## 3 Signal Spatial Reconstruction Method for Relay Cooperative Networks
### Orthogonal Optimization of Signal Space
#### 3.2.3 Optimization of Precoding Vector  
For the solution to the orthogonal alignment scheme of signal subspace, according to the transmission quality of the detection channel, the sequence of the channel matrix from poor to excellent is recorded as \(\mathbf{H}^{[r,1]}\), \(\mathbf{H}^{[r,2]}\), \(\mathbf{H}^{[r,3]}\). At the same time, users are numbered as 1, 2 and 3, in this order. Then, the transmit precoding vector is redesigned to achieve the orthogonal alignment of signal subspace. First, the interactive signals corresponding to the worst transmission channel matrix \(\mathbf{H}^{[r,1]}\) and the subworst channel matrix \(\mathbf{H}^{[r,2]}\) (i.e., the interactive signals between user nodes 1 and 2) are designed by precoding the transmission in the MAC phase. When relay nodes are aligned to the same signal space direction, the conditions for alignment according to the signal space are as follows:  
\[\mathbf{H}^{[r,1]}\cdot\mathbf{v}^{[2,1]}=\mathbf{H}^{[r,2]}\cdot\mathbf{v}^{ [1,2]} \tag{23}\]  
The transmission precoding matrix of the interactive signal pairs of user nodes 1 and 2 is Equation (18), where the transmission precoding vectors \(\mathbf{v}^{[2,1]}\) and \(\mathbf{v}^{[1,2]}\) are \(3\times 1\) dimensions, and the corresponding null space matrix dimensions are \(6\times 3\) dimensions. When solving the transmission precoding vector of interactive signal pair 1 and 2, a column vector can be selected from the 3D space of the null space matrix \(null\left[\mathbf{H}^{[r,1]},\quad-\mathbf{H}^{[r,2]}\right]\). This is used as the transmission precoding vector of the interactive signal pair. The principle for selecting column vectors is to calculate the modulus of each column vector in the null space matrix, and use the column vector with the largest modulus as the transmission precoding vectors \(\mathbf{v}^{[2,1]}\) and \(\mathbf{v}^{[1,2]}\) of user nodes 1 and 2. After the first set of precoding vectors \(\mathbf{v}^{[2,1]}\) and \(\mathbf{v}^{[1,2]}\) are determined, the aligned signal space direction formed by the relay node is also uniquely fixed. Namely:  
\[\mathbf{U}_{12}=\mathbf{H}^{[r,1]}\cdot\mathbf{v}^{[2,1]}=\mathbf{H}^{[r,2]} \cdot\mathbf{v}^{[1,2]} \tag{24}\]  
After the first signal spatial direction formed on the relay node is determined, the second set of transmission precoding vectors and alignment directions are solved according to the first determined spatial alignment direction and orthogonality. The specific solution is as follows: first, define a matrix \(\mathbf{P}\). The initial value of \(\mathbf{P}\) is the determined spatial direction of the first alignment, i.e., \(\mathbf{P}=[\mathbf{U}_{12}]\). Then, using the properties of orthogonal projection, the orthogonal projection matrix of matrix \(\mathbf{P}\) is obtained. \[\mathbf{Q}=\left(\mathbf{I}_{N}-\mathbf{P}*\left(\mathbf{P}^{H}*\mathbf{P} \right)^{-1}*\mathbf{P}^{H}\right) \tag{25}\]  
The second signal space alignment direction can be obtained by using the orthogonal projection matrix \(\mathbf{Q}_{N*N}\). The method is to deconstruct \(\mathbf{Q}_{N*N}\) into eigenvalues and find the eigenvector corresponding to its maximum eigenvalue. This eigenvector is used as the interactive signal of user nodes 1 and 3 in the spatial alignment direction \(\mathbf{U}_{13}\) formed by the relay. After the spatial direction of alignment is determined, the corresponding transmission precoding vector can be inversely solved by using the channel matrix. That is, \(\mathbf{v}^{[3,1]}\) and \(\mathbf{v}^{[1,3]}\) can be obtained from \(\mathbf{U}_{13}=\mathbf{H}^{[r,1]}\cdot\mathbf{v}^{[3,1]}=\mathbf{H}^{[r,3]} \cdot\mathbf{v}^{[1,3]}\). Therefore, the spatial alignment direction formed by the relay for the second group of transmission precoding vectors and interactive signal pairs is determined. The third group of precodingvectors and the corresponding alignment direction are determined according to the second aligned spatial direction.