## 3 Improved Algorithm
### Improved Wavelet Threshold Based on Autocorrelation  
Wavelet transform decomposes the original signal into detail coefficients (cD) and approximate coefficients (cA) [27]. The wavelet decomposition process is shown in Figure 3. In the process, the wavelet coefficients of the signal tend to be stable with the increase in the number of decomposition layers, and the wavelet coefficients of the noise gradually decrease [28].  
Figure 3: Wavelet decomposition.  
Wavelet transform is a linear transformation [29]. The wavelet coefficients of the signal have obvious autocorrelation, while the wavelet coefficients of the noise have weak or no autocorrelation. The autocorrelation of the wavelet coefficients is used to adjust the threshold.  
\(R_{RMS}^{j}\) is the ACF of wavelet coefficients for each frequency band. The RMS of the ACF for wavelet coefficients in different frequency bands \(r_{RMS}^{j}\) is calculated as follows:  
\[r_{RMS}^{j}=\sqrt{\frac{\sum\limits_{\tau=0}^{W}R^{j}(\tau)^{2}}{W}} \tag{7}\]  
where \(W\) is the number of wavelet coefficients, \(j\) is the number of the frequency bands.  
The Universal Threshold was proposed by [PERSON] and [PERSON] [30]. The Universal Threshold is given as follows:  
\[T=\sigma\sqrt{2\ln N} \tag{8}\]  
where \(\sigma\) is the noise standard deviation, \(N\) is the data length.  
The standard deviation of the noise is calculated as follows [31]:  
\[\sigma=\frac{median(|cD(k)|)}{0.6745} \tag{9}\]  
where \(median(|cD(k)|)\) is the median of all detail coefficients at each scale [32].  
We improved the Universal Threshold by exploiting the ACF of the wavelet coefficients. The improved threshold is given as follows:  
\[T_{j}=\left\{\begin{array}{l}e^{-a\cdot r_{RMS}^{j}}\cdot\sigma\sqrt{2\ln N },r_{RMS}^{j}\geq\beta_{2}\\ e^{b/r_{RMS}^{j}}\cdot\sigma\sqrt{2\ln N},r_{RMS}^{j}<\beta_{2}\end{array}\right. \tag{10}\]  
where \(T_{j}\) represents the wavelet threshold of different frequency bands. \(\beta_{2}\) is the set threshold. According to the principle of ACF, when choosing the value of \(\beta_{2}\), these cases should be considered: the amplitude range of the effective signal; mode mixing causes small values of the relevant modes; and the absolute value of the wavelet coefficients of the decomposed effective signal being larger than that of the noise. \(a\) and \(b\) can adjust the exponent to prevent it from being too large or too small, which would increase the computational effort.  
When \(r_{RMS}^{j}>\beta_{2}\), it indicates that there is an effective signal in this frequency band. \(e^{-a\cdot r_{RMS}^{j}}\) and \(e^{b/r_{RMS}^{j}}\) are monotonically decreasing, monotonically increasing exponential functions, respectively. The value of \(e^{-a\cdot r_{RMS}^{j}}\) decreases rapidly as \(r_{RMS}^{j}\) increases, and the value of \(e^{b/r_{RMS}^{j}}\) increases rapidly as \(r_{RMS}^{j}\) decreases. The above-mentioned properties can be used to change the threshold size adaptively in each frequency band. When the autocorrelation of the signal meets the requirement, \(e^{-a\cdot r_{RMS}^{j}}\) is used to adjust the threshold becomes smaller, so that some weak signals can be retained. Otherwise, \(e^{b/r_{RMS}^{j}}\) is used to adjust the threshold to become larger and remove more noise.