## 2 Methodology
### Evaluation Index  
After applying noise reduction methods to process the signal, the results must be evaluated using objective metrics to quantitatively compare the performance differences between the improved and traditional techniques. Five evaluation metrics are introduced in the following sections.  
(1) Root mean square error (RMSE)  
RMSE is a widely used metric in statistics and data analysis to measure the differences between predicted and observed values [27]. RMSE can be defined as follows:  
\[\text{RMSE}=\sqrt{\frac{1}{n}\frac{\sum_{i}\left[y_{i}-\hat{y}_{i}\right]^{2}} } \tag{1}\]  
where \(y_{i}\) is the observed values and \(\hat{y}_{i}\) denotes the predicted values. And \(n\) is the number of observed values. Higher RMSE indicates that the deviation between the two signals is significant, and the de-noising effect is poor. Lower RMSE reflects that the reconstructed signal closely matches the original signal, signifying that de-noising has effectively reduced unwanted noise while retaining the signal's essential features.  
(2) Standard Deviation (STD)  
STD is a statistical measure that quantifies the amount of variation or dispersion in a set of values [27]. It indicates how spread out the data points are around the mean. STD can be defined as follows:  
\[\text{STD}=\sqrt{\frac{1}{n}\sum_{i=1}^{n}(y_{i}-\overline{y})^{2}} \tag{2}\]  
where \(y_{i}\) is the observed values and \(\overline{y}\) denotes the mean of the observed values. And \(n\) is the number of observed values. The smaller the STD, the smaller the fluctuation range of the data, meaning the noise is suppressed and the signal is closer to the target signal. A more significant STD may indicate that additional noise has been introduced during the de-noising process or that the original noise has not been effectively suppressed.  
(3) Signal-to-Noise Ratio (SNR)  
SNR is an indicator that measures the relationship between signal strength and noise strength, representing the ratio of signal power to noise power [27]. The larger the SNR, the better the signal quality and the less interference from noise on the signal. SNR can be defined as follows:  
\[\text{SNR}=10\times\log_{10}\left(\frac{\frac{1}{\text{n}}\left(\text{y}_{i} \right)^{2}}{\frac{1}{\text{n}}\left(\text{y}_{i}-\text{y}_{i}\right)^{2}}\right) \tag{3}\]  
where \(y_{i}\) is the observed values and \(\overline{y}\) denotes the mean of the observed values.  
(4) Signal-to-Noise Ratio (RVR)  
RVR is a ratio of the variance root of the de-noised signal to the variance root of the original signal difference [27]. A smaller RVR indicates a higher level of noise reduction effectiveness.  
\[\text{RVR}=\frac{\sum\limits_{i=1}^{\text{N}-1}\left(\text{x}_{i+1}-\text{x}_ {i}\right)^{2}}{\sum\limits_{i=1}^{\text{N}-1}\left(\text{y}_{i+1}-\text{y}_ {i}\right)^{2}} \tag{4}\]  
where \(y_{i}\) is the observed values and \(x_{i}\) is the de-noised signal.  
(5) Normalized Correlation Coefficient (NCC)  
NCC is a statistical measure used to quantify the similarity or correlation between two signals, datasets, or images, normalized to lie between \(-1\) and \(1\)[20]. NCC can be defined as follows:  
\[NCC=\frac{\sum\limits_{i=1}^{n}\left(y_{i}-\overline{y}\right)\left(x_{i}- \overline{x}\right)}{\sqrt{\sum\limits_{i=1}^{n}\left(y_{i}-\overline{y} \right)^{2}\sum\limits_{i=1}^{n}\left(x_{i}-\overline{x}\right)^{2}}} \tag{5}\]  
where \(y_{i}\) is the observed values and \(\overline{y}\) denotes the mean of the observed values. And \(x_{i}\) is the de-noised signal and \(\overline{x}\) denotes the mean of the de-noised signal.