## 2 Theoretical Background
### Principle of Wavelet Packet Transform  
The principle of wavelet packet transform (WPT) method can be described as a generalization of the wavelet decomposition that offers a richer range of possibilities for signal analysis that can be used in data preprocessing for fault diagnosis [14], damage identification [15], improving the time resolution and noise suppression [27], and signal compression [28]. In wavelet packet analysis, a signal \(S^{0}_{0}\) is split into an approximation \(S^{0}_{1}\) and a detail \(S^{1}_{1}\), which represent the low and high frequencies, respectively. The approximation is then itself split into a second-level approximation and detail, and the process is repeated (Figure 2). This decomposition, in effect, halves the time resolution and doubles the frequency resolution. For an n-level decomposition, there are n+1 possible ways to decompose or encode the signal, yielding more than different ways to encode the signal [29]. The wavelet packet function is a time-frequency function that can be defined as [30],  
\[W^{n}_{ijk}(t)=2^{\frac{j}{2}}W^{n}\big{(}2^{j}t-k\big{)} \tag{7}\]  
where the integers j and k are the index scale and translation operations. The index n is an operation modulation parameter or oscillation parameter. The first two wavelet packet functions are the scaling and mother wavelet functions given by Equations (8) and (9), respectively.  
\[W^{o}_{o,o}(t)=\varnothing(t) \tag{8}\]  
\[W^{1}_{o,o}(t)=\psi(t) \tag{9}\]  
when \(n=2\), \(3\dots\) the function can be defined by the following recursive relationships:  
\[W^{2n}_{o,o}(t)=\sqrt{2}\sum_{k}h(k)W^{n}_{1,k}(2t-k) \tag{10}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Concrete Quality** & **Impact Wave Speed in** & **Compressive Strength,** & **Modulus of Elasticity,** \\  & **Concrete, \(c\) (m/s)** & \(f^{\prime}_{c}\) (MPa)** & \(E=4700\)/\(\bigcup^{0.5}_{\text{\tiny{$\mathcal{O}$}}}\) (MPa)** \\ \hline Excellent & \(>\)4570 & \(>\)73.5 & \(>\)40,294 \\ Good & 3660–4570 & 20.4–73.5 & 21,228–40,294 \\ Not good & 3050–3660 & 8.6–20.4 & 13,783–21,228 \\ Bad & 2130–3050 & 2.4–8.6 & 7281–13,783 \\ Very bad & \(<\)2130 & \(<\)2.4 & \(<\)7281 \\ \hline \hline \end{tabular}
\end{table}
Table 1: The relation between concrete quality and impact wave velocity [24], compressive strength [25], and modulus of elasticity [26].  
\[\mathrm{W}^{2n+1}_{\mathrm{o,o}}(t)=\sqrt{2}\sum_{k}g(k)W^{n}_{1,k}(2t-k) \tag{11}\]  
where h(k) and g(k) are the quadrature mirror filter (QMF) associated with the predefined scaling function and mother wavelet function. The wavelet packet coefficients, \(W^{n}_{j,k}\), are computed by the inner product \(\langle f(t),W^{n}_{j,k}\rangle\) that is defined as,  
\[W^{n}_{j,k}=f(t),W^{n}_{j,k}=\int f(t)W^{n}_{j,k}(t)dt \tag{12}\]  
An example of the framework of the WPT algorithm broken up to three resolution levels is viewed as a complete binary tree, as shown in Figure 2. In the figure, \(S^{0}_{3}\) presents the symbol for a subspace that stands for the third resolution and the 0 th subspace, while \(H\) and \(G\) are the two impulse responses of low-pass and high-pass analyzing filters corresponding to the scaling function and the wavelet function, respectively.  
In pulse echo technique applied to piles, a pulse is emitted, and if a geometric defect, flaw, or material inconsistencies exists, peak reflections or delayed echo is expected. But in any case, the material response introduces grain noise, which hinders echo and defect detection. The main advantage gained by using wavelet packet analysis is the ability to perform a local analysis of a signal, i.e., zooming on any interval of time or space. Wavelet packet analysis is thus capable of revealing some hidden aspects of the data that other signal analysis techniques fail to detect [15], which is a particularly important property for damage and defect detection applications.