# Distributed Satellite Relay Cooperative Communication with Optimized Signal Space Dimension
## 4 Discussion
### Antenna Configuration  
The idea of signal space alignment comes from interference alignment. The beamforming vector is designed to align several signals to the same dimension of the signal space to improve the utilization of space resources. In the co-channel transmission process, mutual interference between transmitted signals is avoided. By reducing the number of independent streams propagating over the air by signal superposition, the requirement of antenna configuration conditions is relieved. \(span\big{(}\mathbf{H}_{r,i}\mathbf{v}_{ij}\big{)}=span\big{(}\mathbf{H}_{r,j} \mathbf{v}_{ij}\big{)}\). If \(M\geq K-1\), \(N\geq K(K-1)/2\) and \(2M\geq N+1\) is achieved, the degrees of freedom of \(K(K-1)\) is satisfied. This is idea of traditional SSA.  
In order to further improve the transmission capacity of the entire cooperative transmission network, the weight of each edge of the weighted directed graph can be extended to \(d\). As long as the number of node antennas is not less than the number of independent streams \(d(K-1)\) to be sent, the relay can implement decoding and forwarding under the condition of \(N\geq d(K^{2}-3K+3)\). In order to further expand the constraints of antenna configuration, a compression matrix \(\mathbf{P}\) is designed to realize the transmission of interactive data in the signal subspace. When the signal space is compressed to \(dK(K-1)/2\) dimensions, the antenna configuration condition can be extended to \(N>2M\).  
Extending this improved SSA to any number of communication nodes \(K\) can more easily realize the switching of communication roles among satellites. As can be seen from Table 4, we not only break through the original antenna constraints \(N\leq 2M\), but also consume fewer antennas. Meanwhile, this breaks the constraints of the number of relay antennas and node antennas, and promotes the engineering application of satellite relay transmission.  
Table 5 lists the comparison of the number of antennas required by the method proposed in this paper and [34] in order to realize a certain transmission tuple. The variable \(T\), defined in Table 2, represents the synthesis of the number of antennas required by all communication nodes and relay nodes in the transmission process. It can be seen that the ring planning and design idea based on this paper can significantly reduce the requirements for antenna resources. For a certain degree of freedom tuple \(\mathbf{d}\), there may be unidirectional transmission, bidirectional transmission and multidirectional transmission in the network  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \(K\) & **Scheme** & \(M\) & \(N\) & **Total Number** \\ \hline \multirow{2}{*}{4} & SSA & 4 & 6 & 22 \\  & Proposed & 3 & 7 & 19 \\ \hline \multirow{2}{*}{5} & SSA & 6 & 10 & 40 \\  & Proposed & 4 & 13 & 33 \\ \hline \multirow{2}{*}{6} & SSA & 8 & 15 & 63 \\  & Proposed & 5 & 21 & 51 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Antenna configuration conditions of different schemes.  
at the same time. A node may be on multiple rings of different lengths or multiple rings of the same length at the same time. Elimination of loops in message flow graphs can significantly improve transmission efficiency.