# Feature Extraction and Signal Enhancement Based on Unsaturation Piecewise Tri-Stable Stochastic Resonance Driven by Lévy Noise

Authors: Shuai Zhao; Peiming Shi
Journal: Security and Communication Networks
 (2023)

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## Abstract

As science advances and machines become larger and more sophisticated, it is vital to determine whether there is a mechanical failure without damaging the device. Stochastic resonance (SR), as a widely used method, can effectively extract the periodic signal in the noise and then realize the identification of mechanical faults, which is very important for safety and property protection. Most studies on SR are based on additive white Gaussian noise (AWGN) as the driving source; however, there are many kinds of noise in reality. In this paper, based on the previously proposed piecewise tri-stable SR (PTSR) model, the influence of parameters on the mean of signal-to-noise ratio gain (SNR-GM) of the system driven by Lévy noise is studied. It is also verified by simulation signals and actual signals that PTSR can also achieve feature extraction and signal enhancement with Lévy noise as the driving source, which proves that PTSR can be applied in a wider range of conditions.


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## Body

## 1. Introduction

Benzi et al. proposed and explained SR in 1981 [1]. As SR is a phenomenon formed by the interaction of periodic force, random force, and nonlinear system, it can be applied in many disciplines [2–8]. Since SR can transfer the energy of noise to periodic signal, it shows excellent performance in the field of feature extraction and signal enhancement [9–15]. The improvement of potential function, the change of SR order, the adjustment of SR parameters, and the extension of SR model have been studied by many scholars [16]. Yan et al. studied the behavior of particles in a fractional SR system, analyzed the first moment of the system, and gave the steady-state conditions [17]. In reference [18], the white noise-driven neural network SR is analyzed at three levels of dynamics, and the network is composed of two modules. Qiao et al. modified the traditional SR model, successfully improved the output saturation problem, and applied it to feature extraction [19]. Yan et al. added a time delay term to the system on the basis of fractional SR and used fractional Gaussian noise to drive it [20]. In reference [21], quantum SR first appeared in real experiments. Yu et al. added a time delay to the neural network SR and investigated the oscillating transitions of the system [22]. Li and Shi studied the method of extracting nonstationary signals using SR and applied it to bearing signals [23].As an important part of SR, noise is of great significance to study the drive of SR system based on different noises. Lévy noise, as a colored noise, is also used by some scholars to drive SR system. Zhu et al. also studied a network-based SR system, but Levy noise was used to drive the SR [24]. In reference [25], a new potential function is presented, using Lévy noise as the driving source, and the characteristics of the system are analyzed. Han and Shi used colored noise to drive multistable SR and studied the first-passage time of the system in detail [26].In order to overcome output saturation, we have previously proposed the PTSR method. In order to make the PTSR model more universal in terms of feature extraction and signal enhancement, a Lévy noise-driven PTSR method has been proposed. Section2 introduces and describes the PTSR model and how the parameters of the PTSR system relate to the shape of the potential function. In Section 3, the Lévy noise generation method is studied and the PTSR model is driven by it. Meanwhile, the PTSR system can still perform feature extraction and signal enhancement with the help of simulation signals. In Section 4, the relationship between parameters and SNR-GM of PTSR system driven by Lévy noise is studied. In Section 5, engineering signals are used to verify the ability of PTSR system to extract features and enhance signals even when Lévy noise is the driving source. Section 6 discusses the possible research direction in the future. Finally, Section 7 gives the conclusions.


## 2. The PTSR System and Parameters

In prior research, a PTSR was invented, and its potential function is shown as follows [27]:(1)Ux=−3mp2q2p2−q2−mp6−q6623q2−p2−12qx+mq43p2−q212−3mp2q3p2−q2−mqp6−q6623q2−p2−12q,x≤−q,m2p2q2x2−m4p2+q2x4+m6x6,q<x<q,3mp2q2p2−q2−mp6−q6623q2−p2−12qx+mq43p2−q212−3mp2q3p2−q2−mqp6−q6623q2−p2−12q,x≥q,where m>0, p>0, and q>0 are positive number and q>p. Figure 1 displays its potential function. It can be seen that the potential function of PTSR has three stable points and two unsteady points, ∆U1=3mp2q2p2−q2+mq6−p6/12 and ∆U2=3mp4q2−mp6/12.Figure 1
Potential functionm=1,p=0.5,q=1.In reference [28], it is found, the adjustment of system parameters has a great effect on the shape of the potential function, because the change of the shape of the potential function will greatly change the transition rate of the internal particles. The study of reference [28] is based on AWGN, and the evaluation standard is the SNR proposed in reference [29], but in this paper, the noise becomes Lévy noise, and the evaluation standard is different from that so it is still important to study the effect of parameters on the shape of the potential function. Now, we redraw the potential for the PTSR system and keep the other two parameters constant and only change the size of the other one at a time, as shown in Figure 2.Figure 2
Potential function. (a) Adjust onlymp=0.5,q=0.8. (b) Adjust only pm=4.0,q=0.8. (c) Adjust only qm=4.0,p=0.5.
(a)(b)(c)It can be seen from Figure2(a) that increasing the size of m will make the left and right potential wells rise and become deeper, and at the same time, the left and right potential walls of the two unsteady points become steeper, which to some extent will prevent particles from jumping between the three potential wells. It can be observed from Figure 2(b) that if the parameter p is increased, the left and right potential wells will also rise, but the depth will be shallower. It can be concluded that the movement of particles between potential wells becomes easier. Finally, it can be seen from Figure 2(c) that with the increase of q, the height and depth of the left and right potential well will decrease and become deeper, while the middle potential well will become steeper, which will reduce the motion rate of the particles.


## 3. PTSR System Driven by Lévy Noise

### 3.1. The Simulation of Lévy Noise

Lévy noise is also known asα stable noise. As a typical non-Gaussian noise, there is usually no explicit expression for its distribution function and probability density function.


#### 3.1.1. Distribution of Lévy Noise

In this paper, in order to simulate the generation of Lévy noise, we use the Chambers-Mallows-Stuck (CMS) [30, 31] algorithm to obtain the random variable of Lévy noise. The characteristic equation and expression of its steady-state distribution are shown in equations (2) and (3):(2)ft=∫−∞+∞eitxlα,βx;σ,μdx,(3)ft=exp−σt1+iβ2πsigntlogt+iμt,α=1,exp−σαtα1−iβsignttanπα2+iμt,α≠1,where α∈0,2, β∈−1,1,σ∈0,+∞, and μ∈RThe random variableX follows the following distribution:(4)X=2σππ2+βVtanV−βlnπ/2WcosVπ/2+βV+μ,α=1,Dα,β,σsinαV+Cα,βcosV1/α×cosV−αV+Cα,βW1−αα+μ,α≠1,where Cα,β and Dα,β,σ satisfy equations (5) and (6), respectively:(5)Cα,β=arctanβtanπα/2α,(6)Dα,β,σ=σcosarctanβtanπα2−1/α.In equation (4), V and W obey interval distribution and exponential distribution, respectively, and they satisfy the interval −π/2,π/2 and the mean 1.


#### 3.1.2. Solving the Langevin Equation

The Langevin equation of SR system driven by Lévy noise is as follows:(7)dxdt=−dUxdx+st+ηtD,where x is the output, st is the periodic signal, ηt is the Lévy noise, and D is the noise amplification factor.The Langevin equation of SR generally needs to be solved by using fourth-order Runge—Kutta. Driven by Lévy noise, the CMS algorithm and Runge—Kutta are combined, as follows:(8)xn+1=xn+16k1+2k2+2k3+k4+h1/αDηn,k1=hfxn+sn,k2=hfxn+k12+sn,k3=hfxn+k22+sn+1,k4=hfxn+k3+sn+1,where fx=−dUx/dt, xn, and sn are the nth sampled values of the output and input signals, respectively, and h is the step size. Since, the final particle will have an infinite trajectory due to oscillation, truncation should be carried out [32, 33], when xn>5, xn=signxn×5 should be implemented. Setting that, β=μ=0 and σ=1, on the premise that only α is changed, Lévy noise is drawn in Figure 3.Figure 3
Lévy noise: (a)α=0.4, (b)α=0.7, (c)α=0.9, (d)α=1.1, (e)α=1.4, and (f)α=1.9.
(a)(b)(c)(d)(e)(f)According to Figure3, when α is relatively small, the number of Lévy noise pulses is relatively small, and the absolute value of the peak value is generally relatively high. With increasing α, the number of Lévy noise pulses increases gradually, and the mean value of the absolute value of the peak decreases gradually. When α approaches 2, the distribution of levy noise is mainly concentrated around the X-axis.


### 3.2. Simulation of Feature Extraction and Signal Enhancement

After simulating Lévy noise, it is necessary to verify whether the PTSR system driven by Lévy noise can perform feature extraction and signal enhancement on input signals. A sinusoidal signal of amplitudeA=0.5 and characteristic frequency fd=0.02hz is mixed with Lévy noise. We set the sampling frequency fs=5hz, h=0.1, α=1.8,β=μ=0, and σ=1 and apply the signal through the PTSR system, The original signal and output signal are shown in Figure 4.Figure 4
Original signal and output signal: (a) input time domain signal, (b) input frequency domain signal, (c) output time domain signal, and (d) output frequency domain signalm=2.0,p=0.5,q=1.2.
(a)(b)(c)(d)From Figures4(a) and 4(b), it can be found that the periodic signal is chaotic after Lévy noise is mixed, and the characteristic frequency is also submerged in other chaotic frequencies, meanwhile, it is also marked that the amplitude is 0.7983. It can be found in Figure 4(c) that the signals become relatively clear and some periodicity is restored, indicating that the PTSR system driven by Lévy noise can filter some chaotic noises from the perspective of filtering. Finally, it is shown in Figure 4(d) that the amplitude of the characteristic frequency is the highest among all frequencies and the value is 0.853. It indicates that the PTSR system successfully carries out feature extraction under Lévy noise and produces signal enhancement.


## 3.1. The Simulation of Lévy Noise

Lévy noise is also known asα stable noise. As a typical non-Gaussian noise, there is usually no explicit expression for its distribution function and probability density function.


### 3.1.1. Distribution of Lévy Noise

In this paper, in order to simulate the generation of Lévy noise, we use the Chambers-Mallows-Stuck (CMS) [30, 31] algorithm to obtain the random variable of Lévy noise. The characteristic equation and expression of its steady-state distribution are shown in equations (2) and (3):(2)ft=∫−∞+∞eitxlα,βx;σ,μdx,(3)ft=exp−σt1+iβ2πsigntlogt+iμt,α=1,exp−σαtα1−iβsignttanπα2+iμt,α≠1,where α∈0,2, β∈−1,1,σ∈0,+∞, and μ∈RThe random variableX follows the following distribution:(4)X=2σππ2+βVtanV−βlnπ/2WcosVπ/2+βV+μ,α=1,Dα,β,σsinαV+Cα,βcosV1/α×cosV−αV+Cα,βW1−αα+μ,α≠1,where Cα,β and Dα,β,σ satisfy equations (5) and (6), respectively:(5)Cα,β=arctanβtanπα/2α,(6)Dα,β,σ=σcosarctanβtanπα2−1/α.In equation (4), V and W obey interval distribution and exponential distribution, respectively, and they satisfy the interval −π/2,π/2 and the mean 1.


### 3.1.2. Solving the Langevin Equation

The Langevin equation of SR system driven by Lévy noise is as follows:(7)dxdt=−dUxdx+st+ηtD,where x is the output, st is the periodic signal, ηt is the Lévy noise, and D is the noise amplification factor.The Langevin equation of SR generally needs to be solved by using fourth-order Runge—Kutta. Driven by Lévy noise, the CMS algorithm and Runge—Kutta are combined, as follows:(8)xn+1=xn+16k1+2k2+2k3+k4+h1/αDηn,k1=hfxn+sn,k2=hfxn+k12+sn,k3=hfxn+k22+sn+1,k4=hfxn+k3+sn+1,where fx=−dUx/dt, xn, and sn are the nth sampled values of the output and input signals, respectively, and h is the step size. Since, the final particle will have an infinite trajectory due to oscillation, truncation should be carried out [32, 33], when xn>5, xn=signxn×5 should be implemented. Setting that, β=μ=0 and σ=1, on the premise that only α is changed, Lévy noise is drawn in Figure 3.Figure 3
Lévy noise: (a)α=0.4, (b)α=0.7, (c)α=0.9, (d)α=1.1, (e)α=1.4, and (f)α=1.9.
(a)(b)(c)(d)(e)(f)According to Figure3, when α is relatively small, the number of Lévy noise pulses is relatively small, and the absolute value of the peak value is generally relatively high. With increasing α, the number of Lévy noise pulses increases gradually, and the mean value of the absolute value of the peak decreases gradually. When α approaches 2, the distribution of levy noise is mainly concentrated around the X-axis.


## 3.1.1. Distribution of Lévy Noise

In this paper, in order to simulate the generation of Lévy noise, we use the Chambers-Mallows-Stuck (CMS) [30, 31] algorithm to obtain the random variable of Lévy noise. The characteristic equation and expression of its steady-state distribution are shown in equations (2) and (3):(2)ft=∫−∞+∞eitxlα,βx;σ,μdx,(3)ft=exp−σt1+iβ2πsigntlogt+iμt,α=1,exp−σαtα1−iβsignttanπα2+iμt,α≠1,where α∈0,2, β∈−1,1,σ∈0,+∞, and μ∈RThe random variableX follows the following distribution:(4)X=2σππ2+βVtanV−βlnπ/2WcosVπ/2+βV+μ,α=1,Dα,β,σsinαV+Cα,βcosV1/α×cosV−αV+Cα,βW1−αα+μ,α≠1,where Cα,β and Dα,β,σ satisfy equations (5) and (6), respectively:(5)Cα,β=arctanβtanπα/2α,(6)Dα,β,σ=σcosarctanβtanπα2−1/α.In equation (4), V and W obey interval distribution and exponential distribution, respectively, and they satisfy the interval −π/2,π/2 and the mean 1.


## 3.1.2. Solving the Langevin Equation

The Langevin equation of SR system driven by Lévy noise is as follows:(7)dxdt=−dUxdx+st+ηtD,where x is the output, st is the periodic signal, ηt is the Lévy noise, and D is the noise amplification factor.The Langevin equation of SR generally needs to be solved by using fourth-order Runge—Kutta. Driven by Lévy noise, the CMS algorithm and Runge—Kutta are combined, as follows:(8)xn+1=xn+16k1+2k2+2k3+k4+h1/αDηn,k1=hfxn+sn,k2=hfxn+k12+sn,k3=hfxn+k22+sn+1,k4=hfxn+k3+sn+1,where fx=−dUx/dt, xn, and sn are the nth sampled values of the output and input signals, respectively, and h is the step size. Since, the final particle will have an infinite trajectory due to oscillation, truncation should be carried out [32, 33], when xn>5, xn=signxn×5 should be implemented. Setting that, β=μ=0 and σ=1, on the premise that only α is changed, Lévy noise is drawn in Figure 3.Figure 3
Lévy noise: (a)α=0.4, (b)α=0.7, (c)α=0.9, (d)α=1.1, (e)α=1.4, and (f)α=1.9.
(a)(b)(c)(d)(e)(f)According to Figure3, when α is relatively small, the number of Lévy noise pulses is relatively small, and the absolute value of the peak value is generally relatively high. With increasing α, the number of Lévy noise pulses increases gradually, and the mean value of the absolute value of the peak decreases gradually. When α approaches 2, the distribution of levy noise is mainly concentrated around the X-axis.


## 3.2. Simulation of Feature Extraction and Signal Enhancement

After simulating Lévy noise, it is necessary to verify whether the PTSR system driven by Lévy noise can perform feature extraction and signal enhancement on input signals. A sinusoidal signal of amplitudeA=0.5 and characteristic frequency fd=0.02hz is mixed with Lévy noise. We set the sampling frequency fs=5hz, h=0.1, α=1.8,β=μ=0, and σ=1 and apply the signal through the PTSR system, The original signal and output signal are shown in Figure 4.Figure 4
Original signal and output signal: (a) input time domain signal, (b) input frequency domain signal, (c) output time domain signal, and (d) output frequency domain signalm=2.0,p=0.5,q=1.2.
(a)(b)(c)(d)From Figures4(a) and 4(b), it can be found that the periodic signal is chaotic after Lévy noise is mixed, and the characteristic frequency is also submerged in other chaotic frequencies, meanwhile, it is also marked that the amplitude is 0.7983. It can be found in Figure 4(c) that the signals become relatively clear and some periodicity is restored, indicating that the PTSR system driven by Lévy noise can filter some chaotic noises from the perspective of filtering. Finally, it is shown in Figure 4(d) that the amplitude of the characteristic frequency is the highest among all frequencies and the value is 0.853. It indicates that the PTSR system successfully carries out feature extraction under Lévy noise and produces signal enhancement.


## 4. SNR-GM of PTSR System Driven by Lévy Noise

There are many evaluation criteria for SR performance. As an indicator of signal enhancement, SNR gain [34] SNRgain is widely concerned. At the same time, due to the randomness of noise, SNR-GM is used as the standard to measure the performance of SR in this paper, and n=100, which is as follows:(9)SNR−GM=1100∑i=1100SNRgaini,where SNRgain is given by the following equation:(10)SNRgain=SNRoutSNRin,where SNRin and SNRout are the input and output SNR, which can be obtained by the following formula:(11)SNRin=2Zk02∑k=0NZk2−2Zk02,SNRout=2Yk02∑k=0NYk2−2Yk02,where Zk is fast Fourier transform (FFT) of input single, Yk is FFT of output single xn,n=0,1,2…N−1, and N is length of sequence.


### 4.1. Effect of System Parameterm, p, and q on SNR-GM

Input periodic signals are set with amplitudeA=0.8 and characteristic fd=0.02hz into the PTSR system with D from 0 to 5 and set α=1.4, β=μ=0, and σ=1. Also, we set the arithmetic parameter to fs=80hz and h=0.125. Under the premise that the value of only one parameter is changed each time and the other two parameters remain unchanged, the SNR-GM curve obtained is shown in Figure 5. It could be seen that all curves in the three images show a trend of first rising and then falling, which is also an important basis for judging whether SR occurs in the system. It can be observed from Figures 5(a) and 5(c) that if other parameters are fixed and only the size of m or q is changed, the SNR-GM of the system will increase with the increase of m or q. On the contrary, it can be seen from Figure 5(b) that the SNR-GM of the system will decrease with the increase of p. The abovementioned conclusion is consistent with the conclusion summarized in Figure 2. Particles are easier to move, so they will generate higher energy and thus have higher SNR-GM. It is worth noting here that the abovementioned analysis was set based on the abovementioned parameters. SR is a complex system, and different results may appear under the setting of other parameters, therefore, it is necessary to continue to study the influence of system parameter m, p, and q on SNR-GM under different values of α. Input the same signal from the abovementioned experiment into the signal, and set parameters fs=80hz, h=0.125, β=μ=0, σ=1, and D=0.6. When the value of m ranges from 1.5 to 3.5, the value of α is changed, and the SNR-GM curve obtained is shown in Figure 6.Figure 5
SNR-GM curve: (a) Adjust onlymp=0.5,q=1.2, (b) adjust only pm=2.5,q=1.2, and (c) adjust only qm=2.5,p=0.5.
(a)(b)(c)Figure 6
SNR-GM curve with respect tom for different values of αp=0.5,q=1.2.As can be seen from Figure6, parameter m has no effect on the rise of SNR when α takes any value, and all curves show a downward or flat trend, which is consistent with the previous judgment on parameter m in some cases. Next, the experiment is conducted for parameters p and q, and the input signal remains unchanged, except for parameter D, which is set as 0.8 and 0.5, respectively, other parameters remain unchanged. SNR-GM curves were obtained under the premise that p values ranged from 0.5 to 0.8 and q values ranged from 0.8 to 1.1, as shown in Figures 7 and 8. It can be seen from Figure 7 that parameter p only has a rising or flat effect on SNR-GM, while Figure 8 shows that parameter q has a falling or flat effect on SNR-GM, which is consistent with the previous conclusion under certain conditions.Figure 7
SNR-GM curve with respect top for different values of αm=2.5,q=1.2.Figure 8
SNR-GM curve with respect toq for different values of αm=2.5,p=0.5.Meanwhile, by observing Figures6–8, it is found that parameter α also has a great influence on SNR-GM, Therefore, we should continue to research the effect of α and β on SNR-GM.


### 4.2. Effects ofα and β on SNR-GM

We input the same signal into the PTSR system as before, with the parameters set toβ=μ=0, σ=1fs=80hz, h=0.125, and D from 0 to 1.5 and draw the SNR curve under different values of αas shown in Figure 9.Figure 9
SNR-GM curve for different values ofαm=1.5,p=0.5,q=0.8.As shown in Figure9, when α increases gradually, SNR-GM will gradually increase; however, when the value of α exceeds 1, SNR-GM will rapidly decrease and further decrease with increasing α. Meanwhile, when the value of α exceeds 1, the value of D, which enables the system to obtain the highest SNR-GM, will obviously change. That is to say, when α is less than 1, a mainly affects the level of SNR; when α is greater than 1, in addition to the size of SNR-GM, it also has a great impact on the position of D. Next, the influence of β on SNR-GM is discussed. The same signal is still used, and parameters=1.8,μ=0, =1fs=80hz, and h=0.125 are set at the same time. When D is from 0 to 6, SNR-GM curves with different values of β are drawn, as displayed in Figure 10. It could be seen from Figure 10 that, like the curves in other figures, all curves show a trend of first rising and then falling, which still proves that SR phenomenon occurs in the PTSR system driven by Lévy noise. At the same time, increasing the absolute value of β, increases the SNR-GM of the system. If SNR-GM is greater when β is negative then when β is positive, β is the same absolute value.Figure 10
SNR-GM curve for different values ofβm=1.5,p=0.5,q=0.8.


## 4.1. Effect of System Parameterm, p, and q on SNR-GM

Input periodic signals are set with amplitudeA=0.8 and characteristic fd=0.02hz into the PTSR system with D from 0 to 5 and set α=1.4, β=μ=0, and σ=1. Also, we set the arithmetic parameter to fs=80hz and h=0.125. Under the premise that the value of only one parameter is changed each time and the other two parameters remain unchanged, the SNR-GM curve obtained is shown in Figure 5. It could be seen that all curves in the three images show a trend of first rising and then falling, which is also an important basis for judging whether SR occurs in the system. It can be observed from Figures 5(a) and 5(c) that if other parameters are fixed and only the size of m or q is changed, the SNR-GM of the system will increase with the increase of m or q. On the contrary, it can be seen from Figure 5(b) that the SNR-GM of the system will decrease with the increase of p. The abovementioned conclusion is consistent with the conclusion summarized in Figure 2. Particles are easier to move, so they will generate higher energy and thus have higher SNR-GM. It is worth noting here that the abovementioned analysis was set based on the abovementioned parameters. SR is a complex system, and different results may appear under the setting of other parameters, therefore, it is necessary to continue to study the influence of system parameter m, p, and q on SNR-GM under different values of α. Input the same signal from the abovementioned experiment into the signal, and set parameters fs=80hz, h=0.125, β=μ=0, σ=1, and D=0.6. When the value of m ranges from 1.5 to 3.5, the value of α is changed, and the SNR-GM curve obtained is shown in Figure 6.Figure 5
SNR-GM curve: (a) Adjust onlymp=0.5,q=1.2, (b) adjust only pm=2.5,q=1.2, and (c) adjust only qm=2.5,p=0.5.
(a)(b)(c)Figure 6
SNR-GM curve with respect tom for different values of αp=0.5,q=1.2.As can be seen from Figure6, parameter m has no effect on the rise of SNR when α takes any value, and all curves show a downward or flat trend, which is consistent with the previous judgment on parameter m in some cases. Next, the experiment is conducted for parameters p and q, and the input signal remains unchanged, except for parameter D, which is set as 0.8 and 0.5, respectively, other parameters remain unchanged. SNR-GM curves were obtained under the premise that p values ranged from 0.5 to 0.8 and q values ranged from 0.8 to 1.1, as shown in Figures 7 and 8. It can be seen from Figure 7 that parameter p only has a rising or flat effect on SNR-GM, while Figure 8 shows that parameter q has a falling or flat effect on SNR-GM, which is consistent with the previous conclusion under certain conditions.Figure 7
SNR-GM curve with respect top for different values of αm=2.5,q=1.2.Figure 8
SNR-GM curve with respect toq for different values of αm=2.5,p=0.5.Meanwhile, by observing Figures6–8, it is found that parameter α also has a great influence on SNR-GM, Therefore, we should continue to research the effect of α and β on SNR-GM.


## 4.2. Effects ofα and β on SNR-GM

We input the same signal into the PTSR system as before, with the parameters set toβ=μ=0, σ=1fs=80hz, h=0.125, and D from 0 to 1.5 and draw the SNR curve under different values of αas shown in Figure 9.Figure 9
SNR-GM curve for different values ofαm=1.5,p=0.5,q=0.8.As shown in Figure9, when α increases gradually, SNR-GM will gradually increase; however, when the value of α exceeds 1, SNR-GM will rapidly decrease and further decrease with increasing α. Meanwhile, when the value of α exceeds 1, the value of D, which enables the system to obtain the highest SNR-GM, will obviously change. That is to say, when α is less than 1, a mainly affects the level of SNR; when α is greater than 1, in addition to the size of SNR-GM, it also has a great impact on the position of D. Next, the influence of β on SNR-GM is discussed. The same signal is still used, and parameters=1.8,μ=0, =1fs=80hz, and h=0.125 are set at the same time. When D is from 0 to 6, SNR-GM curves with different values of β are drawn, as displayed in Figure 10. It could be seen from Figure 10 that, like the curves in other figures, all curves show a trend of first rising and then falling, which still proves that SR phenomenon occurs in the PTSR system driven by Lévy noise. At the same time, increasing the absolute value of β, increases the SNR-GM of the system. If SNR-GM is greater when β is negative then when β is positive, β is the same absolute value.Figure 10
SNR-GM curve for different values ofβm=1.5,p=0.5,q=0.8.


## 5. The Experimental Result

The experimental platform shown in Figure11 is used to extract signals and verify the ability of enhancement for the PTSR system driven by Lévy noise. This equipment is a medium-speed outer ring bearing failure platform of a company and its fs=120000Hz and fd=41.02Hz, and Figure 12 shows the original signal. Figure 12 shows that both the time domain and frequency domain of the original signal are chaotic, and the amplitude at the characteristic frequency is equal to 0.2526. The envelope spectrum of the original signal is given in Figure 13. It can be observed from Figure 13 that the characteristic frequency of the fault can be indicated, but the noise interference is very high. The original signal is then mixed with Lévy noise as shown in Figure 14, and the mixed signal passes through the PTSR system. First, scale transformation [35] is carried out and k=15000 is set; then, the secondary sampling frequency fs′=fs/k and h=1/fs′. For optimal parameters, particle swarm optimization (PSO) is used to search to maximize the output SNR-GM, and the output singles are shown in Figure 15.Figure 11
Machinery fault simulator (MFS)-magnum.Figure 12
Original signal. (a) Time-domain signal. (b) Frequency-domain signal.
(a)(b)Figure 13
Envelope spectrum.Figure 14
Original signal is mixed with Lévy noise. (a) Time-domain signal and (b) frequency-domain signal.
(a)(b)Figure 15
Output signal. (a) Time-domain signal and (b) frequency-domain signal.
(a)(b)It can be seen from Figure13(a) that the time-domain signal recovers part of its periodicity, indicating that the PTSR system driven by Lévy noise can successfully remove noise from the perspective of filtering. Meanwhile, in Figure 13(b), the amplitude of feature frequency is the highest, indicating that feature extraction has been successfully carried out by the system. Finally, the amplitude at the characteristic frequency has increased from 0.2526 to 1.826, indicating successful signal enhancement in the Lévy noise-driven PTSR system.In order to further verify the adaptation degree of Lévy noise in PTSR system, the abovementioned signals are processed by other SR systems. Standard tri-stable SR [36] and classical bistable SR are selected in this paper, and their potentials are Usx=ms/2ps2qs2x2−ms/4ps2+qs2x4+ms/6x6 and Ucx=−c/2x2+d/4x4, respectively. It should be noted that due to the randomness of noise, signals still need to pass through the PTSR system in this experiment. All parameters are consistent with the previous experimental settings. The system parameters of the three SR systems are automatically searched through PSO, and the frequency-domain diagrams of the three output signals are obtained, as shown in Figure 16.Figure 16
The output spectrum of the three systems: (a) classical bistable SR, (b) standard tri-stable SR, and (c) PTSR.
(a)(b)(c)It can be seen from Figure16 that all three SR systems can use Lévy noise to achieve feature extraction and signal enhancement, but the enhancement results are, respectively, from the original 0.2526 to 0.8612, 1.272, and 1.736. The experimental results show that PTSR system has better adaptation to Lévy noise and stronger ability to utilize Lévy noise.


## 6. Discussion

Based on the previously proposed PTSR, Lévy noise is used as the driving source to verify that the PTSR method can still perform feature extraction and signal enhancement. The relationship between system parameters and SNR-GM is also described. In the follow-up study, we can continue to study whether PTSR still has the abovementioned ability under the driving of other noises. However, it is worth investigating whether there are other more effective criteria besides SNR-GM.


## 7. Conclusion

Through the study of the PTSR system driven by Lévy noise, the following conclusions are obtained:(1)
The selection of system parameters will greatly affect the performance of SR system, so the relationship between parameters and SNR-GM is important. When Lévy noise drives, within a certain range, the increase ofp will increase SNR-GM, while m and q have opposite effects, and these effects correspond to the change of potential function.(2)
α and β also have a strong influence on the SNR-GM of the system and will change with their values, plus or minus.(3)
Through the verification of simulation signals and actual engineering signals, it is found that the PTSR system driven by Lévy noise can be used for feature extraction and signal enhancement.(4)
The fault type cannot be predicted in advance, so the capability of feature extraction and signal enhancement is an important criterion to evaluate the performance of SR. When Lévy noise is the driving source, the PTSR system has better adaptation and signal enhancement ability compared to STSR and SR systems.



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*Source: 7071092-2023-01-24.xml*