## 3 Signal Spatial Reconstruction Method for Relay Cooperative Networks
### Heterogeneous Signal Transmission under Multi Beam  
The high-throughput satellite communication system mainly adopts unicast transmission when providing point-to-point services for users. However, this physical layer transmission mode cannot be directly applied to the rapidly growing multicast services. Multicast transmission can use the same wireless resources to save wireless time-frequency resources for users with the same service request and can provide services for users according to their channel characteristics. This paper optimizes the transmission performance of the physical layer, using the same wireless resource to provide transmission services for multiple unicast packets and multicast services simultaneously. The unicast information sent by users to specific users is defined as a private signal, and the multicast information sent by users to specific groups is defined as a common signal. In the MAC phase, the private signal sent by user \(i\) to user \(j\) is \(s_{p}^{[j]}\), and the common signal sent by user \(i\) is \(s_{c}^{[i]}\). The transmitting node uses the beamforming technology of multiple antennas to design precoding vectors \(\mathbf{v}_{p}^{[j,i]}\) and \(\mathbf{v}_{c}^{[i]}\) respectively. Each user designs a private precoding vector to align the interactive pair of user signals to the same signal space. Equation (1) becomes  
\[span\Big{(}\mathbf{H}^{[R,i]}\mathbf{v}_{p}^{[j,i]}\Big{)}=span\Big{(}\mathbf{ H}^{[R,i]}\mathbf{v}_{p}^{[i,i]}\Big{)} \tag{9}\]  
Except for user \(1\) and user \(K\), each user \(i\) sends two common signals with the same content, that is  
\[\mathbf{x}_{c}^{[i]}=\mathbf{v}_{c,1}^{[i]}s_{c}^{[i]}+\mathbf{v}_{c,2}^{[i]} s_{c}^{[i]},\mathbf{v}_{c,1}^{[1]}=\mathbf{v}_{c,2}^{[K]}=\mathbf{0} \tag{10}\]  
The designed common coding rule is that the second common signal of user \(i-1\) must aligned with the first common signal of user \(i\), and the second common signal of user \(i\) must be aligned with the first multicast message of user \(i+1\). \[ \begin{array}{l}span\Big{(}\mathbf{H}^{[R,i-1]}\mathbf{v}_{c,2}^{[i-1]} \Big{)}=span\Big{(}\mathbf{H}^{[R,i]}\mathbf{v}_{c,1}^{[i]}\Big{)}\\ span\Big{(}\mathbf{H}^{[R,i]}\mathbf{v}_{c,2}^{[i]}\Big{)}=span\Big{(} \mathbf{H}^{[R,i+1]}\mathbf{v}_{c,1}^{[i+1]}\Big{)}\end{array}  \tag{11}\]  
Figure 2: Framework diagram of signal space reconstruction system. Taking 4 users as an example, suppose that the user is equipped with 7 antennas and the relay is equipped with 13 antennas. According to Equations (9) and (11), the private alignment direction combination is \(\mathbf{U}_{p}=\left[\mathbf{u}_{p}^{\pi(1,2)}\ \mathbf{u}_{p}^{\pi(1,3)}\ \mathbf{u}_{p}^{\pi(1,4)}\ \mathbf{u}_{p}^{\pi(2,3)}\ \mathbf{u}_{p}^{\pi(2,4)}\ \mathbf{u}_{p}^{\pi(3,4)}\right]\) and the common alignment direction combination is \(\mathbf{U}_{c}=\left[\mathbf{u}_{c,1}\ \ \ \mathbf{u}_{c,2}\ \ \ \mathbf{u}_{c,3}\right]\) which can be obtained by using the method of Formula (4), respectively. Where \(\pi(i,j)\) represents the linear index combination function of unordered pair \((i,j)\). After signal space alignment, the signal received by the relay can be expressed as follows:  
\[ \begin{array}{ll}\mathbf{y}^{[r]}&=\sum\limits_{i=1}^{4}\mathbf{H}^{[r,l]} \left(\sum\limits_{j\
eq i}\mathbf{v}_{p}^{[j,l]}\mathbf{s}_{p}^{[j,l]}+ \mathbf{v}_{c,1}^{[i]}\mathbf{s}_{c}^{[i]}+\mathbf{v}_{c,2}^{[i]}\mathbf{s}_ {c}^{[i]}\right)+\mathbf{n}^{[r]}\\ &=\left[\ \mathbf{U}_{p}\ \ \ \mathbf{U}_{c}\ \right]\left[\begin{array}{c} \mathbf{s}_{p}^{[r]}\\ \mathbf{s}_{c}^{[r]}\end{array}\right]+\mathbf{n}^{[r]}\\ &\text{where}\ \mathbf{v}_{c,1}^{[1]}=\mathbf{v}_{c,2}^{[4]}=\mathbf{0}_{M \times 1}\text{, }\mathbf{s}_{p}^{[r]}=\begin{bmatrix}s_{p}^{[1,2]}+s_{p}^{[2,1]}\\ s_{p}^{[1,3]}+s_{p}^{[3,1]}\\ s_{p}^{[1,4]}+s_{p}^{[4,1]}\\ s_{p}^{[2,3]}+s_{p}^{[4,2]}\\ s_{p}^{[3,4]}+s_{p}^{[4,2]}\\ s_{p}^{[3,4]}+s_{p}^{[4,3]}\end{bmatrix},\mathbf{s}_{c}^{[r]}=\begin{bmatrix}s _{c}^{[1]}+s_{c}^{[2]}\\ s_{c}^{[2]}+s_{c}^{[3]}\\ s_{c}^{[3]}+s_{c}^{[4]}\end{bmatrix}.\end{array}