# Distributed Satellite Relay Cooperative Communication with Optimized Signal Space Dimension
## 3 Results
### Analysis of MIMO Channel Characteristics
In the past, fading was regarded as an unfavorable factor, which was often solved by equalization or diversity techniques. MIMO technology not only does not try to eliminate multi-path signals but makes full use of multi-path vectors in spatial propagation and optimizes transmission and reception as a whole. The signals of the same sub-channel, synthesized under different propagation paths, are extracted from multiple antennas, and the optimal signal is selected as the received signal of this sub-channel through analysis and comparison. This can effectively utilize random channel fading and multipath propagation to increase the transmission rate. When the transmit power of a single antenna remains unchanged, increasing the number of antennas can increase the average signal-to-noise ratio (SNR) at the receiving end by coherently combining multiple signals. The array gain is strongly related to the logarithm of the number of antennas. Array gain can improve system radiated energy efficiency. When the coverage remains unchanged, increasing the number of antennas can reduce the transmit power of the antennas, which in turn can reduce the requirements for the linear range of the power amplifier of the device. If the transmit power of a single antenna remains the same, the total power of massive MIMO will even decrease. The method proposed in this paper utilizes a precoding vector so that pairs of users exchanging information are aligned in the same direction in signal space. The different directions are kept in opposition to each other to avoid interference. The relay performs physical layer network coding on the received superimposed signal. In the design, it is necessary to comprehensively consider whether the constraints of each user on the alignment are satisfied. Take three nodes as an example: \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Channel 1** & **Channel 2** & **Channel 3** & **Channel 4** \\ \hline path 1 & 115.31 & 124.25 & 107.84 & 115.79 \\ path 2 & 4993.75 & 4995.00 & 4995.50 & 4987.51 \\ path 3 & 9991.25 & 9992.50 & 9990.00 & 9981.25 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Measured value of path delay (unit: ns). \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Channel 1** & **Channel 2** & **Channel 3** & **Channel 4** \\ \hline path 1 & \(-\)2.01 & \(-\)2.13 & \(-\)2.00 & \(-\)2.12 \\ path 2 & \(-\)5.13 & \(-\)4.86 & \(-\)5.15 & \(-\)4.89 \\ path 3 & \(-\)11.97 & \(-\)12.11 & \(-\)12.04 & \(-\)12.05 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Measured value of path loss (unit: dB). \[span\Big{(}\mathbf{H}^{[r,1]}\mathbf{v}^{[1,1]}\Big{)} =span\Big{(}\mathbf{H}^{[r,2]}\mathbf{v}^{[1,2]}\Big{)} \tag{17}\] \[span\Big{(}\mathbf{H}^{[r,2]}\mathbf{v}^{[2,2]}\Big{)} =span\Big{(}\mathbf{H}^{[r,3]}\mathbf{v}^{[1,3]}\Big{)}\] \[span\Big{(}\mathbf{H}^{[r,3]}\mathbf{v}^{[2,3]}\Big{)} =span\Big{(}\mathbf{H}^{[r,1]}\mathbf{v}^{[2,1]}\Big{)}\]  
Three network coded signals are obtained from independent codewords at the relay. Consider \(M\times 1\) vector \(\mathbf{q}\) that lies in \(R(\mathbf{A}_{1})\cap R(\mathbf{A}_{2})\), where \(R(\cdot)\) represents the column space of a matrix. \(\mathbf{A}_{1}\) and \(\mathbf{A}_{2}\) are \(M\times M\) random matrices whose entries are drawn from i.i.d. complex Gaussian distribution \(\mathbb{N}_{\mathbb{C}}(0,1)\). There exists \(\mathbf{q}_{i}\in\mathbb{C}^{M\times 1},i=1,2\), such that \(\mathbf{q}=\mathbf{A}_{1}\mathbf{q}_{1}=\mathbf{A}_{2}\mathbf{q}_{2}\). This can be rewritten in matrix form as  
\[ \begin{bmatrix}\mathbf{I}_{M}&-\mathbf{A}_{1}&0\\ \mathbf{I}_{M}&0&-\mathbf{A}_{2}\end{bmatrix}\begin{bmatrix}\mathbf{q}\\ \mathbf{q}_{1}\\ \mathbf{q}_{2}\end{bmatrix} =\mathbf{P}\mathbf{Q}=0 \tag{18}\]  
The cross subspace dimension of \(R(\mathbf{A}_{1})\cap R(\mathbf{A}_{2})\) can be found by computing the null space of \(\mathbf{P}\).