## 2 Summary of Multiwavelet Transform  
Multiwavelet is generated by two or more mother wavelets [28]. Similar to the scalar wavelet transform, the theory of multiwavelet is also based on the concept of multi-resolution analysis (MRA) [28]. Multi-scaling function vector \(\Phi=[\phi_{1},\phi_{2},\ \cdots,\phi_{r}]^{\mathbb{T}}\) and multiwavelet function vector \(\Psi=[\psi_{1},\psi_{2},\cdots,\psi_{r}]^{\mathbb{T}}\) satisfy the following two-scale matrix refinement Equations:  
\[\Phi(t)=\sqrt{2}\sum_{k=o}^{M}H_{k}\Phi(2t-k)\quad\ k\in Z \tag{1}\]  
\[\Psi(t)=\sqrt{2}\sum_{k=o}^{M}G_{k}\Phi(2t-k)\quad\ k\in Z \tag{2}\]The coefficients \(\{H_{k}\}\) and \(\{G_{k}\}\) are \(r\times r\) matrices instead of scalars and \(\Psi=[\psi_{1},\psi_{2},\cdots,\psi_{r}]^{\mathrm{T}}\) denotes the multiwavelet function corresponding to multi-scaling function \(\Phi\). In the frequency domain, Equations (1) and (2) are:  
\[\hat{\Phi}(\omega)=H(e^{-i\omega/2})\hat{\Phi}(\frac{\omega}{2}) \tag{3}\]  
\[\hat{\Psi}(\omega)=G(e^{-i\omega/2})\hat{\Phi}(\frac{\omega}{2}) \tag{4}\]  
\(H(\omega)\) and \(G(\omega)\) are the refinement symbols corresponding to \(\Phi\) and \(\Psi\). The symbols in \(Z\)-domain are determined by:  
\[H(z)=\frac{1}{2}\sum_{k=0}^{M}H_{k}z^{k}\;\;\;\mathrm{and}\;\;G(z)=\frac{1}{2} \sum_{k=0}^{M}G_{k}z^{k} \tag{5}\]  
With the starting vector coefficients \(\lambda_{0,0},\cdots,\lambda_{0,2^{j}-1}\), the decomposition step of multiwavelet transform is:  
\[\lambda_{j-1,n}=\sum_{k}H_{k-2n}\lambda_{j,k}\;\;\mathrm{and}\;\;\gamma_{j-1,n }=\sum_{k}G_{k-2n}\lambda_{j,k} \tag{6}\]  
Low frequency coefficients \(\lambda_{j-1,n}\) and high frequency coefficients \(\gamma_{j-1,n}\) after the decomposition step are vectors of \(r\)-dimension. The reconstruction step of multiwavelet transform is:  
\[\lambda_{j,k}=\sum_{n}H^{*}_{k-2n}\lambda_{j-1,n}+\sum_{n}G^{*}_{k-2n}\gamma_ {j-1,n} \tag{7}\]  
Note that the superscript * means the complex conjugate transpose.  
Due to the translations and dilations operations of multi-scaling and multiwavelet vector functions, multiwavelet can seize the vital vibration data processing properties of orthogonality, symmetry, compact support and higher order of vanishing moments simultaneously [15], which has been proven to be impossible for scalar wavelets, except Haar wavelet. Thus, multiwavelet transform can describe any vibration data more precisely and comprehensively because of its multi-input and multi-output system. In addition, due to the matrix-valued filter-bank, two or more input streams are needed in the process of multiwavelet transform. However, the processing vibration data would be one input stream usually and so some kind of pre-processing should be done before the implementation of multiwavelet transform. Correspondingly, a post-processing step is needed after the multiwavelet transform and it must be the inverse process of the pre-processing step. There are many kinds of pre-filters with different properties [29]. It has also been proven that the pre-filter algorithm called oversampling is more beneficial to vibration data feature identification than critically sampling ones [30]. Therefore, oversampling algorithm is selected as the multiwavelet preprocessing operation in these applications.