# Distributed Satellite Relay Cooperative Communication with Optimized Signal Space Dimension
## 2 Materials and Methods
### Cooperative Transmission of Spatial Information Network
#### 2.2.2 Resource Allocation Optimization of Space Compression  
Each source node \(i\) and the set \(S^{[i]}\) composed of other nodes exchange independent messages, \(S^{[i]}=\{1,2,\ldots,K\}\backslash\{i\}\). The message sent by node \(i\) to node \(j\) is \(\mathbf{s}^{[i,j]}=\left[s_{1}^{[i,j]},s_{2}^{[i,j]},\ldots,s_{d_{i,j}}^{[i,j ]}\right]^{T}\), which contains \(d_{i,j}\) independent information flows. The sum of the number of independent information flows sent in the system is \(d_{total}=\sum\limits_{i=1}^{K}\sum\limits_{j\in S^{[i]}}d_{i,j}\). When \(N<d_{total}\), the relay cannot decode all data streams independently. When the physical layer network coding is used, the relay simply decodes the network coding symbols \(\mathbf{s}_{\oplus}\). The element of vector \(\mathbf{s}_{\oplus}\) is composed of \(\left\{s^{[i,j]}+s^{[j,j]},\forall j\in S^{[i]},\forall i\right\}\). When \(N\geq 2M\), \(\mathbf{s}_{\oplus}\) cannot be directly obtained by signal space alignment using precoding matrices \(\mathbf{V}^{[i,j]}\) and \(\mathbf{V}^{[j,i]}\), \(\mathbf{s}_{\oplus}\) must be obtained by designing the relay compression matrix and aligning the desired interactive signal to the relay compression subspace. suppose that \(J\times N(J\leq N)\)-dimensional matrix \(\mathbf{P}\) is a compressed matrix and full rank, the compressed relay received signal is:  
\[\mathbf{\hat{y}}^{[R]}=\mathbf{P}\mathbf{y}^{[R]}=\sum\limits_{i=1}^{K}\mathbf{ P}\mathbf{H}^{[i,j]}\mathbf{V}^{[i]}\mathbf{g}^{[i]}+\mathbf{P}\mathbf{n}^{[R]} \tag{12}\]  
Thus, the subspace alignment condition of a compressed signal is obtained as follows:  
\[\mathbf{P}\mathbf{H}^{[i,j]}\mathbf{V}^{[i,j]}=\mathbf{P}\mathbf{H}^{[i,j]} \mathbf{V}^{[i,j]}=\mathbf{B}^{[i,j]} \tag{13}\]  
It is required that the compression matrix \(\mathbf{P}\) should not affect the expected decoded signal and that the designed precoding matrix \(\mathbf{V}^{[i,j]}\) has full rank. The equivalent condition is: \( \begin{bmatrix}\mathbf{V}^{[i,j]}\\ \mathbf{V}^{[j,i]}\end{bmatrix} \subseteq Null \begin{bmatrix}\mathbf{P}\mathbf{ H}^{[i,j]}&-\mathbf{P}\mathbf{H}^{[j,j]}\end{bmatrix} \). Align each pair of signals to the same signal subspace. The relay will receive \(\frac{d_{total}}{d_{total}}\) alignment signals. In order to ensure the complete decoding of signal \(\mathbf{s}_{\oplus}\) by the relay, \(J\geq\frac{d_{total}}{2}\) must be guaranteed. The compressed signal processed by the compression matrix is:  
\[\mathbf{\hat{y}}^{[R]}=\mathbf{B}\mathbf{s}_{\oplus}+\mathbf{P}\mathbf{n}^{[R]} \tag{14}\]  
Considering the space spanned by signal dimension, \(span(\mathbf{B})\subseteq span\Big{(}\mathbf{P}\mathbf{H}^{[i,j]}\Big{)}\)\(\cap span\Big{(}\mathbf{P}\mathbf{H}^{[i,j]}\Big{)}\) can be obtained. If \(N\geq 2M\), the intersection subspace is empty between \(span\Big{(}\mathbf{H}^{[i,j]}\Big{)}\) and \(span\Big{(}\mathbf{H}^{[i,j]}\Big{)}\) when the compression method is not exploited. After the compression matrix processing, SSA can be realized only when the intersection subspace exists. This is shown in Figure 3b. If the transmission channel between communicating nodes satisfies the independent Gaussian distribution, the basis vector of the intersection subspace of a pair of source node equivalent channel matrices \(\mathbf{P}\mathbf{H}^{[i,j]}\) and \(\mathbf{P}\mathbf{H}^{[i,j]}\) cannot be located in the intersection subspace of the other pair of source node equivalent channel matrices. Therefore, the matrix \(\mathbf{B}\) is a full rank matrix with full probability, which ensures the decidability of the signal \(\mathbf{s}_{\oplus}\) by the relay. The subspace alignment condition of the compressed signal (Equation (13)) holds if and only if there are at least \(\frac{d_{total}}{2}-2M+d_{i,j}\) basis vectors in the signal space formed by \(\mathbf{P}^{T}\), which are located in the \(\Big{[}\mathbf{H}^{[i,j]}&-\mathbf{H}^{[i,j]}\Big{]}^{T}\) null space of the communication pair \((i,j)\).