## 3 A Collaborative Multi-Satellite Selection Algorithm Based on Maximum SINR
### Proposed Satellite Selection Strategy
\tag{9}\]  
This problem is essentially an NP-hard problem, and according to the method proposed in the literature [25], the square of the norm \(l_{1}\) can be used instead of the cardinality \(\text{card}(\mathbf{W})\). Because when setting the upper square limit parameter of \(l_{1}\) norm, the sparsity of the corresponding vector (the number of 0 component) can be controlled. Therefore, the above optimization problem can be changed to the following:  
\[\left\{\begin{aligned} &\max\limits_{W}&\frac{\mathbf{W}^{H} \mathbf{A}\mathbf{W}}{\mathbf{W}^{H}\mathbf{B}\mathbf{W}+\sigma_{r}^{2}+\sigma _{d}^{2}}\\ & s.t.&\text{$D_{j}|w  
of the optimization problem, matrix \(\mathbf{X}\) can be defined as \(\mathbf{X}\triangleq\mathbf{W}\mathbf{W}^{H}\in H_{N}\). \(\mathbf{X}\) is the semi-positive definite Hermitian matrix of \(K\times K\), \(||\mathbf{W}||_{1}^{2}=(\sum\limits_{i=1}^{K}|w_{i}|)\)\({}^{2}=(1,\ldots,1)_{K}\cdot|\mathbf{X}|\cdot\left( \begin{array}{c}1\\ \ldots\\ 1\end{array} \right)\), where \(|\mathbf{X}|\) represents the absolute value of matrix elements. Then the above optimization problem can be changed to the following:  
\[\left\{ \begin{array}{ll}\underset{W}{\max}&\frac{\operatorname{Tr}\{ \mathbf{A}\mathbf{X}\}}{\operatorname{Tr}\{\mathbf{B}\mathbf{X}\}+\sigma_{r}^ {2}+\sigma_{d}^{2}}\\ &D_{j}|w_{j}|^{2}\leq P_{jth}\quad\forall j=1,\ldots,K\\ &\\ s.t.&(1,\ldots,1)_{K}\cdot|\mathbf{X}|\cdot\left(\begin{array}{c}1\\ \ldots\\ 1\end{array}\right)_{K}\leq\gamma\\ &\mathbf{X}\succcurlyeq 0\\ &\operatorname{rank}(\mathbf{X})=1\end{array} \right. \tag{11}\]  
In Equation (11), \(\operatorname{rank}(\mathbf{X})=1\) condition will destroy the convex property of the optimization problem. This constraint can be removed by relaxation transformation, and Charnes-Cooper transform [26; 27] is introduced as follows: \(\sigma_{sum}^{2}=\sigma_{r}^{2}+\sigma_{d}^{2}\), \(z=\frac{1}{\operatorname{Tr}\{\mathbf{B}\mathbf{X}\}+\sigma_{sum}^{2}}\), \(\mathbf{Y}=\frac{\mathbf{X}}{\operatorname{Tr}\{\mathbf{B}\mathbf{X}\}+\sigma _{sum}^{2}}=z\mathbf{X}\). Then the above problems can be converted into the following convex optimization problems:  
\[\left\{ \begin{array}{ll}\underset{Y,z}{\max}&\operatorname{Tr}\{\mathbf{A} \mathbf{Y}\}\\ &Y_{\eta j}\leq\frac{2P_{jth}}{D_{j}}\quad\forall j=1,\ldots,K\\ &\\ s.t.&(1,\ldots,1)_{K}\cdot|\mathbf{X}|\cdot\left(\begin{array}{c}1\\ \ldots\\ 1\end{array}\right)_{K}\leq\gamma\\ &\operatorname{Tr}\{\mathbf{B}\mathbf{Y}\}+\sigma_{sum}^{2}z=1\\ &\operatorname{\mathbf{Y}\succcurlyeq 0\\ &z\succcurlyeq 0\end{array} \right. \tag{12}\]  
where \(z\) is a positive real number and \(\mathbf{Y}\) is a complex semi-positive definite Hermitian matrix of order \(K\). When the value of parameter \(\gamma\) is determined, the optimization problem can be solved by the CVX toolbox in Matlab. For optimization problem (12), the literature [28] points out that parameter \(\gamma\) is positively correlated with \(\operatorname{card}(\mathbf{W})\), so the value of \(\gamma\) determines the sparsity of the weight vector matrix. Increasing \(\gamma\) will increase the diagonal non-zero elements of \(\mathbf{Y}\) matrix, thus increasing the satellite nodes participating in the collaboration. Therefore, the selection of the forward signal is actually the problem of the value of \(\gamma\). In order to solve this optimization problem, we design an iterative binary search method for \(\gamma\) to obtain the optimal value. After setting the upper limit \(\gamma_{\max}\) and lower limits \(\gamma_{\min}\), the optimal value \(\gamma_{best}\) can be obtained by iteration.