# Algorithm Used to Detect Weak Signals Covered by Noise in PIND

Authors: G. T. Wang; X. W. Liang; Y. Y. Xue; C. Li; Q. Ding
Journal: International Journal of Aerospace Engineering
 (2019)

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## Abstract

Detection of the loose particles is urgently required in the spacecraft production processes. PIND (particle impact noise detection) is the most commonly used method for the detection of loose particles in the aerospace electronic components. However, when the mass of loose particles is smaller than 0.01 mg, the weak signals are difficult to be detected accurately. In this paper, the aperiodic stochastic resonance (ASR) is firstly used to detect weak signals of loose particles. The loose particle signal is simulated by the oscillation attenuation signal. The influences of structure parameters on the potential height and detection performance of ASR are studied by a numerical iteration method. The cross-correlation coefficientC1 between input and output is chosen as a criterion for whether there is an existing a particle or not. Through normalization, the loose particle signal-labeled high frequency of 135 kHz is converted into the low-frequency band, which can be detected by the ASR method. According to the algorithm, weak signals covered by noise could be detected. The experimental results show that the detection accuracy is 66.7%. This algorithm improves the detection range of weak loose particle signals effectively.


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## Body

## 1. Introduction

Loose particles are impurities in aerospace electronic components, which is composed of metal debris, solder slag, conductor skin, etc. In a mechanical environment such as weightlessness and overweight, the loose particles are activated and released. The movable mechanism may be stuck due to the presence of the loose particles, and the action may be rejected. At the same time, the conductive loose particles may cause accidental conduction and malfunction and even lead to serious accidents in the aerospace electronic system. Therefore, the detection of loose particles is of great significance to the aerospace industry [1–3]. Because the loose particle is small and difficult to be detected, it requires high technology for weak signal detection.At present, commonly used loose particle detection methods include microscopic observation, X-ray photography, MATRA detection, and remainder particle impact noise detection (PIND). PIND is widely applied for its fastness, convenience, low cost, and high sensitivity. The PIND method uses the sound signal generated by the collision between loose particles and the device wall to judge the existence of loose particles. The block diagram of the detecting device is shown in Figure1. When the PIND test is carried out, the vibration table outputs the vibration and shock behaviors according to the default plan. The device to be tested is bonded to the sensor surface above the vibration table, which vibrates with the vibration table. The internal loose particle randomly collides with the inner wall of the tested device. The generated sound signal is received by the acoustic emission sensor. The detection waveform is transmitted to a voltage signal and output on the speaker and oscilloscope screen. After 50 years of development, PIND can detect loose particles with a mass of more than 0.01 mg, and the accuracy can reach 90%. However, PIND signals caused by the small particles whose mass is smaller than 0.01 mg are very weak and easily submerged in noise background, which makes it impossible to be detected [4–8].Figure 1
PIND system of MIL-STD-883E.Gao used the wavelet analysis to extract weak signals, and Wang et al. further introduced the stochastic resonance (SR) method, failing to extend the detection limit to 0.02 mg. Zhang used SR to process signals that cannot be extracted by the wavelet analysis, and the recognition probability is 12.5% [9].In linear systems, “noise” is harmful. It destroys the stability, balance, order, certainty, and consistency of the system. It is the biggest obstacle that weakens signal detection. But in a nonlinear system, it is found that after data processing, the useful signal in the output is enhanced as the noise level increases, and the signal-to-noise ratio has been greatly improved, which has some synchronization. Analogous to the phenomenon of resonance in mechanical systems, it is called “stochastic resonance.”An SR system consists of weak signal input, noise input, and a nonlinear system for signal processing. The input signals can be periodic signals, aperiodic signals, random signals, etc. The input noises can be white noises, colored noises, etc.Many scholars have done research related to the SR system. Li et al. used the method of rescaling frequency to solve large-parameter signals, and the improved SR was applied to fault diagnosis of mechanical equipment [10]. Initial research on SR was limited to the detection of periodic sinusoidal signals in the background of white noise using a nonlinear bistable system. That is, Benzi, an Italian scholar, used the phenomenon of “SR” in 1981 to study the nonlinear effects of ancient glaciers under the adiabatic approximation of bistable systems. The SR phenomenon in multistable systems driven by Gaussian noise was investigated theoretically and experimentally [11–13]. Many studies by using SR for weak signal detection have achieved positive research results [14–19]. But more signals are nonperiodic signals in practice [20]. Chow et al. put forward the concept of aperiodic stochastic resonance (ASR), which makes SR from theoretical research to practical application [21–24]. In this paper, the ASR method is used to detect the weak loose particle signals covered in the noise background for the first time.In this paper, the characteristics of the ASR system are analyzed, and the cross-correlation coefficient is chosen as a criterion of the existence of loose particles. The relationship between structural parameters and potential height is obtained, and the effect of these parameters on the detection results are obtained through theoretical calculation. The weak loose particle signal covered in the noise is simulated with the oscillation attenuation signal with white noise. The relationship curve between the cross-correlation coefficient and the structure parameters is calculated numerically. By fine-tuning the structure parameters, when the cross-correlation coefficient is higher than the threshold value, it is determined that the experimental signal contains weak loose particle signals. The experimental results show that even if the particle signals are covered, noise can be detected. The detection accuracy is higher than 66.7%. This algorithm effectively improves the detection capability of weakly loose particles.


## 2. Aperiodic Stochastic Resonance Theory

### 2.1. Response and Characterization of Aperiodic Stochastic Resonance

A bistable Langevin model, which is sensitive to noise, is widely used in stochastic systems. The Langevin equation of the bistable system under the action of aperiodic signalst and Gauss white noise Γt is
(1)ẋ=ax−bx3+st+Γt,a,b>0Γt=0,Γt,Γ0=2Dδt.Here,Γt is Gauss white noise whose mean is 0 and the intensity is D.By using the concept of linear response, the system response function is defined asχω,D, which contains the response information of the weak signal input st and the noise information Γt. Assuming that st and xt are stationary Gauss stochastic processes, the cross-correlation function of input and output is assumed
(2)Kxsτ=∫−∞∞∫−∞∞xtst+τPx,t;s,t+τdxds.Here,Px,t;s,t+τ is the two-dimensional combined probability density of st and xt for stochastic processes, and the cross-power spectral density Gxsω is the Fourier transform of the cross-correlation function Kxsτ.
(3)Gxsω=∫−∞∞Kxsτexp−iωτdτ.The definition of the coherence function is(4)γ2ω=Gxsω2GxxωGssω.The coherence functionγω is a real number between 0 and 1. It can be used to describe the linear coherence degree of input st and output xt in the frequency domain. The statistical characteristics of the weak signal st response of a nonlinear system can be calculated by the linear response theory, and the cross power spectral density Gxsω can be expressed as
(5)Gxsω=χω,DGssω.The power spectral density of output can be expressed as follows:(6)Gxxω=Gxx0ω,D+χω,D2Gssω.Here,Gxx0ω,D is the output power spectral density of nonlinear systems which is not disturbed by external aperiodic signals. Equations (5) and (6) are substituted with Equation (4); then
(7)γ2ω=1−Gxx0ω,DGxx0ω,D+χω,D2Gssω.From Equation (7), the coherence function is less than 1 and varies with the noise intensity D. When the noise intensity is optimized, the consistency between input and output increases and the coherence function increases. The coherence function is the characterization quantity describing the consistency of ASR input and output in the frequency domain. Assuming that the system input is st and the output is xt, the covariance C0 of the input and output is defined as
(8)C0=st−st¯xt−xt¯.Therefore, the cross-correlation coefficientC1 is defined as follows:
(9)C1=−C0xt−xt¯2st−st¯2.From the signal processing point of view, maximizingC1 is equivalent to maximizing the matching degree between input and output. C0 and C1 can also be calculated by Gxsω, Gssω, and Gxxω, and the following relationships are established:
(10)C0=∫0∞ReGxsωdω,C1=C0∫0∞Gssωdω∫0∞Gxxωdω−1/2.ASR’s linear response theory and cross-correlation description are applied to bistable systems. Assuming that the aperiodic inputst is a Gaussian noise signal, the correlation time is τ0, and the root mean square value is Q, the input power spectrum is
(11)Gssω=Qτ01+ω2τ02.According to the linear response theory, the expressions ofGxx0ω,D and χω,D are as follows: when only the inter-well transition is considered, the single exponential term is approximated, and there is no external disturbance.
(12)Gxx0ω,D=2λmx2stλm2+ω2,χω,D=1Dλmx2stλm2+ω2λm−iω.Here,λm is the minimum of nonzero eigenvalue corresponding to the Fokker-Planck operator, which is related to the Kramers transition rate rK.
(13)λm=2rK=2aπexp−ΔVD.In the undisturbed nonlinear bistable system, the steady-state value of the two order matrix isx2st. Equation (12) are substituted with Equation (7), and the expression of the coherence function is obtained [25].
(14)γ2ω=1−1+λmx2stD2Qτ01+ω2τ02.And the cross-correlation coefficientC1 is also obtained at the same time.
(15)C1=λmτ01+λmτ01−D21+λmτ0D21+λmτ0+Qλmτ0x2st1/2.In Equation (15), a=1, b=1, and Q=0.01, 10, and 20 for τ0, respectively, and the curve of C1 with D is shown in Figure 2. The variation curve of C1 with D shows a peak, indicating that ASR occurs in a bistable system. The greater the correlation time of input noise, the higher the matching degree between output and input. Therefore, the use of bistable ASR can detect weak aperiodic signals [26].Figure 2
The curve ofC1 with D.


### 2.2. Mechanism of Stochastic Resonance

The bistable stochastic resonance system for Equation (1) is a non-self-excited system, and the potential function is
(16)Ux=−a2x2+b4x4,a>0,b>0.Whena=1 and b=1 are available, it can be represented in Figure 3. The diagram contains two potential wells under the bistable system, and ΔV indicates potential height.Figure 3
The bistable system potential well.Assuming that it reaches the critical value of double-well SR driven by a signal withAmin amplitude, the potential function Ux satisfies
(17)∂U∂x=−ax+bx3+Amin,∂2U∂x2=−a+3bx2=0.From Equation (17), the threshold A4a3/27bmin of the input signal is obtained; that is to say, when the input signal amplitude A≥4a3/27b, the system can generate double potential well SR.When the input signal strengthA and the noise intensity D are not zero, the original equilibrium of the bistable system will be broken and the potential well will be tilted. When the input amplitude is larger than the threshold value, bistable SR will occur; otherwise, special intrawell resonance will occur, which is called “single potential well approximation model under bistability.” When there is only noise Γt and no periodic excitation, the particle transits between two potential wells according to Kramers transition rate rK. rK depends on the distribution and intensity of noise. The expression is as follows:
(18)rK=a2πexp−ΔVD.For the periodic signal of the input system, the bistable potential well will periodically tilt under its driving and produce SR, whose frequency is the same as that of the input signal; for the nonperiodic signal of the input system, the bistable potential well will also tilt and produce ASR.


## 2.1. Response and Characterization of Aperiodic Stochastic Resonance

A bistable Langevin model, which is sensitive to noise, is widely used in stochastic systems. The Langevin equation of the bistable system under the action of aperiodic signalst and Gauss white noise Γt is
(1)ẋ=ax−bx3+st+Γt,a,b>0Γt=0,Γt,Γ0=2Dδt.Here,Γt is Gauss white noise whose mean is 0 and the intensity is D.By using the concept of linear response, the system response function is defined asχω,D, which contains the response information of the weak signal input st and the noise information Γt. Assuming that st and xt are stationary Gauss stochastic processes, the cross-correlation function of input and output is assumed
(2)Kxsτ=∫−∞∞∫−∞∞xtst+τPx,t;s,t+τdxds.Here,Px,t;s,t+τ is the two-dimensional combined probability density of st and xt for stochastic processes, and the cross-power spectral density Gxsω is the Fourier transform of the cross-correlation function Kxsτ.
(3)Gxsω=∫−∞∞Kxsτexp−iωτdτ.The definition of the coherence function is(4)γ2ω=Gxsω2GxxωGssω.The coherence functionγω is a real number between 0 and 1. It can be used to describe the linear coherence degree of input st and output xt in the frequency domain. The statistical characteristics of the weak signal st response of a nonlinear system can be calculated by the linear response theory, and the cross power spectral density Gxsω can be expressed as
(5)Gxsω=χω,DGssω.The power spectral density of output can be expressed as follows:(6)Gxxω=Gxx0ω,D+χω,D2Gssω.Here,Gxx0ω,D is the output power spectral density of nonlinear systems which is not disturbed by external aperiodic signals. Equations (5) and (6) are substituted with Equation (4); then
(7)γ2ω=1−Gxx0ω,DGxx0ω,D+χω,D2Gssω.From Equation (7), the coherence function is less than 1 and varies with the noise intensity D. When the noise intensity is optimized, the consistency between input and output increases and the coherence function increases. The coherence function is the characterization quantity describing the consistency of ASR input and output in the frequency domain. Assuming that the system input is st and the output is xt, the covariance C0 of the input and output is defined as
(8)C0=st−st¯xt−xt¯.Therefore, the cross-correlation coefficientC1 is defined as follows:
(9)C1=−C0xt−xt¯2st−st¯2.From the signal processing point of view, maximizingC1 is equivalent to maximizing the matching degree between input and output. C0 and C1 can also be calculated by Gxsω, Gssω, and Gxxω, and the following relationships are established:
(10)C0=∫0∞ReGxsωdω,C1=C0∫0∞Gssωdω∫0∞Gxxωdω−1/2.ASR’s linear response theory and cross-correlation description are applied to bistable systems. Assuming that the aperiodic inputst is a Gaussian noise signal, the correlation time is τ0, and the root mean square value is Q, the input power spectrum is
(11)Gssω=Qτ01+ω2τ02.According to the linear response theory, the expressions ofGxx0ω,D and χω,D are as follows: when only the inter-well transition is considered, the single exponential term is approximated, and there is no external disturbance.
(12)Gxx0ω,D=2λmx2stλm2+ω2,χω,D=1Dλmx2stλm2+ω2λm−iω.Here,λm is the minimum of nonzero eigenvalue corresponding to the Fokker-Planck operator, which is related to the Kramers transition rate rK.
(13)λm=2rK=2aπexp−ΔVD.In the undisturbed nonlinear bistable system, the steady-state value of the two order matrix isx2st. Equation (12) are substituted with Equation (7), and the expression of the coherence function is obtained [25].
(14)γ2ω=1−1+λmx2stD2Qτ01+ω2τ02.And the cross-correlation coefficientC1 is also obtained at the same time.
(15)C1=λmτ01+λmτ01−D21+λmτ0D21+λmτ0+Qλmτ0x2st1/2.In Equation (15), a=1, b=1, and Q=0.01, 10, and 20 for τ0, respectively, and the curve of C1 with D is shown in Figure 2. The variation curve of C1 with D shows a peak, indicating that ASR occurs in a bistable system. The greater the correlation time of input noise, the higher the matching degree between output and input. Therefore, the use of bistable ASR can detect weak aperiodic signals [26].Figure 2
The curve ofC1 with D.


## 2.2. Mechanism of Stochastic Resonance

The bistable stochastic resonance system for Equation (1) is a non-self-excited system, and the potential function is
(16)Ux=−a2x2+b4x4,a>0,b>0.Whena=1 and b=1 are available, it can be represented in Figure 3. The diagram contains two potential wells under the bistable system, and ΔV indicates potential height.Figure 3
The bistable system potential well.Assuming that it reaches the critical value of double-well SR driven by a signal withAmin amplitude, the potential function Ux satisfies
(17)∂U∂x=−ax+bx3+Amin,∂2U∂x2=−a+3bx2=0.From Equation (17), the threshold A4a3/27bmin of the input signal is obtained; that is to say, when the input signal amplitude A≥4a3/27b, the system can generate double potential well SR.When the input signal strengthA and the noise intensity D are not zero, the original equilibrium of the bistable system will be broken and the potential well will be tilted. When the input amplitude is larger than the threshold value, bistable SR will occur; otherwise, special intrawell resonance will occur, which is called “single potential well approximation model under bistability.” When there is only noise Γt and no periodic excitation, the particle transits between two potential wells according to Kramers transition rate rK. rK depends on the distribution and intensity of noise. The expression is as follows:
(18)rK=a2πexp−ΔVD.For the periodic signal of the input system, the bistable potential well will periodically tilt under its driving and produce SR, whose frequency is the same as that of the input signal; for the nonperiodic signal of the input system, the bistable potential well will also tilt and produce ASR.


## 3. Establishment of the Bistable Stochastic Resonance System

Bistable systems in SR can be represented by the Langevin equation, which has been described in detail in Section2.1 of this paper. When SR is applied to the detection of weak loose particle signals, practical problems such as the form of loose particle signal and noise and the adjustment of system parameters need to be considered.


### 3.1. The Model of the Particle Signal

Loose particle signal is a typical random and aperiodic signal. The single pulse signal collected in the experiment is the oscillating attenuation signal, which is shown in Figure4(a)). It can be simulated in the following equation [27]:
(19)ut=A0exp−tksin2πft.Figure 4
The real particle pulse signal and its simulating signals.
(a)
The real particle pulse signal
(b)
The simulated particle pulse signalHere,A0 is the maximum amplitude of the input signal, k is the impact attenuation coefficient, and f is the oscillation frequency. Setting the parameters to A0=1.25, k=1.5, and f=5 can simulate the pulse particle signal, as shown in Figure 4(b).To more realistically simulate the actual weak signal of the loose particle, the Gauss white noise with root mean squareσ=1 is superimposed on the above model, and the mixed signal vt is obtained as shown in Figure 5. At this point, the loose particle signal has been completely covered by the noise.Figure 5
The simulated particle pulse signal with noise.


### 3.2. Influence of Structural Parameters on the System

In the process of synergistic effect among signal, noise, and nonlinear systems, potential height and SR peak will change with system structure parameters. This paper analyzes the influence of system parametersa and b on the system and output employing simulation. The qualitative explanation of stochastic resonance is given by using a Langevin equation based on simplification and constraints when uncertainties such as noise are ignored. In this paper, SR is quantitatively analyzed by numerical simulation.Equation (1) is a nonlinear stochastic differential equation, which can be solved only by numerical methods. Based on the Runge-Kutta algorithm, this paper adopts a numerical iteration algorithm for solving nonlinear stochastic differential equations [28, 29].
(20)xn+1=xn+16k1+2k2+2k3+k4,k1=haxn−bxn3+snn,k2=haxn+k12−bxn+k123+snn,k3=haxn+k22−bxn+k223+snn,k4=haxn+k3−bxn+k33+snn.Hereh=1/fs is the simulation step, fs is the sampling frequency, snn is the nth sampling point of the input signal with noise, and x is the output signal. To prevent the divergence of iteration, the simulation condition of the bistable system SR is
(21)ah≤1,x0≤ah+2bh.The results show that the sampling frequency has a great influence on the stability of the system output and SR effect. Generally, the sampling frequency is selected to be more than 50 times of the target detection signal frequency.


#### 3.2.1. Effect ofa and b on Detection Performance

According to Equations (17), the structural parameters a and b of the system have a direct impact on the input threshold of the system. When the input signal amplitude is less than Amin, the single potential well SR will occur in the bistable state; otherwise, the bistable SR will occur.The maximum amplitudeA0=1.25, attenuation coefficient k=1.5, oscillation frequency f=5, and noise variance D=2.5 are selected in the model ut=A0exp−t/ksin2πft of particle signal. At this time, the signal to noise ratio of the input signal is very low, and the particle signal is completely annihilated by noise. The influence of a and b on SR is also studied utilizing control variables, and the potential height and the distance between two wells increase with the increase of a and decrease with the increase of b.C1 is chosen as the characterization parameter to describe the detection performance. Here, we set b=0.8 to study a as an example to discuss. When a=50, the system input and output time-frequency diagram is shown in Figure 6.Figure 6
The time and frequency spectrum of the system input and output whena=50 and b=0.8.
(a)
The time and frequency spectrum of particle signal
(b)
The time spectrum and frequency of particle signal with noise
(c)
The time and frequency spectrum of system outputSince the loose particle signal is a high-frequency signal in most instances, the SR system can only process the low-frequency signal, so the input high signal needs to be normalized; therefore, the obtained output signal is 0~1 Hz. High-frequency signal processing will be explained in Section3.3.Compared with Figures6(a) and 6(c), when a=50 and b=0.8, the output of the system cannot clearly show the characteristics of the input signal, whether in the time domain or frequency domain. Therefore, resonance does not occur because the parameters a and b are not selected properly. Keep the input signal amplitude and noise intensity unchanged, and gradually reduce the value of a. When a is set as 11 and 3, respectively, the output time-frequency diagrams are shown in Figures 7 and 8.Figure 7
The time and frequency spectrum of the system input and output whena=11 and b=0.8.Figure 8
The time and frequency spectrum of the system input and output whena=3 and b=0.8.Compared with Figures6–8, the SR system can extract the loose particle effectively when a is set as 11 and 3, respectively, and the output waveform is closer to the input signal when a=3 than when a=11. It can be concluded that in Figures 7 and 8, due to the reasonable selection of parameters, resonance can be generated and the loose particle signal can be detected. However, since the input time domain signal peak value is at time 0, we get the output frequency domain signal as well and the maximum peak value at time zero. As shown in Figure 9, the parameter b is fixed, and the variation of C1 with the parameter a is calculated and the curve is drawn.Figure 9
The relationship between the cross-correlation coefficientC1 and the structure parameter a.


### 3.3. High-Frequency Signal Processing

The Langevin equation describes the Brownian motion of particles in a fluid, which is the most concise model in the bistable system. The research on the characteristics of SR systems is also mainly dependent on them. According to the SR system described by the model, the Fokker-Planck equation of Equation (22) can be established to analyze the effect of noise on the system.
(22)∂ρx,t∂t=−∂∂xAcos2πftρx,t+D∂2∂x2ρx,t.Because of the time-dependent term−∂/∂xAcos2πftρx,t in the Fokker-Planck equation, the equation can only be approximately solved. The approximate method is the adiabatic approximation theory. Adiabatic approximation theory is the earliest, most systematic, and persuasive approximation theory for describing periodic SR systems. Its application condition is that the signal frequency is far less than the transition rate (2πf<<rK); that is, SR can only process low-frequency signals. However, there are a lot of high-frequency signals in the project, and the particle pulse is about 135 kHz of the narrowband modulation signal. How to deal with high-frequency signal is the key problem of ASR in the loose particle signal detection.The amplitude of the weak signal isA and the frequency is ω>>1. The dynamic model of the bistable system is
(23)ẋ=ax−bx3+Asinωt+φ+Nt.Equation (23) is transformed into a normalized form, and variable substitution z=xb/a,τ=at is introduced.
(24)aabdzdt=aabz−aabz3+Acosωaτ+Nτa.Here,Nτ/a satisfies ENτ/aN0=2Daδτ and defines ετ as Gauss white noise of intensity 0.5; that is Eετ=0,Eετε0=δτ, and Equation (24) is shown as follows:
(25)dzdτ=z−z3+ba3Acosωaτ+φ+2Dba2ετ.Equation (25) is the normalized form of Equation (23). The normalized signal’s frequency becomes 1/a of the original signal frequency, which satisfies the hypothesis of the adiabatic approximation theory. Therefore, a large parameter a can be selected for high-frequency signals to meet the requirements of the SR theory.


### 3.4. The Detection Algorithm for the Weak Particle Signal

Based on the above analysis, this paper presents a detection algorithm for the weak loose particle signal based on ASR. Its flow chart is shown in Figure10.Figure 10
The flow chart of the algorithm for weak particle signal based on ASR.The main steps are as follows:(1)
Signal normalization: the particle signal with noise is normalized to [-1,1] through amplitude and scale transformation, and its frequency is changed into the reciprocal to satisfy the constraints of ASR on frequency
(2)
Parametera selection: according to the law of the loose particle pulse signal width, select the appropriate parameter a. For example, when the pulse width of the loose particle signal is generally concentrated in T=10μs−100μs, the priority is to set a to 1/T of 3-10 times, that is, 30000-1000000
(3)
Parameterb selection: parameter b and parameter a are generally in the same order of magnitude or an order of magnitude different; according to the experience, the value of b is slightly smaller than that of a
(4)
The input is substituted for the ASR model to solve the problem. The output results are analyzed, and the cross-correlation coefficientC1 is calculated to determine whether the set threshold is reached. If not, the parameters are fine-tuned according to the actual situation; then the solution is brought in again until the threshold is reached, and the existence of remainder particle is determined. If the threshold is not reached after 100 iterations, there is no particle. A threshold can be counted by the results of many simulation experiments, and the C1 interval with good output for detection can be obtained. Set the minimum value of the interval to be the threshold value, that is, Ce=0.35


## 3.1. The Model of the Particle Signal

Loose particle signal is a typical random and aperiodic signal. The single pulse signal collected in the experiment is the oscillating attenuation signal, which is shown in Figure4(a)). It can be simulated in the following equation [27]:
(19)ut=A0exp−tksin2πft.Figure 4
The real particle pulse signal and its simulating signals.
(a)
The real particle pulse signal
(b)
The simulated particle pulse signalHere,A0 is the maximum amplitude of the input signal, k is the impact attenuation coefficient, and f is the oscillation frequency. Setting the parameters to A0=1.25, k=1.5, and f=5 can simulate the pulse particle signal, as shown in Figure 4(b).To more realistically simulate the actual weak signal of the loose particle, the Gauss white noise with root mean squareσ=1 is superimposed on the above model, and the mixed signal vt is obtained as shown in Figure 5. At this point, the loose particle signal has been completely covered by the noise.Figure 5
The simulated particle pulse signal with noise.


## 3.2. Influence of Structural Parameters on the System

In the process of synergistic effect among signal, noise, and nonlinear systems, potential height and SR peak will change with system structure parameters. This paper analyzes the influence of system parametersa and b on the system and output employing simulation. The qualitative explanation of stochastic resonance is given by using a Langevin equation based on simplification and constraints when uncertainties such as noise are ignored. In this paper, SR is quantitatively analyzed by numerical simulation.Equation (1) is a nonlinear stochastic differential equation, which can be solved only by numerical methods. Based on the Runge-Kutta algorithm, this paper adopts a numerical iteration algorithm for solving nonlinear stochastic differential equations [28, 29].
(20)xn+1=xn+16k1+2k2+2k3+k4,k1=haxn−bxn3+snn,k2=haxn+k12−bxn+k123+snn,k3=haxn+k22−bxn+k223+snn,k4=haxn+k3−bxn+k33+snn.Hereh=1/fs is the simulation step, fs is the sampling frequency, snn is the nth sampling point of the input signal with noise, and x is the output signal. To prevent the divergence of iteration, the simulation condition of the bistable system SR is
(21)ah≤1,x0≤ah+2bh.The results show that the sampling frequency has a great influence on the stability of the system output and SR effect. Generally, the sampling frequency is selected to be more than 50 times of the target detection signal frequency.


### 3.2.1. Effect ofa and b on Detection Performance

According to Equations (17), the structural parameters a and b of the system have a direct impact on the input threshold of the system. When the input signal amplitude is less than Amin, the single potential well SR will occur in the bistable state; otherwise, the bistable SR will occur.The maximum amplitudeA0=1.25, attenuation coefficient k=1.5, oscillation frequency f=5, and noise variance D=2.5 are selected in the model ut=A0exp−t/ksin2πft of particle signal. At this time, the signal to noise ratio of the input signal is very low, and the particle signal is completely annihilated by noise. The influence of a and b on SR is also studied utilizing control variables, and the potential height and the distance between two wells increase with the increase of a and decrease with the increase of b.C1 is chosen as the characterization parameter to describe the detection performance. Here, we set b=0.8 to study a as an example to discuss. When a=50, the system input and output time-frequency diagram is shown in Figure 6.Figure 6
The time and frequency spectrum of the system input and output whena=50 and b=0.8.
(a)
The time and frequency spectrum of particle signal
(b)
The time spectrum and frequency of particle signal with noise
(c)
The time and frequency spectrum of system outputSince the loose particle signal is a high-frequency signal in most instances, the SR system can only process the low-frequency signal, so the input high signal needs to be normalized; therefore, the obtained output signal is 0~1 Hz. High-frequency signal processing will be explained in Section3.3.Compared with Figures6(a) and 6(c), when a=50 and b=0.8, the output of the system cannot clearly show the characteristics of the input signal, whether in the time domain or frequency domain. Therefore, resonance does not occur because the parameters a and b are not selected properly. Keep the input signal amplitude and noise intensity unchanged, and gradually reduce the value of a. When a is set as 11 and 3, respectively, the output time-frequency diagrams are shown in Figures 7 and 8.Figure 7
The time and frequency spectrum of the system input and output whena=11 and b=0.8.Figure 8
The time and frequency spectrum of the system input and output whena=3 and b=0.8.Compared with Figures6–8, the SR system can extract the loose particle effectively when a is set as 11 and 3, respectively, and the output waveform is closer to the input signal when a=3 than when a=11. It can be concluded that in Figures 7 and 8, due to the reasonable selection of parameters, resonance can be generated and the loose particle signal can be detected. However, since the input time domain signal peak value is at time 0, we get the output frequency domain signal as well and the maximum peak value at time zero. As shown in Figure 9, the parameter b is fixed, and the variation of C1 with the parameter a is calculated and the curve is drawn.Figure 9
The relationship between the cross-correlation coefficientC1 and the structure parameter a.


## 3.2.1. Effect ofa and b on Detection Performance

According to Equations (17), the structural parameters a and b of the system have a direct impact on the input threshold of the system. When the input signal amplitude is less than Amin, the single potential well SR will occur in the bistable state; otherwise, the bistable SR will occur.The maximum amplitudeA0=1.25, attenuation coefficient k=1.5, oscillation frequency f=5, and noise variance D=2.5 are selected in the model ut=A0exp−t/ksin2πft of particle signal. At this time, the signal to noise ratio of the input signal is very low, and the particle signal is completely annihilated by noise. The influence of a and b on SR is also studied utilizing control variables, and the potential height and the distance between two wells increase with the increase of a and decrease with the increase of b.C1 is chosen as the characterization parameter to describe the detection performance. Here, we set b=0.8 to study a as an example to discuss. When a=50, the system input and output time-frequency diagram is shown in Figure 6.Figure 6
The time and frequency spectrum of the system input and output whena=50 and b=0.8.
(a)
The time and frequency spectrum of particle signal
(b)
The time spectrum and frequency of particle signal with noise
(c)
The time and frequency spectrum of system outputSince the loose particle signal is a high-frequency signal in most instances, the SR system can only process the low-frequency signal, so the input high signal needs to be normalized; therefore, the obtained output signal is 0~1 Hz. High-frequency signal processing will be explained in Section3.3.Compared with Figures6(a) and 6(c), when a=50 and b=0.8, the output of the system cannot clearly show the characteristics of the input signal, whether in the time domain or frequency domain. Therefore, resonance does not occur because the parameters a and b are not selected properly. Keep the input signal amplitude and noise intensity unchanged, and gradually reduce the value of a. When a is set as 11 and 3, respectively, the output time-frequency diagrams are shown in Figures 7 and 8.Figure 7
The time and frequency spectrum of the system input and output whena=11 and b=0.8.Figure 8
The time and frequency spectrum of the system input and output whena=3 and b=0.8.Compared with Figures6–8, the SR system can extract the loose particle effectively when a is set as 11 and 3, respectively, and the output waveform is closer to the input signal when a=3 than when a=11. It can be concluded that in Figures 7 and 8, due to the reasonable selection of parameters, resonance can be generated and the loose particle signal can be detected. However, since the input time domain signal peak value is at time 0, we get the output frequency domain signal as well and the maximum peak value at time zero. As shown in Figure 9, the parameter b is fixed, and the variation of C1 with the parameter a is calculated and the curve is drawn.Figure 9
The relationship between the cross-correlation coefficientC1 and the structure parameter a.


## 3.3. High-Frequency Signal Processing

The Langevin equation describes the Brownian motion of particles in a fluid, which is the most concise model in the bistable system. The research on the characteristics of SR systems is also mainly dependent on them. According to the SR system described by the model, the Fokker-Planck equation of Equation (22) can be established to analyze the effect of noise on the system.
(22)∂ρx,t∂t=−∂∂xAcos2πftρx,t+D∂2∂x2ρx,t.Because of the time-dependent term−∂/∂xAcos2πftρx,t in the Fokker-Planck equation, the equation can only be approximately solved. The approximate method is the adiabatic approximation theory. Adiabatic approximation theory is the earliest, most systematic, and persuasive approximation theory for describing periodic SR systems. Its application condition is that the signal frequency is far less than the transition rate (2πf<<rK); that is, SR can only process low-frequency signals. However, there are a lot of high-frequency signals in the project, and the particle pulse is about 135 kHz of the narrowband modulation signal. How to deal with high-frequency signal is the key problem of ASR in the loose particle signal detection.The amplitude of the weak signal isA and the frequency is ω>>1. The dynamic model of the bistable system is
(23)ẋ=ax−bx3+Asinωt+φ+Nt.Equation (23) is transformed into a normalized form, and variable substitution z=xb/a,τ=at is introduced.
(24)aabdzdt=aabz−aabz3+Acosωaτ+Nτa.Here,Nτ/a satisfies ENτ/aN0=2Daδτ and defines ετ as Gauss white noise of intensity 0.5; that is Eετ=0,Eετε0=δτ, and Equation (24) is shown as follows:
(25)dzdτ=z−z3+ba3Acosωaτ+φ+2Dba2ετ.Equation (25) is the normalized form of Equation (23). The normalized signal’s frequency becomes 1/a of the original signal frequency, which satisfies the hypothesis of the adiabatic approximation theory. Therefore, a large parameter a can be selected for high-frequency signals to meet the requirements of the SR theory.


## 3.4. The Detection Algorithm for the Weak Particle Signal

Based on the above analysis, this paper presents a detection algorithm for the weak loose particle signal based on ASR. Its flow chart is shown in Figure10.Figure 10
The flow chart of the algorithm for weak particle signal based on ASR.The main steps are as follows:(1)
Signal normalization: the particle signal with noise is normalized to [-1,1] through amplitude and scale transformation, and its frequency is changed into the reciprocal to satisfy the constraints of ASR on frequency
(2)
Parametera selection: according to the law of the loose particle pulse signal width, select the appropriate parameter a. For example, when the pulse width of the loose particle signal is generally concentrated in T=10μs−100μs, the priority is to set a to 1/T of 3-10 times, that is, 30000-1000000
(3)
Parameterb selection: parameter b and parameter a are generally in the same order of magnitude or an order of magnitude different; according to the experience, the value of b is slightly smaller than that of a
(4)
The input is substituted for the ASR model to solve the problem. The output results are analyzed, and the cross-correlation coefficientC1 is calculated to determine whether the set threshold is reached. If not, the parameters are fine-tuned according to the actual situation; then the solution is brought in again until the threshold is reached, and the existence of remainder particle is determined. If the threshold is not reached after 100 iterations, there is no particle. A threshold can be counted by the results of many simulation experiments, and the C1 interval with good output for detection can be obtained. Set the minimum value of the interval to be the threshold value, that is, Ce=0.35


## 4. Experimental Verification

Considering that the current loose particle detection equipment has a poor detection effect on the weak signal generated by the loose particle of less than 0.01 mg, this is because the signal is too weak and it is covered by noise. In this paper, a tin-containing loose particle with a mass of about 0.01 mg is implanted into 10 models of relays and 5 transistor cavities of a certain type: 100 tests for each group of subjects. The ASR algorithm is used to process the collected acoustic emission signals. Since the frequency of the loose particle signal is about 135 kHz, according to Section3.3 the analysis of the parameter setting of the high-frequency signal processing by the SR system, we can conclude that the bistable system needs to normalize the high frequency loose particle signal to make its frequency become 1/a of the original signal frequency. So we set parameter a to 800000 and parameter b to 50000. In each experiment, the frequency of the input signal fluctuated slightly, so the parameters a and b need to be fine-tuned on a set basis for each experiment. The original input signal is the noise-containing loose particle signal that is amplitude-shifted, scale-scaled, and normalized into the interval [-1,1]. The experimental results are shown in Table 1, and the cross-correlation coefficient is the average of 100 experimental results. The threshold of the cross-correlation coefficient is calculated by the results of multiple simulation experiments, and it is found that it can detect the C1 interval with good output. The minimum value of the interval is set to the threshold Ce; finally, the threshold Ce is set to 0.35 according to the simulation experiment results. Experiment results show that the weak loose particle signal can be detected very well.Table 1
The cross-correlation coefficient was received by averaging 100 tests.
No.
Package types
Particle
C1
Result
1
1/5 crystal cover
0.01 mg tin grain
0.0993
Nonexistent
2
1/5 crystal cover
0.01 mg tin grain
0.2952
Nonexistent
3
1/5 crystal cover
0.01 mg tin grain
0.6410
Existent
4
1/5 crystal cover
0.01 mg tin grain
0.5545
Existent
5
1/2 crystal cover
0.01 mg tin grain
0.6716
Existent
6
1/2 crystal cover
0.01 mg tin grain
0.4590
Existent
7
1/2 crystal cover
0.01 mg tin grain
0.0250
Nonexistent
8
TO-5
0.01 mg tin grain
0.5944
Existent
9
TO-5
0.01 mg tin grain
0.6538
Existent
10
TO-5
0.01 mg tin grain
0.4751
Existent
11
TO-3
0.01 mg tin grain
0.5304
Existent
12
TO-3
0.01 mg tin grain
0.5202
Existent
13
TO-3
0.01 mg tin grain
0.2746
Nonexistent
14
TO-3
0.01 mg tin grain
0.4588
ExistentAs shown in the table, after repeating experiments for 100 times, 10 of the 15 relays with loose particles could accurately detect the existence of particle. The experimental results show that the weak loose particle signal detection algorithm based on ASR can improve the detection level to a certain extent, and the accuracy rate is 66.7%. Moreover, different package types also influence the results of the test. As can be seen in Table1, the TO-5 package has higher accuracy than the other two packages, probably because the structure of this package is simpler.


## 5. Discussion

We introduced a loose particle signal detection algorithm based on the aperiodic stochastic resonance (ASR) system for detecting nonperiodic high-frequency signals and showed this achieved higher classification accuracy. In this study, we can detect the loose particle signal less than 0.01 mg that was not detected before; in other words, the weak loose particle signal is completely covered by the noise signal, which is theoretically and practically meaningful. In our experiment with 15 aerospace relays containing 0.01 mg of the loose particle, we observed that although the signal is completely covered by noise, it can be detected by the method of this paper and the accuracy of the detection obtained is higher than other methods. The main reasons why some aerospace relays cannot be accurately detected are as follows:(1)
Because the input signal amplitude is too small, the system has SR in the well, which results in the change of system characteristics, the “synergistic effect” is affected, and the output characteristics become worse
(2)
Due to the unreasonable selection of parametersa and b, SR is difficult to extract in the system driven by weak signals
(3)
Due to the largeCe setting of the threshold, some of the data that have occurred in SR have been misjudged as nonredundant objectsIn future experiments, while we are working hard to solve the above problems, we should also pursue higher speeds and parameter tuning. We can also use other algorithms to improve the classification accuracy and reduce the error rate.


## 6. Conclusions

In this paper, an ASR system for detecting weak signals is constructed, and the loose particle signal is simulated by oscillating the attenuation signal. The influence of structure parameters on the potential height and detection performance of ASR is studied by the numerical iteration method. Through simulation, the relationship curves between the cross-correlation coefficientC1 and structure parameters are obtained. By normalization, the 135 kHz high-frequency loose particle signal is converted into the low-frequency band, which can be detected by the ASR method. According to the algorithm, a loose particle weak signal detection with a mass of 0.01 mg is performed, with an accuracy of 66.7%. This algorithm improves detecting weak loose particle signal range from 0.02 mg to 0.01 mg effectively and can detect signals that are completely covered by noise.



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*Source: 1637953-2019-12-14.xml*