## 4 Wavelet Energy Evolution Characteristics  
### Wavelet Transform  
The complex frequency component of acoustic emission signals represents multiple fracture modes simultaneously [43]. Different frequency components can reveal the internal damage process within the rock. The high-frequency component is correlated with the strength of the activity of the microscopic pores inside the rock, while the low-frequency component indicates that the rock either remains relatively stable or has large macroscopic cracks. The acoustic emission signals released during the trial belong to the typical non-stationary and discrete signals. For the non-stationary signals, the Fourier transform has a certain limitation, it is impossible to perform frequency domain analysis on the time axis [44]. However, the wavelet transform is a localization process of time-frequency, and the scaling translation operations allow it to gradually multi-scale refine the signal. The wavelet transform can meet the demand of time-frequency signal analysis by the time segmenting at high frequency and frequency segmenting at low frequency, which is more applicable to the analysis of non-stationary processes [45; 46; 47; 48].  
The basic principle of the wavelet transform can be written as [49; 50]:  
The function space \(L^{2}(R)\) contains several functions. One of the functions or signal \(\Psi(x)\) is the wavelet that satisfies the following conditions:  
The wavelet is a function or signal \(\Psi(x)\) in the function space \(L^{2}(R)\) that satisfies the following conditions:  
\[C_{\psi}=\int_{R^{*}}\frac{\big{|}\hat{\psi}(\omega)\big{|}^{2}}{|\omega|}d \omega<\infty, \tag{1}\]  
where \(R^{*}=R\)-[0] expresses all the real numbers, except zero, \(\hat{\Psi}(x)\) is the Fourier transform of \(\Psi(x)\) and \(\Psi(x)\) is called the mother wavelet.  
For the real number pairs (\(a\), \(b\)), the parameter \(a\) is a non-zero real number and the function \(\Psi(a\), \(b)(x)\) is called a continuous wavelet function dependent on the parameter (\(a\), \(b\)) generated by the mother wavelet \(\Psi(x)\), or wavelet, and its formula is:  
\[\psi(a,b)(x)=\frac{1}{\sqrt{|a|}}\psi\bigg{(}\frac{x-b}{a}\bigg{)}, \tag{2}\]  
where \(b\) is called the translation factor, and \(a\) is called the scaling factor.  
The signal \(f(x)\) can implement the continuous wavelet transform by the following:  
\[W_{f}(a,b)=\frac{1}{\sqrt{|a|}}\int_{R}f(x)\psi\bigg{(}\frac{x-b}{a}\bigg{)}dx =\big{<}f(x),\psi_{a,b}(x)\big{>}, \tag{3}\]  
Its inverse transformation (reconstructed signal) can be calculated as:  
\[f(x)=\frac{1}{C_{\psi}}\iint_{R\times R^{*}}W_{f}(a,b,\psi)\bigg{(}\frac{x-b} {a}\bigg{)}dadb, \tag{4}\]  
The signal \(f(x)\) can accomplish discrete wavelet transform by the following:  
\[W_{f}\Big{(}2^{j},2^{j}k\Big{)}=2^{-j/2}\int_{-\infty}^{+\infty}f(x)\psi\Big{(} 2^{-j}x-k\Big{)}dx, \tag{5}\]  
Its inverse transformation (reconstructed signal) can be defined as:  
\[f(t)=C\sum_{j=-\infty}^{+\infty}\sum_{k=-\infty}^{+\infty}W_{f}\Big{(}2^{j},2 ^{j}k\Big{)}\psi_{(2^{j}2^{j}k)}(x), \tag{6}\]  
where \(C\) is a signal-independent constant.  
The wavelet functions are diverse and the appropriate wavelet functions need to be chosen according to the criteria of support length, symmetry, and regularity. Thus, to obtainthe reconstructed sub-signals of different frequency bands, the original signal employed was a wavelet transform to decompose and reconstruct. The dominant frequency band was defined by quantifying the energy value of each frequency segment of the reconstructed sub-signals in this article. Furthermore, the different frequency bands' evolution characteristics of the AE energy proportions were analyzed, revealing the internal damage law of the rock during the rockburst simulation test.