# Distributed Satellite Relay Cooperative Communication with Optimized Signal Space Dimension
## 2 Materials and Methods
### Signal Model  
Consider a satellite relay cooperative transmission network with \(K\) nodes. Each transmission node is configured with \(M_{i},i=1,\ldots,K\) antennas, and the relay forwarding node is configured with \(N\) antennas. For simplicity of description, it is assumed that each transmission node has the same antennas of \(M\). Because the transmission distance is too long and the energy of the transmitted signal is limited, the nodes use the relay forwarding mode to exchange information [26; 27; 28]. Node \(i\) expects to send message \(W^{[i]}\) through the cooperative transport network, and at the same time receives and decodes messages that may be transmitted by any other node except its own. \(\left\{\hat{W}^{[1]},\quad\hat{W}^{[2]},\quad\cdots,\quad\hat{W}^{[K]}\right\} /\left\{\hat{W}^{[i]}\right\}.\)  
The entire transmission process is divided into two stages. In the first stage, all nodes that need to transmit messages send signals to the relay simultaneously. This is called multiple access channel (MAC). It is described by  
\[\mathbf{y}^{[R]}=\sum_{i=1}^{K}\mathbf{H}^{[R,i]}\mathbf{x}^{[i]}+\mathbf{n}^{ [R]} \tag{1}\]  
where, \(\mathbf{H}^{[R,i]}\) is the channel transmission matrix from node \(i\) to relay \(R\), and the size is \(N\times M_{i}\). \(\mathbf{x}^{[i]}\in\mathbb{C}^{M_{i}}\) is the signal vector finally transmitted by node \(i\) after signal processing. \(\mathbf{n}^{[R]}\in\mathbb{C}^{N}\) represents the white Gaussian noise. The signals transmitted by all nodes satisfy the power constraints \(\mathrm{E}\!\left[Tr\!\left(\mathbf{x}^{[i]}\mathbf{x}^{[i]}\right)\right] \leq SNR\). Due to the strong direct component in satellite communication and few scatterers in the channel, the MIMO channel changes slowly. The elements of the channel transmission matrix satisfy the independent and identical distribution under the complex Gaussian random variable, \(\mathbb{NC}(0,1)\). The relay receives and separates the superimposed signal and encodes it into the signal to be sent. The second stage is the broadcast channel (BC) stage, where the relay transmits the encoded signal to all destination nodes in the form of broadcast. The broadcast signal received by node \(i\) is as follows:  
\[\mathbf{y}^{[i]}=\mathbf{H}^{[i,R]}\mathbf{x}^{[R]}+\mathbf{n}^{[i]} \tag{2}\]  
where, \(\mathbf{H}^{[i,R]}\) is the channel transmission matrix from relay \(R\) to node \(i\), and the size is \(M_{i}\times N\). \(\mathbf{x}^{[R]}\in\mathbb{C}^{N}\) is the signal vector finally transmitted by the relay after signal processing. \(\mathbf{n}^{[i]}\in\mathbb{C}^{M}\) represents the white Gaussian noise. The signals transmitted by the relay satisfy the power constraints \(\mathrm{E}\!\left[Tr\!\left(\mathbf{x}^{[R]}\mathbf{x}^{[R]}\right)\right] \leq SNR\). Figure 1: Inter satellite link directed graph. During the MAC phase, \(K-1\) independent messages \(W^{[j,i]}\) are sent from node \(i\) to the relay for node \(j\) where \(i,j\in\{1,2,\ldots K\}\) and \(j\
eq i\). It uses the precoding vector to construct the spatial direction to send the encoded flow. Specifically, it can be expressed as:  
\[\mathbf{x}^{[i]}=\sum_{j=1,j\
eq i}^{K}\mathbf{v}^{[j,i]}s^{[j,i]},i\in\{1,2, \ldots,K\} \tag{3}\]  
The network coding technology of signal space alignment is to design the beam space alignment direction, so that the signals sent by the two nodes that exchange information are aligned to the same signal subspace. At the same time, the different alignment directions are kept independent to avoid mutual interference. The relay performs joint detection of decoding and forwarding on the received superimposed signal. The design of the beamforming vector should satisfy:  
\[ \begin{array}{ll}\mathbf{H}^{[R,1]}\mathbf{v}^{[2,1]}=\mathbf{H}^{[R,2]} \mathbf{v}^{[1,2]}=\mathbf{u}^{[R,1]}\\ \mathbf{H}^{[R,1]}\mathbf{v}^{[3,1]}=\mathbf{H}^{[R,3]}\mathbf{v}^{[1,3]}= \mathbf{u}^{[R,2]}\\ \vdots\\ \mathbf{H}^{[R,K-1]}\mathbf{v}^{[K,K-1]}=\mathbf{H}^{[R,K]}\mathbf{v}^{[K-1,K ]}=\mathbf{u}^{[R,\frac{K(K-1)}{2}]}\end{array}