# Wavelet Packet Method for Locating Critical Slip Surface Using the Strength Reduction Method
## 3 Wavelet and Wavelet Packet Transformation and Reconstruction
### Wavelet Packet Transform and Reconstruct  
In the MRA of signal function, the original signal is first decomposed into two parts, low frequency and high frequency, and then the low-frequency part is further decomposed into low frequency and high frequency, and then it is decomposed gradually. However, in wavelet packet decomposition (WPD), each scale is broken down into high frequency and low frequency simultaneously, so the resolution is higher.  
The wavelet package \(\left\{u_{n}\right\}_{n\in Z^{+}}\) can be created by scale function \(\phi(t)\) and wavelet function \(\psi(t)\). The definition is as follows  
\[\left\{\begin{array}{c}u_{2n}(t)=\sqrt{2}\sum_{k}h(k)u_{n}(2t-k)\\ u_{2n+1}(t)=\sqrt{2}\sum_{k}g(k)u_{n}(2t-k)\end{array}\right. \tag{11}\]  
where \(n\) is a wavelet packet parameter, \(h\) and \(g\) are orthogonal filters [54], \(u_{0}=\phi\), and \(u_{1}=\psi\).  
Binary scaling and translation can also be performed for \(u_{n}\)  
\[u_{j,n,k}(t)=2^{-j/2}u_{n}\Big{(}2^{-j}t-k\Big{)} \tag{12}\]  
For orthogonal wavelet packet analysis, the algorithm is very similar to the Mallat algorithm of orthogonal wavelet analysis. The following recursive formula can achieve the wavelet packet transform  
\[\left\{\begin{array}{c}d_{j+1,2n}(k)=\sum_{l}h(l-2k)d_{j,n}(l)\\ d_{j+1,2n}(k)=\sum_{l}h(l-2k)d_{j,n}(l)\end{array}\right. \tag{13}\]  
where \(d_{j,n}(k)\) is the wavelet packet coefficient, defined by  
\[d_{j,n}(k)=f(t),u_{j,n,k}(t) \tag{14}\]  
The reconstruction wavelet packet procedure is an inverse transform of the decomposition. The reconstruct recursive equation is  
\[d_{j,n}(k)=\sum_{l}h(k-2l)d_{j+1,2n}(l)+\sum_{l}h(k-2l)d_{j+1,2n+1}(l) \tag{15}\]  
The essentials of wavelet packet decomposition are to divide a signal into different frequency spans layer-by-layer. If the original signal contains enough information, the frequency span can be divided very subtly.  
The equation deduces that if the decomposition layer is chosen appropriately, we can obtain the beginning and end of expected frequency spans--and with this method, the original signal when every frequency is wider enough [55]. Thus, there is no need to locate the frequency spans accurately for the ingredients which we needed so that it can adjust frequency drift.  
The advantage of reconstruction is that we can choose all the frequency bins or some parts of them and set the others (noise or random interruption) as zero. Thus, if we can decompose a signal into different frequency bins, it is much easier to extract noise during the reconstruction. Figure 1 illustrates the three-layer wavelet and wavelet packet decomposition, where \(a\) represents the low-frequency part and \(d\) represents the high-frequency part.  
According to the entropy standard, a set of orthogonal bases selected from the wavelet packet is called a wavelet packet basis of \(L^{2}(R)\). Different wavelet packet bases result in different signal decomposition. The optimal wavelet tree can be selected according to the entropy standards [56]. Matlab software provides Shannon entropy, entropy threshold (threshold), norm entropy (norm), Sure entropy, log energy entropy (log energy), and user-defined entropy. Among these standards, Shannon entropy is used to select the optimal wavelet tree in this article.