## 3 Signal Spatial Reconstruction Method for Relay Cooperative Networks
### Orthogonal Optimization of Signal Space  
Take the MIMO multiway channel model of 3 users as an example to introduce the implementation process [25]. In traditional signal space alignment, the number of antennas configured for each user node is 2. The corresponding upstream channel matrix \(\mathbf{H}^{[r,l]}\) is \(3\times 2\) dimensions, and the precoding vector is \(2\times 1\) dimensions. According to the signal space alignment conditions given in Section 2, the three alignment signals at the relay node are uniquely determined. That is, the three spatial directions \(\mathbf{U}_{12}\), \(\mathbf{U}_{13}\) and \(\mathbf{U}_{23}\) are also fixed. In order to realize the orthogonal alignment of signal subspaces, the redundancy of user node antennas is needed. Thus, the flexibility of a users precoding vector selection is increased. Therefore, when the number of antennas of the user node is increased to 3, the number of antennas of the relay node is kept unchanged. Currently, its corresponding uplink channel matrix \(\mathbf{H}^{[r,l]}\) is \(3\times 3\) dimensions, and the precoding vector is \(3\times 1\) dimensions. According to the condition of signal space alignment, each pair of precoding vectors has 6 unknown parameters. From the 3 aligned equality constraints, each pair of direction vectors is determined in a 3D signal space. However, when compared with the traditional MIMO-Y channel model, the selection of \(\mathbf{U}_{12}\), \(\mathbf{U}_{13}\) and \(\mathbf{U}_{23}\) dimensions are more flexible, and the selection gain of the system will increase accordingly. According to the system model of MIMO-Y, the signals received by the relay node in the MAC phase can be expressed as follows:  
\[ \begin{array}{ll}\mathbf{y}^{[r]}=\left(\mathbf{H}^{[r,l]}\cdot\mathbf{v}^ {[2,1]}\cdot s^{[2,1]}+\mathbf{H}^{[r,2]}\cdot\mathbf{v}^{[1,2]}\cdot s^{[1,2 ]}\right)\\ \qquad+\left(\mathbf{H}^{[r,1]}\cdot\mathbf{v}^{[3,1]}\cdot s^{[3,1]}+\mathbf{ H}^{[r,3]}\cdot\mathbf{v}^{[1,3]}\cdot s^{[1,3]}\right)\\ \qquad+\left(\mathbf{H}^{[r,2]}\cdot\mathbf{v}^{[3,2]}\cdot s^{[3,2]}+\mathbf{ H}^{[r,3]}\cdot\mathbf{v}^{[2,3]}\cdot s^{[2,3]}\right)+\mathbf{n}^{[r]}\\ \qquad=\mathbf{U}_{12}\cdot(s_{12})+\mathbf{U}_{13}\cdot(s_{13})+\mathbf{U}_{ 23}\cdot(s_{23})+\mathbf{n}^{[r]}\end{array}  \tag{15}\]  
where \(\mathbf{U}_{ij}\) is the direction of the signal space formed by the relay node for the interaction signal of the user node pair \((i,j)\), and \(s_{ij}\) represents the superposition signal of \(s^{[i,j]}\) and \(s^{[i,j]}\), i.e., \(s_{ij}=s^{[i,j]}+s^{[i,j]}\). Figure 2 is a graph of signal space alignment results formed at the relay nodes. It can be seen from Figure 3 that six signals sent by three user nodes are aligned in two at the relay, forming three signal space directions. The signals in the direction of each signal space are the sum of the interactive signals. In order to make the three spatial directions formed after alignment on the relay node mutually orthogonal, the following requirements shall be met:  
\[\left\{ \begin{array}{ll}\mathbf{U}_{12}{}^{H}\cdot\mathbf{U}_{13}=0\\ \mathbf{U}_{12}{}^{H}\cdot\mathbf{U}_{23}=0\\ \mathbf{U}_{13}{}^{H}\cdot\mathbf{U}_{23}=0\end{array} \right. \tag{16}\]In terms of precoding vector design and channel matrix, it is necessary to achieve:  
\[\left\{ \begin{array}{l}\left(\mathbf{H}^{[r,2]}\cdot\mathbf{v}^{[1,2]}\right)^{ H}\cdot\left(\mathbf{H}^{[r,3]}\cdot\mathbf{v}^{[1,3]}\right)=0\\ \left(\mathbf{H}^{[r,2]}\cdot\mathbf{v}^{[1,2]}\right)^{H}\cdot\left(\mathbf{H} ^{[r,3]}\cdot\mathbf{v}^{[2,3]}\right)=0\\ \left(\mathbf{H}^{[r,3]}\cdot\mathbf{v}^{[1,3]}\right)^{H}\cdot\left(\mathbf{H} ^{[r,3]}\cdot\mathbf{v}^{[2,3]}\right)=0\\ \end{array} \right. \tag{17}\]  
According to the above analysis, the principle of realizing orthogonal alignment of signal subspace is to design the transmission precoding vector in MAC phase. The condition of signal space alignment is \(\left\{