CN113269082A - Partial discharge threshold denoising method based on improved variational modal decomposition - Google Patents

Partial discharge threshold denoising method based on improved variational modal decomposition Download PDF

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CN113269082A
CN113269082A CN202110556432.0A CN202110556432A CN113269082A CN 113269082 A CN113269082 A CN 113269082A CN 202110556432 A CN202110556432 A CN 202110556432A CN 113269082 A CN113269082 A CN 113269082A
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陈波
肖洒
沈道贤
刘冬梅
储昭碧
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Hefei University of Technology
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Abstract

The invention relates to a partial discharge threshold denoising method based on improved variational modal decomposition, which comprises the following steps: determining the decomposition layer number K of the variational modal decomposition; optimizing punishment factors corresponding to each layer of mode of variational mode decomposition; after the decomposition parameters are determined, decomposing the local discharge signal by using an optimized variational modal decomposition algorithm to obtain K intrinsic modal components with limited bandwidth, and calculating kurtosis values of the modal components; defining a significant component and a non-significant component of the kurtosis value; and reconstructing the effective component to obtain a reconstructed signal, and removing low-frequency white noise remained in the reconstructed signal to obtain a final denoised partial discharge signal. The invention effectively inhibits periodic narrow-band interference and white noise, reduces the distortion of the local discharge waveform, and completely retains the characteristic information of the local discharge signal.

Description

Partial discharge threshold denoising method based on improved variational modal decomposition
Technical Field
The invention relates to the field of partial discharge detection of high-voltage electrical equipment, in particular to a partial discharge threshold denoising method based on improved variational modal decomposition.
Background
With the increasing demand of electricity in China, the voltage level required by the power grid in China is higher and higher, and therefore, the high-voltage electrical equipment in the power grid needs to be guaranteed to operate stably and reliably. High voltage electrical equipment in electrical power systems, such as: gas insulated metal enclosed switchgear (GIS), power cable, etc., are in the live operating state for a long time, the inevitable insulating fault phenomenon that can appear. Partial Discharge (PD) is an early expression form of an insulation fault of high-voltage electrical equipment, so that the insulation state of the high-voltage electrical equipment can be effectively evaluated by partial discharge detection, potential faults of the equipment can be found in time, and the occurrence of operation faults is reduced. However, in the partial discharge detection, since the partial discharge signal is very weak and there is a serious noise interference in the detection site, the detected partial discharge signal is often submerged in the noise, which is not favorable for the partial discharge detection. Therefore, an effective noise suppression method is crucial to partial discharge detection of high-voltage electrical equipment.
Disclosure of Invention
The invention aims to provide a partial discharge threshold denoising method based on improved variational modal decomposition, and the algorithm process is clear and intuitive, has a good denoising effect and is high in solving speed.
In order to achieve the purpose, the invention adopts the following technical scheme:
a partial discharge threshold denoising method based on improved variational modal decomposition comprises the following steps:
step S1: determining the decomposition layer number K of the variational modal decomposition by utilizing the number of wave crests of the partial discharge signal spectrogram;
step S2: optimization of penalty factor alpha corresponding to each layer mode of variational modal decomposition by utilizing longicorn beard search algorithmK(k＝1,2,...,K)；
Step S3: after the decomposition parameters are determined, decomposing the local discharge signal by using an optimized variational modal decomposition algorithm to obtain K intrinsic modal components with limited bandwidth, and calculating kurtosis values of the modal components;
step S4: defining components with kurtosis values larger than 10 in the step S3 as valid components, and defining the rest as invalid components;
step S5: and reconstructing the effective components to obtain a reconstructed signal, and removing residual low-frequency white noise in the reconstructed signal by using a lifting wavelet threshold method to obtain a final denoised partial discharge signal.
In order to optimize the technical scheme, the specific measures adopted further comprise:
step S2 specifically includes:
s21: after the number of decomposition layers is determined, defining the search dimension of the longicorn whisker algorithm as K and the position of the longicorn as K
Figure RE-GDA0003146913760000021
Wherein N is the iteration number of the search;
s22: kurtosis is an index for evaluating impact property, and the expression is as follows:
Figure RE-GDA0003146913760000022
wherein: μ represents the mean value of the signal, and x represents the time series value of the signal;
defining an objective function f (x) of the longicorn algorithm as the sum of kurtosis of each modal component of the variational modal decomposition under each group of parameters, wherein the expression is as follows:
Figure RE-GDA0003146913760000023
wherein: k is the number of decomposition layers, Ku, obtained in step S1kIs the kurtosis value of the kth component, K ═ 1, 2.
S23: the longicorn whisker algorithm updating formula is as follows:
xn＝xn-1+δnDsign[f(xr)-f(xl)]
wherein: d is a unitized random vector, xrIs the position of the right beard of the longicorn, xlIs the position of the left beard of the longicorn, deltanFor step size of each iteration, f (x)r) Is the target function value of the right beard of the longicorn, f (x)l) The target function value of the longicorn left tassel is shown, sign is a sign function, and the function of sign is as follows: when x is>0, sign (x) is 1; when x is 0, sign (x) is 0; when x is<At 0, sign (x) is-1.
Further, step S3 is specifically:
s31: the variable modal decomposition is to adaptively decompose the input signal x (t) into a plurality of modal component signals ukContinuously iterating and solving the optimal solution with the minimum sum of the estimation bandwidths of the modal components, wherein the expression is as follows:
Figure RE-GDA0003146913760000031
wherein: u. ofkIs the k-th modal component, ωkK is 1,2, for the respective center frequency, K, δ (t) is a dirac function,
Figure RE-GDA0003146913760000034
for gradient operations, x (t) represents the input signal;
s32: introducing a secondary penalty factor and a Lagrange multiplication operator to convert the secondary penalty factor and the Lagrange multiplication operator into an unconstrained variational problem, solving the problem by an alternating direction multiplier method, converting the quadratic penalty factor and the Lagrange multiplication operator into a frequency domain by utilizing Fourier equidistant transformation, and iteratively updating uk，ωkThe expression is:
Figure RE-GDA0003146913760000032
Figure RE-GDA0003146913760000033
wherein: the power factor represents Fourier transform;
s33: and outputting K modal components through inverse Fourier transform until the iteration stop condition is met.
Further, step S5 is specifically:
s51: reconstructing the effective component in the step S4 to obtain a reconstructed signal;
s52: selecting an optimal lifting wavelet basis function and an optimal decomposition scale;
s53: performing lifting wavelet decomposition on the reconstructed signal to obtain wavelet high-frequency coefficients and wavelet low-frequency coefficients of each layer;
s54: and carrying out soft threshold processing on the wavelet high-frequency coefficient obtained by decomposition, wherein the soft threshold function expression is as follows:
Figure RE-GDA0003146913760000041
wherein: djThe method is a wavelet decomposition coefficient, T is a threshold value, sign is a sign function, and the function of the method is as follows: when x is>0, sign (x) is 1; when x is 0, sign (x) is 0; when x is<0, sign (x) is-1;
s55: and performing lifting wavelet inverse transformation on the low-frequency wavelet coefficient and the high-frequency coefficient subjected to soft threshold processing to obtain a final denoised partial discharge signal.
According to the technical scheme, the partial discharge threshold denoising method based on the improved variational modal decomposition, which is disclosed by the invention, aims at the problem that the variational modal decomposition is difficult to adaptively select decomposition parameters in practical application, and provides that the number of decomposition layers is determined by the number of wave crests of a spectrogram, a punishment factor corresponding to each mode is optimized by adopting a Tianniu whisker search algorithm, and a kurtosis index is introduced to accurately select an effective component for reconstruction. The method effectively inhibits periodic narrow-band interference and white noise, reduces distortion of local discharge waveform, and completely retains characteristic information of the local discharge signal.
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FIG. 1 is a flowchart of a partial discharge threshold denoising method based on improved variational modal decomposition according to an embodiment of the present invention;
FIG. 2 is a time-domain sampling signal diagram of the original partial discharge signal in the present example;
FIG. 3 is a time-domain sampling signal diagram of the noise-contaminated partial discharge signal in the present example;
FIG. 4 is a graph of the spectrum of the noise-contaminated partial discharge signal in this example;
FIG. 5 is an iteration diagram of a Tianniu whisker search in this example;
FIG. 6 is a time domain diagram of modal components decomposed using a variational modal decomposition algorithm according to the present embodiment;
FIG. 7 is a spectrum of modal components decomposed using a variational modal decomposition algorithm according to the present embodiment;
FIG. 8 is a graph of denoising results using lifting db4 wavelet threshold in an example;
FIG. 9 is a graph of denoising results using ensemble empirical mode decomposition in an example;
FIG. 10 is a graph showing the results of the denoising method according to the present invention.
Detailed Description
The invention is further described below with reference to the accompanying drawings:
as shown in fig. 1, the present invention provides a partial discharge threshold denoising method based on improved variational modal decomposition, including the following steps:
step S1: determining the decomposition layer number K of the variational modal decomposition by utilizing the number of wave crests of the partial discharge signal spectrogram;
step S2: optimization of penalty factor alpha corresponding to each layer mode of variational modal decomposition by utilizing longicorn beard search algorithmk(k＝1,2,...,K)；
S21: after the number of decomposition layers is determined, defining the search dimension of the longicorn whisker algorithm as K and the position of the longicorn as K
Figure RE-GDA0003146913760000051
Wherein N is the iteration number of the search;
s22: kurtosis is an index for evaluating impact property, and the expression is as follows:
Figure RE-GDA0003146913760000052
wherein: μ represents the mean value of the signal, and x represents the time series value of the signal;
defining an objective function f (x) of the longicorn algorithm as the sum of kurtosis of each modal component of the variational modal decomposition under each group of parameters, wherein the expression is as follows:
Figure RE-GDA0003146913760000053
wherein: k is the number of decomposition layers, Ku, obtained in step S1kIs the kurtosis value of the kth component, K ═ 1, 2., K;
s23: the longicorn whisker algorithm updating formula is as follows:
xn＝xn-1+δnDsign[f(xr)-f(xl)]
wherein: d is a unitized random vector, xrIs the position of the right beard of the longicorn, xlIs the position of the left beard of the longicorn, deltanFor step size of each iteration, f (x)r) Is the target function value of the right beard of the longicorn, f (x)l) The target function value of the longicorn left tassel is shown, sign is a sign function, and the function of sign is as follows: when x is>0, sign (x) is 1; when x is 0, sign (x) is 0; when x is<At 0, sign (x) is-1.
Step S3: after the decomposition parameters are determined, decomposing the local discharge signal by using an optimized variational modal decomposition algorithm to obtain K intrinsic modal components with limited bandwidth, and calculating kurtosis values of the modal components;
s31: the variable modal decomposition is to adaptively decompose the input signal x (t) into a plurality of modal component signals ukContinuously iterating and solving the optimal solution with the minimum sum of the estimation bandwidths of the modal components, wherein the expression is as follows:
Figure RE-GDA0003146913760000061
wherein: u. ofkIs the k-th modal component, ωkK is 1,2, for the respective center frequency, K, δ (t) is a dirac function,
Figure RE-GDA0003146913760000064
for gradient operations, x (t) represents the input signal;
s32: introducing a secondary penalty factor and a Lagrange multiplication operator to convert the secondary penalty factor and the Lagrange multiplication operator into an unconstrained variational problem, solving the problem by an alternating direction multiplier method, and then utilizing Fourier equidistant variationConversion to frequency domain, iterative update uk，ωkThe expression is:
Figure RE-GDA0003146913760000062
Figure RE-GDA0003146913760000063
wherein: the power factor represents Fourier transform;
s33: outputting K modal components through inverse Fourier transform until an iteration stop condition is met;
step S4: defining components with kurtosis values larger than 10 in the step S3 as valid components, and defining the rest as invalid components;
step S5: reconstructing the effective components to obtain a reconstructed signal, removing residual low-frequency white noise in the reconstructed signal by using a lifting wavelet threshold method to obtain a final denoised partial discharge signal;
s51: reconstructing the effective component in the step S4 to obtain a reconstructed signal;
s52: selecting an optimal lifting wavelet basis function and an optimal decomposition scale;
the optimal wavelet basis function selection process is as follows:
and selecting different wavelet functions as lifting wavelet bases, and performing noise reduction processing comparison on the same signal under the same decomposition scale. White noise is added into the single exponential oscillation attenuation function to serve as a noise-contaminated signal, the signal-to-noise ratio of the noise-contaminated signal is-18.96, and the selected lifting wavelet basis function is as follows: haar, db2, db4, db5, db6, sym4, all with a decomposition scale of 4. The snr after denoising using different lifting wavelet basis functions was calculated, and the results are shown in the following table, where it was found that denoising using wavelet db4 basis function was the best.
Under the same decomposition scale, adopting signal-to-noise ratio after noise reduction processing of different lifting wavelet basis functions
Wavelet basis function haar db2 db4 db5 db6 sym4
Signal to noise ratio 10.96 13.34 17.85 16.67 14.37 17.00
The optimal decomposition scale selection process comprises the following steps:
and decomposing the signal in different scales on the basis of determining the optimal wavelet basis function to determine the optimal decomposition scale. White noise is added into the single exponential oscillation attenuation function to serve as a noise-contaminated signal, the signal-to-noise ratio of the noise-contaminated signal is-18.96, and different decomposition scales J are set to be 3, 4, 5 and 6 respectively. The signal-to-noise ratio after noise reduction processing under different decomposition scales is calculated, and the result is shown in the following table, and the noise reduction processing result is found to be optimal when the decomposition scale is 3.
Under the optimal wavelet basis function, the signal-to-noise ratio after noise reduction processing is carried out by adopting different decomposition scales
Decomposition scale 3 4 5 6
Signal to noise ratio 18.23 16.46 13.24 14.67
S53: performing lifting wavelet decomposition on the reconstructed signal to obtain wavelet high-frequency coefficients and wavelet low-frequency coefficients of each layer;
s54: and carrying out soft threshold processing on the wavelet high-frequency coefficient obtained by decomposition, wherein the soft threshold function expression is as follows:
Figure RE-GDA0003146913760000081
wherein: djThe method is a wavelet decomposition coefficient, T is a threshold value, sign is a sign function, and the function of the method is as follows: when x is>0, sign (x) is 1; when x is 0, sign (x) is 0; when x is<0, sign (x) is-1;
s55: and performing lifting wavelet inverse transformation on the low-frequency wavelet coefficient and the high-frequency coefficient subjected to soft threshold processing to obtain a final denoised partial discharge signal.
The invention discloses a method based on improved variation modal decompositionThe partial discharge threshold denoising method comprises the steps of firstly, using the number of wave crests of a partial discharge signal spectrogram as the decomposition layer number K of variable modal decomposition, and optimizing a penalty factor alpha corresponding to each layer modal of the variable modal decomposition through a Tianniu whisker search algorithmk(K ═ 1, 2.., K). And then, processing the partial discharge signals by using the optimized variation mode decomposition to obtain K finite-bandwidth eigenmode components, and distinguishing effective components and ineffective components by adopting kurtosis index values. And finally reconstructing effective components, and removing residual white noise in the reconstructed signal by using a lifting wavelet threshold method to obtain a final denoised partial discharge signal. The invention can adaptively determine the decomposition parameters of the variational modal decomposition, simultaneously introduces the problem of lifting wavelet algorithm to process the incomplete filtering of low-frequency white noise, effectively filters periodic narrow-band interference and white noise, and better retains the integrity of the local discharge signal.
Examples of the design
Firstly, an ideal partial discharge signal model is established, a single-exponential oscillation attenuation model and a double-exponential oscillation attenuation model can be selected for simulation, periodic narrow-band interference signals and white noise are added into the ideal partial discharge model to simulate noise-polluted signals, and fig. 2 and 3 are graphs of the established ideal partial discharge signals and the noise-polluted partial discharge signals. And performing FFT analysis on the noise-contaminated signals, wherein FIG. 4 shows the frequency spectrum of the noise-contaminated partial discharge signals, and the number of decomposition layers for determining the variational modal decomposition is 5. FIG. 5 is a diagram of a search iteration process of a longicorn whisker, and each layer of penalty factors of variational modal decomposition is determined. And (3) processing the noise-contaminated partial discharge signal by using the optimized variation modal decomposition, wherein fig. 6 is a time domain diagram of each obtained modal component, and fig. 7 is a frequency spectrum corresponding to each modal component.
And calculating kurtosis values of the modal components, namely 18.34, 1.60, 1.56, 17.21 and 1.60 respectively, automatically identifying that the BIMF1 and the BIMF4 are effective components, reconstructing the effective components, and removing narrow-band interference signals and high-frequency white noise.
As can be seen from fig. 7, part of white noise with smaller amplitude still remains in BIMF1 and BIMF4, the white noise remaining in the reconstructed signal is further removed by using the lifting db4 wavelet threshold method, and finally the obtained denoised waveform is shown in fig. 10. Compared with other two traditional denoising methods, fig. 8 is a graph of denoising results by using lifting db4 wavelet threshold, and fig. 9 is a graph of denoising results by using ensemble empirical mode decomposition. As can be seen from the figure, the traditional method has a common denoising effect, the signal has serious waveform distortion and the partial discharge characteristic is lost, and the method provided by the invention has a good denoising effect, completely inhibits the noise and has a small signal distortion degree.
The above-mentioned embodiments are merely illustrative of the preferred embodiments of the present invention, and do not limit the scope of the present invention, and various modifications and improvements of the technical solution of the present invention by those skilled in the art should fall within the protection scope defined by the claims of the present invention without departing from the spirit of the present invention.

Claims (8)

1.一种基于改进变分模态分解的局部放电阈值去噪方法，其特征在于，包括：包括以下步骤：1. a partial discharge threshold denoising method based on improved variational modal decomposition, is characterized in that, comprises: comprise the following steps: (1)确定变分模态分解的分解层数K；(1) Determine the decomposition level K of the variational modal decomposition; (2)优化变分模态分解的各层模态对应的惩罚因子；(2) Optimize the penalty factor corresponding to each layer mode of the variational mode decomposition; (3)确定分解参数后，利用优化后的变分模态分解算法对局部放电信号进行分解，得到K个有限带宽本征模态分量，并计算各模态分量的峭度值；(3) After determining the decomposition parameters, use the optimized variational modal decomposition algorithm to decompose the partial discharge signal, obtain K eigenmode components with limited bandwidth, and calculate the kurtosis value of each modal component; (4)定义所述峭度值的有效分量和无效分量；(4) defining the effective and ineffective components of the kurtosis value; (5)重构有效分量得到重构信号，并去除掉重构信号中残留的低频白噪声，得到最终去噪后的局部放电信号。(5) Reconstructing the effective components to obtain a reconstructed signal, and removing the residual low-frequency white noise in the reconstructed signal to obtain a final denoised partial discharge signal. 2.根据权利要求1所述的基于改进变分模态分解的局部放电阈值去噪方法，其特征在于：步骤(1)中，所述确定变分模态分解的分解层数K，是通过局部放电信号频谱图的波峰个数确定。2. The partial discharge threshold denoising method based on improved variational modal decomposition according to claim 1, characterized in that: in step (1), the determination of the decomposition level K of the variational modal decomposition is performed by The number of peaks of the partial discharge signal spectrogram is determined. 3.根据权利要求1所述的基于改进变分模态分解的局部放电阈值去噪方法，其特征在于：步骤(2)中，利用天牛须搜索算法优化变分模态分解的各层模态对应的惩罚因子。3. the partial discharge threshold denoising method based on improved variational modal decomposition according to claim 1, it is characterized in that: in step (2), utilizes long beetle search algorithm to optimize each layer model of variational modal decomposition The penalty factor corresponding to the state. 4.根据权利要求1所述的基于改进变分模态分解的局部放电阈值去噪方法，其特征在于：步骤(4)中，所述峭度值大于10的分量为有效分量，其余为无效分量。4. The partial discharge threshold denoising method based on improved variational modal decomposition according to claim 1, wherein in step (4), the component with the kurtosis value greater than 10 is an effective component, and the rest are invalid weight. 5.根据权利要求1所述的基于改进变分模态分解的局部放电阈值去噪方法，其特征在于：步骤(5)中，所述去除掉重构信号中残留的低频白噪声是通过提升小波阈值法。5. The partial discharge threshold denoising method based on improved variational modal decomposition according to claim 1, characterized in that: in step (5), the removal of the residual low-frequency white noise in the reconstructed signal is performed by increasing Wavelet threshold method. 6.根据权利要求3所述的基于改进变分模态分解的局部放电阈值去噪方法，其特征在于：步骤2中，所述天牛须搜索公式如下：6. the partial discharge threshold denoising method based on improved variational modal decomposition according to claim 3, is characterized in that: in step 2, described long beetle search formula is as follows: xn＝xn-1+δnDsign[f(xr)-f(xl)]x n =x n-1 +δ n Dsign[f(x r )-f(x l )] 其中：D为单位化的随机向量，xr为天牛右须位置，xl为天牛左须位置，δn为每次迭代的步长，f(xr)为天牛右须目标函数值，f(xl)为天牛左须目标函数值，sign为符号函数，其功能是：当x>0时，sign(x)＝1；当x＝0时，sign(x)＝0；当x<0时，sign(x)＝-1。Among them: D is a unitized random vector, x r is the position of the right whiskers of the beetle, x l is the position of the left whiskers of the beetle, δ n is the step size of each iteration, and f(x r ) is the objective function of the right beard of the beetle value, f(x l ) is the objective function value of the left beard of the beetle, sign is the sign function, and its function is: when x>0, sign(x)=1; when x=0, sign(x)=0 ; when x<0, sign(x)=-1. 7.根据权利要求1所述的基于改进变分模态分解的局部放电阈值去噪方法，其特征在于：所述步骤(3)，具体包括如下步骤：7. The partial discharge threshold denoising method based on improved variational modal decomposition according to claim 1, wherein the step (3) specifically comprises the following steps: (31)求解各模态分量估计带宽之和最小的最优解，其表达式为：(31) Find the optimal solution with the smallest sum of estimated bandwidths of each modal component, and its expression is:
Figure FDA0003077326130000021
Figure FDA0003077326130000021
 其中：uk为第k个模态分量，ωk为相应的中心频率，k＝1,2,...,K，δ(t)是狄拉克函数，
Figure FDA0003077326130000024
为梯度运算，x(t)表示输入信号；Where: u k is the kth modal component, ω k is the corresponding center frequency, k=1,2,...,K, δ(t) is the Dirac function,
Figure FDA0003077326130000024
is the gradient operation, x(t) represents the input signal; (32)引入二次惩罚因子a和Lagrange乘法算子λ将其转换成无约束变分问题，再通过交替方向乘子法求解，然后利用傅里叶等距变换转换到频域，迭代更新uk，ωk，其表达式为：(32) Introduce the quadratic penalty factor a and the Lagrange multiplication operator λ to convert it into an unconstrained variational problem, then solve it by the alternating direction multiplier method, and then use the Fourier isometric transform to convert to the frequency domain, and iteratively update u k , ω k , its expression is:
Figure FDA0003077326130000022
Figure FDA0003077326130000022
Figure FDA0003077326130000023
Figure FDA0003077326130000023
 其中：^表示傅里叶变换；Where: ^ represents the Fourier transform; (33)直至满足迭代停止条件，通过傅里叶逆变换输出K个模态分量。(33) Until the iterative stop condition is satisfied, output K modal components through inverse Fourier transform. 8.根据权利要求1所述的基于改进变分模态分解的局部放电阈值去噪方法，其特征在于：所述步骤(5)，具体包括如下步骤：8. The partial discharge threshold denoising method based on improved variational modal decomposition according to claim 1, wherein the step (5) specifically comprises the following steps: (51)重构步骤(4)中的有效分量得到重构信号；(51) the effective component in the reconstruction step (4) obtains a reconstructed signal; (52)选择最优提升小波基函数和最优分解尺度；(52) Select the optimal boosted wavelet basis function and the optimal decomposition scale; (53)对重构信号进行提升小波分解，得到各层小波高频系数和小波低频系数；(53) Decompose the reconstructed signal by lifting wavelet to obtain high-frequency wavelet coefficients and low-frequency wavelet coefficients of each layer; (54)将分解得到的小波高频系数进行软阈值处理，软阈值函数表达式为：(54) Soft threshold processing is performed on the high-frequency wavelet coefficients obtained by decomposition, and the soft threshold function expression is:
Figure FDA0003077326130000031
Figure FDA0003077326130000031
 其中：dj为小波分解系数，T为阈值，sign为符号函数，其功能是：当x>0时，sign(x)＝1；当x＝0时，sign(x)＝0；当x<0时，sign(x)＝-1；Among them: d j is the wavelet decomposition coefficient, T is the threshold, sign is the sign function, and its function is: when x>0, sign(x)=1; when x=0, sign(x)=0; when x When <0, sign(x)=-1; (55)将低频小波系数和经过软阈值处理后的高频系数进行提升小波逆变换，得到最终去噪后的局部放电信号。(55) Perform inverse lifting wavelet transform on the low-frequency wavelet coefficients and the high-frequency coefficients after soft threshold processing to obtain the final denoised partial discharge signal.
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