## 3 A Collaborative Multi-Satellite Selection Algorithm Based on Maximum SINR
### Proposed Satellite Selection Strategy
According to the processing in packet reconstruction, PK2 and PK3 can be used to demodulate and eliminate the corresponding copies in the forwarded signals of satellites C and D. Therefore, all packets in satellites C and D can be demodulated. ### Performance Analysis

In this section, we will derive the PLR and throughput performance of this scheme based on ACRDA. For the ACRDA scheme, the packet loss ratio comes from the following two sources: (1) Packet loss due to the received packet SINR not reaching the threshold when no deadlock occurs; (2) packet loss due to the inability to handle collision when a deadlock occurs. Where a deadlock is a condition where all copies of a packet have collided (including the packet itself), this is known as a deadlock in ACRDA. Deadlock can take many forms on packets, from multi-packet partial collision deadlock to multi-packet full collision deadlock and so on, which makes it mathematically difficult to describe them directly. Therefore, in order to mathematically derive the PLR in the presence of deadlocks, the literature [29] considers limiting the analysis to the most basic and frequent deadlock approach to obtain a lower bound on the PLR. This basic deadlock approach is that the demodulated packet and its copies are expected to collide with only one packet. In the case of basic deadlock mode, the proposed scheme eliminates the collision that occurs between packets in some time slots. Therefore, the PLR of this scheme comes from the following: (a) packet loss due to the original ACRDA scheme where the bit SNR \(E_{b}/N_{0}\) does not reach the demodulation threshold; (b) packet loss when the scheme cannot achieve sufficient power difference after collaborative beamforming is performed in the presence of deadlock. Assuming that the number of copies sent by the ACRDA scheme is \(N_{cp}\), there are \(N_{sl}\) time slots for a single data frame, and the number of iterative interference cancellations is \(N_{iter}\). The total system packet loss ratio can be expressed as follows:

\[ \begin{split} PLR^{Niter}(G,N_{cp},N_{sl})=PLR_{a}^{Niter}+PLR_{b}^ {Niter}\\ \approx PLR^{Niter}(G,N_{cp})\cdot P_{dl}^{0}+\sum\limits_{n=1}^{ \infty}{(1-\gamma)\cdot\Gamma[10\log_{10}(\frac{a}{1+\beta})]^{N_{cp}}\cdot P_ {dl}^{n}}\end{split}  \tag{15}\]

where \(P_{dl}^{n}\) is the probability of the existence of \(n\) basic deadlock states, \(P_{dl}^{0}\cdot PLR^{Niter}(G,N_{cp})\) is the PLR generated by iterating \(N_{iter}\) times without deadlock, which is the PLR of case (a). \(\sum\limits_{n=1}^{\infty}{(1-\zeta)\cdot\Gamma[10\log_{10}(\frac{a}{1+\beta} )]^{N_{cp}}\cdot P_{dl}^{n}}\) is the PLR for case (b), where \(\zeta\) is the statistical average success rate of relay collaborative beamforming in reaching the demodulation threshold. \(\alpha=E\{(E_{b}/N_{0})_{desire}\}\) is the mean value of the desired packet SNR. \(\beta=E\{(E_{b}/N_{0})_{in}\}\) is

Figure 5: Packet processing flow chart. the mean value of the SNR of the interfering packets. \(\Gamma(x)\) is a polynomial interpolation of the packet error rate (PER) curve in dB for a given channel code and packet SNR \(x\):

\[\Gamma(x)=\left\{ \begin{array}{ll}10(\Sigma_{10}^{10}F_{1}x^{ \prime})&x\geq T_{h}\\ 1&x<T_{h}\end{array} \right. \tag{16}\]

where \(x=10\log_{10}(E_{b}/N_{0})\), \(F_{0}=-0.29846\), \(F_{1}=-0.53778\), \(F_{2}=-0.23827\), \(F_{3}=0.02605\), \(F_{4}=-0.004\), \(F_{5}=-0.01752\), \(F_{6}=3.45\times 10^{-3}\), \(F_{7}=2.02\times 10^{-3}\), \(F_{8}=-3.52\times 10^{-4}\), \(F_{9}=-3.46\times 10^{-5}\), \(F_{10}=0\).