# Bistatic Sea Clutter Suppression Method Based on Compressed Sensing Optimization
## 3 The Proposed Method Description
### Optimal Design of Sensing Matrix  
Designing an effective sensing matrix is a key factor to improve the accuracy of the signal reconstruction in compressed sensing. Recently, various studies have shown that optimizing the sensing matrix by reducing the cross-correlation coefficient between the measurement matrix and the sparse base matrix can significantly improve the quality of signal reconstruction. Therefore, we propose a sensing matrix optimization algorithm based on eigenvalue decomposition and gradient optimization. In compressed sensing, the observation set \(y=\Gamma a\), the sensing matrix \(\Gamma=\phi\psi\). The cross-correlation coefficient can be defined as:  
\[\mu(\Gamma)=\underset{i\
eq j}{\max}\left\{\frac{\left|\Gamma_{i}^{T}\Gamma_{ j}\right|}{\|\Gamma_{i}\|\cdot\|\Gamma_{j}\|}\right\}=max\left\{\left|\overset{ \sim}{\Gamma}_{i}^{T}\overset{\sim}{\Gamma}_{j}\right|\right\} \tag{15}\]  
where \(\overset{\sim}{\Gamma}\) represents \(\Gamma\) after normalization. \(\Gamma_{i}\) and \(\overset{\sim}{\Gamma}_{i}\) represent the column vectors of matrix \(\Gamma\) and \(\overset{\sim}{\Gamma}\), respectively. Let Gram matrix \(G=\Gamma^{T}\Gamma\). The largest off-diagonal element in matrix \(G\) is the required cross-correlation coefficient. The disadvantage of this definition is that it can only reflect local coherence. Thus, one needs to introduce the global cross-correlation coefficient:  
\[\hat{\mu}=\sum_{i\
eq j}(g_{ij})^{2} \tag{16}\]  
where \(g_{ij}\) is the element of Gram matrix \(G\). The definition of the global cross-correlation coefficient is more consistent with RIP criterion. Let the eigenvalue of Gram matrix \(\eta_{i}>0(i=1,\dots\ m)\), according to the trace theory of matrix, \(\sum_{i=1}^{m}\eta_{i}=\sum_{i=1}^{m}g_{ii}\). We can reduce the global cross-correlation coefficient by reducing the sum of non-diagonal elements in the Gram matrix:  
\[\min\sum_{i\
eq j}(g_{ij})^{2}=\sum_{i=1}^{m}(\eta_{i})^{2}-\sum_{i=1}^{m}(g_ {ii})^{2},\ \ \text{s.t.}\sum_{i=1}^{m}\eta_{i}=n \tag{17}\]  
Because all elements of \(g_{ii}\) are 1, problem (17) can be solved by eigenvalue decomposition of the Gram matrix. The optimal solution can be obtained by making the non-zero terms in the eigenvalue \(n/m\), where \(n\) is the number of rows in the Gram matrix, and \(m\) is the rank of the Gram matrix. In order to make the signal reconstruction by the above Gram matrix more accurate, the sensing matrix can be further optimized by gradient descent algorithm:  
\[\min\|I_{N}-G\|_{F}^{2}=\min\|I_{N}-\psi^{T}\phi^{T}\phi\psi\|_{F}^{2}=\min\|I_ {N}-\Gamma^{T}\Gamma\|_{F}^{2} \tag{18}\]  
where \(I_{N}\) is the identity matrix, \(T\) means transposition. The error is defined as:  
\[\varepsilon=\left\|I_{N}-\Gamma^{T}\Gamma\right\|_{F}^{2}=\text{tr}\bigg{(} \Big{(}I_{N}-\Gamma^{T}\Gamma\Big{)}\Big{(}I_{N}-\Gamma^{T}\Gamma\Big{)}^{T} \bigg{)} \tag{19}\]  
After taking the derivative of \(\varepsilon\), one obtains \(\
abla\varepsilon=\frac{\partial\varepsilon}{\partial\Gamma}=4\Gamma\Big{(} \Gamma^{T}\Gamma-I_{N}\Big{)}\). We iteratively update the sensing matrix by setting the step size:  
\[\Gamma^{i+1}=\Gamma^{i}-\delta\Gamma^{i}\bigg{(}\Big{(}\Gamma^{i} \Big{)}^{T}\Gamma^{i}-I_{N}\bigg{)} \tag{20}\]  
After each iteration, the sensing matrix \(G^{i+1}\) column is unitized. Algorithm 2 presents a description of the sensing matrix optimization algorithm based on eigenvalue decomposition and gradient descent. ```
Input: Optimized sparse basis matrix \(\psi\), measurement matrix \(\phi\), identity matrix \(I\), step \(\mu\), the maximum iterations \(I_{\text{max}}\) and iteration threshold \(\varepsilon_{\text{th}}\). 1. Initial sensing matrix \(\Gamma=\phi\psi\), unitize \(\Gamma\) to get \(\overset{\sim}{\Gamma}\)
2. Construction Gram matrix \(G=\Gamma^{T}\Gamma\), eigenvalue decomposition of \(G\) to get \(G=SVS^{T}\);
3. Replace all non-zero elements in matrix \(V\) with \(n/m\) to get a matrix \(\widetilde{V}\)
4. Decompose matrix \(\widetilde{V}\) to get \(D_{m\times n}\), \(\widetilde{V}=D^{T}D\);
5.