## 1 Introduction
### Wavelet Denoising  
To solve the problem that the transient signal distorts in the process of transmission and cannot reflect the characteristics of the original waveform of the primary side traveling wave, an optimized variable step size Least Mean Squares (LMS) adaptive filtering algorithm is introduced in this study to achieve accurate detection of the transient waveform.  
The process of wavelet threshold denoising is shown in Figure 1. By using an adaptive filter to learn and obtain the black box model parameter matrix A, the black box model is constructed.  
The inversion process is shown in Figure 2. In the figure, y' is the secondary signal after preprocessing and \(W_{opt}\) is the best weight at time m. The schematic diagram of the adaptive filter is shown in Figure 3.  
Figure 1: Process of wavelet denoising.  
Figure 3: Inversion of LMS adaptive training model flow with multiple samples.  
Figure 2: Inversion process.  
In the application of the LMS adaptive filter, a complete signal sequence \(Y_{m+N}\) is introduced as the input signal, and an expected signal \(d_{m}\) is set as the reference. At a given time \(i\), the filter precisely extracts a specified interval \(Y_{i}\) from the input signal, which is defined as the receptive field region. Then, the matched weight vector \(W_{i}(n)\) obtained during the nth time iteration is used to weight the signal in the receptive field region, and the output signal \(x_{i}(n)\) is estimated:  
\[x_{i}(n)=\sum^{N}y_{i-k+1}w_{i,k}=Y_{i}(W_{i}(n))^{\mathrm{T}} \tag{8}\]  
where \(y_{i-k+1}\) is the \(k\) value of the \(Y_{i}\) interval at the \(i\) time; \(w_{i,k}\) is the \(k\) value in \(W_{i}(n)\) at time \(i\); and \(n\) is the number of filter taps.  
The error value \(e_{i}(n)\) at time \(i\) is the difference between the expected signal \(d_{i}\) and the estimated signal at time \(i\), as shown below.  
\[e_{i}(n)=d_{i}-x_{i}(n) \tag{9}\]  
In the input signal processing at subsequent time points, the signal in the current receptive field interval will be synchronously compared with the receptive field interval signal defined at the \(i\) time; this process adopts the sliding window mechanism, in which the window definition length is a specific value. The weight vector update formula of the \(n\) iteration is as follows:  
\[W_{i}(n+1)=W_{i}(n)+2\mu e_{i}(n)Y_{i} \tag{10}\]  
where \(\mu\) is the step factor of the LMS algorithm, which determines the convergence speed and stability of the algorithm.  
The weight vector is updated by using the update rule of the LMS algorithm, and the optimal weight vector \(W_{opt}\) is obtained until the error signal is no longer reduced, so that the output of the filter is as close as possible to the expected signal.  
The advantages of wavelet denoising  
(1) Multi-scale analysis: By decomposing sub-bands of different frequencies, noise and signals are precisely separated.  
(2) Localization feature: It can simultaneously locate noise in both the time domain and the frequency domain, making it suitable for handling non-stationary signals.  
(3) Detail retention: Avoids edge blurring or excessive smoothing caused by traditional filtering methods.  
(4) Flexibility and controllability: The effect can be optimized by choosing wavelet basis functions, the number of decomposition layers, and threshold strategies.  
(5) Low computing cost: Compared with deep learning, it is suitable for embedded systems or real-time processing.