## 4 Numerical Results
### Channel Quality Analysis  
In order to reflect the channel quality of the communication link of a satellite communication system, the satellite communication model is equivalent to the baseband transmission model. Suppose that the communication satellite is basically aligned in the relay signal space, the communication frequency is 6 GHz, and the modulation mode is QPSK. The difference that channel quality makes on the propagation path has a very important impact on the channel selection and precoding design. The solution for the aforementioned signal subspace orthogonal alignment scheme should be designed according to the  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \(\boldsymbol{x_{1}+x_{2}}\) & \(-2L+2\) & \(-2L+4\) & \(-2L+6\) & \(\cdots\) & \(\boldsymbol{0}\) & \(\boldsymbol{2}\) & \(\cdots\) & \(\boldsymbol{2L-6}\) & \(\boldsymbol{2L-4}\) & \(\boldsymbol{2L-2}\) \\ \hline \(\left(c_{1},c_{2}\right)\) & (0,0) & (0,1) & (0,2) & & \(\left(0,L-1\right)\) & (1,\(L-1\)) & & \(\left(L-3,L-1\right)\) & \(\left(L-2,L-1\right)\) & \(\left(L-1,L-1\right)\) \\  & & (1,0) & (1,1) & & \(\left(1,L-2\right)\) & \(\left(2,L-2\right)\) & & \(\left(L-2,L-2\right)\) & \(\left(L-1,L-2\right)\) & \\  & & & (2,0) & \(\vdots\) & \(\vdots\) & \(\vdots\) & \(\vdots\) & \(\vdots\) & \(\left(L-1,L-3\right)\) & & \\  & & & & (\(L-2,1\)) & \(\left(L-2,2\right)\) & & & & & \\  & & & & & \(\left(L-1,0\right)\) & \(\left(L-1,1\right)\) & & & & & \\ \hline \(s_{r}=G(c_{1}+c_{2})\) & 0 & 1 & 2 & \(\cdots\) & \(L-1\) & 0 & \(\cdots\) & \(L-4\) & \(L-3\) & \(L-2\) \\ \(M_{R,L}(s_{r})\) & \(-2L+2\) & \(-2L+6\) & \(-2L+10\) & \(\cdots\) & \(2L-2\) & \(-2L+2\) & \(\cdots\) & \(2L-14\) & \(2L-10\) & \(2L-6\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: PNC remapping scheme under \(M\)-QAM.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline user antennas & \(M\) \\ relay antennas & \(N\) \\ \(K\) & 10 \\ angle & \(\theta\in\left[-\pi,\pi\right)\) \\ number of channels & \(M\times N\) \\ frequency & 6 GHz \\ FFT Length & 128 \\ modulation & BPSK/16 QAM/64 QAM \\ band width & 25 MHz \\ multipath & 3 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Simulation Parameters.  
different channel qualities. For this reason, Figure 4 simulates three different quality transmission channels for simulating various communication environments.  
Then, we evaluate how the correlation index of large-scale fading changes on the array effects of the channel estimation. The average correlation factor r of the antenna is shown in Figure 5. The fixed position of the target node is 30 degrees, and the effective SNR is 10 dB. The variable position of the interfering node covers \(\theta\in[-\pi,\pi)\), and the effective SNR = 0 dB. Figure 5 shows that the larger the antenna number \(M\), the lower the correlation between nodes. The larger the rank equivalent to the channel matrix, the more dispersed the eigenvalues are. In addition, it can be seen that whenever the interfering node has the same angle as the desired node, the correlation coefficient is equal to 1. That is, the estimates are completely correlated. This means that the estimates are parallel, and they can only be distinguished by scaling factors. This usually happens in the case of uncorrelated Rayleigh fading and a single antenna. If each node has a different angle, the correlation level between nodes will decrease. This highlights the fact that the completely different characteristic directions of nodes help to reduce the correlation between channel estimates. Another interesting phenomenon is related to correlation factors: when correlation factors increase, more spatial dimensions are obtained. This increases the difference between channels and reduces the impact between them.  
Figure 4: Channel transmission loss under different propagation paths.