## 2 Materials and Methods
### Methods of Signal Analysis
#### 2.2.3 WT  
Since the development of signal processing technology, many new methods have been developed for single- or two-dimension signal analysis [33; 34; 35; 36]. Among these methods, WT is a widely used one [37]. WT appeared in the 1970s, and in recent years, it has been used for various types of signal analysis, such as signal and image processing [38; 39]. The principle of WT is to use a scalable and displaceable finite-length base wavelet inner-product time domain data. The function of WT can be written as  
\[X_{\text{WT}}(a,\tau)=\frac{1}{\sqrt{a}}{\int_{-\infty}^{\infty}x(t)\varphi^{ *}\left(\frac{n-\tau}{a}\right)dn}, \tag{2}\]  
where \(x(t)\) is the value of the time domain data at the moment of \(t\); \(\varphi(t)\) is a scalable and displaceable finite-length base wavelet; the positive coefficient \(a\) controls the scaling of \(\varphi(t)\); and coefficient \(\tau\) controls the displacement of \(\varphi(t)\). The settings of WT in this research can be seen in Table 3.  
The scales controlling the window sizes can be calculated as  
\[S=\frac{2 fT}{A},A=[T,T-1,T-2,\ldots,2,1], \tag{3}\]  
where \(S\) is the scale, \(f\) is the sample rate of the original signal, and \(T\) is the total scale. [PERSON] et al. introduced a wavelet tool and windowed scale index to solve a particular case of Haar wavelet [41]. [PERSON] et al. also introduced a scale index in their research [42]. The scale index is necessary for orthogonal decomposition. The Cmor wavelet is a Gaussian envelope function and is related to non-orthogonal decomposition without a scale index. Both WT and STFT use the infinitesimal element method, in which the original data are divided into successive windows, while each window is transformed into the time-frequency domain, and the results are sorted in chronological order. Obviously, the shorter the window length is, the higher the temporal resolution will be. This is the uncertainty principle. We chose an original whistle signal without a filter and processed it using different window lengths to demonstrate the uncertainty principle (Figure 2). It can be seen that when the window length was short, the temporal resolution was high, with a low-frequency resolution. When the window length was longer, the temporal resolution became lower, with a higher frequency resolution. In essence, temporal resolution and frequency resolution are negatively correlated. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Setting** \\ \hline Software & SciPy 1.7.1 1 \\ Window & Hamming \\ Window length & 500 sampling points \\ Overlap & 250 sampling points \\ \hline \hline \end{tabular} 1  
\end{table} 
Table 2: Parameters of STFT. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Setting** \\ \hline Software & PyWavelets v1.1.1 1 \\ Base wavelet & Cmor100-100 \\ Total scale & 1000 \\ Sampling period & Decided by sample rate of original signal \\ Scales & Calculated by Function (3) \\ \hline \hline \end{tabular} 1  
\end{table} 
Table 3: Parameters of WT. As Figure 3 shows, the window length and overlap of STFT are constant values, i.e., they uniformly divide the entire signal into several parts and can be employed to calculate each part using FT. This data division method determines that STFT cannot increase the temporal resolution and frequency resolution at the same time, being affected by the uncertainty principle. In WT, each window is obtained by scaling and displacing the base wavelet. This data division method determines that WT has a higher temporal resolution with a lower frequency resolution for the high-frequency band and a lower temporal resolution with a higher frequency resolution for the low-frequency band. The click signals have a short duration, high upper-frequency limit, and wide frequency band. Therefore, the detection of clicks requires a relatively high temporal resolution. The frequency resolutions of STFT are exactly the same in any frequency band. It is difficult for STFT to detect catastrophe signals (e.g., clicks).