## 3 A Collaborative Multi-Satellite Selection Algorithm Based on Maximum SINR
### Proposed Satellite Selection Strategy  
Construction of Optimization Problem for Maximizing SINR  
As shown in Figure 3, if packets from different terminals collide, all the satellite nodes covering these terminals are able to receive the collided packets. Our scheme aims to select the appropriate nodes from these satellites to collaborate and eliminate the collided packets. In the scenario shown above, where terminals 1-M send a packet at the same time, then in the RA protocol with no resource allocation, assume that all packets sent by \(M\) terminals collide at \(K\) satellite nodes. The signal received at a single satellite node can be modeled as follows:  
\[z_{i}=\sqrt{P_{i}}h_{1,i}S_{1}+\sum_{j=2}^{M}\sqrt{P_{i}}h_{j,i}S_{j}+n_{i}(1 \leq i\leq K) \tag{1}\]  
where \(S_{1}\) is the expected signal vector, \(S_{j},j\in[2,M]\) is the interference signal vector, \(n_{i}\) is the noise signal component and \(\sqrt{P_{i}}\) is the transmitting power. Beamforming is performed on the forwarded signal at the gateway station, and the receiver's signal expression obtained is as follows:  
\[y=\sum_{i=1}^{K}\sqrt{P_{i}}w_{i}h_{1,i}S_{1}+\sum_{i=1}^{K}\sum_{j=2}^{M} \sqrt{P_{i}}w_{i}h_{j,i}S_{j}+\sum_{i=1}^{K}w_{i}n_{i} \tag{2}\]  
Then, the useful signal power of the receiver can be expressed as follows:  
\[P_{S}=E\{|\sqrt{P_{i}}\sum_{i=1}^{K}w_{i}h_{1,i}S_{1}|^{2}\}=\mathbf{W}^{H} \mathbf{A}\mathbf{W} \tag{3}\]  
Figure 3: The relationship between terminals and satellite nodes when sending packets. where \(\mathbf{W}=[w_{1},\dots,w_{K}]^{H}\), \(\mathbf{A}=P_{t}E(\mathbf{H_{1}H_{1}}^{H})\), \(\mathbf{H_{1}}=[h_{1,1},h_{1,2},\dots,h_{1,K}]^{T}\). The power of interference and noise received at the receiver is as follows:  
\[P_{IN}=E\{|\sum\limits_{i=1}^{K}\sum\limits_{j=2}^{M}\sqrt{P_{i}}w_{i}h_{j,i}S _{j}|^{2}+|\sum\limits_{i=1}^{K}w_{i}n_{i}|^{2}\}+\sigma_{d}^{2}=\mathbf{W}^{H} \mathbf{B}\mathbf{W}+\sigma_{r}^{2}+\sigma_{d}^{2} \tag{4}\]  
where \(\mathbf{B}=P_{t}E(\mathbf{H_{2}H_{2}}^{H})\), \(\mathbf{H_{2}}=[h_{2,1},h_{2,2},\dots,h_{2,K}]^{T},\sigma_{r}^{2}\) is the noise variance between sender and satellites, and \(\sigma_{d}^{2}\) is the noise variance between satellites and gateway station. Then the SINR of the receiver can be written as the following:  
\[SINR=\frac{P_{S}}{P_{IN}}=\frac{\mathbf{W}^{H}\mathbf{A}\mathbf{W}}{\mathbf{ W}^{H}\mathbf{B}\mathbf{W}+\sigma_{r}^{2}+\sigma_{d}^{2}} \tag{5}\]  
Assuming that the number of satellites expected to participate in the process is \(N\). Selecting \(N\) suitable satellite nodes from \(K\) satellites, the non-convex optimization problem with Equation (5) as the objective function can be formulated as the following:  
\[\left\{ \begin{aligned} &\max\limits_{W}&\frac{\mathbf{W}^{H} \mathbf{A}\mathbf{W}}{\mathbf{W}^{H}\mathbf{B}\mathbf{W}+\sigma_{r}^{2}+\sigma _{d}^{2}}\\ & s.t.&\text{card}(\mathbf{W})=N\end{aligned} \right. \tag{6}\]  
where \(\text{card}(\mathbf{W})=N\) represents the cardinality of the weight vector (the number of non-zero parameters in \(\mathbf{W}\)). Moreover, when the gateway station processes the forwarded signal, a power processing upper limit needs to be set. The upper limit will make \(|\mathbf{W}|^{2}\leq 1\) and the individual component values \(|w_{j}|^{2}\) as small as possible, ensuring the overall low energy consumption and physical realization possibilities of the system. It is assumed that the power processing upper limit for the selected \(j\) (\(1\leq j\leq N\)) forwarding signal is denoted \(P_{j}\), which can be defined as follows:  
\[P_{j}=E\{|r_{j}|^{2}\}|w_{j}|^{2}=D_{j}|w_{j}|^{2} \tag{7}\]  
\[\mathbf{D}=[D_{1},\dots,D_{N}]=P_{t}diag(E\{|h_{1,1}|^{2}\}+E\{|h_{2,1}|^{2} \},\dots,\]  
\[E\{|h_{1,N}|^{2}\}+E\{|h_{2,N}|^{2}\})+\sigma_{r}^{2}I \tag{8}\]  
Therefore, optimization problems can be change 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{card}(\mathbf{W})=N\end{aligned} \right.