Abstract

Wheelset bearing is a critical and easily damaged component of a high-speed train. Wheelset bearing fault diagnosis is of great significance to ensure safe operation of high-speed trains and realize intelligent operation and maintenance. The convolutional sparse coding technique based on the dictionary learning algorithm (CSCT-DLA) provides an effective algorithm framework for extracting the impulses caused by bearing defect. However, dictionary learning is easily affected by foundation vibration and harmonic interference and cannot learn the key structure related to fault impulses. At the same time, the detection performance of fault impulse heavily depends on the selection of parameters in this approach. Union of convolutional dictionary learning algorithm (UC-DLA) is an efficient algorithm in CSCT-DLA. In this paper, UC-DLA is introduced and improved for wheelset bearing fault detection. Finally, a novel bearing fault detection method, adaptive UC-DLA combined with bandwidth optimization (AUC-DLA-BO), is proposed. The mathematical formulation of AUC-DLA-BO is a sort of constrained optimization problem, which can overcome foundation vibration and harmonic interference and adaptively determine parameters related to UC-DLA. The proposed method can detect the fault resonance band adaptively, eliminate the noise with the same frequency band as the fault resonance band, and highlight the bearing fault impulses. Simulated signals and bench tests are used to verify the effectiveness of the proposed method. The results show that AUC-DLA-BO can effectively detect bearing faults and realize the refined analysis of fault behavior.

1. Introduction

The high-speed train is a typical representative of modern advanced transportation tools and has been paid attention to by many countries (such as Japan, Germany, Spain, China, etc.). High-speed train realizes high-speed movement by driving wheelset rotation by the traction motor. The power system of a high-speed train is a typical rotating mechanical system. Wheelset bearing is a critical and easily damaged mechanical component in the power system of the high-speed train. It is used to support the rotating system of high-speed trains to resist static and dynamic external forces and to transfer power. Unexpected wheelset bearing faults can lead to reduced efficiency of a power system and equipment shutdown and damage and in some cases may lead to safety accidents [1–3]. Therefore, a powerful wheelset bearing fault diagnosis technique is very important to avoid catastrophic failures in high-speed trains.

The strong nonlinear and nonstationary vibration signal containing impulses will be generated when a local defect appears on wheelset bearing [4, 5]. The generated impulses usually exhibit modulation phenomenon [6, 7] and are often submerged in strong background noise and irrelevant interferences [8–10]. Thus, it is a challenge to extract impulses from the vibration signal for bearing fault diagnosis.

In recent years, the compressed sensing (CS) technique has attracted widespread attention and attention made remarkable achievements in information theory, image processing, pattern recognition, and biomedical engineering [11–13]. CS is also used for bearing fault feature extraction recently with its features of flexibility, sparsity, and superresolution [10, 12, 14, 15]. The essence of processing bearing vibration signals with CS is to obtain sparse representation of the vibration signal in the redundant dictionary. The redundancy dictionary design is the key issue of sparse representation [14, 16], which mainly includes analytic dictionary and learning dictionary [10, 17–20]. Analytic dictionary needs to be constructed with existing predefined dictionary in advance. In most cases, the impulses related to bearing fault have different vibration waveform features. Even for the same bearing, the vibration waveform features excited by different fault modes are different. There is always inevitable dissimilarity between atoms of analytic dictionary and real vibration signal. So, sparse representation based on analytic dictionary might fail to capture fault features in the target signal. Learning dictionary (including regular dictionary learning and shift-invariant dictionary learning (SIDL)) are learned directly from target signals by machine learning techniques and could match the local structures of signals [19]. Many advanced and effective dictionary learning methods such as K-SVD [14], shift-invariant sparse coding-based feature-sign search (SISC-FSS) [17], shift-invariant K-SVD (SI-K-SVD) [21], and convolutional sparse coding based on ADMM (CSC-ADMM) [22] have been proposed and successfully applied to the learning of bearing fault feature structures. A recent contribution is that SISC-FSS was introduced into mechanical fault diagnosis for adaptive feature extraction of bearing fault in [23]. SI-K-SVD was used to identify the double-impact characteristics of rolling bearing vibration signal in [19]. CSC-ADMM was introduced to extract impulse responses and time location coefficients in [24, 25]. Although sparse representations based on learning dictionary have been applied successfully in the field of fault detection, there are still two major shortcomings which are difficult to solve in practical engineering application:(1)In the process of dictionary learning and sparse coding, the number of dictionaries, length of dictionary atom, and target sparsity seriously affect the performance of fault diagnosis, and it is hard to interpret the physical meaning of the learned waveform in the dictionary.(2)Dictionary learning is very susceptible to strong interference signals including foundation vibration and harmonic interference [26]. If the vibration signals contain strong interference signals, the impulse responses related to the fault might not be learned from the vibration signals.

A union of convolutional dictionary learning algorithm (UC-DLA) was proposed by Cristian Rusu [27], which can efficiently implement the convolutional sparse representations of vibration signals. UC-DLA has the clear physical meaning of representing the impulses generated by the circular interactions of faults and their matching surfaces. The learned circular structured dictionaries can describe the shift-invariance of the fault-related impulses; the sparse representations matrix can describe the time and intensity of fault-related impulses occurrence. As such, this paper introduces UC-DLA to the field of wheelset bearing fault diagnosis. Like other dictionary learning methods, UC-DLA does not work well for vibration signals with strong interference signals and inappropriate selections of the related parameters also seriously affect the performance of wheelset bearing fault diagnosis. Therefore, a novel fault detection method, adaptive UC-DLA combined with bandwidth optimization (AUC-DLA-BO), is proposed for wheelset bearing fault diagnosis in this paper. AUC-DLA-BO can eliminate interference signals and adaptively determine parameters related to UC-DLA. The new method not only realizes fault detection of wheelset bearing but also realizes fault behavior analysis.

2. Theoretical Background

2.1. The Convolutional Sparse and Circular Structure Characteristics of Impulses

When a local defect appears on wheelset bearing, bearing rollers will strike the defect reciprocally, and the impulses , shown in Figure 1(a), are repeatedly generated.

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Figure 1 

(a) Dynamic generating process of the impulses induced by faults; (b) shock responses; and (c) location coefficients.

Taking a defect of the outer race as an example to illustrate the dynamic generating process of fault impulses . When the roller passes over the defect on the outer race for the first time, the first impulse is generated. Due to the continuous rotation of the wheelset, other impulses, , , , , and so on, will be generated in turn. Considering the influence of acquisition noises and foundation vibration of the mechanical system, the vibration signals (plotted in Figure 1(a)) collected by accelerometer will be the mixtures of impulses , acquisition noises , and foundation vibration . The shapes of different impulses in Figure 1 are similar. If the types and sizes of defects, the rotating speeds of wheelset bearings, and the type of wheelset bearings and loads are different, the shapes of impulses will be different. The time when locations of the impulses appear depends on the rotating speed and the geometric parameters of wheelset bearing. Therefore, impulses can be regarded as the translation of a general shock response with different intensities along the time axis. The convolutional operation can well describe the translation-invariant characteristics of impulses . Impulses , plotted in Figure 1(a), are expressed aswhere is the convolution operator, is the shock responses, and denotes the locations where the shock responses appear.

Impulses , plotted in Figure 1(a), are the result of convolution of shock responses and location coefficients . Shock responses and location coefficients are shown in Figures 1(b) and 1(c), respectively. Thus, the convolution operator well describes the dynamic generating process of wheelset bearing fault impulse. The natural frequency and damping characteristics of the system can be derived from shock responses . The amplitudes of location coefficients are the intensities of shock responses, which are related to rotation speeds and loads of wheelset bearing. The interval between nonzero coefficients is related to rotation speeds and geometric parameters of wheelset bearing. In practical engineering, only impulses can be collected. Shock responses and location coefficients can only be obtained by solving the inverse problems of equation (1).

Convolution operator is not conducive to the expression of matrix operation. Convolution calculation is converted to matrix multiplication by employing the Toeplitz matrix.where the vector is shock responses, the vector is location coefficients, and the vector is the result of their convolution, where and . and denote the data lengths of shock responses and convolution result, respectively. denotes the Toeplitz operator.

Circular matrices are completely defined by their first column : every column is a circular shift of the first one. The example circular matrix can be expressed aswhere is the operator that construct vectors into circular matrices. The main property of circular matrices is their eigenvalue factorization:where denote the unitary Fourier matrix and the diagonal , .

Convolutional matrices can be reduced to circulant matrices by observing, and equation (2) can be rewritten aswhere are convolutional matrices, is zero padding operator, and is location coefficients (also called sparse representations) over convolutional matrices.

In equation (5), the fault impulses of wheelset bearing are described with the convolution matrix multiplied by the locations coefficients. By using dictionary learning and sparse coding technique, the inverse problem of equation (1) can be solved, and the fault information of the wheelset bearing can be well revealed. Therefore, in this paper, UC-DLA is introduced into the field of wheelset bearing fault diagnosis.

2.2. Basic Theory and Discussion of UC-DLA

UC-DLA provides heuristics that approximate solutions to the following problem: given a signal , a target sparsity , and the number of dictionaries , a dictionary and a sparse representation are computed so that the following equation holds [27]:where , , , , and . is an operator that transforms a matrix into a vector, and is the length of the target signal. Since the target signal has columns, the sparse coefficients . If the shock response length is , then and . Equation (5) shows that the solution of equation (6) is the solution of the following convolution sparse problem:

An iterative alternating optimization algorithm could be used to solve this optimization problem: keep dictionary fixed, update sparse representation , then keep sparse representation fixed, update dictionary , and perform this update process in a loop until convergence.

2.2.1. Updating the Convolutional Dictionary

When both target signals and the associated sparse representation are fixed, the updating process of a convolutional dictionary is an optimization problem:

Using equation (4), we can getwhere , , and . According to the invariance of the Frobenius norm under unitary transformations, equation (9) could be rewritten aswhere is Fourier transform of , and are Fourier transform of and , respectively. denotes the Kronecker product, and is the first columns of the Fourier matrix. Due to, the matrix approximation problem in equation (8) could be converted to linear least squares problem in equation (10). The solution of equation (10) is given as

The final, convolutional dictionary is .

2.2.2. Updating the Sparse Representation

When convolutional dictionaries are known, updating sparse representation in equation (6) is a typical sparsity-constrained optimization problem, which has been widely discussed in compressed sensing theory, and many methods can accomplish this optimization task. This sparse representation problem is expressed as

The orthogonal matching pursuit (OMP) algorithm [28] is used to solve this optimization problem, but any other sparse approximation method could be chosen. Repeatedly update dictionaries and sparse representations until , the UC-DLA converges, where is the error of the th loop and is the convergence condition. In summary, the algorithm flow of UC-DLA is described in Figure 2.

Figure 2 

The schematic diagram of UC-DLA algorithm.

2.3. Discussion of UC-DLA

UC-DLA provides a good framework for learning convolution dictionary related to shock responses, obtaining the locations where coefficients of shock responses appear and detecting fault impulses of wheelset bearing. In bearing fault signals, impulses are always the key structure indicating defective faults. The convolution sparse coding based on UC-DLA is used to detect the impulse structure in the signal, which provides a good idea for wheelset bearing fault diagnosis.

Taking the simulation signal in Figure 1 as an example, the first signal contains impulses and Gaussian noise , and the second signal is processed by UC-DLA. Results of the UC-DLA processing signal and signal are plotted in Figures 3 and 4, respectively. UC-DLA can effectively recover the impulse signal when the impulse signal only contains Gauss noise, as shown in Figures 3(a) and 3(b). The shock responses and location coefficients obtained from the signal are shown in Figures 3(c) and 3(d); the shape of learned shock responses plotted in Figure 3(c) is very similar to the real shock responses and almost coincides with the first three oscillations of the responses. The time of active nonzero location coefficients inferred from simulation signals completely corresponds to the real situation (see Figure 3(d)). However, if the impulses contain periodic fundamental vibration signal with large energy, a large error will occur when the signal is processed by UC-DLA. The processing results are shown in Figure 4. There are great differences in the shape of shock responses learned from signal in Figure 4(c) and real shock response, especially in the low oscillation region. Compared with Figures 3(c) and 4(c), the low oscillation part is most susceptible to noise interference because of its small amplitude. However, due to the independent and identical distribution of each sampling point of Gauss noise, the dictionary learning method can better fit the original low-frequency oscillation from the signal with Gauss noise. However, the sampling points of the base vibration do not satisfy the distribution characteristics, so the learned shock response includes the fluctuation components of the foundation vibration, as shown in Figures 3(a), 3(b), 4(a), and 4(b). Because of the large errors in the shock responses learned from the signal, the location coefficients learned from the signal and the real location coefficients are dislocated, as shown in Figure 4(d). In practice, although the whole time domain of shock responses is always disturbed by noise and foundation vibration, the signal-to-noise ratio (SNR) in the high-oscillation region should be much larger than that in the low-oscillation region. In order to describe the shock responses completely, the high-oscillation region of shock responses should contain at least two oscillation periods.

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Figure 3 

The result of UC-DLA processing signal . (a) Impulses obtained by convolution sparse reconstruction; (b) comparison of reconstructed impulses and real impulses; (c) shock responses learned from signal; and (d) location coefficients obtained from signals.

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Figure 4 

The result of UC-DLA processing signal . (a) Impulses obtained by convolution sparse reconstruction; (b) comparison of reconstructed impulses and real impulses; (c) shock responses learned from signal; and (d) location coefficients obtained from signals.

According to the theory of UC-DLA, there are three parameters, length of shock response , a target sparsity , and the number of dictionaries , that need to be predefined and the quality of the algorithm is closely related to these parameters. The too large parameter will lead to more noise, and the too small one cannot fully describe the shock response characteristics. Parameter reflects the number of impulses included in the target signal, which is related to the type of bearing fault and rotation speed of wheelset. If the parameter is too large, the false impulses will be absorbed into the reconstructed impulses. If the parameter is too small, some of the impulses with smaller oscillation amplitude are thrown away and the reconstructed impulses cannot represent all the impulses in the target signal. The parameter indicates the number of resonance frequencies induced by defects of the wheelset bearing. In practical engineering, the measured vibration signals of bearing may contain different shock response types, shock response numbers, and shock response lengths. Therefore, improper parameter selection will have adverse effects on the extraction of shock responses caused by wheelset bearing. In conclusion, the convolution sparse coding method used in bearing fault diagnosis needs to overcome the following two kinds of problems:(1)In dictionary learning, how to overcome the influence of foundation vibration and harmonic interference on a dictionary. In practical engineering, these interferences are inevitable, such as low-frequency vibration caused by track irregularity and power-line interference.(2)In dictionary updating and sparse representation updating, how to automatically select the length of shock responses, target sparsity, and number of a dictionary to obtain optimal impulses detection performance.

These two problems are very significant for the application of sparse representation based on the learning dictionary in bearing fault diagnosis.

3. The Proposed AUC-DLA-BO Method

In this section, automatic parameter determination UC-DLA method for fault diagnosis of wheelset bearing is proposed. Firstly, we will discuss the solutions to the two problems mentioned in Section 2.3. To distribute the operation process of the proposed method, we establish a wheelset bearing fault model based on the wheelset bearing system plotted in Figure 5. This model considers the periodic impulses caused by wheelset defects, harmonic interference generated by wheelset polygon, power-line interference, and Gaussian noise generated by measurement. A set of simulation signals generated by the wheelset bearing fault model are designed and expressed as follows:where denotes the impulses caused by wheelset defects with characteristic frequency , means the impulses caused by the bearing outer race fault with characteristic frequency , denotes power-line interference with characteristic frequency 50 Hz, is harmonic interference, and is Gaussian noise generated at a −5 dB SNR of the sum of , , , , and are, respectively, expressed as follows:where and are the amplitudes of the impulses, and are the structural damping coefficient, and denote the excited resonance frequencies, and denote the number of impulses, is a unit step function, Hz denotes the rotational frequency, , and . The parameters related to and are listed in Table 1. The sampling frequency of the simulation signal is kHz.

Figure 5 

Wheelset bearing system.

3.1. Elimination Method of Foundation Vibration and Harmonic Interference

If there is a large amount of fundamental vibration and harmonic interference in the analyzed signal, the impulse response learned by UC-LDA method contains part of the fluctuation component of the fundamental vibration (shown in Figures 4(a)–4(c)), which is obviously not conducive to the application of sparse representation method in bearing fault impulses extraction. Ideally, the fundamental vibration and harmonic interference should be eliminated as much as possible before the dictionary learning. As we all know, signals with different vibration modes have different frequencies. It is a natural idea to distinguish different vibration signals by the difference of frequency spectrum of different vibration signals. Figure 6(a) is the simulation signal of wheelset bearing fault represented by equation (13), and Figure 6(b) is the Fourier spectrum of the simulation signal. Different frequency interval (FI) corresponds to different vibration components in Figure 6(b). FI1, FI2, FI3, and FI4 represent the frequency interval of signals , , , and , respectively. Therefore, it is an essential process before the dictionary learning that filtering the signal can eliminate the influence of fundamental vibration and harmonic interference on the algorithm.

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Figure 6 

(a) The simulation signal . (b) The Fourier spectrum of simulation signal.

In practical engineering applications, the frequency interval of the real fault signal is not constant, and the bandwidth and center frequency of the filter should be adjusted arbitrarily to suit different application scenarios. In this paper, empirical wavelet transform is used as a variable filter, but any other wavelet transforms with similar properties can be selected such as Morlet wavelet [29], harmonic wavelet [30], and tunable Q-factor wavelet transform [31]. For any given frequency interval , the empirical scaling function or the empirical wavelet function can be used to filter the signal. is used when , is used in other cases. Frequency response function of scale function and wavelets are, respectively, designed as follows [32]:where , is sampling frequency, ; and should be satisfied. It is very easy to construct an appropriate filter with EWT when given the cut-off frequency. The filtered result can be calculated by the following formula:where is the complex conjugate operator, and and represent Fourier transform and inverse Fourier transform, respectively.

3.2. Automatic Selection of the Length of Shock Responses

The length of shock responses is closely related to the structural damper coefficient. To obtain parameters , or of the impulse response, the previous two maximum values and must be known in advance, as seen in Figures 7(a) and 7(b). The time difference between the two maximum values is (take the shock response in Figure 7(b) as an example). Assuming that the shock response length is , , and the resonance frequency should satisfy the following relationship:where is the minimum frequency of shock responses. Equation (17) can be converted into the following form:

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Figure 7 

(a) The shock response in signal . (b) The shock response in signal .

According to equation (18), the higher vibration frequency implies the shorter shock response length . The value range determined by equation (18) is too wide to be effectively used in practical projects. That is because we suggest that the target signal should be filtered before the dictionary learning, and the frequency range of the filter is . should satisfy the following conditions:

Finally, we can determine that the length of the shock response should meet the following equation:

In general, the central frequency of the resonance band is equal to the resonance frequency , so is a reasonable estimate of shock response length. Note that this value is the minimum value of the shock response length. For a resonance frequency of kHz ( kHz), exceeds 20. Therefore, can be initialized to 20. When the central frequency of the filter is less than kHz, the length should be increased to represent the low-frequency resonance frequency. Finally, is the reliable value of the length of shock responses.

3.3. Automatic Selection of the Target Sparsity

The target sparsity directly reflects the number of shock responses contained in the sparse reconstruction signal. An inappropriate target sparsity may result in the loss of valuable impulses or detection of spurious impulses, either of which is not conducive to wheelset bearing fault detection. The UC-DLA method with different target sparsity is used to analyze simulation signal in Figure 8(a), and the results are shown in Figures 8(c)–8(h). Obviously, when the target sparsity is less than the real number of impulses (50) in , some of the impulses with fault information are lost in the sparse reconstruction signal (such as Figures 8(c)–8(e)); otherwise, some false impulses without any effective information are introduced (such as Figures 8(g) and 8(h)). When the target sparsity is exactly equal to the number of real impulses, the reconstructed signal is closest to the real impulses signal in Figure 8(b). Fortunately, if the characteristic frequency can be known in advance, the target sparsity of the fault impulses signal can be calculated by . So, the target sparsity can be set to directly.

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Figure 8 

Impulses extracted using UC-DLA for different values of . (a) The simulation signal ; (b) real impulses signal in ; and (c–h) impulses extracted using UC-DLA for different values of target sparsity.

3.4. Automatic Selection of the Number of Dictionaries

According to the physical meaning of shock response, a shock response is mainly determined by resonance frequency and damping coefficient. The number of dictionaries depends on the number of resonance frequencies excited by the fault. In order to analyze conveniently, the UC-DLA method is used to process signal , the length of shock responses , target sparsity , and the number of dictionaries . The simulation signal (blue line) and sparse reconstruction signal (red line) are plotted in Figure 9(a). UC-DLA method can accurately extract the impulses of two different structures and has very strong denoising ability. The shock responses learned from the signal in Figure 9(a) are plotted in Figures 9(c) and 9(d). Obviously, the shape of the shock responses is similar to real shock responses. If the number of dictionaries is appropriate, UC-DLA has a good effect on impulses extraction. In fact, we do not know in advance the number of resonance frequencies excited by the fault, which makes it very difficult to choose the number of dictionaries. The spectrums of shock responses are plotted in Figure 9(b). Because the structure of the shock response is related to the resonance frequency, the dominant frequencies of the shock responses can be used to identify different shock responses. It can be seen from Figure 9(b) that the dominant frequencies of shock responses coincide with the center frequency of resonance band of simulation signal. Because different shock response has different resonance band (such as Filter1 and Filter2), a suitable filter can always be found, so the filtered signal only contains one type of shock response. So, the number of dictionaries can always be set to .

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Figure 9 

(a) The simulation signal (blue line) and sparse reconstruction signal (red line); (b) the spectrums of simulation signal, shock response 1 and shock response 2; (c) shock response 1 learned from signal; and (d) shock response 2 learned from signal.

3.5. AUC-DLA-BO Method of Fault Impulses Detection

Operator is defined as UC-DLA processing of the signal , and operator is defined as empirical wavelet filtering of the signal . According to the discussion in the above four sections, a new method of fault impulses detection for wheelset bearing named AUC-DLA-BO can be expressed as the following constrained optimization problem:where is an index used for quantification of the fault signal with impulses characteristics. In this paper, the Hilbert envelope spectrum fault feature ratio (HESFFR) is used to determine whether a measured signal is informative, which is given as [33]where , , and are envelope spectrum amplitudes corresponding to frequency points , , and , and is the total amplitudes of the envelope spectrum. Any of the following metaheuristic optimization algorithms, such as genetic algorithm [34], particle swarm optimization [35, 36], quantum inspired differential evolution [37, 38], etc., is capable of solving the constrained optimization problem in equation (21). The new method is a coupling method of filter optimization and convolutional sparse coding and its procedure is illustrated in Figure 10. To sum up, the main procedure of AUC-DLA-OB is as follows:(1)Use EWT filter with the band-pass to filter the original signal.(2)The length of shock response is determined according to the pass-band of the filter, the number of dictionaries is set to 1, then the filtered signal is processed by UC-DLA, and the convolution sparse reconstruction signal is obtained.(3)The filter parameters are continuously updated to maximize the HESFFR value of the reconstructed signal and obtain the corresponding parameters .(4)Repeat steps 1 and 2 according to parameters to detect fault impulses.

Figure 10 

The procedure for the proposed AUC-DLA-OB.

4. A Case Study: Fault Simulation Signal

To illustrate the effectiveness of the proposed method, AUC-DLA-OB is used to process the simulation signal in Figure 6(a). First, we try to detect the fault impulses with the fault characteristic frequency of Hz, in which the fault characteristic frequency is a prior parameter. In practical engineering applications, it can be calculated according to structural parameters of bearing and rotation speed. The results obtained using AUC-DLA-OB are shown in Figure 11. It can be seen from Figure 11(b) that the proposed method accurately finds out the resonance frequency band of shock response 2 and recovers the fault impulses from the resonance band, as shown in Figure 11(d). The shock responses and location coefficients obtained by AUC-DLA-OB are plotted in Figures 11(e) and 11(f). It is obvious that the learned shock response and the real shock response have similar vibration shapes, and the time interval between the shock coefficients is exactly the real fault impulses interval. The signal in Figure 11(c) is a filtered directly without sparse coding. Compared with Figures 11(c) and 11(d), the characteristics of the impulses in AUC-DLA-OB processed signals are more obvious and contain fewer noise signals. The Fourier spectrums and envelope spectrums of the signals in Figures 11(c) and 11(d) are shown in Figure 12. The noise energy in Figure 12(b) is significantly smaller than that in Figure 12(a), and the harmonic components in the envelope spectrum of Figure 12(d) are significantly richer and more obvious than those in Figure 12(c). Figure 12(e) further confirms this conclusion and the envelope spectrum of signal in Figure 12(d) is closest to the real envelope spectrum.

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Figure 11 

AUC-DLA-OB for detection of fault impulses with the fault characteristic frequency of 100 Hz. (a) Convergence process of solving constrained optimization problems; (b) filtering frequency band related to fault impulses; (c) filtered signal by filters in Figure 11(b); (d) sparse reconstruction signal; (e) learned shock response 2; and (f) location coefficients obtained from simulation signals.

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Figure 12 

(a) Fourier spectrum of signal in Figure 11(c); (b) Fourier spectrum of signal in Figure 11(d); (c) and (d) envelope spectrums of signals in Figures 11(c) and 11(d), respectively; and (e) comparison between envelope spectrum of real impulses and obtained envelope spectrum (envelope spectrums in Figures 12(c) and 12(d)).

Second, we try to detect the fault impulses with the fault characteristic frequency of . Obviously, this shock response corresponding to this fault feature is shock response 1. The results obtained using AUC-DLA-OB are shown in Figure 13. The proposed method can accurately detect the shock response 1, and the learned shock response 1 can be regarded as the translation of the real shock response 1, as shown in Figure 13(e). Comparing Figures 13(c) and 13(d), and the corresponding Fourier spectrum and envelope spectrum (plotted in Figures 14(a)–14(d)), obviously, the proposed method not only accurately finds the resonance band where the fault feature, but also directly extracts the fault impulses from it, which greatly improves the SNR of the fault impulses. Figure 14(e) shows that the envelope spectrum obtained by AUC-DLA-OB is the closest to the envelope spectrum of real impulses.

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Figure 13 

AUC-DLA-OB for detection of fault impulses with the fault characteristic frequency of 26 Hz. (a) Convergence process of solving constrained optimization problems; (b) filtering frequency band related to fault impulses; (c) filtered signal by filters in Figure 13(b); (d) sparse reconstruction signal; (e) learned shock response 1; and (f) location coefficients obtained from simulation signals.

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Figure 14 

(a) Fourier spectrum of signal in Figure 13(c); (b) Fourier spectrum of signal in Figure 13(d); (c) and (d) envelope spectrums of signals in Figure 13(c) and in Figure 13(d), respectively; and (e) comparison between envelope spectrum of real impulses and obtained envelope spectrum (envelope spectrums in Figures 14(c) and 14(d)).

It can be seen from the simulation signal composition in Figure 6(a) that the signal contains harmonic interference and power-line interference signals. The proposed method avoids the influence of these interference signals and successfully detects two fault impulses signals. In conclusion, the proposed method can learn the fault shock response and extract the impulses signal related to the learned shock response under the condition of strong interference.

In order to illustrate the effectiveness of the proposed method, the UC-DLA method and the well-known spectral kurtosis method are used to analyze the simulation signals in Figure 6(a). The results obtained using the UC-DLA method are plotted in Figure 15. The parameters used in UC-DLA method are and the original signals are divided into five segments, which form the input matrix of dictionary learning. Because the impulse signal corresponding to impulse response 1 has large energy, so it is less affected by other interference signals, UC-DLA method can effectively extract the relevant impulse signal, as shown in Figures 15(a) and 15(c). The impulses signal corresponding to shock response 2 can hardly be detected directly, as shown in Figures 15(b) and 15(c), but the proposed method can detect the impulse signal accurately, so the AUC-DLA-OB has strong adaptability. UC-DLA still has the disadvantage that parameters cannot be estimated adaptively, which seriously limits the application of this method, and overcoming this disadvantage is exactly one of the remarkable highlights of the proposed method in this paper.

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Figure 15 

Processing results obtained using UC-DLA. (a) Reconstruction impulses corresponding to the learned shock response 1; (b) reconstruction impulses corresponding to the learned shock response 2; (c) learned shock response 1; and (d) learned shock response 2.