## 3 Classification Algorithm Design  
### Noise Reduction Based on Wavelet Transform  
The acquisition of partial discharge and corona discharge signals under controlled laboratory conditions is a reliable method for minimizing extraneous noise; however, the existence of white noise can still distort the original signals. Traditional denoising methods include various types, such as the empirical wavelet and wavelet transform (WT) denoising methods [72; 73; 74; 75; 76]. WT achieves optimal signal processing results for different signal types by selecting different wavelet basis functions, effectively removing noise from the signal. Addressing this problem, this research employs a wavelet denoising technique to eliminate white noise interference from the signals gathered in the laboratory. The wavelet base function is selected as sym4, and the decomposition level is 8, and white noise displays distinct characteristics. Specifically, the maximal modulus of wavelet transforms for partial discharge and corona discharge tend to increase with the scale, whereas background noise demon  
Figure 4: Waveform of single discharge signal. (**a**) Partial discharge. (**b**) Corona discharge.  
Figure 5: Waveform of mixed discharge signals. (**a**) Situation 1. (**b**) Situation 2.  
Figure 3: Circuit diagram of partial discharge testing platform.  
strates contrasting behavior. Through successive wavelet decompositions, the magnitude of the wavelet coefficients associated with the noise diminishes. By establishing appropriate thresholds for high-frequency constituents, wavelet coefficients below the threshold are nullified. Subsequently, the denoised partial discharge and corona discharge signals are reconstructed using the processed wavelet coefficients, resulting in enhanced signal fidelity.  
This paper employs discrete wavelet conversion, with the transformation formula depicted as:  
\[\Psi_{j,k}(t)=a_{0}^{-\frac{j}{2}}\psi\left(\frac{t-ka_{0}^{j}b_{0}}{a_{0}^{j}} \right)=a_{0}^{-\frac{j}{2}}\psi\left(a_{0}^{-j}t-kb_{0}\right) \tag{1}\]  
where \(a=a_{0}^{j}\) and \(b=ka_{0}^{j}b_{0}\) represent the scale parameter and shift parameter, respectively, \(j,k\in Z\).  
The discrete wavelet coefficients that correspond to the transformation can be depicted as:  
\[C_{j,k}=\int_{-\infty}^{+\infty}f(t)\psi_{j,k}^{*}(t)dt=<f,\psi_{j,k}> \tag{2}\]  
This study employs the median threshold determined through empirical Bayesian estimation, where \(\sigma_{n}\) represents the variance of the interference signals, \(\sigma_{s}\) represents the variance of the raw signals, and \(\sigma\) represents the variance of the contaminated signals, and their relationship is as follows:  
\[\sigma^{2}=\sigma_{s}^{2}+\sigma_{n}^{2} \tag{3}\]  
Estimate the noise variance generated by high-frequency wavelet transforms with Formula (5), and Formula (6) is used to estimate the noisy signal variance.  
\[\sigma_{n}=\frac{median(|\omega_{i}(j,k)|)}{0.6745} \tag{4}\]  
\[\sigma^{2}=\frac{1}{n^{2}}\sum_{j,k=1}^{n}\omega_{i}^{2}(j,k) \tag{5}\]  
From Equations (3)-(5), the variance of raw signals can be expressed as Equation (6):  
\[\sigma_{s}=\sqrt{max[\sigma_{n}^{2}-\sigma^{2},0]} \tag{6}\]  
Thus, the Bayesian threshold formula can be expressed as depicted in Equation (7):  
\[\lambda=\frac{\sigma_{n}^{2}}{\sigma^{2}} \tag{7}\]  
This study employs one median threshold formula for selecting wavelet coefficients. Above all, the denoising process of wavelet transform is shown in Figure 6.  
Figure 6: Flowchart of wavelet noise reduction.