## 3 Signal Spatial Reconstruction Method for Relay Cooperative Networks
### Heterogeneous Signal Transmission under Multi Beam
 \tag{12}\]  
Using ZF decoding, the relay can obtain six private superimposed signals and three common superimposed signals, respectively. In the BC stage, the BPSK modulated signal modulates and maps the superimposed signal according to the physical layer network coding principle [41] to obtain the signal to be transmitted \(\mathbf{s}_{r}=\left[s_{r,p}^{1},s_{r,p}^{2},s_{r,p}^{3},s_{r,p}^{4},s_{r,p}^{5 },s_{r,p}^{6},s_{r,p}^{1},s_{r,c}^{2},s_{r,c}^{3}\right]^{T}\). According to Equation (6), the null space precoding vector \(\mathbf{v}_{r}=\left[\mathbf{v}_{r,p}^{1},\mathbf{v}_{r,p}^{2},\mathbf{v}_{r, p}^{3},\mathbf{v}_{r,p}^{4},\mathbf{v}_{r,p}^{5},\mathbf{v}_{r,p}^{6}, \mathbf{v}_{r,p}^{1},\mathbf{v}_{r,c}^{1},\mathbf{v}_{r,c}^{2},\mathbf{v}_{r, c}^{3}\right]\) is designed to eliminate unwanted interference signals. In order to eliminate interference signals \(s_{r,p}^{4}\), \(\mathbf{s}_{r,p}^{5}\) and \(\mathbf{s}_{r,p}^{6}\), user 1 needs to make precoding vectors \(\mathbf{v}_{r,p}^{4}\), \(\mathbf{v}_{r,p}^{5}\) and \(\mathbf{v}_{r,p}^{6}\) lie in the null space of the channel transmission matrix. Similarly, the precoding vectors \(\mathbf{v}_{r,p}^{2}\), \(\mathbf{v}_{r,p}^{3}\) and \(\mathbf{v}_{r,p}^{6}\) need to be located in the null space of the channel transmission matrix \(\mathbf{H}^{[2,r]}\) to eliminate the interference of user 2. The precoding vectors \(\mathbf{v}_{r,p}^{1}\), \(\mathbf{v}_{r,p}^{3}\) and \(\mathbf{v}_{r,p}^{5}\) need to be located in the null space of the channel transmission matrix \(\mathbf{H}^{[3,r]}\) to eliminate the interference of user 3. The precoding vectors \(\mathbf{v}_{r,p}^{1}\), \(\mathbf{v}_{r,p}^{2}\) and \(\mathbf{v}_{r,p}^{4}\) need to be located in the null space of the channel transmission matrix \(\mathbf{H}^{[4,r]}\) to eliminate the interference of user 4. It can be found that the precoding vector \(\mathbf{v}_{r,p}^{6}\) needs to be located in the null space of \(\mathbf{H}^{[1,r]}\) and \(\mathbf{H}^{[2,r]}\). That is, \(\mathbf{v}_{r,p}^{6}\in null\left(\mathbf{H}^{[1,r]}\right)\cap null\left( \mathbf{H}^{[2,r]}\right)\). Therefore, 6 antennas from 7 antennas of each user are selected as receiving antennas to form channel transmission matrix \(\hat{\mathbf{H}}^{[ir]}\) with \(6\times 13\) dimensions to meet the null space existence condition \(N<2M\). Using the same method, the design for the remaining receive precoding vectors is as follows:  
\[ \begin{array}{ll}\left[\begin{array}{c}\hat{\mathbf{H}}^{[3,r]}\\ \hat{\mathbf{H}}^{[4,r]}\end{array}\right]\mathbf{v}_{r,p}^{1}=0,\left[ \begin{array}{c}\hat{\mathbf{H}}^{[2,r]}\\ \hat{\mathbf{H}}^{[4,r]}\end{array}\right]\mathbf{v}_{r,p}^{2}=0\\ \left[\begin{array}{c}\hat{\mathbf{H}}^{[2,r]}\\ \hat{\mathbf{H}}^{[3,r]}\end{array}\right]\mathbf{v}_{r,p}^{3}=0,\left[ \begin{array}{c}\hat{\mathbf{H}}^{[1,r]}\\ \hat{\mathbf{H}}^{[4,r]}\end{array}\right]\mathbf{v}_{r,p}^{4}=0\\ \left[\begin{array}{c}\hat{\mathbf{H}}^{[1,r]}\\ \hat{\mathbf{H}}^{[1,r]}\\ \hat{\mathbf{H}}^{[2,r]}\end{array}\right]\mathbf{v}_{r,p}^{6}=0\end{array}  \tag{13}\]The signals received by user 1 is as follows:  
\[ \begin{array}{ll}\mathbf{y}^{[1]}&=\mathbf{\hat{H}}^{[1,r]}\left(\sum\limits_{ m=1}^{6}\mathbf{v}_{R,p}^{m}s_{R,p}^{m}+\sum\limits_{n=1}^{3}\mathbf{v}_{R,c}^{n}s_{R,c}^{ n}\right)+\mathbf{n}^{[1]}\\ &=\mathbf{\hat{Q}}^{[1,r]}\Big{[}\ s_{R,p}^{1}\ s_{R,p}^{2}\ s_{R,p}^{3}\ s_{R,p}^{1}\ s_{R,c}^{1}\ s_{R,c}^{2}\ s_{R,c}^{3}\ \Big{]}^{T}+\mathbf{n}^{[1]}\end{array}  \tag{14}\]  
where \(\mathbf{\hat{Q}}^{[1,r]}\) represents the equivalent transmission channel after antenna selection. User 1 adopts ZF decoding. According to Equation (8), the desired signal can be obtained by using the signal sent by user 1. In particular, user 1 can obtain \(s_{c}^{[2]}\) by using its own signals \(s_{c}^{[1]}\) and \(s_{R,c}^{1}\) for multicast signals. \(s_{c}^{[3]}\) can be obtained by reusing \(s_{c}^{[2]}\) and \(s_{R,c}^{2}\). Finally, \(s_{c}^{[4]}\) can be obtained by using \(s_{c}^{[3]}\) and \(s_{R,c}^{3}\).