Micro-Vibration Signal Denoising Algorithm of Spectral Morphology Fitting Based on Variational Mode Decomposition

Abstract

Environmental micro-vibration has a significant impact on the proper functioning of semiconductor production and testing equipment such as the Czochralski growth furnace, polishing machine, photoetching machine, scanning electron microscope, etc. Low-frequency micro vibration has a significant influence on the normal operation of high-precision machining and testing equipment, and even causes irreversible damage to the equipment. Therefore, the micro-vibration test has important theoretical significance and engineering value for guiding the vibration isolation design of an electronic industrial workshop and ensuring the stable operation of various precision equipment in the workshop. As the observed acceleration signal is affected by noise introduced by the acceleration sensor itself, the signal processing circuit, the external power supply and interference from environmental factors, direct integration operations can lead to problems such as baseline drift and signal distortion in the calculation results. Aiming at the problem of noise interference in the micro-vibration measurement process, this paper proposed a micro-vibration signal denoising algorithm of spectral morphology fitting based on variational mode decomposition. The observed acceleration signal is decomposed into several orders of finite bandwidth intrinsic mode function components. The asymmetric Gaussian mixture model is used to complete the fitting of the spectral curves of each order of intrinsic mode function components. We can obtain the relevant parameters of the asymmetric Gaussian mixture model curves to complete the division of the effective information frequency band and finally achieve the denoising of the observed acceleration signal. Finally, the algorithm of this paper is compared with traditional denoising algorithms through numerical simulation examples and comparative experiments on denoising effects. The results show that the proposed algorithm has higher accuracy and anti-noise ability.

1. Introduction

With the rapid growth in the field of microelectronics production and testing, industrial manufacturing is increasingly developing in the direction of high precision and ultra-high precision, and the demand for high-precision electronic industrial workshops (microelectronics, optical instruments, lasers, nanomaterials, etc.) is on the rise with each passing day [1]. Precision equipment in electronic industry workshops such as microelectronics, semiconductors and precision instruments have extremely demanding requirements for micro-vibration [2]. As stipulated in the national standard document GB51076-2015, the allowable vibration velocity value of the photoetching equipment and etching equipment in the 5 nm production line in the frequency band of 1–100 Hz is only 1.60 $\mathsf{\mu }$m/s [3]. Micro-vibration measurement and evaluation have strict requirements for measuring equipment. For example, Gao Guangyun used the TROMINO microseismograph of Micromed Company to realize the evaluation of micro-vibration characteristics of the workshop [4,5]. Vladimir measured external background noise vibrations using an eight-channel vibration analyzer LMS Scadas Mobile-I, and highly sensitive ICP accelerometers AP2006 and BC-130 [6]. lgen et al. built a single-degree-of-freedom vibration isolation system (SDOF) to assess the vibration isolation performance of a specially designed structure [7]. Masoud et al. at Tufts University used MEGGITT’s piezoelectric accelerometer 731A-P31 to measure environmental micro-vibration of building foundations caused by ground trains and subways [8].

Vibration displacement and velocity are among the key indicators in environmental micro-vibration evaluation of electronic industrial workshop [3]. For example, VC vibration standard, which is widely used in the field of microelectronics production, uses the root mean square (RMS) value of vibration velocity as an evaluation indicator [9,10]. In the micro-vibration test, due to transportation, maintenance, installation, parameter limitation and other factors, the vibration sensor which directly measures the vibration velocity cannot be applied to the actual measurement of environmental micro-vibration signals. Acceleration sensors are used to measure environmental micro-vibration signals. The micro-vibration velocity and displacement signals are obtained by means of analogue circuits or digital signal processing. However, the raw acceleration signals are often mixed with multiple types of noise. As a result, the velocity and displacement signals obtained through integration contain serious drift and trend term errors [11,12,13]. The micro-vibration measurement system established in this paper is used to obtain the observed acceleration signal. Figure 1 shows the velocity spectrum of micro-vibration with low-frequency noise obtained by direct integration. How to overcome the signal drift problem after integration while extracting and suppressing the trend term error is an important research direction of integration algorithms [14].

The integration methods of the micro-vibration signal denoising algorithm can be divided into two categories: a time-domain integration method and a frequency-domain integration method [15]. Eliminating trend term error is an important technique to suppress noise interference in time-domain integration method. The main methods for eliminating trend term error in vibration signals mainly include the difference method, the low-pass filtering method, the least square method, the wavelet transform method and empirical mode decomposition method [16,17,18]. Using the difference method, low-pass filtering method and least square method to eliminate trend term errors usually requires a prior assumption about the type of trend term in the signal, such as linear trend terms, polynomial and exponential trends, etc. We need to understand the characteristics of the trend term contained in the mic-vibration signal in advance. Eliminating trend term error by wavelet transform has strict requirements on the selection of wavelet basis and the determination of decomposition depth. Compared with the previous methods, the empirical mode decomposition method does not consider the trend term type and is more adaptive [19]. The variational mode decomposition algorithm (VMD) is introduced in this paper to avoid problems such as endpoint effect, over-envelope, under-envelope and false component in the iterative process of the traditional empirical mode decomposition method [20]. The algorithm can decompose a signal into a set of finite bandwidth intrinsic mode function components. Compared with the traditional empirical mode decomposition method, the VMD algorithm is more robust to noise.

It is found that although the time-domain integration method is more intuitive in the form of expression, the denoising effect of the time-domain integration method is not ideal when the complex noise is aliased in the acceleration signal. Therefore, in engineering practice, the frequency domain integration method is widely employed to complete micro-vibration signal denoising, among which the frequency cutoff algorithm, low frequency attenuation algorithm and effective frequency band selection algorithm are more typical. As the trend term error is mostly presented as low-frequency and slowly changing signals, the frequency cutoff algorithm [21] and low-frequency attenuation algorithm [22] both select the upper frequency of the trend term error and then nullify the frequency information between the zero frequency and the upper frequency (frequency cutoff) or attenuation suppression (low-frequency attenuation). Both algorithms have significant denoising effects for trend term errors, but the processing effect is not ideal for the noise randomly distributed in other frequency bands. To solve this problem, Chen Taicong [23] proposed an acceleration integration method based on spectral energy morphology fitting. He used Gaussian curve to fit the spectrum curve of the frequency band near the main frequency in the spectrum curve and determined the effective frequency band range of the main frequency according to the principle of triple standard deviation to achieve the denoising effect. However, there are some problems in the symmetric Gaussian curve spectrum shape fitting denoising algorithm, such as artificially dividing the peak acceleration interval, the great influence of noise outside the main frequency, the poor flexibility of the algorithm and the lack of effective information of the main frequency signal.

Aiming at the problem of micro-vibration measurement in the vibration-sensitive area of the electronic industrial workshop, this paper accurately measured the micro-vibration in the vibration-sensitive area to guide the vibration isolation design. To address the problem that trend term errors and random noise interfere with micro-vibration measurements, this paper proposed a micro-vibration signal denoising algorithm of spectral morphology fitting based on variational mode decomposition. The spectral curve of the effective frequency band is fitted by the asymmetric Gaussian mixture model curve and finally complete the denoising of the vibration signal. The algorithm can effectively suppress the trend term error and random noise interference, and better integration accuracy and anti-noise ability for asymmetric acceleration spectrum.

2. Micro-Vibration Measurement System

Figure 2 shows the schematic diagram of the triaxial micro-vibration real-time measurement system. In view of the characteristics of low amplitude of micro-vibration and the main energy being concentrated in the low-frequency band, three uniaxial acceleration sensors are selected to form a triaxial sensor module.

Due to the low amplitude of the micro-vibration signal, the peak-to-peak value of the acceleration is generally around 200 μg. The triaxial micro-vibration real-time measurement system includes a signal conditioning module composed of a filter circuit, an active amplifier circuit and relay components. The digital circuit used FPGA to complete the three-channel AD conversion control. It achieved synchronous acquisition of each channel and improved the real-time performance of the measurement system. The measurement system can accurately measure the micro-vibration signal within the frequency range of 0.5–200 Hz.

The measurement signal is interfered with by the noise introduced by amplifying the circuit and external power supply. Figure 3 shows the noise randomly distributed over the measurement band. In order to solve the problem that the original acceleration signal is disturbed by noise, this paper proposed a micro-vibration signal denoising algorithm of spectral morphology fitting based on variational mode decomposition and applied it to the micro-vibration measurement system. We can complete the accurate measurement and grade evaluation of the micro-vibration signal.

3. Methodology

When the VMD algorithm is used to preprocess the micro-vibration signal, the VMD algorithm is first applied to the modal decomposition of the acceleration data. According to the improved VMD algorithm mentioned in Reference [20], the measured micro-vibration acceleration signal is decomposed into several orders of finite bandwidth eigenmode function component. Then, the frequency of each order of intrinsic mode function components is extracted by power spectral density to determine the frequency range of each order of intrinsic mode function components and eliminate the noisy modes. Figure 4 shows the measured micro-vibration acceleration signal, which is decomposed into several order intrinsic mode function components by the variational mode decomposition algorithm. Figure 5 shows the spectral diagram of each order of intrinsic mode function signals.

As can be seen from Figure 5, the essence of the VMD algorithm is to decompose the acceleration signal into several intrinsic mode function components of different frequency bands. Each order of intrinsic mode function components obtained by the VMD decomposition algorithm have noise in other frequency bands with small amplitude. In this paper, we proposed a micro-vibration signal denoising algorithm of spectral morphology fitting based on variational mode decomposition (AGMD). The steps of the algorithm are as follows.

3.1. Segmentation of Peak Region of Acceleration Spectrum

Adaptive recognition and segmentation of peak regions is the basis of the algorithm applied to real-time measurement systems. Considering the noise characteristics of environmental micro-vibration signals and the real-time requirements of the measurement system, an efficient and automatic signal contraction peak detection algorithm is adopted [24]. The signal contraction peak detection algorithm can effectively suppress the fluctuation noise interference in low-amplitude signals. The algorithm can avoid the influence of “burr” and “false peak” phenomena in the observed acceleration spectrum curve on the segmentation of the peak region [25].

Assuming the given acceleration signal time series is $S\left(n\right)=\{S_1,S_2,S_3,\dots ,S_N\}$, $n=1,2,3,\dots ,N$. Firstly, we need to find the local minimum and local maximum values of the acceleration signal $S\left(n\right)$ and their corresponding positions. The condition for wave peak points is as follows:

$$P\left(n\right)=S\left(n\right)ifS(n-1)<S\left(n\right)>S(n+1)$$

(1)

$$L_p\left(i\right)=nifS(n-1)<S\left(n\right)>S(n+1)$$

(2)

where $P\left(n\right)$ is the wave peak at the time series $S\left(n\right)$ of the nth acceleration signal, $L_p\left(i\right)$ is the position corresponding to the local wave peak $P\left(n\right)$, $i=1,2,3,\dots ,m$, and m is the number of local wave peaks (valleys).

The condition for wave valley points is as follows:

$$V\left(n\right)=S\left(n\right)ifS(n-1)>S\left(n\right)<S(n+1)$$

(3)

$$L_v\left(j\right)=nifS(n-1)<S\left(n\right)>S(n+1)$$

(4)

where $V\left(n\right)$ is the wave valley at the time series $S\left(n\right)$ of the nth acceleration signal, $L_v\left(j\right)$ is the position corresponding to the local wave valley $V\left(n\right)$, and $j=1,2,3,\dots ,m$;

Secondly, the value of valley peak difference ($VPD$) was calculated. The false peak points were screened out by judging the value of $VPD$. When calculating the value of $VPD$, make sure that the processing starts from the wave valley. The position of the first wave peak and the first wave valley are compared. If the wave peak appears first, the data are discarded to ensure that the vibration signal is obtained from the wave valley first. This means that the discarded wave peaks have no corresponding wave valleys to ensure that the numbers of wave peaks and wave valleys are the same. The expression of $VPD$ are as follows:

$$VPD\left(k\right)=P\left(k\right)-V\left(k\right)$$

(5)

where $P\left(k\right)=\{P_1,P_2,P_3,\dots ,P_m\}$ is the set of all local wave peaks, $V\left(k\right)=\{V_1,V_2,V_3,\dots ,V_m\}$ is the set of all local wave valleys, $VPD\left(k\right)$ is the difference between the value of the kth wave peak and the value of the kth wave valley, and $k=1,2,3,\dots ,m$.

After the calculation of $VPDs$, the algorithm searches the value of $VPDs$. If the value of $VPDs$ satisfies the relationship shown in Equation (6), this wave peak is removed. Then, the corresponding local wave peak $P\left(n\right)$ and the position information corresponding to the wave peak $L_p\left(i\right)$ is removed. Repeating this $VPD$ processing until the number of wave peaks in two consecutive iterations remains the same.

$$VPD\left(k\right)<\theta \left\{VPD\right(k-1)+VPD(k)+VPD(k+1\left)\right\}/3$$

(6)

where $\theta$ is the weight parameter, the value of $\theta$ is confirmed by reference or experimental data.

In order to facilitate the subsequent analysis and reduce the influence of noise fluctuations on the fitting results, the amplitude and frequency of the micro-vibration in the N-segment peak region were normalized, respectively, to obtain the normalized acceleration spectrum ${\overline{A}}_i$($\overline{f}$), ${\overline{A}}_i=\frac{A_i}{{\sum }_{i=1}^n{A}_i\in (0,1)}$, $i=1,2,3,\dots ,n$, and $\overline{f}$ is the normalized frequency.

3.2. Assumption of Curve Morphology in Peak Region of Acceleration Spectrum

The Gaussian distribution curve can represent the data by the mean and covariance of the data. However, because the data are symmetric according to the mean and have a fixed shape [26], Gaussian distribution is often unsuitable for fitting the real vibration signal data. In order to overcome the limitations of Gaussian distribution curves, the asymmetric generalized Gaussian distribution (AGGD) is proposed to represent asymmetric Gaussian data [27]. It also has a good effect in simulating the asymmetric pattern and properties of the curve in the peak region of the acceleration spectrum. The expression of the asymmetric generalized Gaussian curve is as follows:

$$f\left(X\right|\mu ,{\sigma }_l,{\sigma }_r,\lambda )=\frac{\lambda {\left[\frac{\mathrm{\Gamma }(3/\lambda )}{\mathrm{\Gamma }(1/\lambda )}\right]}^{1/2}}{\left({\sigma }_l+{\sigma }_r\right)\mathrm{\Gamma }(1/\lambda )}\left\{\begin{array}{c}\exp \left[-A\left(\lambda \right){\left(\frac{\mu -X}{{\sigma }_l}\right)}^{\lambda }\right]ifX<\mu \hfill \\ \exp \left[-A\left(\lambda \right){\left(\frac{X-\mu }{{\sigma }_r}\right)}^{\lambda }\right]ifX\ge \mu \hfill \end{array}\right.$$

(7)

where $\mu$ is the mean, ${\sigma }_l$ is the left standard deviation, ${\sigma }_r$ is the right standard deviation, $\lambda$ is the shape parameter, $A\left(\lambda \right)$ is the function of $\lambda$ and the expression is as follows:

$$A\left({\lambda }_{ij}\right)={\left[\frac{\mathrm{\Gamma }(3/\lambda )}{\mathrm{\Gamma }(1/\lambda )}\right]}^{\lambda /2}$$

(8)

where $\mathrm{\Gamma }(.)$ is the gamma function, and the expression is as follows:

$$\mathrm{\Gamma }\left(x\right)={\int }_0^{\infty }{t}^{x-1}{e}^{-t}dtx>0$$

(9)

The asymmetric generalized Gaussian distribution curve introduces three parameters: shape parameter $\lambda$, left standard deviation ${\sigma }_l$ and right standard deviation ${\sigma }_r$ to enhance the flexibility of fitting the shape of the curve, which can better fit the asymmetric data. The shape of the curve is altered by changing the value of the shape parameter to make it flatten or sharpen. Figure 6 shows the influence of different parameters of asymmetric generalized Gaussian curve distribution on the shape of the curve.

In the actual acceleration signal data processing, the acceleration spectrum curve cannot be simply represented by a single asymmetric generalized Gaussian distribution curve. The asymmetric generalized Gaussian distribution lacks flexibility in fitting the real acceleration signal data. The asymmetric Gaussian mixture model [28] not only effectively solves the problem of the limitation of the curve shape of Gaussian distribution, but is also more flexible in fitting the actual acceleration signal data and can fit the asymmetric data better and faster. The expression of the asymmetric Gaussian mixture model is as follows.

$$f\left(X\right|{\xi }_j)=\prod _{i=1}^n\prod _{j=1}^m\frac{{\lambda }_{ij}{\left[\frac{\mathrm{\Gamma }(3/{\lambda }_{ij})}{\mathrm{\Gamma }(1/{\lambda }_{ij})}\right]}^{1/2}}{\left({\sigma }_{l_{ij}}+{\sigma }_{r_{ij}}\right)\mathrm{\Gamma }(1/{\lambda }_{ij})}\left\{\begin{array}{c}\exp \left[-A\left({\lambda }_{ij}\right){\left(\frac{\mu -X_i}{{\sigma }_{l_{ij}}}\right)}^{{\lambda }_{ij}}\right]if{X}_i<\mu \hfill \\ \exp \left[-A\left({\lambda }_{ij}\right){\left(\frac{X_i-\mu }{{\sigma }_{r_{ij}}}\right)}^{{\lambda }_{ij}}\right]if{X}_i\ge \mu \hfill \end{array}\right.$$

(10)

where ${\xi }_j=\{{\mu }_j,{\sigma }_{l_i},{\sigma }_{r_i},{\lambda }_j\}$ is the set of parameters of the jth term mixture distribution, $j=1,2,3,,\dots ,,m$, ${\mu }_j=\{{\mu }_{j1},{\mu }_{j2},{\mu }_{j3},\dots ,{\mu }_{ij}\}$ is the set of means of the jth term mixture distribution, $i=1,2,3,\dots ,n$, ${\sigma }_{l_i}=\{{\sigma }_{l_{i1}},{\sigma }_{l_{i2}},{\sigma }_{l_{i3}},\dots ,{\sigma }_{l_{ii}}\}$ is the set of left standard deviations of the jth term mixed distribution, ${\sigma }_{r_i}=\{{\sigma }_{r_{i1}},{\sigma }_{r_{i2}},{\sigma }_{r_{i3}},\dots ,{\sigma }_{r_{ii}}\}$ is the set of right standard deviations of the jth term mixed distribution, ${\lambda }_j=\{{\lambda }_{j1},{\lambda }_{j2},{\lambda }_{j3},\dots ,{\lambda }_{ij}\}$ is the set of shape parameters of the jth term mixture distribution, $A\left({\lambda }_{ij}\right)$ is the function of ${\lambda }_{ij}$ and the expression is as follows:

$$A\left({\lambda }_{ij}\right)={\left[\frac{\mathrm{\Gamma }(3/{\lambda }_{ij})}{\mathrm{\Gamma }(1/{\lambda }_{ij})}\right]}^{{\lambda }_{ij}/2}$$

(11)

3.3. Parameter Estimation for Curve Fitting of Asymmetric Gaussian Mixture Model

Optimization methods such as least squares, Newton’s method, gradient descent and conjugate gradient have been developed to solve the problem of fitting non-linear curves. However, these methods have drawbacks such as a large amount of computation, requirement for gradient information and unsatisfactory fitting effect for high-dimensional complex functions. In this paper, the whale swarm optimization algorithm [29] is introduced to fit the parameters.The whale swarm optimization algorithm has the advantages of simple mechanism and few parameters and strong optimization ability. It has higher accuracy and optimization speed when fitting parameters of non-linear complex functions.

As shown in Figure 7, the whale swarm optimization algorithm is prone to fall into the local optimal solution when searching for the optimal solution, which leads to poor curve fitting effect. In this paper, according to the properties of the asymmetric Gaussian mixture model, adaptive weights are introduced to improve the convergence speed and global optimization ability of the algorithm.

By introducing the weight parameter $\omega$, according to the properties of acceleration micro-vibration signal data and Equation (9), the normalized acceleration spectrum curve expression can be obtained as follows:

$$\left\{\begin{array}{c}f\left(X\right|{\xi }_k,{\omega }_k)={\displaystyle \prod _{k=1}^n}\frac{{\omega }_k{\lambda }_k{\left[\frac{\mathrm{\Gamma }(3/{\lambda }_k)}{\mathrm{\Gamma }(1/{\lambda }_k)}\right]}^{1/2}}{\left({\sigma }_{l_k}+{\sigma }_{r_k}\right)\mathrm{\Gamma }(1/{\lambda }_k)}\left\{\begin{array}{c}\exp \left[-{\left[\frac{\mathrm{\Gamma }(3/{\lambda }_k)}{\mathrm{\Gamma }(1/{\lambda }_k)}\right]}^{{\lambda }_k/2}{\left(\frac{\mu -X}{{\sigma }_{l_k}}\right)}^{{\lambda }_k}\right]ifX<\mu \hfill \\ \exp \left[-{\left[\frac{\mathrm{\Gamma }(3/{\lambda }_k)}{\mathrm{\Gamma }(1/{\lambda }_k)}\right]}^{{\lambda }_k/2}{\left(\frac{X-\mu }{{\sigma }_{r_k}}\right)}^{{\lambda }_k}\right]ifX\ge \mu \hfill \end{array}\right.\hfill \\ \sum _{k=1}^n{\omega }_k=1\hfill \end{array}\right.$$

(12)

where ${\xi }_k=\{{\mu }_k,{\sigma }_{l_k},{\sigma }_{r_k},{\lambda }_k\}$ represents the set of parameters of the kth-term mixture distribution, $k=1,2,3,\dots ,n$, ${\mu }_k$ represents the mean of the kth-term mixture distribution, ${\sigma }_{l_k}$ represents the left standard deviations of the kth-term mixed distribution, ${\sigma }_{r_k}$ represents the right standard deviations of the kth-term mixed distribution and ${\lambda }_k$ represents the shape parameters of the kth term mixture distribution.

The first absolute moment is defined as $m_a^1\triangleq E\left\{\right|A_i\left|\right\}$. The second origin moment is defined as $m_0^2\triangleq E\left\{A_i^2\right\}$. The ratio of the two is defined as $p=\frac{{\left(m_a^1\right)}^2}{m_0^2}$. Lasmar [30] summarized the relationship between statistical data and various parameters of the curve and obtained the relationship between ratio p and curve parameters, as follows:

$$q\left(\lambda \right)=\frac{({\gamma }^3+1)(\gamma +1)}{{({\gamma }^2+1)}^2}\rho \left(\lambda \right)$$

(13)

where $\gamma =\frac{{\sigma }_l}{{\sigma }_r}$, and $q\left(\lambda \right)=\frac{\mathrm{\Gamma }{(2/\lambda )}^2}{\mathrm{\Gamma }(1/\lambda )\mathrm{\Gamma }(3/\lambda )}$ is the generalized Gaussian ratio function. The shape parameter $\lambda$ has an estimated value $\widehat{\lambda }$ of two, according to the properties of the vibration signal. The estimated value of the weight parameter ${\omega }_k$ is expressed as follows:

$${\widehat{\omega }}_k={\int }_{-1}^1{\overline{A}}_k\left(\overline{f}\right)d\overline{f}$$

(14)

where ${\widehat{\omega }}_k$ is the estimated value of the weight parameter ${\omega }_k$ of the kth term, ${\omega }_k$ is the normalized acceleration spectrum curve of the kth term and $\overline{f}$ is the normalized frequency. The estimated values of left standard deviation ${\sigma }_{l_k}$ and right standard deviation ${\sigma }_{r_k}$ of Equation (15) are expressed as follows:

$$\widehat{{\sigma }_{l_k}}=\widehat{{\sigma }_{r_k}}=\frac{1}{2}\frac{{\widehat{\omega }}_k{\widehat{\lambda }}_k}{\max \left({\overline{A}}_k\left(\overline{f}\right)\right)\mathrm{\Gamma }(1/{\widehat{\lambda }}_k)}{\left[\frac{\mathrm{\Gamma }(3/{\widehat{\lambda }}_k)}{\mathrm{\Gamma }(1/{\widehat{\lambda }}_k)}\right]}^{1/2}$$

(15)

where $\widehat{{\sigma }_{l_k}}$ is the estimated value of left standard deviation, $\widehat{{\sigma }_{r_k}}$ is the estimated value of right standard deviation, ${\widehat{\lambda }}_k$ is the estimated value of shape parameter and $\max \left({\overline{A}}_k\left(\overline{f}\right)\right)$ is the maximum of normalized acceleration spectrum curve. The estimated values of the above parameters were used as adaptive weights to narrow the retrieval range. The convergence speed and global optimization ability of the algorithm are effectively improved. Figure 8 shows the fitting effect of the improved algorithm.

3.4. Determine the Effective Master Frequency Band

According to the estimation method of non-linear curve fitting parameters based on whale swarm optimization algorithm described in this paper, the peak signal of acceleration spectrum can be fitted. In order to ensure that the information of the effective frequency band is not lost, the effective main frequency band should be delimited first. The upper and lower frequencies of the effective main frequency band should be defined as follows:

$$\left\{\begin{array}{c}c·\max \left({\overline{A}}_k\right)=\max \left({\overline{A}}_k\right)·\exp \left[-{\left(\frac{-\left(\overline{f}-X\right)}{{\sigma }_{l_k}}\right)}^{{\lambda }_k}\right]X<\overline{f}\hfill \\ c·\max \left({\overline{A}}_k\right)=\max \left({\overline{A}}_k\right)·\exp \left[-{\left(\frac{X-\overline{f}}{{\sigma }_{r_k}}\right)}^{{\lambda }_k}\right]X\ge \overline{f}\hfill \end{array}\right.$$

(16)

where c is the frequency band division coefficient. Generally, it will be selected in the range of 0.05–0.5% according to the main frequency signal shape. The upper and lower limit frequency of the effective main frequency band are as follows:

$$\left\{\begin{array}{c}\overline{f_l}=\xi \left(c;{\lambda }_k,{\sigma }_{l_k}\right)X<\overline{f}\hfill \\ \overline{f_r}=-\xi \left(c;{\lambda }_k,{\sigma }_{r_k}\right)X\ge \overline{f}\hfill \end{array}\right.$$

(17)

where $\xi \left(•\right)$ is the monotonically decreasing function of the frequency band division coefficient, including parameters ${\lambda }_k$, ${\sigma }_{l_k}$ and ${\sigma }_{r_k}$. The actual integrated frequency band is as follows:

$$\left[\underline{\underline{f_k}}+\frac{1}{2}(\overline{\overline{f_k}}-\underline{\underline{f_k}})·(1-\overline{f})),\overline{\overline{f_k}}+\frac{1}{2}(\overline{\overline{f_k}}-\underline{\underline{f_k}})·(\overline{f}-1))\right]$$

(18)

where $\overline{\overline{f_k}}$ is the upper limit frequency of the peak region, and $\underline{\underline{f_k}}$ is the lower limit frequency of the peak region.

Figure 9 shows the flow chart of the denoising algorithm of spectral shape fitting of asymmetric Gaussian mixture model curves. The algorithm can automatically identify and divide the non-noise frequency band according to the curve fitting of the main frequency signal of the observed acceleration spectrum, thereby suppressing interference from measurement noise outside the main frequency.

4. Experiment and Result

4.1. Numerical Example Simulation Experiments

To verify the denoising effect of the algorithm, 5%, 10%, 15%, 20%, 25%, 30% and 35% white noise were added to the acceleration signal, respectively. The observed acceleration was denoised using the denoising algorithm of the symmetric Gaussian distribution curve spectral morphology fitting(GD), the denoising algorithm of the asymmetric Gaussian distribution curve spectral morphology fitting (AGGD) and the denoising algorithm proposed in this paper (AGMD). The observed acceleration after denoising is integrated in the frequency domain to investigate the denoising effect of different denoising algorithms and the influence on the signal accuracy after integration. The relative error of integration peak (Erp), the relative error of integration difference (Erd) and the relative error of integration average (Era) are introduced as evaluation metrics. The expressions are as follows:

$${\delta }_{Erp}=\frac{1}{2}\times \left[\left|\frac{\max \left(v\left(t\right)\right)-\max \left(v_{real}\left(t\right)\right)}{\max \left(v_{real}\left(t\right)\right)}\right|+\left|\frac{\min \left(v\left(t\right)\right)-\min \left(v_{real}\left(t\right)\right)}{\min \left(v_{real}\left(t\right)\right)}\right|\right]$$

(19)

$${\delta }_{Erd}=\frac{1}{2}\times \left[\left|\frac{\max \left(v\left(t\right)-v_{real}\left(t\right)\right)}{\max \left(v_{real}\left(t\right)\right)}\right|+\left|\frac{\min \left(v\left(t\right)-v_{real}\left(t\right)\right)}{\min \left(v_{real}\left(t\right)\right)}\right|\right]$$

(20)

$${\delta }_{Era}=\frac{{\sum }_{i=1}^N{\left(v_{real}\left(t_i\right)-v\left(t_i\right)\right)}^2}{{\sum }_{i=1}^N{\left(v\left(t_i\right)\right)}^2}$$

(21)

where ${\delta }_{Erp}$ is the relative error of integration peak, ${\delta }_{Erd}$ is the relative error of integration difference, ${\delta }_{Era}$ is the relative error of integration average, $v_{real}$ is the true value of vibration velocity at the measuring point position and v is the reconstructed vibration velocity calculated by denoising algorithm and frequency domain integration.

Figure 10 shows the integration accuracy comparison between the denoising algorithm of spectral morphology fitting based on variational mode decomposition and the denoising algorithm of the symmetric Gaussian curve spectral morphology fitting. Figure 10a shows that the relative error of integration peak of the denoising algorithm proposed in this paper is significantly lower than the denoising algorithm of the symmetric Gaussian curve spectral morphology fitting. The relative error of integration peak of the former is always less than 2.6%. The results show that compared with the other two denoising algorithms, the denoising algorithm proposed in this paper is more accurate in the reduction of signal peak.

Figure 10b shows that the trend in the relative error of the integration difference is essentially the same for both algorithms. When the relative noise energy is relatively low, the relative error of the integral difference of the proposed algorithm is slightly lower than the other two denoising algorithms. The results show that the difference between the denoised signal and the real signal by the two algorithms is relatively stable. The denoising stability of the algorithm is good.

Figure 10c shows that the denoising algorithm proposed in this paper has lower relative error of integration average and higher integration accuracy when integrating the acceleration under different noise relative energy. Compared with the other two denoising algorithms, the average relative error of the proposed algorithm is basically below 2%, which has better integration accuracy.

Figure 10 shows the error between the integral result and the real value. In order to further analyze the denoising effect of the three algorithms, the acceleration spectrum after denoising by the three algorithms and the velocity time history curve after integration can be selected for comparative analysis when the noise relative energy ratio is 30%. Figure 11 shows the comparison results of acceleration spectra. Figure 12 shows the comparison results of velocity time history curves.

4.2. Comparative Experiment of Denoising Effect

In order to verify whether the algorithm can effectively solve the problem that the micro-vibration signal is disturbed by noise, the linear sinusoidal excitation system is used to sinusoidally excite the sensor to output the standard sinusoidal signal. In order to verify whether the algorithm can effectively solve the problem of noise interference in micro-vibration signals, in this paper, we used a linear sinusoidal excitation system to excite the transducer so that it can output a standard sinusoidal signal. The measurement signal is compared and analyzed with that of the laser Doppler vibrometer. Figure 13 shows the schematic diagram of the comparative experimental scheme.

In this paper, the original acceleration signal is denoised by using the denoising algorithm of the symmetric Gaussian curve spectral morphology fitting and the denoising algorithm proposed in this paper. The denoised signal is compared with the output signal of the laser Doppler vibriometer. Because the sampling frequency of the output signal of the laser Doppler vibrometer is different from the sampling frequency of the measurement system set up in this paper, the average error of peak-to-peak vibration velocity is used as the evaluation criterion for the denoising effect. Signal comparison results before and after denoising are shown in

Table 1. Signal comparison results before and after denoising.

Signal Type	Peak-to-Peak Value of Acceleration	Peak-to-Peak Error of Acceleration
Original signal	26.50 mg	27.51%
GD	21.88 mg	5.30%
AGMD	21.09 mg	1.52%
Laser signal	20.78 mg

.

Figure 14 shows the acceleration spectrum of the vibration signal before denoising. In addition to the main frequency signal, it is also mixed with broadband noise randomly distributed in the whole spectrum range. Figure 15 shows the comparison results of acceleration time curves after algorithm denoising. As shown in Table 1, the original signal acceleration time-domain waveform is obviously disturbed by complex noises which are aliased by various frequency noises. The relative error of peak-to-peak acceleration is 27.51%. Using the denoising algorithm of the symmetric Gaussian curve, spectral morphology fitting can complete the separation of the main frequency signal from the noise signal. The fitting curve is symmetric and can only be fitted to the frequency band with a single peak. Therefore, it is inevitable that the radius of the fitted Gaussian curve is different from the spectrum curve of the real signal when the asymmetric main frequency signal is fitted. As a result, the effective frequency band information is lost or redundant noise information is introduced. The peak-to-peak error of acceleration is 5.30%. The proposed denoising algorithm in this paper uses an asymmetric Gaussian mixture model to fit the curve when fitting to an asymmetric main frequency signal. According to the shape of the main frequency signal, the left and right radius of the fitting curve are automatically adjusted, which is more flexible in the curve fitting of the spectrum peak region, so as to achieve better denoising effect. Comparing the acceleration signals obtained before and after denoising with the standard acceleration signals obtained by the laser Doppler vibriometer, it is obvious that the proposed algorithm has a better denoising effect than other denoising algorithms. The peak-to-peak error of acceleration decreased from 27.51% to 1.52% after denoising. The algorithm is applied to the micro-vibration measurement system, which can effectively solve the problem of micro-vibration signal interference by noise.

5. Conclusions

In this paper, we study the signal processing algorithm of micro-vibration measurement, and propose a micro-vibration signal denoising algorithm of spectral morphology fitting based on variational mode decomposition. The variational mode decomposition algorithm cannot effectively identify the low-amplitude noise information. Therefore, the intrinsic mode function components are mixed with low-amplitude noise which is randomly distributed in the whole spectrum range. In this paper, the spectral morphology fitting method of the asymmetric generalized Gaussian mixture model is used to separate the main frequency information from the secondary frequency information and the measurement noise on it, so as to improve the denoising performance of the algorithm. In order to address the problems of nonlinear curve fitting algorithms that tend to fall into local optimum solutions and convergence speed, this paper improves the whale optimization algorithm. The adaptive weight is introduced to improve the convergence speed and global optimization ability of the algorithm. When the parameters of nonlinear complex functions are fitted, the algorithm has higher accuracy. In the fitting of complex micro-vibration acceleration signal data, the algorithm proposed in this paper has higher flexibility. It effectively solves the problem that the effective frequency band information is lost or the excess noise information is introduced due to the inaccurate parameter fitting of the traditional curve fitting algorithm.

Through numerical simulation examples and comparative experimental analysis, the superiority of the proposed algorithm in the denoising effect is verified. The acceleration peak-to-peak error was decreased from 27.51% to 1.52%. The denoising algorithm can effectively solve the problem of noise interference. The system and the denoising algorithm in this paper can realize the real-time and accurate grade evaluation of environmental micro-vibration. It has important significance for the scientific design of semiconductor production line and the safety production of semiconductor products.