# Elimination of End effects in LMD Based on LSTM Network and Applications for Rolling Bearing Fault Feature Extraction

Authors: Jianhong Liang; Liping Wang; Jun Wu; Zhigui Liu
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

Local mean decomposition (LMD) is widely used in the area of multicomponents signal processing and fault diagnosis. One of the major problems is end effects, which distort the decomposed waveform at each end of the analyzed signal and influence feature frequency. In order to solve this problem, this paper proposes a novel self-adaptive waveform point extended method based on long short-term memory (LSTM) network. First, based on existing signal points, the LSTM network parameters of right and left ends are trained; then, these parameters are used to extend the waveform point at each end-side of signal; furthermore, the corresponding parameters are adaptively updated. The proposed method is compared with the characteristic segment extension and the traditional neural network extension methods through a simulated signal to verify the effectiveness. By combing the proposed method with LMD, an improved LMD algorithm is obtained. Finally, application of rolling bearing fault signal is carried out by the improved LMD algorithm, and the results show that the feature frequencies of the rolling bearing’s ball and inner and outer rings are successfully extracted.


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## Body

## 1. Introduction

Local mean decomposition (LMD), one of the popular self-adaptive time-frequency processes, was presented by Smith [1]. LMD can decompose a complex signal into a series of product functions (PFs), each of which is result of an envelope signal multiplied by a pure frequency signal, making each PF contain amplitude-modulated (AM) and frequency-modulated (FM) components [2], also making each PF represent practical physical significance [3]. Based on these features, considerable studies have used LMD in the area of multicomponents signal processing and fault diagnosis, such as voice signal processing [4], EEG signal processing [1], fault diagnosis [5–7], etc.Although LMD has been successfully applied, one of the major problems is end effects [2, 8, 9], which distort the decomposed waveform at each end of the analyzed signal and influence feature frequency. The main reason is that LMD needs to find out all extrema of analyzed signal itself. The interior extrema can be accurately identified, but it is difficult to find out rightful extrema at each end-side due to not enough comparison of effective waveform points.Some methods have been presented to solve the problem of end effects, one is adding the sliding window [10], which has been successfully applied in Fourier analysis and continuous Wavelet Transform. However, the result is irrelevant to the analyzed signal itself, and its causes loss of valuable information near the signal endpoints [11].Another method is extending the analyzed signal, which is the most effective method to solve end effects. These extended methods are used by the characteristics of signal segment [8, 12] and predicted extrapolation [13–15]. For characteristics of segment extension, existing signal is divided into n (n>3) segments; then the local characteristic values are calculated between endpoint segments and other segments by Revised Spectral Coherence (RSC) [8] or Hilbert Transform [12] approach. All calculated values are searched and compared, and the segment, which has similar characteristic values, will be used as the extended endpoint waveform segment to inhibit end effects. Although the literature [8, 12] also adopted adaptive search similar characteristic segment, the length of each segment is hardly determined from the analyzed signal. For the extrapolation methods, the drawback of selected segment length can be avoided because its extended basic unit is waveform point. Moreover, the LMD only needs fewer points to identify the extremums near the end-side of the signal. These extrapolation methods have been applied by extracting characteristics of signal points or by intelligently learning characteristics of existed points. The extracted characteristic points methods mainly include envelope straight line [11], similarity searching [13], Hermitian polynomial function [14], and mirror image methods [15]. Although all of these methods can effectively eliminate the end effects, these methods are based on the feature points by trendy signal (envelope waveform) rather than the signal itself. Learning all existed signal characteristics, the traditional neural network predictions (e.g., artificial neural network based on radial basis function [16], support vector machine regression [17–19]) are mostly suitable choice. Traditional neural network methods can extrapolate waveform point in accordance with the characteristics of the existed signal. However, these predicted methods only build the relationship between the input and output, and it causes the extended points irrelevant to previous input.As a consequence, the characteristics of segment extended method have a drawback for selecting each segment length, and the existing predictive extrapolation method does not consider the correlation of sequence signal. The rapid development of deep learning [20, 21] models have been integrated into sequence predication, and one of the models is the Recurrent Neural Network (RNN), which considers current and previous input information, but it suffers from the problem of exploding or vanishing gradients. The long short-term memory (LSTM) network is one type of RNN model. The LSTM elementary unit is memory cell, which can avoid the aforementioned problem. Additionally, the LSTM network has been successfully used in circuit current sequence predication [22], predicted excess vibration events [23], and machine vibration sequence predication [24, 25]. Thus, it has a great potential to complete signal extension by LSTM network. However, the number of the related publications to eliminate the end effects in LMD by LSTM network is still limited.Hence, this study presents a novel self-adaptive waveform point extended method based on LSTM network to eliminate the problem of right and left end effects. This method consists of three steps. First, LSTM network parameters of each end-side are trained by existing waveform points. Then, the waveform points are extended by these parameters. Lastly, each extended point adaptively updates corresponding LSTM network parameters. The proposed method not only learns characteristics of the existing waveform points, but also updates parameters adaptively by extended waveform points to ensure the validity of the extension. By using the proposed method to extend signal points, the end effects can be eliminated due to enough comparison of effective waveform points.The remainder of the literature is organized as follows: Section2 introduces related methodologies, reviews the LMD algorithm, explains the problem of end effects, and then describes the architecture of LSTM network; after that, a simulated signal is used to explain and compare the proposed method with two popularly extended methods; additionally, the proposed method is incorporated into the LMD to get an improved LMD algorithm. In Section 3, using the proposed method to extend real fault signal is discussed, then the signal is decomposed by the improved LMD algorithm, and extracting the fault feature frequency is finally illustrated. The conclusions are provided in Section 4.


## 2. Related Methodologies

### 2.1. LMD Algorithm and Its End Effects

#### 2.1.1. LMD Algorithm

By using the LMD, the multicomponent signalxt can be self-adaptive decomposed as(1)xt=∑p=1kPFkt+utt,where xt is analyzed signal, PFpt is product function, and uit is residual function. Each PFpt contains envelope and frequency information of analyzed signal and also exhibits practical physical significance [3]. First make sijt=xt, where i=j=1, and the algorithm steps are described as follows [1]:(1)
Identify all local extremumsnijk (include maxima and minima) from sijt, and the k-th mean values mijk and the k-th envelope values aijk are given by(2)mijk=nijk+nijk+12,(3)aijk=nijk−nijk+12,wherenijk and nijk+1 represent two successive extremums. All mean values mijk are connected by a straight line and then smoothened by spline interpolation to obtain mean function mijt. The same way is used for aijk to get the envelope function aijt.(2)
Subtractingmijt from signal sijt leads to(4)hijt=sijt−mijt,hijt divided by aijt leads to(5)sij+1t=hijtaijt.Usesij+1t as new sijt; repeat step (1) again to obtain the corresponding new envelope function aij+1t; if aij+1t=1, stop decomposed processing and the sijt as the j-th purely frequency single. Otherwise, set j=j+1; repeat aforementioned steps until sijt as purely frequency single, namely, the envelope function aij+1t of sijt equals 1. This research does not adopt such a harsh condition, and equation (6) is used to replace aij+1=1, where ε=0.00001.(6)aij+1t−1<ε,therefore, this iterative processing can be expressed as(7)hi1t=xt−mi1t,hi2t=hi1t−mi2t,⋮hijt=hij−1t−mijt,where(8)si1t=hi1tai1t,si2t=hi2tai2t,⋮sijt=hijtaijt.(3)
Thei-th envelope signal ait is obtained by multiplying all envelope functions aijt, and it can be written as(9)ait=ai1tai2t,…,aijt=∏q=1jaiqt,then thei-th product function PFit can be obtained by(10)PFit=aitsijt.(4)
SubtractPFit from the signal sijt to get si+1jt, and si+1jt is replaced as sijt; then make j=1 and i=i+1 and repeat upper preceding steps i times until the residual function uit is a monotonic or constant, and it can be written as(11)duitdt≡0,so, all the above decomposed results can be expressed as(12)u1t=xt−PF1t,u2t=u1t−PF2t,⋮uit=ui−1t−PFit.Thus, equation (12) is simplified to equation (1). Obviously, equation (6) is directly related to each PF component’s amplitude, and equation (11) will influence the mumble of PF.The first step of LMD identifies all extrema from the analyzed signal itself. If using wrong extrema, the performance of decomposed results will be distorted after iterative operations. The next subsection explains why end effects are imported and their relationship to erroneous extremums.


#### 2.1.2. End Effects of the LMD Algorithm

In Section2.1, LMD is described as a self-adaptive processing with double loops, which the inner and outer loops are restrained by equations (6) and (11), respectively. LMD initially needs to find out all extremums of the analyzed signal before proceeding the subsequent processes; thus, determining the right extremum is crucial. In case of misidentification of extremums at the beginning, a recurring error will be observed in the double loops, and it will introduce error information into the finally decomposed results.Generally, the extremums near each endpoint are misidentified, since given analyzed signal with finite length, and no sufficient signal points are used to contrast. For example, a sequence finite signal is shown in Figure1(a), in which the blue line is the simulated signal, and the red origin points are the extremums. Except those near the endpoints, the other extremums are accurately identified. As shown in Figures 1(b) and 2(b), all red origin points are correct extremums, except for E2 and E4. If using the E2 and E4 as extremums, the corresponding mean values m1j and envelope values a1j are calculated by equations (2) and (3). Then, m1j and a1j are connected by the red dashed straight line and are shown in Figures 1(a) and 2(a). However, the right extremums are E1 and E5 points, both of them are found by the extended simulated signal. The real local means and envelopes (green dashed straight line) are obtained as shown in Figures 1(b) and 2(b). The differences between the two local means, i.e., Figures 1(a) and 1(b) and envelopes, i.e., Figures 2(a) and 2(b) are shown in Figures 1(c) and 2(c), respectively. The local mean/envelope is the same as the real waveform, while the end-side of mean/envelope waveform is difference.Figure 1
Analysis of local mean for a simulated signal (solid blue line): (a) using the endpoint as the extremum of the local mean waveform (dashed red line); (b) real local mean waveform (dashed green line); (c) difference between the two local mean waveforms.
(a)
(b)
(c)Figure 2
Analysis of local envelope for a simulated signal (solid blue line): (a) using the endpoint as the extremum of the local envelope waveform (dashed red line); (b) real local mean waveform obtained using the extension signal (dashed green); (c) difference between the two local envelope waveforms.
(a)
(b)
(c)In the analyzed signal, neither the left nor the right endpoint is the extremum. Moreover, the left endpoint is farther from the real extremum than the right endpoint and is shown in Figures1(b) and 2(b). Consequently, the local means/envelopes of the left waveform are larger from real waveform than those of the right endpoint, as shown in Figures 1(c) and 2(c). Thus, the difference in the local endpoints will lead to the difference in the errors of m1j and a1j. So, the different error is solved only by extending to a new extremum, and the end effects can be eliminated.In fact, the left endpoint may randomly fall in [E1, E3] or [E3, E6] for the simulated waveform, and three are three assumptions. The first assumption is that the endpoint is extremum, i.e., E1, E3, or E6, but the validity of extremum cannot be directly determined. All points must be extended until a new extremum appears. For example, point E3 will extend until a new extremum E1 appears, only then E3 can be considered as an effective extremum. The second assumption is that the endpoint falls within (E3, E6), where E6 is a known extremum (minimum point) that must be extended until the maximum point E3 appears, and after then, E3 can be judged as a new effective extremum. The third assumption is that the endpoint falls within (E1, E3), where E3 is the maximum point and also the extremum. Similarly, E3 must extend several points until the minimum point E1 appears; after then, E1 can be judged as a new extremum. Therefore, local endpoints must extend several signal points to identify a new extremum to eliminate the end effects of the LMD algorithm, which indicate advantage of the waveform point rather than segment extension. The extension of the original waveform method, like neural networks and spectral coherence, and the limitations of these methods are explained in Section 1. This study proposes a novel self-adaptive waveform point extended method based on LSTM network. An improved LMD algorithm, which connected the proposed extended method with LMD, is explained in next section.


### 2.2. Waveform Point Extension Method and an Improved LMD Algorithm

For a nonlinear and nonstationary signal, it stochastically varies with time and makes its accurate prediction difficult [9]. Fault signals generated by rolling bearings are inherently periodic feature [26]. LSTM network, one of the sequence-to-sequence models, has been successfully applied to predict vibration sequences [23]. Based on this, a novel self-adaptive waveform point extended method would be developed in this section and then incorporated into the LMD algorithm.


#### 2.2.1. LSTM Architecture

The elementary unit of the LSTM network is memory cell, which not only prevents the explosion or disappearance of gradients, but also controls the memory length of the sequences to avoid memory loss after a long time [23, 25]. This network has ability of short-term sensitive memory because the unit cell possesses three gates: the forget, input, and output gates.As shown in Figure3, the memory cell has three input and two outputs at t-time. The xt, ht, and Ct are denoted as current input, output, and cell state output, respectively. The previous output of cell is ht−1, the previous output of cell is Ct−1. The memory cell has four weight parameters, namely, Wf, Wi, Wc, and Wo, and corresponding bias matrixes are bf, bi, bc, and bo. The relationship between parameters and cell inputs and outputs are described as follows [25]:(1)
Forget gate: control level of cell state reset (forget) and can be written as(13)ft=σWf∗ht−1,xt+bf,where theσ represents the sigmoid function and [.] denotes the concatenate operation. The ft value is limited between 0 and 1 because of sigmoid function, and the value of ft influences the ht−1 output to the current memory cell, if ft=0 means no information influence, and ft=1 means all information are contained.(2)
Input gate: control level of cell state update can be expressed as(14)it=σWi∗ht−1,xt+bi.The input gate also takes the sigma function, and determining the percentage ofxt is used for update, and the updated current memory states C^ can be written as(15)C^t=tanhWc∗ht−1,xt+bc,wheretanh is the hyperbolic tangent function. Then, update the previous cell state Ct−1 to the current cell state Ct, and it can be expressed as(16)Ct=ft∗Ct−1+it∗C^t.Equations (12)–(14) are used to restrict Ct by previous state Ct−1, previous output ht−1, and current input xt. The forget gate can save previous sequences feature information, and the input gate avoids the insertion of irrelevant contents into memory cell.(3)
Output gate: control level of cell state added to hidden state and can be given by(17)ot=σWo∗ht−1,xt+bo.Figure 3
Memory cell of LSTM network unit.The outputot, which determines the percentage of cell input will be updated, is restricted by the sigma function. The final memory cell unit output ht can be written as(18)ht=ot∗tanhCt.LSTM network uses the memory cell as elementary unit to replace the neuron [26], which is the elementary unit of traditional neural network, and finally, all memory cells fully connect as shown in Figure 4. Moreover, the error back-propagation (BP) algorithm is adopted to optimize network parameters and effectively realize sequence-to-sequence prediction and is applied to the extension of the waveform points in the study.Figure 4
Block diagram of the LSTM network parameters, the network parameters of right extension are obtained by the left figure, and the network parameters of left extension are obtained by the right figure.


#### 2.2.2. Waveform Point Extension Method Based on LSTM Network

When using sufficient signal points, the LSTM network can accurately predict nonexistent sequence points [23], but the real signal is a finite length sequence. Moreover, learning long sequence samples requires a long training time. Guo et al. [8] stated that rotating fault information will periodically appear within the sampling period; as long as the sampling exceeds the rotation period, fault information will reappear; however, once it appears again, the fault information will be covered with differential noise signals. Hence, the length of extended signal needs to exceed the rotation period of several times. Using LSTM network predicts waveform points, and the extended method needs to answer two questions: (1) what is the effective training method to solve the left and right end effects? (2) How to promise the adaptability of the extended network after each point of expansion?For the first problem, LSTM network can predict the sequence points in accordance with all the existing waveform features. If all training is conducted in time order, then only the right end effect be solved. Therefore, LSTM network parameters must be obtained in two different orders. Suppose the fixed length sequence isX=x1,x2,…,xt, and it is shown in Figure 4. The network parameters of right end-side are obtained directly through sequence X. To obtain the network parameters of the left end-side, the reverse time sequence X′=xt,xt−1,…,x2,x1 be used as training samples.For the second problem, as described in Section2.2, solving end effects requires extend points to find a new extremum. Since the LSTM network is a memory sequence model, the extended points must change the memory states and it affects the accuracy of subsequently extended points. Only by updating the network status and amending parameters, the validity of all extended points is guaranteed.Based on the above, this paper proposes a three-step self-adaptive waveform point extended method based on LSTM network, and the steps are described as follows:Step (1): using existing waveform points feature, LSTM network parameters of left and right ends are trained. The LSTM network’s input and target are continuous two-point information of the existing signal. For example, the right end network’s input isx1 and target is x2, while the left end network’s input is xt and target is xt−1.Step (2): extending the waveform points through the Step (1) parameters, the last point of the existing waveform is used as input element of the first extension point, and for subsequent extend point, use the previous extended point used as input to the LSTM network. For example, for the right end extended points, use existed last pointxt and get the first extended point xt+1 and the input xt+1 to get second extended point xt+2.Step (3): extending one waveform point, corresponding network parameters (left and right end-side parameters) are updated. As shown in Figure4, when one point extends completely, and getting xt+1 point, xt+1 needs to be used to update the right parameters. These parameters are used as the newest parameters to predict extension point xt+2, until finding out new extremum at the right end-side of signal.Step (3) enhances the self-adaptive ability of the proposed method, which is different from the traditional neural network extended method. The basic extended unit of proposed method is one point; thus, it avoids the determination of each extended segment length. To illustrate the proposed method, a simulated signal would be presented in the next subsection.


#### 2.2.3. Simulation and Comparisons

A simulated signal will be used to compare the proposed method with two popular methods. The simulated signal is a random signal; it includes multiple impulse response functions and two low frequency signals. Moreover, each of impulse functions has a high-frequency modulation component and can be described as follows [8, 27]:(19)yk=∑rexp−γ×k−r×Fs/fmFs×sin2πf0×k−r×Fs/fm−τrFs+0.7×sin2πf1×kFs+sin2πf2×kFs,where the frequencies of the two sin signals, f1 and f2, are 50 Hz and 100 Hz, respectively; the damping coefficient γ = 900; the sampling frequency Fs = 1200 Hz; the characteristic frequency fm = 100 Hz; τr is a random number constrained by the uniform distribution; and the resonant frequency f0 = 2600 Hz. In the simulation, 1200 points are selected from the simulated signal, and the time range is [0, 0.1] s, and the waveform is shown in Figure 5.Figure 5
A simulated signal waveform.Firstly, the proposed method will be compared with the traditional back-propagation neural network (BPNN). To quantitatively evaluate the performances of the two methods, the Root Mean Squared Error (RMSE) [19] measure is adopted. The RMSE is the square root of the average of the square of all of the errors, and the corresponding equation is written as(20)RMSE=1n∑i=1nyi∗−yi2,where yi∗ are extension points and yi are real points.The LSTM network uses one input/output layer, which means only one element is input or target, so each of the existing waveform points will be trained. In the simulation, 1200 points are used as training samples and extended 120 points are used as testing samples. For the right end-side extended networks, the number of hidden layers is 200, the maximum epoch is 500, and the initial learning rate is 0.001. For the left end-side extended networks, the hidden layer number is 200, the maximum epoch is 500, and the initial learning rate is 0.0018.Similarly, 1200 points of simulated signal are used as training samples for the BPNN, which is a three-layer network that includes an input layer, a hidden layer, and an output layer. A total of 200 neurons are set in the hidden layer, and also extended 120 points are used as testing samples. The BPNN input and target vectors are formed by a section of the simulated signal. For example, for the right end extended network, during the training process, the input vector isP=yk+1,yk+2,…,yk+120, and the corresponding target is yk+121. During the testing process, the first extension point, i.e. yk+121, will be used as the new element of the input vector for the new round of predict extension.As shown in Table1, although BPNN training time is little shorter than LSTM network training time, the values of RMSE training and testing are very larger than LSTM network. As a result, the BPNN extended points exhibit large error deviations as shown in Figures 6(b) and 7(b). The reason for upper phenomenon is that the BPNN output is only correlated to the current input, while the LSTM network output is related to the current and previous inputs, and this condition also increases the complexity of the LSTM network and causes a longer training time. As shown in Figures 6(a) and 7(a), although the LSTM extended waveform points are inconsistent with the real signal points, the waveform overall trend is the same. In the beginning of the LSTM extended waveform points, the errors (seen in Figures 6(b) and 7(b)) are relatively small, and then it increases with extended time. The waveform point has a weak follow-up in the high-frequency segment but has a reliable follow up in the smooth sin wave segment. This indicates that the LSTM network extended point is favorable for signals which appear periodically, and it can efficaciously extend waveform points for periodic rolling bearing fault signal.Table 1
Performance comparison of the LSMT and BPNN for the simulated signal.
Right extension networks
Left extension networks
Training time (s)
Training RMSE
Testing RMSE
Training time (s)
Training RMSE
Testing RMSE
SLTM
59
0.0013
0.1393
63
0.0518
0.1928
BPNN
52
0.1750
0.2320
51
0.2267
0.2757Figure 6
Comparison of LSTM and BPNN left waveform point extension for simulation signal: (a) waveform extension; (b) waveform extension error.
(a)
(b)Figure 7
Comparison of LSTM and BPNN right waveform point extension for simulation signal: (a) waveform extension; (b) waveform extension error.
(a)
(b)The proposed method will be further compared with the characteristic of segment extended method, which has a drawback due to determination of the length of each segment. To express the influence of each segment length, RSC [8] method is used to find extended segments. Firstly, the simulated signal is divided into 8, 10, and 15 segments. When C=8, the RSC values between the first-segment and other segments and between the last-segment and other segments need to be calculated. Comparison of calculated results showed that the maximum RSC values are γ1,5=0.8669 and γ8,1=0.9916, which indicates that the 1-st segment and the 5-th segment have the most similar characteristic spectrum and the 1-st segment and the 8-th segment have the most similar characteristic spectrum. Therefore, the 5-th and 1-st segments are set as the left and right end extended segments, respectively, and waveform shown in Figure 8(b). Using same way to process when C = 10 (γ1,3=γ10,2=1) and C = 15 (γ1,4=0.8639 and γ15,14=0.8592), as illustrated in Figures 8(c) and 8(d). It can be seen that setting difference of unit segment may lead to different waveforms, and the extended result is consistent with the real signal only C = 10, while the other length of segment cause error.Figure 8
Comparison of LSTM and different length segment RSC extended results for simulated signal (the red line represents the extended waveform points, and the blue line is the simulated signal): (a) the real simulated waveform; (b) 8 segments; (c) 10 segments; (d) 15 segments; (e) based on proposed method.
(a)
(b)
(c)
(d)
(e)If utilizing LMD process the upper extended signals, i.e., Figures8(b), 8(d), and 8(e), the final result PF1 and PF2 waveform shown in Figures 9(a)–9(c), respectively. As shown in Figures 9(a) and 9(b), it can be seen the large end swings near each end of the two PFs. As shown in Figure 9(c), if using the proposed method will get well-extended signal, because large end swings are not found at the starting or ending of each PFs. As shown in Figures 8 and 9, only C = 10 and the proposed method are similar with the real signal waveform, and the large end swings do not appear in decomposed waveforms.Figure 9
Comparisons of decomposition waveform obtained by LMD based on three extended methods: (a) 8 segments; (b) 15 segments; (c) proposed waveform points extended method.
(a)
(b)
(c)The comparison results confirm the superior performance of the proposed method over the traditional neural network and characteristic segment extended methods, because the former lacks the processing for the previous input, and the latter hardly determines the segment length, and different lengths cause difference of extended errors except for suitable segment length.


#### 2.2.4. Improved LMD Algorithm

An improved LMD algorithm, which connects proposed method with LMD algorithm, is presented in the subsection. Additionally, the improved LMD algorithm considers the feature of LMD, and the extended point includes a new extremum; thus, it is more effective to solve end effects of LMD, and its process is shown in Figure10.Figure 10
Processing of improve LMD algorithm.After identifying the interior extremumsnijk, the existing waveform points must continuously extend to find out a new extremum near each end-side of analyzed signal. As shown in Figure 10, signal sijz must extend one point sijz+1 by the right parameters, then judge whether sijz is a new extremum. If not true, then a new point needs to extend and judge again until a new extremum is identified, and the same way is used to process the left extension. After both end-sides are extended, the original extremums nijk will be added one extremum near each end of the analyzed signal, but the analyzed signal has a fixed range. Hence, the added extremums are only used to complete interpolation, and the subsequent operations will be performed within the same as the original signal time range.The improved LMD algorithm can judge new extremum rather than fixed extended points, i.e., as shown in Figures6(a) and 7(a), the left end extends only 3 points, but the right end needs to extend 25 points. The improved LMD algorithm will be applied for rolling bearing fault feature extraction, and its further explanation that the proposed method is used to eliminate end effects is given in the next section.


## 2.1. LMD Algorithm and Its End Effects

### 2.1.1. LMD Algorithm

By using the LMD, the multicomponent signalxt can be self-adaptive decomposed as(1)xt=∑p=1kPFkt+utt,where xt is analyzed signal, PFpt is product function, and uit is residual function. Each PFpt contains envelope and frequency information of analyzed signal and also exhibits practical physical significance [3]. First make sijt=xt, where i=j=1, and the algorithm steps are described as follows [1]:(1)
Identify all local extremumsnijk (include maxima and minima) from sijt, and the k-th mean values mijk and the k-th envelope values aijk are given by(2)mijk=nijk+nijk+12,(3)aijk=nijk−nijk+12,wherenijk and nijk+1 represent two successive extremums. All mean values mijk are connected by a straight line and then smoothened by spline interpolation to obtain mean function mijt. The same way is used for aijk to get the envelope function aijt.(2)
Subtractingmijt from signal sijt leads to(4)hijt=sijt−mijt,hijt divided by aijt leads to(5)sij+1t=hijtaijt.Usesij+1t as new sijt; repeat step (1) again to obtain the corresponding new envelope function aij+1t; if aij+1t=1, stop decomposed processing and the sijt as the j-th purely frequency single. Otherwise, set j=j+1; repeat aforementioned steps until sijt as purely frequency single, namely, the envelope function aij+1t of sijt equals 1. This research does not adopt such a harsh condition, and equation (6) is used to replace aij+1=1, where ε=0.00001.(6)aij+1t−1<ε,therefore, this iterative processing can be expressed as(7)hi1t=xt−mi1t,hi2t=hi1t−mi2t,⋮hijt=hij−1t−mijt,where(8)si1t=hi1tai1t,si2t=hi2tai2t,⋮sijt=hijtaijt.(3)
Thei-th envelope signal ait is obtained by multiplying all envelope functions aijt, and it can be written as(9)ait=ai1tai2t,…,aijt=∏q=1jaiqt,then thei-th product function PFit can be obtained by(10)PFit=aitsijt.(4)
SubtractPFit from the signal sijt to get si+1jt, and si+1jt is replaced as sijt; then make j=1 and i=i+1 and repeat upper preceding steps i times until the residual function uit is a monotonic or constant, and it can be written as(11)duitdt≡0,so, all the above decomposed results can be expressed as(12)u1t=xt−PF1t,u2t=u1t−PF2t,⋮uit=ui−1t−PFit.Thus, equation (12) is simplified to equation (1). Obviously, equation (6) is directly related to each PF component’s amplitude, and equation (11) will influence the mumble of PF.The first step of LMD identifies all extrema from the analyzed signal itself. If using wrong extrema, the performance of decomposed results will be distorted after iterative operations. The next subsection explains why end effects are imported and their relationship to erroneous extremums.


### 2.1.2. End Effects of the LMD Algorithm

In Section2.1, LMD is described as a self-adaptive processing with double loops, which the inner and outer loops are restrained by equations (6) and (11), respectively. LMD initially needs to find out all extremums of the analyzed signal before proceeding the subsequent processes; thus, determining the right extremum is crucial. In case of misidentification of extremums at the beginning, a recurring error will be observed in the double loops, and it will introduce error information into the finally decomposed results.Generally, the extremums near each endpoint are misidentified, since given analyzed signal with finite length, and no sufficient signal points are used to contrast. For example, a sequence finite signal is shown in Figure1(a), in which the blue line is the simulated signal, and the red origin points are the extremums. Except those near the endpoints, the other extremums are accurately identified. As shown in Figures 1(b) and 2(b), all red origin points are correct extremums, except for E2 and E4. If using the E2 and E4 as extremums, the corresponding mean values m1j and envelope values a1j are calculated by equations (2) and (3). Then, m1j and a1j are connected by the red dashed straight line and are shown in Figures 1(a) and 2(a). However, the right extremums are E1 and E5 points, both of them are found by the extended simulated signal. The real local means and envelopes (green dashed straight line) are obtained as shown in Figures 1(b) and 2(b). The differences between the two local means, i.e., Figures 1(a) and 1(b) and envelopes, i.e., Figures 2(a) and 2(b) are shown in Figures 1(c) and 2(c), respectively. The local mean/envelope is the same as the real waveform, while the end-side of mean/envelope waveform is difference.Figure 1
Analysis of local mean for a simulated signal (solid blue line): (a) using the endpoint as the extremum of the local mean waveform (dashed red line); (b) real local mean waveform (dashed green line); (c) difference between the two local mean waveforms.
(a)
(b)
(c)Figure 2
Analysis of local envelope for a simulated signal (solid blue line): (a) using the endpoint as the extremum of the local envelope waveform (dashed red line); (b) real local mean waveform obtained using the extension signal (dashed green); (c) difference between the two local envelope waveforms.
(a)
(b)
(c)In the analyzed signal, neither the left nor the right endpoint is the extremum. Moreover, the left endpoint is farther from the real extremum than the right endpoint and is shown in Figures1(b) and 2(b). Consequently, the local means/envelopes of the left waveform are larger from real waveform than those of the right endpoint, as shown in Figures 1(c) and 2(c). Thus, the difference in the local endpoints will lead to the difference in the errors of m1j and a1j. So, the different error is solved only by extending to a new extremum, and the end effects can be eliminated.In fact, the left endpoint may randomly fall in [E1, E3] or [E3, E6] for the simulated waveform, and three are three assumptions. The first assumption is that the endpoint is extremum, i.e., E1, E3, or E6, but the validity of extremum cannot be directly determined. All points must be extended until a new extremum appears. For example, point E3 will extend until a new extremum E1 appears, only then E3 can be considered as an effective extremum. The second assumption is that the endpoint falls within (E3, E6), where E6 is a known extremum (minimum point) that must be extended until the maximum point E3 appears, and after then, E3 can be judged as a new effective extremum. The third assumption is that the endpoint falls within (E1, E3), where E3 is the maximum point and also the extremum. Similarly, E3 must extend several points until the minimum point E1 appears; after then, E1 can be judged as a new extremum. Therefore, local endpoints must extend several signal points to identify a new extremum to eliminate the end effects of the LMD algorithm, which indicate advantage of the waveform point rather than segment extension. The extension of the original waveform method, like neural networks and spectral coherence, and the limitations of these methods are explained in Section 1. This study proposes a novel self-adaptive waveform point extended method based on LSTM network. An improved LMD algorithm, which connected the proposed extended method with LMD, is explained in next section.


## 2.1.1. LMD Algorithm

By using the LMD, the multicomponent signalxt can be self-adaptive decomposed as(1)xt=∑p=1kPFkt+utt,where xt is analyzed signal, PFpt is product function, and uit is residual function. Each PFpt contains envelope and frequency information of analyzed signal and also exhibits practical physical significance [3]. First make sijt=xt, where i=j=1, and the algorithm steps are described as follows [1]:(1)
Identify all local extremumsnijk (include maxima and minima) from sijt, and the k-th mean values mijk and the k-th envelope values aijk are given by(2)mijk=nijk+nijk+12,(3)aijk=nijk−nijk+12,wherenijk and nijk+1 represent two successive extremums. All mean values mijk are connected by a straight line and then smoothened by spline interpolation to obtain mean function mijt. The same way is used for aijk to get the envelope function aijt.(2)
Subtractingmijt from signal sijt leads to(4)hijt=sijt−mijt,hijt divided by aijt leads to(5)sij+1t=hijtaijt.Usesij+1t as new sijt; repeat step (1) again to obtain the corresponding new envelope function aij+1t; if aij+1t=1, stop decomposed processing and the sijt as the j-th purely frequency single. Otherwise, set j=j+1; repeat aforementioned steps until sijt as purely frequency single, namely, the envelope function aij+1t of sijt equals 1. This research does not adopt such a harsh condition, and equation (6) is used to replace aij+1=1, where ε=0.00001.(6)aij+1t−1<ε,therefore, this iterative processing can be expressed as(7)hi1t=xt−mi1t,hi2t=hi1t−mi2t,⋮hijt=hij−1t−mijt,where(8)si1t=hi1tai1t,si2t=hi2tai2t,⋮sijt=hijtaijt.(3)
Thei-th envelope signal ait is obtained by multiplying all envelope functions aijt, and it can be written as(9)ait=ai1tai2t,…,aijt=∏q=1jaiqt,then thei-th product function PFit can be obtained by(10)PFit=aitsijt.(4)
SubtractPFit from the signal sijt to get si+1jt, and si+1jt is replaced as sijt; then make j=1 and i=i+1 and repeat upper preceding steps i times until the residual function uit is a monotonic or constant, and it can be written as(11)duitdt≡0,so, all the above decomposed results can be expressed as(12)u1t=xt−PF1t,u2t=u1t−PF2t,⋮uit=ui−1t−PFit.Thus, equation (12) is simplified to equation (1). Obviously, equation (6) is directly related to each PF component’s amplitude, and equation (11) will influence the mumble of PF.The first step of LMD identifies all extrema from the analyzed signal itself. If using wrong extrema, the performance of decomposed results will be distorted after iterative operations. The next subsection explains why end effects are imported and their relationship to erroneous extremums.


## 2.1.2. End Effects of the LMD Algorithm

In Section2.1, LMD is described as a self-adaptive processing with double loops, which the inner and outer loops are restrained by equations (6) and (11), respectively. LMD initially needs to find out all extremums of the analyzed signal before proceeding the subsequent processes; thus, determining the right extremum is crucial. In case of misidentification of extremums at the beginning, a recurring error will be observed in the double loops, and it will introduce error information into the finally decomposed results.Generally, the extremums near each endpoint are misidentified, since given analyzed signal with finite length, and no sufficient signal points are used to contrast. For example, a sequence finite signal is shown in Figure1(a), in which the blue line is the simulated signal, and the red origin points are the extremums. Except those near the endpoints, the other extremums are accurately identified. As shown in Figures 1(b) and 2(b), all red origin points are correct extremums, except for E2 and E4. If using the E2 and E4 as extremums, the corresponding mean values m1j and envelope values a1j are calculated by equations (2) and (3). Then, m1j and a1j are connected by the red dashed straight line and are shown in Figures 1(a) and 2(a). However, the right extremums are E1 and E5 points, both of them are found by the extended simulated signal. The real local means and envelopes (green dashed straight line) are obtained as shown in Figures 1(b) and 2(b). The differences between the two local means, i.e., Figures 1(a) and 1(b) and envelopes, i.e., Figures 2(a) and 2(b) are shown in Figures 1(c) and 2(c), respectively. The local mean/envelope is the same as the real waveform, while the end-side of mean/envelope waveform is difference.Figure 1
Analysis of local mean for a simulated signal (solid blue line): (a) using the endpoint as the extremum of the local mean waveform (dashed red line); (b) real local mean waveform (dashed green line); (c) difference between the two local mean waveforms.
(a)
(b)
(c)Figure 2
Analysis of local envelope for a simulated signal (solid blue line): (a) using the endpoint as the extremum of the local envelope waveform (dashed red line); (b) real local mean waveform obtained using the extension signal (dashed green); (c) difference between the two local envelope waveforms.
(a)
(b)
(c)In the analyzed signal, neither the left nor the right endpoint is the extremum. Moreover, the left endpoint is farther from the real extremum than the right endpoint and is shown in Figures1(b) and 2(b). Consequently, the local means/envelopes of the left waveform are larger from real waveform than those of the right endpoint, as shown in Figures 1(c) and 2(c). Thus, the difference in the local endpoints will lead to the difference in the errors of m1j and a1j. So, the different error is solved only by extending to a new extremum, and the end effects can be eliminated.In fact, the left endpoint may randomly fall in [E1, E3] or [E3, E6] for the simulated waveform, and three are three assumptions. The first assumption is that the endpoint is extremum, i.e., E1, E3, or E6, but the validity of extremum cannot be directly determined. All points must be extended until a new extremum appears. For example, point E3 will extend until a new extremum E1 appears, only then E3 can be considered as an effective extremum. The second assumption is that the endpoint falls within (E3, E6), where E6 is a known extremum (minimum point) that must be extended until the maximum point E3 appears, and after then, E3 can be judged as a new effective extremum. The third assumption is that the endpoint falls within (E1, E3), where E3 is the maximum point and also the extremum. Similarly, E3 must extend several points until the minimum point E1 appears; after then, E1 can be judged as a new extremum. Therefore, local endpoints must extend several signal points to identify a new extremum to eliminate the end effects of the LMD algorithm, which indicate advantage of the waveform point rather than segment extension. The extension of the original waveform method, like neural networks and spectral coherence, and the limitations of these methods are explained in Section 1. This study proposes a novel self-adaptive waveform point extended method based on LSTM network. An improved LMD algorithm, which connected the proposed extended method with LMD, is explained in next section.


## 2.2. Waveform Point Extension Method and an Improved LMD Algorithm

For a nonlinear and nonstationary signal, it stochastically varies with time and makes its accurate prediction difficult [9]. Fault signals generated by rolling bearings are inherently periodic feature [26]. LSTM network, one of the sequence-to-sequence models, has been successfully applied to predict vibration sequences [23]. Based on this, a novel self-adaptive waveform point extended method would be developed in this section and then incorporated into the LMD algorithm.


### 2.2.1. LSTM Architecture

The elementary unit of the LSTM network is memory cell, which not only prevents the explosion or disappearance of gradients, but also controls the memory length of the sequences to avoid memory loss after a long time [23, 25]. This network has ability of short-term sensitive memory because the unit cell possesses three gates: the forget, input, and output gates.As shown in Figure3, the memory cell has three input and two outputs at t-time. The xt, ht, and Ct are denoted as current input, output, and cell state output, respectively. The previous output of cell is ht−1, the previous output of cell is Ct−1. The memory cell has four weight parameters, namely, Wf, Wi, Wc, and Wo, and corresponding bias matrixes are bf, bi, bc, and bo. The relationship between parameters and cell inputs and outputs are described as follows [25]:(1)
Forget gate: control level of cell state reset (forget) and can be written as(13)ft=σWf∗ht−1,xt+bf,where theσ represents the sigmoid function and [.] denotes the concatenate operation. The ft value is limited between 0 and 1 because of sigmoid function, and the value of ft influences the ht−1 output to the current memory cell, if ft=0 means no information influence, and ft=1 means all information are contained.(2)
Input gate: control level of cell state update can be expressed as(14)it=σWi∗ht−1,xt+bi.The input gate also takes the sigma function, and determining the percentage ofxt is used for update, and the updated current memory states C^ can be written as(15)C^t=tanhWc∗ht−1,xt+bc,wheretanh is the hyperbolic tangent function. Then, update the previous cell state Ct−1 to the current cell state Ct, and it can be expressed as(16)Ct=ft∗Ct−1+it∗C^t.Equations (12)–(14) are used to restrict Ct by previous state Ct−1, previous output ht−1, and current input xt. The forget gate can save previous sequences feature information, and the input gate avoids the insertion of irrelevant contents into memory cell.(3)
Output gate: control level of cell state added to hidden state and can be given by(17)ot=σWo∗ht−1,xt+bo.Figure 3
Memory cell of LSTM network unit.The outputot, which determines the percentage of cell input will be updated, is restricted by the sigma function. The final memory cell unit output ht can be written as(18)ht=ot∗tanhCt.LSTM network uses the memory cell as elementary unit to replace the neuron [26], which is the elementary unit of traditional neural network, and finally, all memory cells fully connect as shown in Figure 4. Moreover, the error back-propagation (BP) algorithm is adopted to optimize network parameters and effectively realize sequence-to-sequence prediction and is applied to the extension of the waveform points in the study.Figure 4
Block diagram of the LSTM network parameters, the network parameters of right extension are obtained by the left figure, and the network parameters of left extension are obtained by the right figure.


### 2.2.2. Waveform Point Extension Method Based on LSTM Network

When using sufficient signal points, the LSTM network can accurately predict nonexistent sequence points [23], but the real signal is a finite length sequence. Moreover, learning long sequence samples requires a long training time. Guo et al. [8] stated that rotating fault information will periodically appear within the sampling period; as long as the sampling exceeds the rotation period, fault information will reappear; however, once it appears again, the fault information will be covered with differential noise signals. Hence, the length of extended signal needs to exceed the rotation period of several times. Using LSTM network predicts waveform points, and the extended method needs to answer two questions: (1) what is the effective training method to solve the left and right end effects? (2) How to promise the adaptability of the extended network after each point of expansion?For the first problem, LSTM network can predict the sequence points in accordance with all the existing waveform features. If all training is conducted in time order, then only the right end effect be solved. Therefore, LSTM network parameters must be obtained in two different orders. Suppose the fixed length sequence isX=x1,x2,…,xt, and it is shown in Figure 4. The network parameters of right end-side are obtained directly through sequence X. To obtain the network parameters of the left end-side, the reverse time sequence X′=xt,xt−1,…,x2,x1 be used as training samples.For the second problem, as described in Section2.2, solving end effects requires extend points to find a new extremum. Since the LSTM network is a memory sequence model, the extended points must change the memory states and it affects the accuracy of subsequently extended points. Only by updating the network status and amending parameters, the validity of all extended points is guaranteed.Based on the above, this paper proposes a three-step self-adaptive waveform point extended method based on LSTM network, and the steps are described as follows:Step (1): using existing waveform points feature, LSTM network parameters of left and right ends are trained. The LSTM network’s input and target are continuous two-point information of the existing signal. For example, the right end network’s input isx1 and target is x2, while the left end network’s input is xt and target is xt−1.Step (2): extending the waveform points through the Step (1) parameters, the last point of the existing waveform is used as input element of the first extension point, and for subsequent extend point, use the previous extended point used as input to the LSTM network. For example, for the right end extended points, use existed last pointxt and get the first extended point xt+1 and the input xt+1 to get second extended point xt+2.Step (3): extending one waveform point, corresponding network parameters (left and right end-side parameters) are updated. As shown in Figure4, when one point extends completely, and getting xt+1 point, xt+1 needs to be used to update the right parameters. These parameters are used as the newest parameters to predict extension point xt+2, until finding out new extremum at the right end-side of signal.Step (3) enhances the self-adaptive ability of the proposed method, which is different from the traditional neural network extended method. The basic extended unit of proposed method is one point; thus, it avoids the determination of each extended segment length. To illustrate the proposed method, a simulated signal would be presented in the next subsection.


### 2.2.3. Simulation and Comparisons

A simulated signal will be used to compare the proposed method with two popular methods. The simulated signal is a random signal; it includes multiple impulse response functions and two low frequency signals. Moreover, each of impulse functions has a high-frequency modulation component and can be described as follows [8, 27]:(19)yk=∑rexp−γ×k−r×Fs/fmFs×sin2πf0×k−r×Fs/fm−τrFs+0.7×sin2πf1×kFs+sin2πf2×kFs,where the frequencies of the two sin signals, f1 and f2, are 50 Hz and 100 Hz, respectively; the damping coefficient γ = 900; the sampling frequency Fs = 1200 Hz; the characteristic frequency fm = 100 Hz; τr is a random number constrained by the uniform distribution; and the resonant frequency f0 = 2600 Hz. In the simulation, 1200 points are selected from the simulated signal, and the time range is [0, 0.1] s, and the waveform is shown in Figure 5.Figure 5
A simulated signal waveform.Firstly, the proposed method will be compared with the traditional back-propagation neural network (BPNN). To quantitatively evaluate the performances of the two methods, the Root Mean Squared Error (RMSE) [19] measure is adopted. The RMSE is the square root of the average of the square of all of the errors, and the corresponding equation is written as(20)RMSE=1n∑i=1nyi∗−yi2,where yi∗ are extension points and yi are real points.The LSTM network uses one input/output layer, which means only one element is input or target, so each of the existing waveform points will be trained. In the simulation, 1200 points are used as training samples and extended 120 points are used as testing samples. For the right end-side extended networks, the number of hidden layers is 200, the maximum epoch is 500, and the initial learning rate is 0.001. For the left end-side extended networks, the hidden layer number is 200, the maximum epoch is 500, and the initial learning rate is 0.0018.Similarly, 1200 points of simulated signal are used as training samples for the BPNN, which is a three-layer network that includes an input layer, a hidden layer, and an output layer. A total of 200 neurons are set in the hidden layer, and also extended 120 points are used as testing samples. The BPNN input and target vectors are formed by a section of the simulated signal. For example, for the right end extended network, during the training process, the input vector isP=yk+1,yk+2,…,yk+120, and the corresponding target is yk+121. During the testing process, the first extension point, i.e. yk+121, will be used as the new element of the input vector for the new round of predict extension.As shown in Table1, although BPNN training time is little shorter than LSTM network training time, the values of RMSE training and testing are very larger than LSTM network. As a result, the BPNN extended points exhibit large error deviations as shown in Figures 6(b) and 7(b). The reason for upper phenomenon is that the BPNN output is only correlated to the current input, while the LSTM network output is related to the current and previous inputs, and this condition also increases the complexity of the LSTM network and causes a longer training time. As shown in Figures 6(a) and 7(a), although the LSTM extended waveform points are inconsistent with the real signal points, the waveform overall trend is the same. In the beginning of the LSTM extended waveform points, the errors (seen in Figures 6(b) and 7(b)) are relatively small, and then it increases with extended time. The waveform point has a weak follow-up in the high-frequency segment but has a reliable follow up in the smooth sin wave segment. This indicates that the LSTM network extended point is favorable for signals which appear periodically, and it can efficaciously extend waveform points for periodic rolling bearing fault signal.Table 1
Performance comparison of the LSMT and BPNN for the simulated signal.
Right extension networks
Left extension networks
Training time (s)
Training RMSE
Testing RMSE
Training time (s)
Training RMSE
Testing RMSE
SLTM
59
0.0013
0.1393
63
0.0518
0.1928
BPNN
52
0.1750
0.2320
51
0.2267
0.2757Figure 6
Comparison of LSTM and BPNN left waveform point extension for simulation signal: (a) waveform extension; (b) waveform extension error.
(a)
(b)Figure 7
Comparison of LSTM and BPNN right waveform point extension for simulation signal: (a) waveform extension; (b) waveform extension error.
(a)
(b)The proposed method will be further compared with the characteristic of segment extended method, which has a drawback due to determination of the length of each segment. To express the influence of each segment length, RSC [8] method is used to find extended segments. Firstly, the simulated signal is divided into 8, 10, and 15 segments. When C=8, the RSC values between the first-segment and other segments and between the last-segment and other segments need to be calculated. Comparison of calculated results showed that the maximum RSC values are γ1,5=0.8669 and γ8,1=0.9916, which indicates that the 1-st segment and the 5-th segment have the most similar characteristic spectrum and the 1-st segment and the 8-th segment have the most similar characteristic spectrum. Therefore, the 5-th and 1-st segments are set as the left and right end extended segments, respectively, and waveform shown in Figure 8(b). Using same way to process when C = 10 (γ1,3=γ10,2=1) and C = 15 (γ1,4=0.8639 and γ15,14=0.8592), as illustrated in Figures 8(c) and 8(d). It can be seen that setting difference of unit segment may lead to different waveforms, and the extended result is consistent with the real signal only C = 10, while the other length of segment cause error.Figure 8
Comparison of LSTM and different length segment RSC extended results for simulated signal (the red line represents the extended waveform points, and the blue line is the simulated signal): (a) the real simulated waveform; (b) 8 segments; (c) 10 segments; (d) 15 segments; (e) based on proposed method.
(a)
(b)
(c)
(d)
(e)If utilizing LMD process the upper extended signals, i.e., Figures8(b), 8(d), and 8(e), the final result PF1 and PF2 waveform shown in Figures 9(a)–9(c), respectively. As shown in Figures 9(a) and 9(b), it can be seen the large end swings near each end of the two PFs. As shown in Figure 9(c), if using the proposed method will get well-extended signal, because large end swings are not found at the starting or ending of each PFs. As shown in Figures 8 and 9, only C = 10 and the proposed method are similar with the real signal waveform, and the large end swings do not appear in decomposed waveforms.Figure 9
Comparisons of decomposition waveform obtained by LMD based on three extended methods: (a) 8 segments; (b) 15 segments; (c) proposed waveform points extended method.
(a)
(b)
(c)The comparison results confirm the superior performance of the proposed method over the traditional neural network and characteristic segment extended methods, because the former lacks the processing for the previous input, and the latter hardly determines the segment length, and different lengths cause difference of extended errors except for suitable segment length.


### 2.2.4. Improved LMD Algorithm

An improved LMD algorithm, which connects proposed method with LMD algorithm, is presented in the subsection. Additionally, the improved LMD algorithm considers the feature of LMD, and the extended point includes a new extremum; thus, it is more effective to solve end effects of LMD, and its process is shown in Figure10.Figure 10
Processing of improve LMD algorithm.After identifying the interior extremumsnijk, the existing waveform points must continuously extend to find out a new extremum near each end-side of analyzed signal. As shown in Figure 10, signal sijz must extend one point sijz+1 by the right parameters, then judge whether sijz is a new extremum. If not true, then a new point needs to extend and judge again until a new extremum is identified, and the same way is used to process the left extension. After both end-sides are extended, the original extremums nijk will be added one extremum near each end of the analyzed signal, but the analyzed signal has a fixed range. Hence, the added extremums are only used to complete interpolation, and the subsequent operations will be performed within the same as the original signal time range.The improved LMD algorithm can judge new extremum rather than fixed extended points, i.e., as shown in Figures6(a) and 7(a), the left end extends only 3 points, but the right end needs to extend 25 points. The improved LMD algorithm will be applied for rolling bearing fault feature extraction, and its further explanation that the proposed method is used to eliminate end effects is given in the next section.


## 2.2.1. LSTM Architecture

The elementary unit of the LSTM network is memory cell, which not only prevents the explosion or disappearance of gradients, but also controls the memory length of the sequences to avoid memory loss after a long time [23, 25]. This network has ability of short-term sensitive memory because the unit cell possesses three gates: the forget, input, and output gates.As shown in Figure3, the memory cell has three input and two outputs at t-time. The xt, ht, and Ct are denoted as current input, output, and cell state output, respectively. The previous output of cell is ht−1, the previous output of cell is Ct−1. The memory cell has four weight parameters, namely, Wf, Wi, Wc, and Wo, and corresponding bias matrixes are bf, bi, bc, and bo. The relationship between parameters and cell inputs and outputs are described as follows [25]:(1)
Forget gate: control level of cell state reset (forget) and can be written as(13)ft=σWf∗ht−1,xt+bf,where theσ represents the sigmoid function and [.] denotes the concatenate operation. The ft value is limited between 0 and 1 because of sigmoid function, and the value of ft influences the ht−1 output to the current memory cell, if ft=0 means no information influence, and ft=1 means all information are contained.(2)
Input gate: control level of cell state update can be expressed as(14)it=σWi∗ht−1,xt+bi.The input gate also takes the sigma function, and determining the percentage ofxt is used for update, and the updated current memory states C^ can be written as(15)C^t=tanhWc∗ht−1,xt+bc,wheretanh is the hyperbolic tangent function. Then, update the previous cell state Ct−1 to the current cell state Ct, and it can be expressed as(16)Ct=ft∗Ct−1+it∗C^t.Equations (12)–(14) are used to restrict Ct by previous state Ct−1, previous output ht−1, and current input xt. The forget gate can save previous sequences feature information, and the input gate avoids the insertion of irrelevant contents into memory cell.(3)
Output gate: control level of cell state added to hidden state and can be given by(17)ot=σWo∗ht−1,xt+bo.Figure 3
Memory cell of LSTM network unit.The outputot, which determines the percentage of cell input will be updated, is restricted by the sigma function. The final memory cell unit output ht can be written as(18)ht=ot∗tanhCt.LSTM network uses the memory cell as elementary unit to replace the neuron [26], which is the elementary unit of traditional neural network, and finally, all memory cells fully connect as shown in Figure 4. Moreover, the error back-propagation (BP) algorithm is adopted to optimize network parameters and effectively realize sequence-to-sequence prediction and is applied to the extension of the waveform points in the study.Figure 4
Block diagram of the LSTM network parameters, the network parameters of right extension are obtained by the left figure, and the network parameters of left extension are obtained by the right figure.


## 2.2.2. Waveform Point Extension Method Based on LSTM Network

When using sufficient signal points, the LSTM network can accurately predict nonexistent sequence points [23], but the real signal is a finite length sequence. Moreover, learning long sequence samples requires a long training time. Guo et al. [8] stated that rotating fault information will periodically appear within the sampling period; as long as the sampling exceeds the rotation period, fault information will reappear; however, once it appears again, the fault information will be covered with differential noise signals. Hence, the length of extended signal needs to exceed the rotation period of several times. Using LSTM network predicts waveform points, and the extended method needs to answer two questions: (1) what is the effective training method to solve the left and right end effects? (2) How to promise the adaptability of the extended network after each point of expansion?For the first problem, LSTM network can predict the sequence points in accordance with all the existing waveform features. If all training is conducted in time order, then only the right end effect be solved. Therefore, LSTM network parameters must be obtained in two different orders. Suppose the fixed length sequence isX=x1,x2,…,xt, and it is shown in Figure 4. The network parameters of right end-side are obtained directly through sequence X. To obtain the network parameters of the left end-side, the reverse time sequence X′=xt,xt−1,…,x2,x1 be used as training samples.For the second problem, as described in Section2.2, solving end effects requires extend points to find a new extremum. Since the LSTM network is a memory sequence model, the extended points must change the memory states and it affects the accuracy of subsequently extended points. Only by updating the network status and amending parameters, the validity of all extended points is guaranteed.Based on the above, this paper proposes a three-step self-adaptive waveform point extended method based on LSTM network, and the steps are described as follows:Step (1): using existing waveform points feature, LSTM network parameters of left and right ends are trained. The LSTM network’s input and target are continuous two-point information of the existing signal. For example, the right end network’s input isx1 and target is x2, while the left end network’s input is xt and target is xt−1.Step (2): extending the waveform points through the Step (1) parameters, the last point of the existing waveform is used as input element of the first extension point, and for subsequent extend point, use the previous extended point used as input to the LSTM network. For example, for the right end extended points, use existed last pointxt and get the first extended point xt+1 and the input xt+1 to get second extended point xt+2.Step (3): extending one waveform point, corresponding network parameters (left and right end-side parameters) are updated. As shown in Figure4, when one point extends completely, and getting xt+1 point, xt+1 needs to be used to update the right parameters. These parameters are used as the newest parameters to predict extension point xt+2, until finding out new extremum at the right end-side of signal.Step (3) enhances the self-adaptive ability of the proposed method, which is different from the traditional neural network extended method. The basic extended unit of proposed method is one point; thus, it avoids the determination of each extended segment length. To illustrate the proposed method, a simulated signal would be presented in the next subsection.


## 2.2.3. Simulation and Comparisons

A simulated signal will be used to compare the proposed method with two popular methods. The simulated signal is a random signal; it includes multiple impulse response functions and two low frequency signals. Moreover, each of impulse functions has a high-frequency modulation component and can be described as follows [8, 27]:(19)yk=∑rexp−γ×k−r×Fs/fmFs×sin2πf0×k−r×Fs/fm−τrFs+0.7×sin2πf1×kFs+sin2πf2×kFs,where the frequencies of the two sin signals, f1 and f2, are 50 Hz and 100 Hz, respectively; the damping coefficient γ = 900; the sampling frequency Fs = 1200 Hz; the characteristic frequency fm = 100 Hz; τr is a random number constrained by the uniform distribution; and the resonant frequency f0 = 2600 Hz. In the simulation, 1200 points are selected from the simulated signal, and the time range is [0, 0.1] s, and the waveform is shown in Figure 5.Figure 5
A simulated signal waveform.Firstly, the proposed method will be compared with the traditional back-propagation neural network (BPNN). To quantitatively evaluate the performances of the two methods, the Root Mean Squared Error (RMSE) [19] measure is adopted. The RMSE is the square root of the average of the square of all of the errors, and the corresponding equation is written as(20)RMSE=1n∑i=1nyi∗−yi2,where yi∗ are extension points and yi are real points.The LSTM network uses one input/output layer, which means only one element is input or target, so each of the existing waveform points will be trained. In the simulation, 1200 points are used as training samples and extended 120 points are used as testing samples. For the right end-side extended networks, the number of hidden layers is 200, the maximum epoch is 500, and the initial learning rate is 0.001. For the left end-side extended networks, the hidden layer number is 200, the maximum epoch is 500, and the initial learning rate is 0.0018.Similarly, 1200 points of simulated signal are used as training samples for the BPNN, which is a three-layer network that includes an input layer, a hidden layer, and an output layer. A total of 200 neurons are set in the hidden layer, and also extended 120 points are used as testing samples. The BPNN input and target vectors are formed by a section of the simulated signal. For example, for the right end extended network, during the training process, the input vector isP=yk+1,yk+2,…,yk+120, and the corresponding target is yk+121. During the testing process, the first extension point, i.e. yk+121, will be used as the new element of the input vector for the new round of predict extension.As shown in Table1, although BPNN training time is little shorter than LSTM network training time, the values of RMSE training and testing are very larger than LSTM network. As a result, the BPNN extended points exhibit large error deviations as shown in Figures 6(b) and 7(b). The reason for upper phenomenon is that the BPNN output is only correlated to the current input, while the LSTM network output is related to the current and previous inputs, and this condition also increases the complexity of the LSTM network and causes a longer training time. As shown in Figures 6(a) and 7(a), although the LSTM extended waveform points are inconsistent with the real signal points, the waveform overall trend is the same. In the beginning of the LSTM extended waveform points, the errors (seen in Figures 6(b) and 7(b)) are relatively small, and then it increases with extended time. The waveform point has a weak follow-up in the high-frequency segment but has a reliable follow up in the smooth sin wave segment. This indicates that the LSTM network extended point is favorable for signals which appear periodically, and it can efficaciously extend waveform points for periodic rolling bearing fault signal.Table 1
Performance comparison of the LSMT and BPNN for the simulated signal.
Right extension networks
Left extension networks
Training time (s)
Training RMSE
Testing RMSE
Training time (s)
Training RMSE
Testing RMSE
SLTM
59
0.0013
0.1393
63
0.0518
0.1928
BPNN
52
0.1750
0.2320
51
0.2267
0.2757Figure 6
Comparison of LSTM and BPNN left waveform point extension for simulation signal: (a) waveform extension; (b) waveform extension error.
(a)
(b)Figure 7
Comparison of LSTM and BPNN right waveform point extension for simulation signal: (a) waveform extension; (b) waveform extension error.
(a)
(b)The proposed method will be further compared with the characteristic of segment extended method, which has a drawback due to determination of the length of each segment. To express the influence of each segment length, RSC [8] method is used to find extended segments. Firstly, the simulated signal is divided into 8, 10, and 15 segments. When C=8, the RSC values between the first-segment and other segments and between the last-segment and other segments need to be calculated. Comparison of calculated results showed that the maximum RSC values are γ1,5=0.8669 and γ8,1=0.9916, which indicates that the 1-st segment and the 5-th segment have the most similar characteristic spectrum and the 1-st segment and the 8-th segment have the most similar characteristic spectrum. Therefore, the 5-th and 1-st segments are set as the left and right end extended segments, respectively, and waveform shown in Figure 8(b). Using same way to process when C = 10 (γ1,3=γ10,2=1) and C = 15 (γ1,4=0.8639 and γ15,14=0.8592), as illustrated in Figures 8(c) and 8(d). It can be seen that setting difference of unit segment may lead to different waveforms, and the extended result is consistent with the real signal only C = 10, while the other length of segment cause error.Figure 8
Comparison of LSTM and different length segment RSC extended results for simulated signal (the red line represents the extended waveform points, and the blue line is the simulated signal): (a) the real simulated waveform; (b) 8 segments; (c) 10 segments; (d) 15 segments; (e) based on proposed method.
(a)
(b)
(c)
(d)
(e)If utilizing LMD process the upper extended signals, i.e., Figures8(b), 8(d), and 8(e), the final result PF1 and PF2 waveform shown in Figures 9(a)–9(c), respectively. As shown in Figures 9(a) and 9(b), it can be seen the large end swings near each end of the two PFs. As shown in Figure 9(c), if using the proposed method will get well-extended signal, because large end swings are not found at the starting or ending of each PFs. As shown in Figures 8 and 9, only C = 10 and the proposed method are similar with the real signal waveform, and the large end swings do not appear in decomposed waveforms.Figure 9
Comparisons of decomposition waveform obtained by LMD based on three extended methods: (a) 8 segments; (b) 15 segments; (c) proposed waveform points extended method.
(a)
(b)
(c)The comparison results confirm the superior performance of the proposed method over the traditional neural network and characteristic segment extended methods, because the former lacks the processing for the previous input, and the latter hardly determines the segment length, and different lengths cause difference of extended errors except for suitable segment length.


## 2.2.4. Improved LMD Algorithm

An improved LMD algorithm, which connects proposed method with LMD algorithm, is presented in the subsection. Additionally, the improved LMD algorithm considers the feature of LMD, and the extended point includes a new extremum; thus, it is more effective to solve end effects of LMD, and its process is shown in Figure10.Figure 10
Processing of improve LMD algorithm.After identifying the interior extremumsnijk, the existing waveform points must continuously extend to find out a new extremum near each end-side of analyzed signal. As shown in Figure 10, signal sijz must extend one point sijz+1 by the right parameters, then judge whether sijz is a new extremum. If not true, then a new point needs to extend and judge again until a new extremum is identified, and the same way is used to process the left extension. After both end-sides are extended, the original extremums nijk will be added one extremum near each end of the analyzed signal, but the analyzed signal has a fixed range. Hence, the added extremums are only used to complete interpolation, and the subsequent operations will be performed within the same as the original signal time range.The improved LMD algorithm can judge new extremum rather than fixed extended points, i.e., as shown in Figures6(a) and 7(a), the left end extends only 3 points, but the right end needs to extend 25 points. The improved LMD algorithm will be applied for rolling bearing fault feature extraction, and its further explanation that the proposed method is used to eliminate end effects is given in the next section.


## 3. Application for Rolling Bearing Fault Feature Extraction

In this section, the improved LMD algorithm would be applied for the rolling bearing fault feature extraction, and fault signal comes from Case Western Reserve University [28]. In the bearing fault database, a single-point fault is added to the ball and inner and outer rings by electrodischarge machining technology, and three accelerometers are placed near the drive, fan, and base end (with magnetic bases). The acceleration signals are collected at the sampling rate of 12 kHz. The drive end of the acceleration signal is applied to the experimental application, and its waveform is shown in Figure 11, and the bearing type is SKF 6205-2RS-JEM with fault point diameter of 7 mils and rotating speed of 1997 RMP/60 = 29.95 Hz. Based on the rolling bearing information and running condition, the theoretical fault feature frequency is calculated as 70.6 Hz, 107.4 Hz, and 162.2 Hz for the ball and inner and outer rings, respectively.Figure 11
Fault acceleration signal for this experimental application.


### 3.1. Fault Signal Extension

As mentioned in Section 3.4, using the proposed method may be different initial training set-parameters because of different signal features. For limited-length signal sequences, a segment of the existing signal points can be used as the testing sample to select the initial training set-parameters. After selecting the suitable set-parameters, the training result matrix will be used as the initial matrix and then retrain testing samples to obtain more effective extended waveform points parameters. For example, 1200 points are selected as training samples, and among these points, 50 points are used as testing samples to select the initial training set-parameters as shown in Figure11. The optimal initial set-parameter after several testings is shown in Table 2, and it also shows that the higher complexity of the experimental signal than former simulated signal.Table 2
Initial set-parameters of the training LSMT network for the ball and inner and outer rings.
Right extension networks
Left extension networks
Hidden layer
Learning rate
Hidden layer
Initial learning rate
Ball
301
0.00011
303
0.00025
Outer
362
0.00016
362
0.00010
Inner
293
0.00015
297
0.00010For these signals, the left and right end parameters of the LSTM network is trained by upper initial set-parameters; the performance of training and testing pressing can be obtained as shown in Table3. As shown in Table 3, the training time is approximately 100 s, the RMSE value of the training samples is approximately 0.1, and the RMSE value of the testing samples is between 0.1 and 0.21. Relatively, the extension effect of the outer ring signal is the best among the three (RMSE = 0.1398); this finding indicates that the periodicity of the outer ring fault signal is stronger than the two places fault signal.Table 3
Performance of LSMT network in the acceleration signal extension for the ball and inner and outer rings.
Right extension networks
Left extension networks
Training time (s)
Training RMSE
Testing RMSE
Training time (s)
Training RMSE
Testing RMSE
Ball
114
0.1032
0.2503
112
0.0965
0.2074
Outer
121
0.1134
0.1989
126
0.1013
0.1398
Inner
96
0.1193
0.2072
102
0.1018
0.1779The extended waveform points and real signal are shown in Figure12, there are also little differences in each left and right extended waveform points, but the trend between the extended and real waveform is the same, which promises less error in the LMD process. After obtaining the initial set-parameters and the training result matrix, the 50 waveform points are retrained to obtain the new LSTM network parameters. The improved LMD algorithm is applied to decomposed the acceleration signal and then extract the feature frequency in the next subsection.Figure 12
Comparison of the signal extension (dotted red line) and real (solid blue line) waveform points: (a) ball fault signal left extended points; (b) ball fault signal right extended points; (c) outer ring fault signal left extended points; (d) outer ring fault signal right extended points; (e) inner ring fault signal left extended points; (f) inner ring fault signal right extended points.
(a)
(b)
(c)
(d)
(e)
(f)


### 3.2. Feature Extraction

After processing the rolling bearing fault signal by the improved LMD algorithm, the decomposed waveforms comprise multiple and oneut as shown in Figure 13. Decomposition of the ball and outer ring fault has six PFs components and one ut, while the inner ring fault only has five PFs. Regardless of the different decomposed results, the complex waveform is gradually simplified from the PF1 to end PFs. The first two PFs have more useful values than the other PFs, both of which have larger correlation coefficients about fault feature from the analyzed signal [29]. Therefore, the two PFs (PF1 and PF2) are selected for further fault feature extraction.Figure 13
Decomposition results obtained using the improved LMD algorithm: (a) ball fault decomposition result; (b) outer ring fault decomposition result; (c) inner ring fault decomposition result.
(a)
(b)
(c)By extracting the upper envelopes ofFP1 and FP2 signals, and then applying Fourier transform [30], the envelope spectrum waveform within 1–1000 Hz is obtained as shown in Figure 14; the ball, outer ring, and inner ring faults envelope spectrum waveform are shown in Figures 14(a)–14(c), respectively. It can be seen that the PF1 spectrum has a high pulse at 70 Hz, 110 Hz, and 160 Hz, all of which are close to the respective theoretical values of ball fault characteristic (70.6 Hz) and the outer and inner ring fault frequencies (107.4 Hz and 162.2 Hz). The minor error between the theoretical and extracted feature values can be attributed to the insufficient number of samples (only 1200 sample points).Figure 14
Envelope spectra waveform ofPF1 and PF2 within 1–2000 Hz for the rolling bearing signal: (a) ball fault envelope spectra waveform; (b) outer ring fault envelope spectra waveform; (c) inner ring fault envelope spectra waveform.
(a)
(b)
(c)As shown in Figure 14(b), thePF1 spectrum waveform has multiple octave pulses, and it indicates that the fault feature frequency of the outer ring is stronger than other two places fault feature. Comparison of the three waveforms shows that the peak envelope spectra frequency of ball fault signal appears at 70 Hz, but larger disturbances occur around. However, the feature frequency amplitudes of outer and inner ring fault signals are relatively large compared with the other fault signal frequency amplitudes.According to the above analysis, the fault characteristic frequency can be obtained from the envelope spectrum waveform ofPF1 signal, while the rotation frequency can be obtained from the envelope spectrum waveform of PF2 signal. As a consequence, the envelope spectrum waveform of each fault signal successfully extracts feature frequency and rotation frequency, and these values considerably approximate the theoretical values.As shown in Figure14, it can be seen that the feature frequency of outer ring fault signal is better than other two places (inner ring and ball) fault signal. Therefore, the outer ring fault signal is taken as analysis object to further dominate the advantage of improved LMD algorithm. Four methods (EMD [31, 32], LMD and EMD without end effects (EEMD, using proposed method eliminate end effects), and improved LMD algorithm) are compared. The comparison performance extracted characteristic frequency, amplitude of corresponding characteristic frequency, time consumption and number of decomposition components, respectively, and the final results are shown in Table 4.Comparison results show that time consumption based on the EMD method is higher than that based on LMD approach; additionally, the number of decomposed components by EMD or EEMD is more than LMD or improved LMD algorithm, and it casuses the amplitude of feature frequency by the improved LMD algorithm is bigger than the EEMD method [9]. But, on the whole, both EEMD and improved LMD algorithm can effectively extract feature frequency, and it further demonstrates that the advantage of the proposed method can eliminate end effects.Table 4
Performance comparison of EMD, EEMD, LMD, and improved LMD algorithm for outer ring fault signal.
Frequency (Hz)
Amplitude (mm2/s)
Time (s)
Number
EMD
100
284.0
0.6229
8
EEMD
110
306.2
1.0231
8
LMD
120
284.1
0.3025
5
Improved LMD algorithm
110
315.2
0.5632
5


## 3.1. Fault Signal Extension

As mentioned in Section 3.4, using the proposed method may be different initial training set-parameters because of different signal features. For limited-length signal sequences, a segment of the existing signal points can be used as the testing sample to select the initial training set-parameters. After selecting the suitable set-parameters, the training result matrix will be used as the initial matrix and then retrain testing samples to obtain more effective extended waveform points parameters. For example, 1200 points are selected as training samples, and among these points, 50 points are used as testing samples to select the initial training set-parameters as shown in Figure11. The optimal initial set-parameter after several testings is shown in Table 2, and it also shows that the higher complexity of the experimental signal than former simulated signal.Table 2
Initial set-parameters of the training LSMT network for the ball and inner and outer rings.
Right extension networks
Left extension networks
Hidden layer
Learning rate
Hidden layer
Initial learning rate
Ball
301
0.00011
303
0.00025
Outer
362
0.00016
362
0.00010
Inner
293
0.00015
297
0.00010For these signals, the left and right end parameters of the LSTM network is trained by upper initial set-parameters; the performance of training and testing pressing can be obtained as shown in Table3. As shown in Table 3, the training time is approximately 100 s, the RMSE value of the training samples is approximately 0.1, and the RMSE value of the testing samples is between 0.1 and 0.21. Relatively, the extension effect of the outer ring signal is the best among the three (RMSE = 0.1398); this finding indicates that the periodicity of the outer ring fault signal is stronger than the two places fault signal.Table 3
Performance of LSMT network in the acceleration signal extension for the ball and inner and outer rings.
Right extension networks
Left extension networks
Training time (s)
Training RMSE
Testing RMSE
Training time (s)
Training RMSE
Testing RMSE
Ball
114
0.1032
0.2503
112
0.0965
0.2074
Outer
121
0.1134
0.1989
126
0.1013
0.1398
Inner
96
0.1193
0.2072
102
0.1018
0.1779The extended waveform points and real signal are shown in Figure12, there are also little differences in each left and right extended waveform points, but the trend between the extended and real waveform is the same, which promises less error in the LMD process. After obtaining the initial set-parameters and the training result matrix, the 50 waveform points are retrained to obtain the new LSTM network parameters. The improved LMD algorithm is applied to decomposed the acceleration signal and then extract the feature frequency in the next subsection.Figure 12
Comparison of the signal extension (dotted red line) and real (solid blue line) waveform points: (a) ball fault signal left extended points; (b) ball fault signal right extended points; (c) outer ring fault signal left extended points; (d) outer ring fault signal right extended points; (e) inner ring fault signal left extended points; (f) inner ring fault signal right extended points.
(a)
(b)
(c)
(d)
(e)
(f)


## 3.2. Feature Extraction

After processing the rolling bearing fault signal by the improved LMD algorithm, the decomposed waveforms comprise multiple and oneut as shown in Figure 13. Decomposition of the ball and outer ring fault has six PFs components and one ut, while the inner ring fault only has five PFs. Regardless of the different decomposed results, the complex waveform is gradually simplified from the PF1 to end PFs. The first two PFs have more useful values than the other PFs, both of which have larger correlation coefficients about fault feature from the analyzed signal [29]. Therefore, the two PFs (PF1 and PF2) are selected for further fault feature extraction.Figure 13
Decomposition results obtained using the improved LMD algorithm: (a) ball fault decomposition result; (b) outer ring fault decomposition result; (c) inner ring fault decomposition result.
(a)
(b)
(c)By extracting the upper envelopes ofFP1 and FP2 signals, and then applying Fourier transform [30], the envelope spectrum waveform within 1–1000 Hz is obtained as shown in Figure 14; the ball, outer ring, and inner ring faults envelope spectrum waveform are shown in Figures 14(a)–14(c), respectively. It can be seen that the PF1 spectrum has a high pulse at 70 Hz, 110 Hz, and 160 Hz, all of which are close to the respective theoretical values of ball fault characteristic (70.6 Hz) and the outer and inner ring fault frequencies (107.4 Hz and 162.2 Hz). The minor error between the theoretical and extracted feature values can be attributed to the insufficient number of samples (only 1200 sample points).Figure 14
Envelope spectra waveform ofPF1 and PF2 within 1–2000 Hz for the rolling bearing signal: (a) ball fault envelope spectra waveform; (b) outer ring fault envelope spectra waveform; (c) inner ring fault envelope spectra waveform.
(a)
(b)
(c)As shown in Figure 14(b), thePF1 spectrum waveform has multiple octave pulses, and it indicates that the fault feature frequency of the outer ring is stronger than other two places fault feature. Comparison of the three waveforms shows that the peak envelope spectra frequency of ball fault signal appears at 70 Hz, but larger disturbances occur around. However, the feature frequency amplitudes of outer and inner ring fault signals are relatively large compared with the other fault signal frequency amplitudes.According to the above analysis, the fault characteristic frequency can be obtained from the envelope spectrum waveform ofPF1 signal, while the rotation frequency can be obtained from the envelope spectrum waveform of PF2 signal. As a consequence, the envelope spectrum waveform of each fault signal successfully extracts feature frequency and rotation frequency, and these values considerably approximate the theoretical values.As shown in Figure14, it can be seen that the feature frequency of outer ring fault signal is better than other two places (inner ring and ball) fault signal. Therefore, the outer ring fault signal is taken as analysis object to further dominate the advantage of improved LMD algorithm. Four methods (EMD [31, 32], LMD and EMD without end effects (EEMD, using proposed method eliminate end effects), and improved LMD algorithm) are compared. The comparison performance extracted characteristic frequency, amplitude of corresponding characteristic frequency, time consumption and number of decomposition components, respectively, and the final results are shown in Table 4.Comparison results show that time consumption based on the EMD method is higher than that based on LMD approach; additionally, the number of decomposed components by EMD or EEMD is more than LMD or improved LMD algorithm, and it casuses the amplitude of feature frequency by the improved LMD algorithm is bigger than the EEMD method [9]. But, on the whole, both EEMD and improved LMD algorithm can effectively extract feature frequency, and it further demonstrates that the advantage of the proposed method can eliminate end effects.Table 4
Performance comparison of EMD, EEMD, LMD, and improved LMD algorithm for outer ring fault signal.
Frequency (Hz)
Amplitude (mm2/s)
Time (s)
Number
EMD
100
284.0
0.6229
8
EEMD
110
306.2
1.0231
8
LMD
120
284.1
0.3025
5
Improved LMD algorithm
110
315.2
0.5632
5


## 4. Conclusions

To eliminate the problem of end effects in LMD algorithm, this study proposes a novel self-adaptive waveform point extended method. This method mainly consists of three steps: (1) training the LSTM network parameters; (2) extending the waveform points; and (3) updating corresponding network parameters. The proposed method not only intelligently learns the characteristics of existing waveform points, but also adaptively updates parameters through extended waveform points, so that the validity of extension is guaranteed.Comparison of a simulated signal extended result demonstrates that the proposed method is better than the traditional BPNN using similar training samples and easier to implement than the characteristic of segment waveform extended method.The improved LMD algorithm can extract the fault feature frequency of the rolling bearing’s ball and inner and outer rings from eachPF1 envelope spectrum waveform, and the rotation feature frequency from each PF2 envelope spectrum waveform. Additionally, the extracted values considerably approximate the theoretical ones. Thus, the proposed method can eliminate the influence of end effects for the LMD algorithm, and it also provides a new method for a finite sequence extension.



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*Source: 7293454-2020-01-09.xml*