# Blasting Vibration Control and Signal Analysis of Adjacent Existing Deterioration Tunnels
## 5 Wavelet Packet Analysis of Measured Data
In practical engineering, the blasting scheme can be adjusted to reduce the low-frequency vibration effect produced by blasting and the distribution of dispersed energy in the low-frequency band. Thus, the damage of blasting vibration to the deterioration tunnel can be reduced. Take measuring point 1 as an example to show the decomposition and reconstruction process of MATLAB wavelet packet analysis. The original signal diagram and spectrum diagram of measuring point 1 are shown in Figure 15. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Layer** & **Si,0** & **Si,1** &... & **Si,j - 1** & **Si,j** \\ \hline
1 & 0–1000 & data &... & & 1000–2000 \\
2 & 0–500 & 500–1000 &... & 1000–1500 & 1500–2000 \\
3 & 0–250 & 250–500 &... & 1500–1750 & 1750–2000 \\
4 & 0–125 & 125–250 &... & 1750–1875 & 1875–2000 \\
5 & 0–62.5 & 62.5–125 &... & 1875–1937.5 & 1937.5–2000 \\
6 & 0–31.25 & 31.25–62.5 &... & 1937.5–1968.75 & 1968.75–2000 \\
7 & 0–15.625 & 15.625–31.25 &... & 1968.75–1984.375 & 1984.375–2000 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Wavelength band distribution table after wavelet packet decomposition. Figure 15: _Cont._  
is at node (7,3). The vibration velocity and spectrum in three directions of 16 nodes at measuring points 1 and 2 are shown in Figure 18.  
Figure 17: Reconstructed node frequency spectrum diagram. (**a**) Node (7,0); (**b**) Node (7,1); (**c**) Node (7,2); (**d**) Node (7,3). Figure 18: The relationship between three-way main frequency and peak vibration velocity of each node after wavelet packet reconstruction. (**a**) X direction; (**b**) Y direction; (**c**) Z direction. As can be seen from Figure 18, the peak vibration velocity in three directions is distributed at Node 9 (125-140.625 Hz) and Node 10 (140.625-156.25 Hz). The peak frequency is distributed at Node 3 (31.25-46.875 Hz) and Node 7 (93.75-109.375 Hz). The blasting vibration signal \(x(t)\) is decomposed by an i-layer wavelet packet; that is, the signal is decomposed into 2i components. The signal is projected onto the db8 wavelet base, and 2i wavelet packet coefficients are obtained. Then, each coefficient is reconstructed to reflect the different characteristics of the original signal. The expression for signal \(x(t)\)[29; 30] is  
\[x(t)=\sum\limits_{k=0}^{j-1}x_{i,k}= x_{i,0}+x_{i,1}+\cdots+x_{i,j-1} \tag{6}\]  
\(x_{i,k}\) is the reconstructed signal decomposed to the node K of layer \(i,j=2^{i}\), \(k=0\), 1, 2, \(\ldots\),\(j-1\). The time-frequency relation in the K frequency band after wavelet packet reconstruction is  
\[W(t,f_{k})=\big{|}x_{i,k}(t)\big{|}^{2} \tag{7}\]  
\(f_{k}\) is the center frequency of the reconstructed node \((i,k)\),  
According to [PERSON]'s theorem and Equation (1), the energy in the k frequency band after the reconstruction of the wavelet packet is  
\[E_{k}=\int W(t,f_{k})df=\int\big{|}x_{i,k}(t)\big{|}^{2}dt=\sum\limits_{l}^{n }\big{|}v_{k,m}\big{|}^{2} \tag{8}\]  
\(v_{k,m}\) is the amplitude corresponding to the discrete point \(x_{i,k}\) of the reconstructed signal, \(m=1\), 2, \(\ldots\), \(n\); \(n\) indicates the number of sampling data points. The total energy of the blasting vibration signal is \(E=\sum\limits_{k=0}^{j-1}E_{i,k}\). The energy percentage in the \(k\) frequency band after wavelet packet reconstruction is  
\[\eta=\frac{E_{k}}{E}\times 100\% \tag{9}\]  
The energy distribution of each frequency band after wavelet packet reconstruction is shown in Figure 19.  
As can be seen from Figure 19, the blasting vibration signal energy is widely distributed in each frequency band. It mainly concentrated in the range of frequency band 3 to band 8, in which the X-direction of test point 1 accounted for 75.6%, the X-direction of test point 2  
Figure 15:0 Energy proportion diagram. (**a**) Measuring point 1. (**b**) Measuring point 2.  
accounted for 85.6%, and the other directions accounted for a higher proportion. It shows that the energy of the blasting vibration signal is mainly concentrated in the low frequency band.