of the N-th layer wavelet packet.
The decomposition structure of wavelet packet is shown in Figure 1 [21].

Fault Feature Extraction Model of Check Valve Based on Local Mean Decomposition Wavelet Packet Denoising
Based on the above theoretical analysis, a check valve fault feature extraction method based on local mean decomposition wavelet packet denoising is proposed in this paper. The research outline of the proposed method is shown in Figure 2, and the method implementation process is shown in Figure 3. The specific steps are as follows: (1) The experimental failure data of wear breakdown of check valve are selected, and then the above samples are decomposed into several PF components by LMD decomposition.

Fault Feature Extraction Model of Check Valve Based on Local Mean Decomposition Wavelet Packet Denoising
Based on the above theoretical analysis, a check valve fault feature extraction method based on local mean decomposition wavelet packet denoising is proposed in this paper. The research outline of the proposed method is shown in Figure 2, and the method implementation process is shown in Figure 3. The specific steps are as follows: Coatings 2022, 12, x FOR PEER REVIEW 7 of 16 (2) According to the calculation method of K-L divergence, the K-L divergence between the original signal and each PF component obtained by decomposition is calculated. Then, all K-L divergence values are normalized.
(3) According to the set threshold, the calculated K-L divergence value is compared with it, and then any PF component less than the threshold is filtered.
(4) The filtered PF components are reconstructed, and then wavelet packet denoising is carried out.
(5) Hilbert envelope demodulation is performed on the denoised signal.

Experimental Analysis
The field data of the high-pressure diaphragm pump check valve obtained by t data acquisition system of a slurry transmission pipeline in Western China were used the experiment. The diaphragm pump used in the slurry transmission pipeline pump s tion was of the TZPM series, and the data acquisition card used was the PXIe-3342 eig channel acquisition card. Through the acceleration sensor arranged outside the d phragm pump, the vibration signal was collected and transmitted to the computer. Th the collected data of the check valve in the wear fault state were analyzed to extract fault characteristics. The Schematic diagram of high-pressure diaphragm pump and fa check valve is shown in Figure 4 [22]. The vibration signal acquisition system diagram check valve is shown in Figure 5 [22]. (1) The experimental failure data of wear breakdown of check valve are selected, and then the above samples are decomposed into several PF components by LMD decomposition.
(2) According to the calculation method of K-L divergence, the K-L divergence between the original signal and each PF component obtained by decomposition is calculated. Then, all K-L divergence values are normalized.
(3) According to the set threshold, the calculated K-L divergence value is compared with it, and then any PF component less than the threshold is filtered.
(4) The filtered PF components are reconstructed, and then wavelet packet denoising is carried out.
(5) Hilbert envelope demodulation is performed on the denoised signal.

Experimental Analysis
The field data of the high-pressure diaphragm pump check valve obtained by the data acquisition system of a slurry transmission pipeline in Western China were used in the experiment. The diaphragm pump used in the slurry transmission pipeline pump station was of the TZPM series, and the data acquisition card used was the PXIe-3342 eight-channel acquisition card. Through the acceleration sensor arranged outside the diaphragm pump, the vibration signal was collected and transmitted to the computer. Then, the collected data of the check valve in the wear fault state were analyzed to extract its fault characteristics. The Schematic diagram of high-pressure diaphragm pump and fault check valve is shown in Figure 4 [22]. The vibration signal acquisition system diagram of check valve is shown in Figure 5 [22].   In the check valve, wear breakdown caused by coarse particles stuck in the valve is very common. The check valve of the high-pressure diaphragm pump has a cone-valvetype structure, and the main components are the valve body, valve core, and spring. Among them, the "valve core spring" constitutes a low-damping oscillation system, and its frequency is: where k is the stiffness value of the spring; and ms is the equivalent mass value. Set the spring stiffness as follows according to the activity of the plunger of the pump Where f is the frequency of normal operation of the high-pressure diaphragm pump. Its normal operating frequency is 0.5~0.517 Hz, and the frequency of the spring valve core system fp = 2f, namely, 1~1.034 Hz. The check valve will show the corresponding fault fundamental frequency and double frequency when it fails.
The check valves of the high-pressure diaphragm pump are matched by the feed valve and discharge valve in pairs. Therefore, the normal operation vibration signal and fault vibration signal of a group of feed and discharge check valves were randomly selected as a group of experimental data for analysis. The fault data of the check valve used in the experiment were the data when a wear breakdown fault occurred. The sampling frequency of this experiment was 2560 Hz, and the sampling data length was 10,240. After A/D conversion, the collected vibration acceleration signal was input into the computer through the controller.   In the check valve, wear breakdown caused by coarse particles stuck in the valve is very common. The check valve of the high-pressure diaphragm pump has a cone-valvetype structure, and the main components are the valve body, valve core, and spring. Among them, the "valve core spring" constitutes a low-damping oscillation system, and its frequency is: where k is the stiffness value of the spring; and ms is the equivalent mass value. Set the spring stiffness as follows according to the activity of the plunger of the pump Where f is the frequency of normal operation of the high-pressure diaphragm pump. Its normal operating frequency is 0.5~0.517 Hz, and the frequency of the spring valve core system fp = 2f, namely, 1~1.034 Hz. The check valve will show the corresponding fault fundamental frequency and double frequency when it fails.
The check valves of the high-pressure diaphragm pump are matched by the feed valve and discharge valve in pairs. Therefore, the normal operation vibration signal and fault vibration signal of a group of feed and discharge check valves were randomly selected as a group of experimental data for analysis. The fault data of the check valve used in the experiment were the data when a wear breakdown fault occurred. The sampling frequency of this experiment was 2560 Hz, and the sampling data length was 10,240. After A/D conversion, the collected vibration acceleration signal was input into the computer through the controller. In the check valve, wear breakdown caused by coarse particles stuck in the valve is very common. The check valve of the high-pressure diaphragm pump has a conevalve-type structure, and the main components are the valve body, valve core, and spring. Among them, the "valve core spring" constitutes a low-damping oscillation system, and its frequency is: where k is the stiffness value of the spring; and m s is the equivalent mass value. Set the spring stiffness as follows according to the activity of the plunger of the pump where f is the frequency of normal operation of the high-pressure diaphragm pump. Its normal operating frequency is 0.5~0.517 Hz, and the frequency of the spring valve core system fp = 2f, namely, 1~1.034 Hz. The check valve will show the corresponding fault fundamental frequency and double frequency when it fails. The check valves of the high-pressure diaphragm pump are matched by the feed valve and discharge valve in pairs. Therefore, the normal operation vibration signal and fault vibration signal of a group of feed and discharge check valves were randomly selected as a group of experimental data for analysis. The fault data of the check valve used in the experiment were the data when a wear breakdown fault occurred. The sampling frequency of this experiment was 2560 Hz, and the sampling data length was 10,240. After A/D conversion, the collected vibration acceleration signal was input into the computer through the controller.
Next, the normal operation and fault operation data of 10,240 check valves were taken for analysis. Figure 6 is the time-domain waveform diagram of the normal operation of the check valve. Figures 7 and 8 are the time-domain diagram and frequency-domain diagram when the check valve fails, respectively. Through comparative analysis of Figures 6 and 7, it can be seen that when the check valve was in normal operation, the vibration signal contains prominent impact components, and the amplitude was relatively more apparent. It can be seen from Figure 8 that the operating frequency was mainly within 200 Hz, but it was impossible to conclude whether it had a fault. To further analyze the signal, LMD decomposition was carried out first. As shown in Figure 9, the original signal was decomposed into several PF components and a residual component. To compare the decomposition effect, empirical mode decomposition (EMD) was used to decompose the signal. The results are shown in Figure 10. After EMD decomposition, more component signals were obtained, and there was a certain degree of modal aliasing. Through comparison, it can be seen that the effect of signal decomposition using the LMD method was better, as it reduced the phenomenon of mode aliasing to a great extent. Therefore, the obtained signal component contained more information. At the same time, LMD had fewer iterations. Next, we calculated the K-L divergence of all component signals decomposed by LMD, EMD, and the original signal. The results are shown in Tables 1 and 2. diagram when the check valve fails, respectively. Through c ures 6 and 7, it can be seen that when the check valve was in n tion signal contains prominent impact components, and th more apparent. It can be seen from Figure 8 that the opera within 200 Hz, but it was impossible to conclude whether it ha the signal, LMD decomposition was carried out first. As sho signal was decomposed into several PF components and a re pare the decomposition effect, empirical mode decomposition pose the signal. The results are shown in Figure 10. After EMD ponent signals were obtained, and there was a certain degree comparison, it can be seen that the effect of signal decomposit was better, as it reduced the phenomenon of mode aliasing t the obtained signal component contained more information. A fewer iterations. Next, we calculated the K-L divergence of all posed by LMD, EMD, and the original signal. The results are sh  taken for analysis. Figure 6 is the time-domain waveform diagram of th of the check valve. Figures 7 and 8 are the time-domain diagram and diagram when the check valve fails, respectively. Through comparat ures 6 and 7, it can be seen that when the check valve was in normal o tion signal contains prominent impact components, and the amplit more apparent. It can be seen from Figure 8 that the operating freq within 200 Hz, but it was impossible to conclude whether it had a fault. the signal, LMD decomposition was carried out first. As shown in Fig  signal was decomposed into several PF components and a residual co pare the decomposition effect, empirical mode decomposition (EMD) w pose the signal. The results are shown in Figure 10. After EMD decomp ponent signals were obtained, and there was a certain degree of moda comparison, it can be seen that the effect of signal decomposition usin was better, as it reduced the phenomenon of mode aliasing to a grea the obtained signal component contained more information. At the sa fewer iterations. Next, we calculated the K-L divergence of all compon posed by LMD, EMD, and the original signal. The results are shown in T      As shown in Table 1, the K-L divergence values of PF1, PF2, and PF3 were relatively small and less than the set threshold. Because the discrimination of K-L divergence was relatively obvious, it can be seen that their correlation with the original signal was relatively high. From the calculation results of kurtosis values, the kurtosis values of PF1 to PF5 were all greater than 3. Although it can be seen that the above component signals contained more impact components, this led to some difficulties in the signal screening. Therefore, the filtered PF1, PF2, and PF3 components were reconstructed according to the K-L divergence value. It can be seen from Table 2 that the K-L divergence values of IMF2, IMF 3, IMF4, and IMF 5 were less than the set threshold, which shows that their correlation with the original signal is relatively high. At the same time, by analyzing the calculation results of the kurtosis values from IMF 1 to IMF 10, it can be seen that the kurtosis values of other IMF components were greater than 3, except IMF 6, IMF 9, and IMF 10, which also made it difficult to screen effective signals. Similarly, the above component signal was selected for reconstruction according to the K-L divergence value calculation result. Although the above filtered signal components were highly correlated with the original signal, they contained a large number of fault signal features, and a large number of noise interference components. Therefore, to reduce the impact of noise on fault feature extraction, it was necessary to denoise the reconstructed signal by using the wavelet packet. In this experiment, the sym5 wavelet was used to decompose and reconstruct the reconstructed signal. The reconstructed signal waveform is shown in Figure 11, and the signal time-domain waveform after noise reduction is shown in Figures 12 and 13.      Next, the signals based on LMD wavelet packet denoising, EMD wavelet packet denoising, and wavelet packet direct denoising were demodulated by the Hilbert envelope to compare and analyze the experimental results. The results are shown in Figures 14-16. It is evident from Figure 14 that there was a fundamental frequency (0.3125 Hz) and a second to sixth doubling frequency of the fundamental frequency components (0.625, 0.9375, 1.25, 1.563 and 1.875 Hz) in the envelope spectrum of the signal after noise reduction based on the LMD wavelet packet, which has become the dominant frequency of the vibration signal, indicating that a fault occurred at this time. As shown in Figure 15, fundamental frequency (0.3125 Hz) and other doubling frequency components appeared in the envelope spectrum of the signal denoised based on the EMD wavelet packet. Still, the overall amplitude was less than the result seen in Figure 14. As shown in Figure 16, frequency components such as fundamental frequency (0.3125 Hz) and the second doubling frequency (0.625 Hz) cannot be found in the envelope spectrum of the signal after wavelet packet noise reduction. Therefore, using the proposed method for fault feature extraction can achieve better results.
2, 12, x FOR PEER REVIEW 1 through the Hilbert envelope spectrum analysis, it can be seen that the characterist quency in the Hilbert envelope spectrum obtained by the proposed method was rela apparent. Meanwhile, six frequency components such as the fundamental frequen the fault and the second to sixth doubling frequency of the fundamental frequency c extracted.  through the Hilbert envelope spectrum analysis, it can be seen that the characterist quency in the Hilbert envelope spectrum obtained by the proposed method was rela apparent. Meanwhile, six frequency components such as the fundamental frequen the fault and the second to sixth doubling frequency of the fundamental frequency c extracted.  The above qualitative analysis shows that the proposed method has better fault feature extraction advantages than the other two methods. Firstly, the fault signal of the check valve was decomposed by the time-frequency analysis method, and the complete timefrequency distribution information of the signal was obtained. Compared with EMD, LMD suppresses the endpoint effect brought about by EMD, eliminates the problems of over envelope, under envelope, and mode aliasing caused by EMD, and the decomposed signal can retain the information of the original signal. Secondly, this research introduces K-L divergence as the screening criterion in the signal screening link. Through the calculation and comparative analysis of the K-L divergence value of each signal component, it can be seen that the discrimination of K-L divergence was relatively more significant, which is helpful to better select the signal component. Then, the subsequent reconstructed signal was denoised by a wavelet packet to filter out the noise interference. Finally, through the Hilbert envelope spectrum analysis, it can be seen that the characteristic frequency in the Hilbert envelope spectrum obtained by the proposed method was relatively apparent. Meanwhile, six frequency components such as the fundamental frequency of the fault and the second to sixth doubling frequency of the fundamental frequency can be extracted.

Conclusions
Aiming at the non-stationary characteristics of the fault vibration signal of the check valve of a high-pressure diaphragm pump, a fault feature extraction method for the check valve based on LMD and wavelet packet analysis is proposed. Finally, the fault features were extracted by Hilbert envelope spectrum analysis. The following conclusions were obtained: (1) The LMD method was used to decompose the original signal adaptively, which overcomes the problems of modal aliasing and endpoint effect caused by the EMD method, and there were fewer iterations. The component signal decomposed by this method only included five PF components and one residual signal component. In comparison, the EMD obtained more signal components (ten IMF components and one residual component). The results show that LMD can extract the time-frequency information of the signal more effectively and provide a guarantee for the screening of subsequent signal components.
(2) By calculating the K-L divergence value of each signal component and selecting the signal component whose K-L divergence value is less than the set threshold as the effective component signal, the problem of poor discrimination caused by the traditional kurtosis method can be avoided. Because the original signal contains more impact components, it is difficult to screen the signal. The experimental results in the signal screening link show that three signal components, PF1, PF2, and PF3, can be extracted as effective signals by using K-L divergence. If the kurtosis criterion is used for filtering, all signal component values are more significant than the set threshold and cannot be selected. At the same time, in the comparative experiment using EMD, four signal components were successfully selected as effective signals by using K-L divergence. When kurtosis was used as the screening criterion, it also showed the defect of poor discrimination.
(3) By further comparing and analyzing the envelope spectra after wavelet packet denoising based on different methods, it can be seen that the peak of characteristic frequency in the Hilbert envelope spectrum obtained by using the method proposed in this paper