## 2 Methods
### Signal Overshoot and Smoothing Signal with Window Function  
When there are abrupt changes in the signal, directly applying the Hilbert transform to extract the envelope results in overshoot phenomena in the envelope signal. As shown in Figure 4, this is a function with abrupt changes, as well as the envelope signal obtained after applying the Hilbert transform. The envelope signal shows overshoot at two positions, which correspond exactly to the locations where the data in the original signal undergoes abrupt changes.  
The reasons for this phenomenon are as follows. First of all, Hilbert transform is actually a convolution operation; that is, the convolution of the original function and \(\frac{1}{nt}\). It has a strong long-tail effect, meaning that the energy distribution at the points of signal abrupt changes cannot be fully smoothed by the filter, leading to oscillations near the abrupt  
Figure 4: A signal with abrupt and its envelope.  
changes. Secondly, the edge effect caused by these abrupt changes leads to an excessive concentration of signal energy at the points of abrupt changes, further exacerbating the overshoot phenomenon in the transformed signal. In addition, the Gibbs phenomenon also plays a crucial role, as it describes that the ideal filter cannot completely suppress the oscillation when dealing with the sudden change signal, which is consistent with the wideband filtering property of the Hilbert transform. This results in the spectral energy near the abrupt changes not being evenly distributed, causing significant spectral leakage. Finally, the existence of spectrum leakage makes part of the energy overflow into the adjacent frequency components, which shows the oscillatory behavior of the transformed signal in the time domain.  
Combining the above reasons, we believe that smoothing the original signal is an effective way to solve this problem. In our study, we consider using the method of window functions to smooth the original signal. The principle of window functions is to apply weighting to the signal, causing it to decay smoothly at the edges, thereby avoiding discontinuities in the signal. This smoothing process effectively reduces abrupt changes in the signal, thereby lowering high-frequency components in the spectrum and mitigating spectral leakage issues. Therefore, in signal processing, selecting an appropriate window function can significantly enhance the quality and accuracy of spectral analysis. Commonly used window functions include the Hanning window, Hamming window, Blackman window, and Gaussian window. In our experiment, to facilitate the adjustment of window function parameters, we selected the Gaussian window for improvement attempts. The Gaussian window demonstrates excellent smoothness in both the time domain and frequency domain. It can effectively suppress abrupt changes and side-lobe effects in the signal, reduce the interference from high-frequency components, significantly decrease spectral leakage, and thereby improve resolution. It is defined as  
\[g(x)=a\times\exp\frac{-{(x-b)}^{2}}{2c^{2}} \tag{12}\]  
where \(a\) is the amplitude, which determines the maximum height of the curve; \(c\) is the standard deviation, which controls the width of the curve; and the Gaussian function is symmetric with respect to \(x=b\). After the signal is smoothed by the Gaussian window, the envelope can be obtained to effectively avoid the overshoot; as shown in Figure 5, the initial signal remains the one shown in Figure 4, which contains the mutant signal. The images in Figure 5 present, respectively, the envelope signal directly obtained through Hilbert transform and the envelope signal obtained after windowing processing.  
Figure 5: The envelope signal obtained directly and the envelope signal obtained after windowing.