## 2 Methods
### Improved Morlet Wavelet Transform
#### 2.1.3 Morlet Wavelet Improvement  
In Morlet wavelet given by Equation (5), taking parameter \(\omega_{0}=2\pi\), the improved Morlet fundamental wavelets are obtained as follows:  
\[\varphi(t)=\frac{1}{\sqrt{2\pi}}\exp\left(-\frac{t^{2}}{2}+j2\pi t\right) \tag{8}\]  
In Equation (8), \(\frac{1}{\sqrt{2\pi}}\) is the wavelet amplitude parameter such that  
\[\int_{R}|\varphi(t)|dt=1 \tag{9}\]  
The modified Morlet wavelet of Formula (8) is transformed by \"expansion\" and \"translation\", and the improved Morlet wavelet function family can be obtained:  
\[\varphi_{f,\tau}(t)=\frac{|f|}{\sqrt{2\pi}}\exp\left[-\frac{f^{2}(t-\tau)^{2 }}{2}+j2\pi f(t-\tau)\right] \tag{10}\]In Equation (10), the parameters \(t\) and \(\tau\) are time, whose variable unit is \(s\), and \(f\) is the wavelet dominant frequency, whose unit is Hz.  
Morlet wavelet kernel function transform can be used for signal time-frequency analysis. A Morlet wavelet is a double-window function, and the time window center \(t^{*}\) of \(\varphi_{f,\tau}(t)\), frequency window center \(f^{*}\), time window radius \(\Delta t\), and frequency window radius \(\Delta f\) are, respectively,  
\[t^{*}=\tau,\ f^{*}=f,\ \Delta t=\frac{1}{\sqrt{2\pi}},\ \Delta f=\frac{f}{2 \sqrt{2}\pi} \tag{11}\]  
The time-frequency window is  
\[\left[\tau-\frac{1}{\sqrt{2}f},\ \tau+\frac{1}{\sqrt{2}f}\right]\times\left[f- \frac{f}{2\sqrt{2}\pi},\ f+\frac{f}{2\sqrt{2}\pi}\right] \tag{12}\]  
and there are  
\[\Delta t\Delta f=\frac{1}{\sqrt{2}f}\frac{f}{2\sqrt{2}\pi}=\frac{1}{4\pi} \tag{13}\]  
According to [15], the improved Morlet wavelet might satisfy the admissibility condition, and \(\varphi_{f,\tau}(t)=0\) when \(f=0\). The function space generated by the improved Morlet wavelet function is the same as that generated by the original Morlet wavelet function.  
The real part of complex Morlet wavelets:  
\[\varphi(t)=\left(\frac{\beta}{\sqrt{2\pi}}e^{-\beta^{2}t^{2}/2}\right)\cos( \omega_{0}t) \tag{14}\]  
The bandwidth parameter \(\beta\) controls the rate at which the envelope decays over time, thereby influencing the temporal resolution of \(\beta\). And \(\beta\) is also the frequency bandwidth of the Gaussian filter, which determines the resolution in the frequency domain. Therefore, by adjusting the time-frequency resolution of the continuous wavelet transform to accommodate the attenuation rate of the pulse to be extracted, the optimal \(\beta\) value for a given signal can be found.  
Let the mean function be set as \(\mu_{\varphi}=f(\beta,\omega_{c})\), in order to achieve the zero-value condition \(\mu_{\varphi}\to 0\), appropriate parameter values need to be selected. Figure 1 is a binary thresholding of the function \(\mu_{\varphi}\), dividing the graph into two regions. The nonshaded region corresponds to approximately satisfying the zero-mean condition \(\left|\mu_{\varphi}\right|\leq 0.0025\), that is, approximately satisfying the wavelet admissibility condition.  
Table 1 shows the modified Morlet wavelet parameters. Includes the normalization factor, center frequency \(\omega_{c}\), parameter range \(\beta\), and \(\beta\) number of test values.  
Figure 1: Approximate satisfaction of the zero-mean condition.