# Underwater Target Recognition Based on Multi-Decision
## 2 System Model
### Signal Decomposition Algorithm Based on Resonance
#### 2.1.1 Morphological Component Analysis  
Morphological component analysis is usually used to decompose signals with different morphological characteristics [29]. The ship radiated noise with oscillating and non-oscillating component has different morphological characteristics. So, the MCA algorithm can be used to separate and extract the ship radiated noise to construct the optimal sparse representation for its high-resonance and low-resonance component.  
Considering the discrete ship radiated noise sequence, the signal can be sparsely expressed as:  
\[x=\Phi_{h}w_{h}+\Phi_{l}w_{l}+n, \tag{1}\]  
where \(w_{h}\), \(w_{l}\) are the wavelet coefficients corresponding to the high resonant component \(x_{h}\) and the low resonant component \(x_{l}\). \(\Phi_{h}\), \(\Phi_{l}\) are wavelet basis functions corresponding to \(x_{h}\), \(x_{l}\). \(n\) represents the residual components of the signal which removes first two.  
The purpose of MCA is to obtain an optimal representation \(w_{h}\), \(w_{l}\) of the high-resonance component and low-resonance component of the signal. This problem can be solved by minimizing the following objective function:  
\[J(w_{l},w_{h})=\|x-\Phi_{h}w_{h}-\Phi_{l}w_{l}\|_{2}^{2}+\sum_{j=1}^{I_{h}+1} \lambda_{h,j}\|w_{h}^{j}\|_{1}+\sum_{j=1}^{I_{l}+1}\lambda_{l,j}\|w_{l}^{j}\|_ {1}. \tag{2}\]  
Here, \(J_{h}\) and \(J_{l}\) represent the number of decomposition layers of \(x_{h}\) and \(x_{l}\). \(w_{h}^{j}\) and \(w_{l}^{j}\) are the wavelet coefficients of the high-resonance component and the low-resonance component of the \(j\)th layer, respectively. \(\lambda_{h,j}\), \(\lambda_{l,j}\) are the normalized coefficients of \(w_{h,j}\), \(w_{l,j}\) and their values are related to energy of \(\Phi_{h,j}\), \(\Phi_{l,j}\):  
\[\lambda_{l,j}=k_{l,j}\|\Phi_{l,j}\|_{2},j=1,2,\cdots,J_{l}+1, \tag{3}\]  
\[\lambda_{h,j}=k_{l,j}\|\Phi_{h,j}\|_{2},j=1,2,\cdots,J_{h}+1, \tag{4}\]  
where \(k_{l,j}\), \(k_{h,j}\), \((k_{l,j}+k_{h,j}=1)\) are the proportionality coefficient of the energy distribution of the high-resonance component and the low-resonance component. \(k_{l,j}=k_{h,j}=0.5\) are selected to balance the energy distribution of the two components.  
Through decomposition of the Augmented Lagrangian Shrinkage Algorithm (SALSA) [26], the optimal wavelet coefficients can be obtained by solving the optimization problem of the formula. Therefore, the optimal expressions for the high-resonance component and the low-resonance component obtained by the MCA algorithm are:  
\[x_{h}^{*}=\Phi_{h}w_{h}^{*}, \tag{5}\]  
\[x_{l}^{*}=\Phi_{l}w_{l}^{*}. \tag{6}\]  
In summary, the purpose of the RSSD algorithm is to construct the optimal sparse representation of the high and low-resonance components of the ship radiated noise. The specific steps can be expressed as follows:  
Figure 2: Signal decomposition algorithm based on resonance.  
1. Select the appropriate filter scaling factor \(\alpha\), \(\beta\) according to the waveform characteristics of the signal. Then calculate the parameters \(Q_{h}\), \(r_{h}\), \(J_{h}\) corresponding to the high-resonance component, and the parameters \(Q_{l}\), \(r_{l}\), \(J_{l}\) corresponding to the low-resonance component. At last, construct the corresponding wavelet basis function \(\Phi_{h}\), \(\Phi_{l}\).
2. Reasonably set the weighting coefficient \(\lambda_{h,j}\), \(\lambda_{l,j}\) of the L1 norm of the wavelet coefficients of each layer. Obtain the optimal wavelet coefficient \(w_{h}^{*}\), \(w_{l}^{*}\) by minimizing the objective function through the SALSA algorithm.
3. Reconstruct the optimal sparse representation \(x_{h}^{*}\), \(x_{l}^{*}\) of high-resonance components and low-resonance components.