## 2 The Theory of Multiwavelets and the Symmetric Lifting Schemes  
### Multiwavelet Transform  
Like scalar wavelets, the theory of multiwavelets is based on multiple resolution analysis (MRA) [PERSON] and [PERSON] (2008). The difference is that multiwavelets are generated by dilations and translations of the vector functions \(\mathbf{\Phi}=\left[\varphi_{1},\varphi_{2,}\cdots,\varphi_{r}\right]^{\mathrm{T}}\), where \(r\in N\) and the space \(V_{j}\) at scale \(j\) is:  
\[V_{{}_{j}}=clos\{2^{/i/2}\varphi_{{}_{i}}(2^{j}t-k):1\leq i\leq r,k\in  
where \(\mathbf{\Psi}\!=\! [\psi_{1},\psi_{2},\cdots,\psi_{r}]^{\mathrm{T}}\) are multiwavelet functions, \(\mathbf{H}_{k}\) and \(\mathbf{G}_{k}\) are lowpass and highpass matrix filter banks, respectively, \(k=0,1,\ldots,\)\(N\) is the number of filter banks. In this paper, the case multiplicity \(r=2\) is under study. By the dilations of Equations (2) and (3), the following recursive relationship between coefficients \((c_{1,j,k},\)\(c_{2,j,k})^{\mathrm{T}}\) and \((d_{1,j,k},\)\(d_{2,j,k})^{\mathrm{T}}\) can be obtained:  
\[ \begin{pmatrix}c_{1,j-1,k}\\ c_{2,j-1,k}\end{pmatrix} \!=\!\sqrt{2}\!\sum_{n=0}^{K}H_{n}\! \begin{pmatrix}c_{ 1,j,2k+n}\\ c_{2,j,2k+n}\end{pmatrix} ,\qquad j,k\!\in Z \tag{4}\] \[ \begin{pmatrix}d_{1,j-1,k}\\ d_{2,j-1,k}\end{pmatrix} \!=\!\sqrt{2}\!\sum_{n=0}^{K}G_{n}\! \begin{pmatrix}c_{ 1,j,2k+n}\\ c_{2,j,2k+n}\end{pmatrix} ,\qquad j,k\!\in Z \tag{5}\]  
Similarly, multiwavelet reconstruction can be obtained by:  
\[ \begin{pmatrix}c_{1,j,n}\\ c_{2,j,n}\end{pmatrix} \!=\!\sqrt{2}\!\sum_{n=0}^{K}\!\left(H_{k}^{*}\! \begin{pmatrix} c_{1,j-1,2k+n}\\ c_{2,j-1,2k+n}\end{pmatrix} \!+\!G_{k}^{*}\! \begin{pmatrix}d_{1,j-1,2k+n}\\ d_{2,j-1,2k+n}\end{pmatrix} \right)\right] \tag{6}\]  
where \(\mathbf{H_{k}}^{*}\) and \(\mathbf{G_{k}}^{*}\) are dual matrices of \(\mathbf{H_{k}}\) and \(\mathbf{G_{k}}\), respectively. Decomposition and reconstruction of MWT can be represented in Figure 2. In view of the matrix filter banks, preprocessing is necessary for the one stream input signal \(x[n]\). Correspondingly, a postprocessing method is the inverse process of preprocessing. \(Q\) represents a prefilter while \(P\) is a post-filter. \(\left\{H,G,H^{*},G^{*}\right\}\) represent low-pass filter banks, dual low-pass filter banks, high-pass filter banks, and dual filter banks of multiwavelets. \(2\)\(\bigvee\) and \(2\)\(\bigvee\) represent decimation and zero-padding, respectively. \(\left\{c_{1,0,k},c_{1,-1,k},\cdots c_{1,i_{k},k},d_{1,i_{k},k}\right\}\) are the first branch and of the decomposition result and \(\left\{c_{2,0,k},c_{2,-1,k},\cdots c_{2,i_{k},k},d_{2,i_{k},k}\right\}\) are the second branch. Figure 2: **(a)** Decomposition (\(r=2\)) of MWT. **(b)** Reconstruction (\(r=2\)) of MWT. Different from scalar wavelets, multiwavelets require two or more input streams because of their matrix filter banks. Usually, there is only one input stream \(x[n]\) and therefore some kind of preprocessing method must be performed before multiwavelet decomposition [29]. Then a postprocessing method is needed to be performed after multiwavelet reconstruction and it is the inverse process of preprocessing. Moreover, different preprocessing methods have significantly different influences on performances of MWT. The most obvious way to get the second input row was just to repeat the input stream with a factor, namely the repeated row as shown in Equation (7). Here \(x_{k}\) is the input signal and \(\alpha\) is a constant which makes the system hold approximation order higher than zero. [PERSON] [30] claimed that the \"repeated row\" was effective for feature extraction. Therefore, it is adopted in this paper:  
\[\mathbf{s}_{0,n}= \begin{pmatrix}s_{0,n}^{(0)}\\ s_{0,n}^{(1)}\end{pmatrix} = \begin{pmatrix}x_{k}\\ \alpha x_{k}\end{pmatrix}  \tag{7}\]  
Although multiwavelets have several advantages over scalar wavelets, fixed or standard multiwavelets are usually not the optimal for specified applications because they are usually independent of a measured signal [22].