Application of an Improved Laplacian-of-Gaussian Filter for Bearing Fault Signal Enhancement of Motors

: The presence of strong noise and vibration interference in fault vibration signals poses challenges for extracting fault features from motor bearings. Therefore, appropriate pre-filtering procedures can effectively suppress the impact of the noise interference and further enhance fault-related signals. In this work, an improved Laplacian-of-Gaussian (ILoG) filter is proposed to enhance the fault-related signal. The proposed ILoG approach employs an enhanced Kurtosis-based indicator known as Correlated Kurtosis (CK). The CK capitalizes on the cyclostationarity of fault-related impulses and mitigates the random nature of impulse noise. Subsequently, an objective function, based on CK statistics, is suggested to iteratively update LoG coefficients by maximizing the CK value of the output signal. Therefore, the ILoG filter can better highlight the fault cyclic impulses associated with bearing faults. Furthermore, the ILoG filter is capable of attenuating impulsive noise, a feature that is absent in the original LoG filter. The simulation and experimental results demonstrate that the proposed ILoG method provides a remarkable capability to effectively enhance the fault-induced components, thereby improving the diagnostic accuracy. Consequently, the ILoG filter holds great potential for application in motor bearing fault diagnosis.


Introduction
As one of the most pivotal driving forces in modern industry, motors play a prominent role and hold significant importance across various domains, including household appliances, transportation, manufacturing, aerospace, and other sectors.Motors, being mechanical equipment, are subject to inevitable failures during operation due to factors such as friction, vibration, and insulation aging [1,2].The bearing is a crucial component of the motor, and during prolonged operation, it is susceptible to wear, cracking, and other failures in its inner ring, rolling elements, and outer ring.The presence of these faults may adversely impact motor performance, resulting in significant consequences such as malfunctioning, economic losses, or casualties [3].Therefore, it is imperative to deeply investigate the fault diagnosis methods of motor bearings.
Vibration signals can capture crucial dynamic information that reflects the operational states of machines [4,5].Hence, vibration analysis is a commonly used method in the field of motor bearing fault diagnosis.Nevertheless, the vibration signals extracted from a faulty bearing in the motor are generally corrupted by strong noise and various forms of interference such as random impulsive components and periodic vibration components owing to the harsh operational circumstances.As a consequence, the fault diagnosis procedures of motor bearings mainly involve two stages, namely pre-filtering and feature detection.The pre-filtering stage is very important since it effectively removes undesired components from the raw signals, thus providing a solid foundation for accurate feature detection.Filtering techniques have been extensively used in the motor diagnosis of bearing faults.Zhou et al. [6] used a Wiener filter to cancel out the ambient noise and other components unrelated to motor bearing faults.Zhao et al. [7] proposed a feature detection approach combining Ensemble Empirical Mode Decomposition (EEMD) and multi-scale fuzzy entropy to diagnose motor bearing faults.In this approach, EEMD was used to separate noise and other interference from the raw signal.Wu et al. [8] introduced a novel approach combining the Empirical Wavelet Transform (EWT), fuzzy entropy, and a Support Vector Machine (SVM) named EWTFSFD for diagnosing motor bearing faults.In this method, the EWT was employed for fault signal enhancement.Tang et al. [9] designed a fault signal enhancement and compression (SEC) method which was implemented on an IoT platform for motor bearing fault diagnosis.Kumar et al. [10] presented a method using the Minimum Entropy Deconvolution (MED) technique and zero-frequency filter to identify the incipient motor bearing faults.Li et al. [11] proposed a bi-filter multi-scale diversity entropy method (BMDE) for effectively identifying fault information in a rotor-bearing system.Hao et al. [12] proposed an enhanced filtering and feature enhancement approach for the bearing diagnosis of spindle motors.
In this work, we deeply explore the Laplacian-of-Gaussian (LoG) filtering technique.The LoG filter is widely recognized as a powerful tool in the realm of image processing, particularly for its ability to identify and highlight the boundaries within an image.The image processing field defines 'edge' as a sudden change within an image, whereas in vibration signal analysis, a sudden change from noise to an impact characterizes a faultinduced impulse [13].The LoG filter is very sensitive to edge (sudden changes) in the image, so its one-dimensional (1D) version can be used to highlight the impulses (sudden changes) related to bearing faults in 1D vibration signals.Revealing this inherent characteristic of the filtering method, some scholars have employed it for bearing fault diagnosis.Feng et al. [13] developed an optimized LoG filtering tool to enhance bearing fault signals.Liu et al. [14] combined the Teager energy operator (TEO) with the Modified Laplacian-of-Gaussian (MLoG) filtering tool to improve the effectiveness of fault detection under varying rotational speeds.The MLoG technique is developed based on the LoG filter, ensuring that the sum of its taps equals zero.This characteristic enables effective edge extraction.In comparison to the LoG filter, the MLOG filter not only captures transient impulse components but also significantly mitigates background noise interference.However, in cases where a fault signal is contaminated with random impulsive noise, the LoG and MLoG filtering tools may fail to enhance the fault-related impulses since the random impulses may exhibit a higher level of energy compared to the cyclic impulses caused by the fault.This can result in an inaccurate determination of the fault state [15,16].
To address this issue, this paper proposes an improved LoG (ILoG) filter.The proposed ILoG approach utilizes an enhanced Kurtosis-based indicator known as Correlated Kurtosis (CK).CK takes advantage of the cyclostationarity of fault-related impulses and avoids the randomness of impulse noise [17,18].An objective function based on the CK statistics is proposed to iteratively update LoG coefficients by maximizing the CK of the output signal.Consequently, in comparison with the LoG filter, the ILoG filtering technique is more suitable for enhancing the cyclic fault signals.Furthermore, it can attenuate the random impulse noise to a certain extent.
The rest of this paper is organized as follows: In Section 2, the methodology of the proposed filtering tool is introduced in detail.In Section 3, we analyze a set of synthetic vibration signals with varying noise levels to evaluate the performance of our filtering tool.Section 4 focuses on applying the ILoG filter to real vibration data collected from different test rigs.Next, in Sections 3 and 4, the ILoG filtering technique is demonstrated using synthetic and real vibration data.The last section concludes this work.

Theoretical Basis LoG Filter Coefficient
The LoG filter coefficients can be obtained by computing the second-order derivative of a Gaussian kernel.The expression for the Gaussian kernel is as follows [19]: where σ is the standard deviation and l is the Gaussian index.
The first derivative of the Gaussian kernel in Equation ( 1) can be expressed as: The coefficients of the LoG filter are derived by obtaining the derivative of Equation ( 2): Normalizing the LoG filter coefficients can be achieved by dividing Equation ( 3) by the sum of exponential yields: It is noteworthy that an optimal edge detection filter for identifying abrupt changes is a high-pass Finite Impulse Response (FIR) filter with taps summing up to zero.In practical applications, the LoG filter can be considered a FIR filtering technique for highlighting impulsive signals.As a consequence, a modified expression for calculating the LoG filter coefficients in [20] is used in this study, which is shown as follows: In Equation ( 5), the variable N stands for the number of taps.
To enhance the LoG filtering technique, Correlated Kurtosis (CK), which can effectively measure the periodicity of bearing faults, is used to integrate with the original LoG filter.The definition of the CK is shown as follows [18]: where s is the filtered signal, M is the umber of the samples in the collected signal, and T is the fault period.
The following section delves into the ILoG filter.As previously stated, the LoG filter can be regarded as a tool for filtering with FIR.In terms of a discrete FIR model, one can describe the system as: where L denotes the filter length, f denotes the FIR filter coefficients, and x denotes the input signals.The FIR filter coefficients can be directly calculated by Equation (5) CK(T) = 0 to solve the maximization problem and obtain the numerator: Similarly, we can obtain the denominator as follows: Based on Equations ( 8) and ( 9), Equation ( 10) is rewritten in matrix form as where in which ∥•∥ 2 denotes the Euclidian norm.The expression in Equation ( 7) can be represented with a matrix multiplication, The iterative process for determining f is outlined as follows: Step 1: Obtain X T , X T 0 and X 0 X T 0 −1 using the raw signal x.
Step 3: Obtain the filtered signal s using Equation (12).
Step 4: Use the filtered signal s to calculate a 0 , a 1 , and b.
Step 6: If the ∆CK 1 (T) is larger than ε, terminate the iteration process.Otherwise, repeat Step 2 to Step 5.
Step 7: Obtain the final filtered signal.

Simulation
In this section, the ILoG method will be thoroughly examined and compared with LoG methods and other methods in order to evaluate its effectiveness in dealing with simulated faulty bearing signals that are accompanied by high levels of noise, vibration interferences, and random impulsive noise.The objective is to reveal whether the ILoG method performs better under these challenging conditions.
The composition of the signal y(t), which includes a simulated defect in the bearing, various interferences, background noise, and random impulse noise, can be represented as where A n is the amplitude of the faulty signal, α represents the structural damping characteristic, T p is the fault period, τ = 0.01T p ∼ 0.02T p denotes the amount of deviation from T p , ω k is the excited resonance frequency, e(•) represents the decay of a unit impulse, and u(•) denotes a unity step function.Table 1 shows the values of these parameters.In the second part, L 1 = L 2 = 1 and θ 1 = 33Hz and θ 2 = 165Hz are the amplitude and frequencies of the vibration interferences, respectively.η(t) (Signal-to-Noise Ratio, SNR = −10 dB) represents the background noise and γ(t) is the random noise, which is generated by using the "stablernd" MATLAB 2022b command.
The four sub-components of the signal mixture are shown in Figure 1.

Simulation
In this section, the ILoG method will be thoroughly examined and compared with LoG methods and other methods in order to evaluate its effectiveness in dealing with simulated faulty bearing signals that are accompanied by high levels of noise, vibration interferences, and random impulsive noise.The objective is to reveal whether the ILoG method performs better under these challenging conditions.
The composition of the signal y(t), which includes a simulated defect in the bearing, various interferences, background noise, and random impulse noise, can be represented as where An is the amplitude of the faulty signal, α represents the structural damping char- acteristic, Tp is the fault period,  1 shows the values of these parameters.
Table 1.Values of the parameters in Equation (14).

Parameter
An   Figure 2 gives the signal mixture and its envelope spectrum.From an analysis of Figure 2a, it becomes evident that the presence of interferences significantly overwhelms the signal mixture, resulting in the masking of useful fault information.Similarly, upon examining Figure 2b, we observe that even in the envelope spectrum, vibration interference frequencies at 33 Hz and 165 Hz continue to dominate over the fault characteristic frequency (FCF).The heavily corrupted signal was then enhanced by the proposed ILoG filter.The enhanced signal and its corresponding envelope spectrum are illustrated in Figure 3. From the time−domain signal, we can see that this proposed filtering technique effectively reduces the interferences and improves the overall quality of the signal.The enhanced signal exhibits clearer fault−related characteristics with reduced interference.Meanwhile, the FCF and associated harmonics dominate the envelope spectrum of the enhanced signal, as shown in Figure 3b.The heavily corrupted signal was then enhanced by the proposed ILoG filter.The enhanced signal and its corresponding envelope spectrum are illustrated in Figure 3. From the time-domain signal, we can see that this proposed filtering technique effectively reduces the interferences and improves the overall quality of the signal.The enhanced signal exhibits clearer fault-related characteristics with reduced interference.Meanwhile, the FCF and associated harmonics dominate the envelope spectrum of the enhanced signal, as shown in Figure 3b.ure 2a, it becomes evident that the presence of interferences significantly overwhelms the signal mixture, resulting in the masking of useful fault information.Similarly, upon examining Figure 2b, we observe that even in the envelope spectrum, vibration interference frequencies at 33 Hz and 165 Hz continue to dominate over the fault characteristic frequency (FCF).The heavily corrupted signal was then enhanced by the proposed ILoG filter.The enhanced signal and its corresponding envelope spectrum are illustrated in Figure 3. From the time−domain signal, we can see that this proposed filtering technique effectively reduces the interferences and improves the overall quality of the signal.The enhanced signal exhibits clearer fault−related characteristics with reduced interference.Meanwhile, the FCF and associated harmonics dominate the envelope spectrum of the enhanced signal, as shown in Figure 3b.The classical Minimum Entropy Deconvolution [21,22] (MED) serves as the second method for comparison, as the proposed ILoG filtering technique can also be considered a deconvolution method, as discussed in Section 2. Figure 5 exhibits the MED−filtered signal and its envelope spectrum.Similar to the results shown in Figure 4, it is observed The classical Minimum Entropy Deconvolution [21,22] (MED) serves as the second method for comparison, as the proposed ILoG filtering technique can also be considered a deconvolution method, as discussed in Section 2. Figure 5 exhibits the MED-filtered signal and its envelope spectrum.Similar to the results shown in Figure 4, it is observed that the MED filter also fails to completely eliminate the presence of random impulsive components.Consequently, this limitation hinders the accurate detection of FCF and its harmonics in the envelope spectrum.The classical Minimum Entropy Deconvolution [21,22] (MED) serves as the second method for comparison, as the proposed ILoG filtering technique can also be considered a deconvolution method, as discussed in Section 2. Figure 5 exhibits the MED−filtered signal and its envelope spectrum.Similar to the results shown in Figure 4, it is observed that the MED filter also fails to completely eliminate the presence of random impulsive components.Consequently, this limitation hinders the accurate detection of FCF and its harmonics in the envelope spectrum.Based on the previous comparative analysis, it is evident that the proposed ILoG filtering technique effectively reduces unwanted interference components in the raw signal.This successful attenuation of undesired interferences lays a solid foundation for conducting further envelope analysis.Moreover, when operating under challenging working conditions, the ILoG−based bearing fault diagnosis method outperforms similar methods like MLoG−based and MED−based approaches.

Experimental Results
In this section, the effectiveness and practicality of the ILoG technique in real-world scenarios are thoroughly investigated.Two case studies on the fault detection of motor bearings are conducted to demonstrate its capabilities.Additionally, comparative analysis is carried out to evaluate the performance of the ILoG technique against other advanced techniques commonly used in bearing fault detection.Based on the previous comparative analysis, it is evident that the proposed ILoG filtering technique effectively reduces unwanted interference components in the raw signal.This successful attenuation of undesired interferences lays a solid foundation for conducting further envelope analysis.Moreover, when operating under challenging working conditions, the ILoG-based bearing fault diagnosis method outperforms similar methods like MLoG-based and MED-based approaches.

Experimental Results
In this section, the effectiveness and practicality of the ILoG technique in real-world scenarios are thoroughly investigated.Two case studies on the fault detection of motor bearings are conducted to demonstrate its capabilities.Additionally, comparative analysis is carried out to evaluate the performance of the ILoG technique against other advanced techniques commonly used in bearing fault detection.At the heart of the test rig was a 1/2 HP motor, which provided the necessary power for conducting experiments.To control and adjust the speed of the motor, a 1/2 HP variable speed controller was employed.The two 6″ aluminum rotors were used to control forces and speeds.To support precise alignment during testing procedures, two shaft alignment jack bolts were incorporated into the setup.These bolts enabled fine adjustments to ensure the optimal positioning of all components involved in transmitting vibrations from one end to another.Lastly, a sturdy 15″ rotor deck served as a stable platform At the heart of the test rig was a 1/2 HP motor, which provided the necessary power for conducting experiments.To control and adjust the speed of the motor, a 1/2 HP variable speed controller was employed.The two 6 ′′ aluminum rotors were used to control forces and speeds.To support precise alignment during testing procedures, two shaft alignment jack bolts were incorporated into the setup.These bolts enabled fine adjustments to ensure the optimal positioning of all components involved in transmitting vibrations from one end to another.Lastly, a sturdy 15 ′′ rotor deck served as a stable platform for mounting all elements securely.ER-16K bearings are a prevalent type of rolling bearing, characterized by their simplistic structure, exceptional load-carrying capacity, and seamless operation.Consequently, they find extensive utilization across diverse machinery and industrial applications, such as high-speed motors, machine tools, aircraft, etc.Therefore, an ER-16K bearing with a fault in the inner ring was used in this test.Table 2 lists the specifications of the faulty bearing.It is important to mention that the inner fault signal measured did not include any random impulse noise due to the experimental constraints.Hence, to increase the difficulty of the fault detection, an additional signal with random impulse noise was added to the raw signal.In the signal mixture with random impulse noise shown in Figure 7a, it is evident that the fault-related cyclic impulses are heavily distorted by both ambient and impulsive noise.This makes it difficult to identify the inner race fault characteristic frequency at f i = 157.4Hz in the envelope spectrum of Figure 7b due to the negative influence of noise interference.The ILoG filtering technique was then employed to enhance the heavily corrupted signal.Figure 8a plots the enhanced signal.As can be seen, the ILoG filter can not only remove most of the impulse noise but also highlight the fault−induced periodic impulses.As anticipated, the fault signature fi and its harmonics can clearly be identified in the envelope spectrum as depicted in Figure 8b.The ILoG filtering technique was then employed to enhance the heavily corrupted signal.Figure 8a plots the enhanced signal.As can be seen, the ILoG filter can not only remove most of the impulse noise but also highlight the fault-induced periodic impulses.As anticipated, the fault signature f i and its harmonics can clearly be identified in the envelope spectrum as depicted in Figure 8b.
As before, to validate the superiority of the proposed filtering technique, a blind deconvolution based on based on cyclostationarity maximization (CYCBD) [23] was utilized for comparison with the ILoG method.Similar to the ILoG, the CYCBD method is also immune to random impulsive components and has the ability to enhance fault-induced impulses.The results obtained by the CYCBD method are shown in Figure 9. From Figure 9a, it is evident that the CYCBD method effectively eliminates a significant portion of impulse noise.However, despite this improvement, the fault-induced periodic impulses still remain concealed by the presence of strong background noise.Consequently, this poses a challenge in detecting the fault characteristic frequency f i from the envelope spectrum, as depicted in Figure 9b.The ILoG filtering technique was then employed to enhance the heavily corrupted signal.Figure 8a plots the enhanced signal.As can be seen, the ILoG filter can not only remove most of the impulse noise but also highlight the fault−induced periodic impulses.As anticipated, the fault signature fi and its harmonics can clearly be identified in the envelope spectrum as depicted in Figure 8b.As before, to validate the superiority of the proposed filtering technique, a blind deconvolution based on based on cyclostationarity maximization (CYCBD) [23] was utilized for comparison with the ILoG method.Similar to the ILoG, the CYCBD method is also immune to random impulsive components and has the ability to enhance fault-induced impulses.The results obtained by the CYCBD method are shown in Figure 9. From Figure 9a, it is evident that the CYCBD method effectively eliminates a significant portion of impulse noise.However, despite this improvement, the fault-induced periodic impulses still remain concealed by the presence of strong background noise.Consequently, this poses a challenge in detecting the fault characteristic frequency fi from the envelope spectrum, as depicted in Figure 9b.

Case 2: Outer Race Fault Signal Measured from the Motor of a Vibrating Screen
In this case study, a vibrating screen test bench was used to verify the effectiveness of the proposed technique, as shown in Figure 10.The SDM00 vibrating screen is a versatile and efficient piece of equipment that falls under the category of biaxial and dual−motor multifunctional vibrating screens.Its design consists of several key components, including a screen box, four eccentric blocks (exciters), two transmission shafts, and two motors.In the real test, an outer ring fault signal was recorded from the defective bearing of the motor of the vibrating screen.The rotating speed of the vibrating screen is a critical factor that significantly impacts the screening effectiveness.Typically, the frequency is

Case 2: Outer Race Fault Signal Measured from the Motor of a Vibrating Screen
In this case study, a vibrating screen test bench was used to verify the effectiveness of the proposed technique, as shown in Figure 10.The SDM00 vibrating screen is a versatile and efficient piece of equipment that falls under the category of biaxial and dual−motor multifunctional vibrating screens.Its design consists of several key components, including a screen box, four eccentric blocks (exciters), two transmission shafts, and two motors.In the real test, an outer ring fault signal was recorded from the defective bearing of the motor of the vibrating screen.The rotating speed of the vibrating screen is a critical factor that significantly impacts the screening effectiveness.Typically, the frequency is adjusted within the range of 16-24 Hz.An insufficient frequency may impede material In the real test, an outer ring fault signal was recorded from the defective bearing of the motor of the vibrating screen.The rotating speed of the vibrating screen is a critical factor that significantly impacts the screening effectiveness.Typically, the frequency is adjusted within the range of 16-24 Hz.An insufficient frequency may impede material passage through screen apertures, thereby compromising the screening efficiency.Conversely, an excessive frequency can lead to more severe wear and even the breakage of screen apertures.As a result, the rotating frequency was set to 16 Hz (960 rpm).A sampling frequency of 20 kHz was used in this test.Due to its low noise, excellent stability, minimal temperature increase, and longer service life, bearing type 1308 is highly suitable for use in vibrating screens.Accordingly, the bearing type used in the experiment was 1308.
In Table 3, the detailed specifications of this bearing are provided.By utilizing the rotating speed and bearing specification data, we were able to calculate this frequency to be 104.24Hz. Figure 11 shows the outer ring fault signal and its envelope spectrum.Compared with the inner ring fault signal used in Case 1, the waveform of the outer ring fault signal in Figure 11a is more complicated.In addition to the obvious random impulse noise, it includes a periodic component caused by the exciting force.It should also be noted that unlike Case 1, the random impulse noise in this test is not synthetic but rather a result of the time-varying impacts generated by the material stream continuously falling into the vibrating screen.From the envelope spectrum in Figure 11b, it can be found that the envelope analysis cannot directly extract the fault characteristic frequency from the complicated signal.Figure 11 shows the outer ring fault signal and its envelope spectrum.Compared with the inner ring fault signal used in Case 1, the waveform of the outer ring fault signal in Figure 11a is more complicated.In addition to the obvious random impulse noise, it includes a periodic component caused by the exciting force.It should also be noted that unlike Case 1, the random impulse noise in this test is not synthetic but rather a result of the time−varying impacts generated by the material stream continuously falling into the vibrating screen.From the envelope spectrum in Figure 11b, it can be found that the envelope analysis cannot directly extract the fault characteristic frequency from the complicated signal.Subsequently, the signal mixture was enhanced using the ILoG filter.Figure 12 displays the enhanced signal and its envelope spectrum.According to Figure 12a, most of the random impulse noise and the periodic component are effectively removed.The cyclic impulsive components related to the outer ring fault are obviously present in the filtered signal.Thus, the elimination of interfering components facilitates the envelope analysis in accurately identifying the fault characteristic frequency fo and its harmonics from the filtered signal, as illustrated in Figure 12b.Subsequently, the signal mixture was enhanced using the ILoG filter.Figure 12 displays the enhanced signal and its envelope spectrum.According to Figure 12a, most of the random impulse noise and the periodic component are effectively removed.The cyclic impulsive components related to the outer ring fault are obviously present in the filtered signal.Thus, the elimination of interfering components facilitates the envelope analysis in accurately identifying the fault characteristic frequency f o and its harmonics from the filtered signal, as illustrated in Figure 12b.
plays the enhanced signal and its envelope spectrum.According to Figure 12a, most of the random impulse noise and the periodic component are effectively removed.The cyclic impulsive components related to the outer ring fault are obviously present in the filtered signal.Thus, the elimination of interfering components facilitates the envelope analysis in accurately identifying the fault characteristic frequency fo and its harmonics from the filtered signal, as illustrated in Figure 12b.As before, a comparative analysis is conducted here.A fault detection method designed for the motor bearings of vibrating screens proposed in [24] was employed in this comparative analysis.The fault detection method first used a filtering technique named αstable filter to remove the random impulsive noise.The α-stable-filtered signal is exhibited in Figure 13, in which we can see that a substantial portion of impulse noise is removed.However, the periodic interfering component still exists in the filtered signal, since the α-stable filter cannot handle such interfering components.As before, a comparative analysis is conducted here.A fault detection method designed for the motor bearings of vibrating screens proposed in [24] was employed in this comparative analysis.The fault detection method first used a filtering technique named α−stable filter to remove the random impulsive noise.The α−stable−filtered signal is exhibited in Figure 13, in which we can see that a substantial portion of impulse noise is removed.However, the periodic interfering component still exists in the filtered signal, since the α−stable filter cannot handle such interfering components.Subsequently, an advanced fast−resampled iterative filtering decomposition (FRIFD) [25] technique was used to separate the periodic component from the filtered signal.The decomposed signals are shown in Figure 14.We can see that IMF5 is the periodic interfering component.Finally, an enhanced signal was constructed by combining IMF1, IMF2, and IMF3. Figure 15 gives the final enhanced signal and its envelope spectrum.In Figure 15a, it is evident Subsequently, an advanced fast-resampled iterative filtering decomposition (FRIFD) [25] technique was used to separate the periodic component from the filtered signal.The decomposed signals are shown in Figure 14.We can see that IMF5 is the periodic interfering component.As before, a comparative analysis is conducted here.A fault detection method designed for the motor bearings of vibrating screens proposed in [24] was employed in this comparative analysis.The fault detection method first used a filtering technique named α−stable filter to remove the random impulsive noise.The α−stable−filtered signal is exhibited in Figure 13, in which we can see that a substantial portion of impulse noise is removed.However, the periodic interfering component still exists in the filtered signal, since the α−stable filter cannot handle such interfering components.Subsequently, an advanced fast−resampled iterative filtering decomposition (FRIFD) [25] technique was used to separate the periodic component from the filtered signal.The decomposed signals are shown in Figure 14.We can see that IMF5 is the periodic interfering component.Finally, an enhanced signal was constructed by combining IMF1, IMF2, and IMF3. Figure 15 gives the final enhanced signal and its envelope spectrum.In Figure 15a, it is evident that the presence of random impulse noise and vibration interference has been significantly reduced, allowing for the clear observation of fault cyclic impulses.Furthermore, Finally, an enhanced signal was constructed by combining IMF1, IMF2, and IMF3. Figure 15 gives the final enhanced signal and its envelope spectrum.In Figure 15a, it is evident that the presence of random impulse noise and vibration interference has been significantly reduced, allowing for the clear observation of fault cyclic impulses.Furthermore, Figure 15b presents the corresponding envelope spectrum in which the outer ring fault characteristic frequency f o and its first two harmonics 2f o and 3f o are easily identifiable.
Machines 2024, 12, x FOR PEER REVIEW 13 of 14 offers a straightforward and efficient solution for removing undesired interfering components.With its simple implementation process, it eliminates the need for multiple steps or complex algorithms.This makes it highly accessible and user−friendly.

Conclusions
In this paper, an improved Laplacian−of−Gaussian filter (ILoG) based on Correlated Kurtosis (CK) is introduced.To demonstrate the advantages of our proposed filter, we first conducted an experiment using a complicated synthetic signal.The synthetic signal was carefully designed to mimic complex patterns and structures commonly found in real−world data.By applying the ILoG filter, we effectively extracted fault features from the synthetic signal.Furthermore, we tested our filter on two real signals obtained from two different test rigs.The proposed method also achieved remarkable results in terms of interference removal and feature extraction.In addition, the comparative analysis clearly demonstrates that our proposed ILoG filter outperforms existing methods in handling complicated signals.
The key findings are concisely summarized as follows: (1) In comparison to the original iterative LoG filter, the ILoG filtering method offers significant improvements in attenuating random impulsive components present in bearing signals.
(2) The ILoG filtering tool is equipped with an impressive filtering capacity that effectively eliminates various types of unwanted noise and interferences, thereby facilitating its implementation.
Nevertheless, the proposed ILoG filter still retains some drawbacks.For example, the performance of the ILoG filtering technique is contingent upon the selection of crucial pre−defined parameters.In this study, a trial−and−error approach was employed to determine the optimal parameter values, but this method is time−consuming.Therefore, a method for selecting optimal parameters will be investigated in future work.In addition, in the case of varying speeds, the methodology may not be applicable.The research will also emphasize the integration of this proposed method with other approaches to apply to working conditions of varying speeds.

Data Availability Statement:
The data presented in this study are available in this article.

Conflicts of Interest:
The authors declare no conflicts of interest.The fault detection method, however, is a complex multi-step procedure that may impose limitations on its practical applications.By contrast, the proposed ILoG technique offers a straightforward and efficient solution for removing undesired interfering components.With its simple implementation process, it eliminates the need for multiple steps or complex algorithms.This makes it highly accessible and user-friendly.

Conclusions
In this paper, an improved Laplacian-of-Gaussian filter (ILoG) based on Correlated Kurtosis (CK) is introduced.To demonstrate the advantages of our proposed filter, we first conducted an experiment using a complicated synthetic signal.The synthetic signal was carefully designed to mimic complex patterns and structures commonly found in real-world data.By applying the ILoG filter, we effectively extracted fault features from the synthetic signal.Furthermore, we tested our filter on two real signals obtained from two different test rigs.The proposed method also achieved remarkable results in terms of interference removal and feature extraction.In addition, the comparative analysis clearly demonstrates that our proposed ILoG filter outperforms existing methods in handling complicated signals.
The key findings are concisely summarized as follows: (1) In comparison to the original iterative LoG filter, the ILoG filtering method offers significant improvements in attenuating random impulsive components present in bearing signals.
(2) The ILoG filtering tool is equipped with an impressive filtering capacity that effectively eliminates various types of unwanted noise and interferences, thereby facilitating its implementation.
Nevertheless, the proposed ILoG filter still retains some drawbacks.For example, the performance of the ILoG filtering technique is contingent upon the selection of crucial predefined parameters.In this study, a trial-and-error approach was employed to determine the optimal parameter values, but this method is time-consuming.Therefore, a method for selecting optimal parameters will be investigated in future work.In addition, in the case of varying speeds, the methodology may not be applicable.The research will also emphasize the integration of this proposed method with other approaches to apply to working conditions of varying speeds.
of deviation from Tp, k ω is the excited resonance frequency, e(•) represents the decay of a unit impulse, and u(•) denotes a unity step function.Table and frequencies of the vibration interferences, respectively.( ) t η (Signal−to−Noise Ratio, SNR = −10 dB) represents the background noise and ( ) t γ is the random noise, which is generated by using the "stablernd" MATLAB 2022b command.The four sub-components of the signal mixture are shown in Figure 1.

Figure 2
Figure2gives the signal mixture and its envelope spectrum.From an analysis of Figure2a, it becomes evident that the presence of interferences significantly overwhelms the signal mixture, resulting in the masking of useful fault information.Similarly, upon examining Figure2b, we observe that even in the envelope spectrum, vibration interference frequencies at 33 Hz and 165 Hz continue to dominate over the fault characteristic frequency (FCF).

Figure 3 .
Figure 3. ILoG−enhanced signal.(a) Time−domain signal and (b) envelope spectrum of the enhanced signal.For comparison, the MLoG filter[14] was used to analyze the signal mixture.The results of the MLoG filter are plotted in Figure4.According to Figure4a, the vibration interferences have been effectively eliminated, but some obvious random noise still exists in the filtered signal.Moreover, the fault−related impulsive components are masked by the background noise.As expected, the corresponding envelope spectrum is dominated by excessive noise frequency, rendering the extraction of fault features unfeasible.

Figure 3 .
Figure 3. ILoG−enhanced signal.(a) Time−domain signal and (b) envelope spectrum of the enhanced signal.For comparison, the MLoG filter[14] was used to analyze the signal mixture.The results of the MLoG filter are plotted in Figure4.According to Figure4a, the vibration interferences have been effectively eliminated, but some obvious random noise still exists in the filtered signal.Moreover, the fault−related impulsive components are masked by the background noise.As expected, the corresponding envelope spectrum is dominated by excessive noise frequency, rendering the extraction of fault features unfeasible.

Figure 3 .
Figure 3. ILoG-enhanced signal.(a) Time-domain signal and (b) envelope spectrum of the enhanced signal.For comparison, the MLoG filter[14] was used to analyze the signal mixture.The results of the MLoG filter are plotted in Figure4.According to Figure4a, the vibration interferences have been effectively eliminated, but some obvious random noise still exists in the filtered signal.Moreover, the fault-related impulsive components are masked by the background noise.As expected, the corresponding envelope spectrum is dominated by excessive noise frequency, rendering the extraction of fault features unfeasible.Machines 2024, 12, x FOR PEER REVIEW 7 of 14

Figure 5 .
Figure 5. MED−enhanced signal.(a) Time−domain signal and (b) envelope spectrum of the iLoG−enhanced signal.

Figure 5 .
Figure 5. MED-enhanced signal.(a) Time-domain signal and (b) envelope spectrum of the ILoGenhanced signal.

4. 1 .
Case 1: Inner Race Fault Signal Measured from Spectra Quest Motor Fault Simulator In this case study, the Spectra Quest Motor Fault Simulator (MFS-LT) test rig was utilized to capture and analyze the vibration signals of inner race bearing faults.The test rig, as depicted in Figure 6, consists of various components.Machines 2024, 12, x FOR PEER REVIEW 8 of 144.1.Case 1: Inner Race Fault Signal Measured from Spectra Quest Motor Fault SimulatorIn this case study, the Spectra Quest Motor Fault Simulator (MFS−LT) test rig was utilized to capture and analyze the vibration signals of inner race bearing faults.The test rig, as depicted in Figure6, consists of various components.

Figure 7 .
Figure 7. Inner fault signal with additional impulse noise.(a) Time domain and (b) its envelope spectrum.

Figure 7 .
Figure 7. Inner fault signal with additional impulse noise.(a) Time domain and (b) its envelope spectrum.

Figure 7 .
Figure 7. Inner fault signal with additional impulse noise.(a) Time domain and (b) its envelope spectrum.

Figure 11 .
Figure 11.Outer ring fault signal.(a) Time domain and (b) its envelope spectrum.

Figure 11 .
Figure 11.Outer ring fault signal.(a) Time domain and (b) its envelope spectrum.

Author Contributions:
Conceptualization, methodology, software, validation, funding acquisition, and writing-original draft preparation, D.T. and Y.X.; resources, software, writing-review and editing, D.T. and X.L.All authors have read and agreed to the published version of the manuscript.Funding: This study was supported by the National Natural Science Foundation of Shaanxi Province [No.2023-JC-YB-075].
. To make the LoG filter more suitable for analyzing bearing fault signals, a CK-based objective function method is introduced to iteratively update the FIR filter coefficients.The goal of the ILoG is to maximize the CK by updating the FIR filter coefficients until the iterations terminate.

Table 1 .
Values of the parameters in Equation (14).

Table 2 .
Specifications of the test rig.

Table 3 .
Parameters related to the tests.

Table 3 .
Parameters related to the tests.