Extracting non-stationary signal under strong noise background: Time-varying system analysis

The extraction of non-stationary feature information under strong noise background is a difficult problem. In this paper, a novel general time-varying scale transformation aperiodic stochastic resonance is proposed to extract and enhance the weak non-stationary signal under strong noise background. The theoretical framework of a parameters time-varying Duffing system is built for aperiodic stochastic resonance. By studying the resonance region migration when scale coefficient takes different values, an optimal scale transformation is achieved. Also, the time-varying system is optimized with cross-correlation coefficient as the index. Compared with the existing methods, the proposed method can be applied to stronger noise background and has stronger noise robustness. When under the same noise background, the proposed method can provide output with higher signal-to-noise ratio and higher cross-correlation coefficient. Finally, experimental analysis of faulty bearing vibration signal verifies the high accuracy, which indicates a good signal extraction and enhancement ability of the proposed method.


Introduction
In signal processing, the extraction and enhancement of non-stationary signal under strong noise is an important and difficult problem. Therefore, various signal processing methods have been proposed and applied to the engineering practice. These methods are mainly divided into two types (Lu et al., 2019). One is the method based on digital filter, which can retain signal and attenuate noise (Hu et al., 2018;Lu and Wang, 2018). Another is signal decomposition method, including empirical mode decomposition (Zhang et al., 2018b(Zhang et al., , 2020a, variational modal decomposition (Gai et al., 2020;Li et al., 2019a), wavelet decomposition (Chen et al., 2019;Liang et al., 2020;Lonare et al., 2021;Qin et al., 2019), and signal sparse decomposition (Fan et al., 2021;Li et al., 2018Li et al., , 2019c. However, when processing noisy signal with very low signal-to-noise ratio (SNR), the above-mentioned methods fail to provide good results (Cheng et al., 2021;Gong et al., 2019;Wang et al., 2019a;Zhang et al., 2018a). In fact, the extraction and enhancement of non-stationary signal is still a long-term problem to be solved.
Noise, which is usually considered as a negative factor in signal processing, can play a positive role under special circumstances. For example, when nonlinear system, signal, and noise reach a best match, the noise energy will be transferred to the signal; hence, the SNR increases significantly. This phenomenon is called stochastic resonance (SR) (Li et al., 2019b;Qiao et al., 2019). According to the resonance theory, this phenomenon only occurs when small parameter requirements are met; hence, the signal frequency is lower than 1 Hz and the system parameter has an order of magnitude of 1. However, the signals in engineering practice that can reflect the fault characteristics of rotating machinery is mostly in the range of tens to hundreds of Hertz. Also, it often has non-stationary characteristics, including frequency and amplitude modulation.
To break through the limitation that SR can only process low-frequency signals, a series of scale transformation are proposed. Leng et al. proposed twice sampling technology (Leng et al., 2004;Wang and Leng, 2003). Lu et al. pro-posed the normalized variable scale technology (Lu et al., 2014). Huang et al. proposed the general scale transformation. Compared to the normalized scale transformation stochastic resonance, the general scale transformation stochastic resonance can obtain a better output with higher SNR, and it has been applied in bearing fault diagnosis (Yang et al., 2020a). However, the parameters in the above methods are fixed, which determines that it is more suitable for stationary signals processing, such as vibration signal of rotating machinery with fixed speed. However, because of unstable loads and other factors, rotation speed often changes in real time, causing characteristic frequency of rotating parts changes in real time (Li et al., 2019d;Wang et al., 2019b;Yu, 2019).
To realize the aperiodic stochastic resonance (ASR), researchers have done a lot of research. Yang et al. successfully realized the SR of the variable frequency signal by using the piecewise scale transformation (Yang et al., 2020b). In that paper, original signal was cut into several sub-segments and each sub-segment was regarded as stationary. Subsequently, by optimizing the parameters of each sub-segment, suboutput of piecewise scale transformation SR for each subsegment was obtained. At last, all sub-outputs were spliced into the final output. Similar segmentation processing methods appeared in vibration resonance (Jia et al., 2019). However, such methods have great human factors when dividing signal. More extremely, Zhang et al. cut each data point into separate sub-segment, and proposed a new varying parameters and normalization stochastic resonance (VPNSR), which was based on the normalized transformation (Zhang et al., 2016). This method maps the time-varying frequency to a constant low frequency with the help of time-varying parameters, and successfully realizes the ASR of linear frequency modulation (LFM) signal. However, in this work, the noise intensity is weak, and the signal frequency modulation mode is linear. So, faced with nonlinear frequency modulation signal and strong noise background, whether the method is effective remains further discussion.
In fact, most of the methods mentioned above are designed for stationary signals processing, but not for nonstationary signals processing. Under non-stationary conditions, such as rotating machinery, the time-varying speed causes the instantaneous characteristic frequencies of various components to have time-varying characteristics. Also, strong noise and low SNR exist in non-stationary conditions. These negative factors bring great challenges to signal processing, and cause signal processing methods based on fixed parameter not to work in non-stationary signal processing.
To solve the above problems, a new general time-varying scale transformation aperiodic stochastic resonance (GTVST-ASR) is proposed. Firstly, in Duffing system, a new general time-varying scale transformation (GTVST) is needed. Then, parameter optimization should be carried out to obtain the optimal ASR. Also, comparison between the proposed method and existing methods is of great importance. To verify the effectiveness of the GTVST-ASR, faulty bearing signals under strong noise and complex variable speed condition should be processed with GTVST-ASR. The innovations of this paper are summarized as follows: • A novel time-varying parameter nonlinear system is built, and aperiodic stochastic resonance can occur in it under strong noise background. • Compared with the previous aperiodic resonance methods, this proposed method provides higher quality output with higher SNR, higher cross-correlation coefficient (C). Also, the proposed method can be applied to stronger noise background and has stronger noise robustness. • The proposed method can process more complex nonstationary signal, it is superior to the previous stochastic resonance in scope of application.
The structure of this paper is arranged as follows. In Section 2, the theoretical framework of GTVST-ASR is established, including derivation and dynamic analysis. In Section 3, the migration process of resonance region under different parameters is studied, based on which, the parameters are optimized. In Section 4, the experimental validation is carried out, where GTVST-ASR is employed to process the faulty bearing signal under the condition of variable speed and strong noise, and hence further verify the practical value. Finally, the discussion and conclusions are provided in Section 5 and Section 6, respectively.

Methodology of general time-varying scale transformation aperiodic stochastic resonance
The time-varying parameter bistable system excited by noise and variable frequency signal can be expressed by Here, a(t) and b(t) are large parameters that change with time, the time-varying instantaneous frequency f(t) is greater than 1. A is signal amplitude of the pure signal u(t), ξ(t) is the standard Gaussian white noise with zero mean, where hξðtÞi ¼ 0, hξðtÞ, ξð0Þi ¼ δðtÞ. D is the noise intensity, and δ(t) represents the Dirac equation.
Considering the small parameter requirements of the resonance theory, classical SR excited by fixed frequency signal and noise in Duffing system can be expressed as equation (2) Here, the order of magnitude of fixed system parameters a 1 and b 1 is 1, and the magnitude of the constant signal frequency f 1 is less than 1. This is the small parameter requirement in resonance theory. As is shown in equation (2), stochastic resonance that occurs in bistable system meets the small parameters requirement. Where the system parameters a 1 and b 1 is in order of 1, and the signal frequency f 1 is far less than 1 Hz, and the signal amplitude A is much less than 1. However, equation (1) has large parameters, hence does not meet the small parameters requirements. So, equation (1) cannot directly realize resonance. To make equation (1) meet the small parameters requirements and realize aperiodic resonance, GTVST is proposed in this paper.
Firstly, variable substitution is introduced in equation (3) xðtÞ Here, t and τ denote different time scales, x(t) and z(τ) represent signal s in different time scales, a(t) and b(t) stand for the parameters in time-varying Duffing system. The scale coefficient m 0 is a fixed positive value, the timevarying instantaneous frequency f(t) is greater than 1, so m(t) is always greater than 0.
The above derivation mainly includes two parts. One is the time scale transformation for equation (1), where the time scale is switched from t to τ. The second is the amplitude transformation in scale τ. In fact, the above two parts can be finished in one part, that is the amplitude transformation in scale t, and get equation (6) Here, a(t), b(t), and f(t) are time-varying large parameters.
In this section, based on GTVST, the dynamic characteristics of Langevin equation of large parameter variable frequency signal are deduced. The deduction shows that the time-varying system excited by aperiodic signal has the same dynamics characteristics as the classical Duffing system, which proves the realization possibility of GTVST-ASR.
3. Optimal general time-varying scale transformation aperiodic stochastic resonance 3.1. Framework of optimization in general time-varying scale transformation aperiodic stochastic resonance In ASR, parameter optimization often plays an important role when massive parameters are to be determined. Here, some commonly used optimization algorithms include genetic algorithm and quantum swarm algorithms. However, parameter optimization based on above algorithms may fall into local optimum (Cao et al., 2018;Chen et al., 2018). Also, these methods are based on single thread and much computing power are wasted. To solve it, based on global parameter search method and parallel computing, this section introduces the optimal GTVST-ASR. Figure 1 illustrates the flow chart of GTVST-ASR, which includes three parts, namely, GTVST, parameters optimization, and the optimal GTVST-ASR.
Step 1. General time-varying scale transformation. According to the frequency range of signal, decompose the original signal s(t) into filtered pure signal u(t) and noise n(t) by band-pass filter. Then, amplify u(t) and n(t) by m(τ) and ffiffiffiffiffiffiffiffiffi ffi mðτÞ p times, respectively.
Step 2.1. Optimization of m 0 . Based on different m 0 , calculate the GTVST-ASR output with different a 1 and b 1 in all cores, and obtain the cross-correlation coefficient (C) between the output x(t) and pure signal u 0 (t). Chose the m 0 as the optimal parameters when double well resonance area becomes the main part. The C is defined as follows Here, u0(i) and x(i) represent the pure signal and the resonance output, respectively, u 0 and x represent their mean value.
Step 2.2. Optimization of a 1 and b 1 . Based on the optimal m 0 and all combinations of a 1 and b 1 , obtain the outputs and C, choose the a 1 and b 1 as the optimal when C reaches the peak in double well resonance region.
Step 3. Obtain the optimal GTVST-ASR output. Based on the optimal m 0 , a 1 , and b 1 , obtain the optimal GTVST-ASR output.

Parameter optimization
3.2.1. General time-varying scale transformation of simulation signal. Firstly, the pure signal u 0 (t) is defined as follows Here, A and f(t) represent the amplitude and instantaneous frequency, respectively. The amplitude A is 0.5, the sampling frequency f s is 10,000 Hz, the signal length N is 20,000, and f(t) is defined in equation (9) f ðtÞ ¼ 20t þ 50 (9) To simulate a strong noise background, add white Gaussian noise into u 0 (t) and obtain a noisy signal s(t) with SNR of À25 dB.
According to f(t), decompose the noisy signal s(t) into filtered pure signal u(t) and noise n(t) with band-pass filter, the filter center frequency of which is equal to f(t).  firstly. The indicators used here is C, which has been defined in equation (7). Figure 3 shows the migration of resonance region when m 0 takes 5, 20, 50, and 100. In Figure 3(a), m 0 = 5, the verification shows that resonance occurs in single potential well when C reaches the peak. In Figures 3(b) and (a) similar result occurs when m 0 = 20, but one thing different is that another resonance region appears when a 1 →0 + , which is proved to be double well resonance region. In Figure 3(c), m 0 = 50, the single potential well resonance region migrates in the negative direction of the b 1 -axis, and double potential well resonance region takes more region of 0 < a 1 < 1 and b 1 > 2. In Figure 3(d), m 0 = 100, single potential well resonance region shrinks to the finite region of 0.5 <a 1 < 1.5, b 1 < 2, but double potential well resonance region takes up most of the region of 0 < a 1 < 1, b 1 > 2, and it can ensure C > 0.9.
To limit the divergence within a finite parameter region, take 100 as the optimal m 0 .
3.2.3. Optimization of a 1 and b 1 . In ahead section, m 0 has been optimized and set to 100, when the double well resonance region mainly exists in 0 < a 1 < 1, 2 < b 1 < 10. Within this region, take 0.01 and 0.1 as the step of a 1 and b 1 , respectively. Figure 4 shows the surf plot of C in the a 1 -b 1 plane and its projection at C = À0.5. Resonance region distribution of system parameters a 1 and b 1 presents two peaks, namely, double well resonance region and signal well resonance region. Previous section has shown the shortcomings of signal well resonance. So, parameters in double well resonance region are more recommended. In double well resonance region, the contour of C = 0.9 divides a parameter area, C reaches the maximum of 0.92,511 when a 1 = 0.1, b 1 = 6.3.
Based on the optimal parameters that a 1 = 0.1, b 1 = 6.3, m 0 = 100, GTVST-ASR of s(t) occurs, and the output x(t) is drawn in Figure 5. Compared to Figure 2(b), the waveform  amplitude in Figure 5(a) is more stable. Also, the characteristic curve in Figure 5(b) is more centralized and continuous than that of Figure 2(c) and Figure 2(d). Therefore, a conclusion is drawn that GTVST-ASR can enhance the time-frequency characteristics of non-stationary signal under strong noise.

Comparison of aperiodic resonance methods
The above chapters introduce the theoretical framework and implementation method of optimal GTVST-ASR, and the advantages of this method need to be proved. In this section, the proposed method is used, and classical SR and VPNSR are selected as comparison methods to process the frequency conversion signal under strong noise background. The indexes SNR input and the SNR output are defined in equation (10) 8 > > > > > > > > > > > > > < > > > > > > > > > > > > > : where u 0 , s, x are pure signal, noisy signal, and resonance output, respectively.
The signal processed in this section is linear frequency modulation signal, which is defined in equation (8) and equation (9). The system parameter a 1 of GTVST-ASR and classical SR is set to 0.1, and a = b = b 1 f(t) in VPNSR. Here, 0.1≤b 1 ≤10, step size is 0.1. Figure 6 (a) and (b) show the curves of C and SNR output versus the parameter b 1 when the SNR input = À10 dB. It shows that the proposed method can obtain the maximal value C = 0.93 and max SNR output = 8 dB, which are always greater than that of classical SR and VPNSR. Figure 6(c) shows the maximal value C when SNR input takes different values in [À25 dB, 0 dB]. In Figure 6(c), GTVST-ASR can always obtain a well output with C > 0.9. However, the classical SR and VPNSR are greatly affected by noise and the output quality is unstable.

Experimental verification
Fault bearing vibration signal mainly includes bearing natural vibration signal, fault characteristic signal, and environmental noise. Based on stochastic resonance, fault diagnosis of bearing under stable working condition has been realized, but the fault diagnosis of bearing at variable speed is still difficult. To verify the value of the GTVST-ASR in engineering application, faulty bearing vibration signal is processed, which is collected from bearing testbed of China University of Mining and Technology. Figure 7 shows the testbed, it includes driving and loading parts, bearing experiment module, signal measurement, acquisition system, etc. The driving motors model are 198BGL-H5P515/120 and the rated power is 5.5 kW. The models of signal acquisition card and acceleration sensor are NI-9234 and DH-1A206 E, respectively. As shown in Figure 7(c), on the outer ring of N306 E faulty bearing, the scratch fault is artificially machined, the circumferential width, axial width, and radial depth of which is 1.2 mm, 19 mm, and 0.5 mm, respectively.
Types of speed change include linear acceleration and nonlinear speed modulation. Its instantaneous rotation frequency f r1 and f r2 follow equation (10) and equation (11), respectively.
To extract the fault characteristic frequency f c , fault order is necessary. Fault order is the ratio of the fault characteristic frequency f c to the rotation frequency f r , which is defined in equation (12) (Huang et al., 2018;Yang et al., 2018) order Here, structural parameters z, d, D m , and α stand for number of rolling elements, rolling element diameter,   1,10]. In (c), SNR2[À25 dB, 0 dB], a 1 = 0.1. Here, C 1 , C 2 , C 3 , SNR 1 , SNR 2 , and SNR 3 are cross-correlation coefficient and signal-to-noise ratio of the proposed general time-varying scale transformation aperiodic stochastic resonance, varying parameters and normalization stochastic resonance, and classical stochastic resonance, respectively.
bearing raceway pitch diameter, and contact angle, respectively. These parameters of N306 E are listed in Table 1.
After calculation, the order of outer ring fault equals 4.44, substitute it into equation (7), the characteristic frequency f c of outer ring fault can be obtained in equation (13) and equation (14).
2t þ 100 t 2 ½0; 1:5 133:3 t 2 ½1:5, 2 (14) Figure 8 shows the GTVST-ASR outputs of faulty bearing signal under linear acceleration working conditions. In Figure 8(a), bearing signal contains strong noise and there is no obvious regular impact. In Figure 8(c), the timefrequency information of bearing fault is completely submerged by strong noise. After processed by GTVST-ASR, the output shows obvious and regular time domain waveform in Figure 8(b). Also, Figure 8(d) shows a clear curve in time-frequency diagram, and the time-frequency characteristics of this curve is exactly corresponded to the f c1 of outer ring fault. Figure 9 shows the GTVST-ASR outputs of faulty bearing signal under nonlinear speed modulation working condition. Although the working condition is different from that in Figure 8, the conclusions are similar. GTVST-ASR also shows excellent results in processing signals under complex working condition.

Discussions
In GTVST-ASR, instantaneous frequency of the target signal is necessary. It means that the instantaneous rotational speed is required when GTVST-ASR is applied to signal analysis of rotating machinery. But in some cases, this requirement cannot be met because not every machine has a speed measuring device. Fortunately, the measurement of mechanical instantaneous rotation frequency is not difficult, which is a very mature technology. Also, for many large machines, instantaneous rotation frequency measuring device is a common configuration. Therefore, the above requirements are reasonable and achievable, and it will not limit the application and promotion of GTVST-ASR.
In GTVST-ASR, the signal and noise are separated and multiplied by different multiples. It is undeniable that this operation will directly improve the SNR, weaken the strong noise. However, the above operations are completely based on the derivation, not opportunism. The results also show that this method can significantly enhance the signal and obtain a output with more clearer fault characteristics.

Conclusions
In this paper, GTVST-ASR is proposed to extract and enhance aperiodic signals under strong noise. Compared with our previous research work on ASR, some theoretical innovation and interesting improvement are obtained.
Theoretical derivation shows this proposed GTVST can realize ASR in parameter time-varying nonlinear system, its dynamic characteristics are similar to those of traditional  SR. Then, the numerical analysis results show that GTVST-ASR can enhance noisy variable frequency signal. Comparison with the existing ASR methods shows that the proposed method can provide higher quality output with higher SNR and C under the same noise background. At the same time, the proposed method can be applied to stronger noise background and has higher noise robustness. Although the strong noise covers the signal in the spectrum and time-frequency diagram, the proposed method can still accurately recover the signal, but other existing ASR methods cannot achieve the same effect.
In addition, experimental verification shows that this method can be applied to rotating machinery signal processing under complex variable speed conditions. The time domain and time-frequency domain features of bearing fault signal can be extracted and obviously enhanced.