## 3 A Collaborative Multi-Satellite Selection Algorithm Based on Maximum SINR
### Proposed Satellite Selection Strategy
The pseudo-code of the Algorithm 1 is as follows:```
Step 1: Initialization Set the desired number of collaboration nodes: \(N\); Set initial \(\gamma\) upper limit \(\gamma_{\max}=\sum\limits_{1}^{K}\frac{Np_{\min}}{D_{i}}\); Set initial \(\gamma\) lower limit \(\gamma_{\min}=0\); Set the initial value \(\gamma=(\gamma_{\max}+\gamma_{\min})/2\); Define \(num\) as the number of non-zero diagonal elements of matrix \(\mathbf{Y}\); Substitute \(\gamma\) into Equation (12), calculate matrix \(\mathbf{Y}\) and count \(num\). ``` Step 2: Iterative selection of optimal \(\gamma\) (1) while (\(num\
eq N\)) if\(num>N//\gamma\) greater than optimal value \(\gamma_{best}\), so reduce the upper limit \(\gamma_{\max}\) if (\(|\gamma-\gamma_{\min}|<0.1\)) \(\gamma_{\min}=\gamma_{\min}/2\); \(\gamma=(\gamma_{\max}+\gamma_{\min})/2\); else \(\gamma_{\max}=\gamma\); \(\gamma=(\gamma_{\max}+\gamma_{\min})/2\); end if if\(num<N//\gamma\) less than optimal value \(\gamma_{best}\), so increase the lower limit \(\gamma_{\min}\) if (\(|\gamma_{\max}-\gamma|<0.1\)) \(\gamma_{\max}=\gamma_{\max}/2\); \(\gamma=(\gamma_{\max}+\gamma_{\min})/2\); else \(\gamma_{\min}=\gamma\); \(\gamma=(\gamma_{\max}+\gamma_{\min})/2\)  end end end ```

**Algorithm 1**Iterative binary search method of parameter \(\gamma\)

It should be noted that this algorithm is only used to obtain the optimal parameter \(\gamma_{best}\), the subset \(R=\{R_{1},R_{2},\ldots,R_{N}\}\in\{1,2,\ldots K\}\) of the optimal collaboration and make the constraint condition \(\text{card}(\mathbf{W})=N\) hold. The weight vector \(\mathbf{W}\) obtained by directly substituting \(\gamma_{best}\) into Equation (12) is not accurate because the variance constraint of the \(l_{1}\) norm restricts the value of \(\mathbf{W}\). In order to compute the optimal weight vector \(\mathbf{W}^{\star}\), this constraint and the subset of relays that do not participate in the collaboration must be removed from this problem. Thus, when the optimal collaborative subset \(R=\{R_{1},R_{2},\ldots,R_{N}\}\) is obtained, the optimal weight vector \(\mathbf{W}^{\star}\) needs to be derived from the following equation:

\[\left\{ \begin{array}{ll}\underset{\mathbf{\widehat{Y}},z}{\max}&\text{Tr} \{\mathbf{A\widehat{Y}}\}\\ &\quad\quad\widehat{Y}_{jj}\leq\frac{z\gamma_{\min}}{D_{i}}&\forall j=1,\ldots,N\\ s.t.&\text{Tr}\{\mathbf{BY}\}+\sigma_{\text{sum}}^{2}z=1\\ &\quad\quad\widehat{\mathbf{Y}}\succeq 0\\ &\quad\quad z\succeq 0\end{array} \right. \tag{13}\]

where both \(\mathbf{\widehat{Y}}\) and \(\mathbf{\widehat{D}}\) are matrices corresponding to the optimal subset \(R=\{R_{1},R_{2},\ldots,R_{N}\}\). The optimal \(\mathbf{\widehat{Y}}\) matrix can be obtained by CVX. Assuming that \(\mathbf{\widehat{Y}}^{\star}\) is the optimal solution matrix and \(z^{*}\) is the optimal value, then \(\widehat{\mathbf{X}}^{*}=\frac{\widehat{\mathbf{Y}}^{*}}{z^{*}}\). The optimal weight vector \(\mathbf{W}^{*}\) can be solved by the following:

\[\mathbf{W}^{*}=\mathbf{v_{max}}\sqrt{\lambda_{\max}} \tag{14}\]

where \(\lambda_{\max}\) is the maximum eigenvalue of the \(\widehat{\mathbf{X}}^{*}\) matrix and \(\mathbf{v_{max}}\) is the eigenvector corresponding to \(\lambda_{\max}\). An Asynchronous Collision-Tolerant ACRDA Scheme Based on Satellite-Selection Collaboration-Beamforming

This section introduces the design of the SC-ACRDA scheme, the packet processing flow in SC-ACRDA and its performance analysis. ### Design of SC-ACRDA

When a satellite node signal is forwarded to the gateway station, the gateway station will process each signal in parallel using multiple sliding windows. Figure 4 shows the basic flowchart of the SC-ACRDA scheme. For packets being processed:

A. If the packet did not collide:

(1) Packet demodulation: Obtain the RM sequences and demodulate the packet directly. In ACRDA protocol, the packet generates multiple time domain copies within a virtual frame. The packet preamble contains the slot offsets of the time domain copies.