## 3 Signal Spatial Reconstruction Method for Relay Cooperative Networks
### Orthogonal Optimization of Signal Space
\begin{array}{l}\mathbf{H}^{[r,1]}\cdot\mathbf{v}^{[2,1]}=\mathbf{H} ^{[r,2]}\cdot\mathbf{v}^{[1,2]}\\ \mathbf{H}^{[r,1]}\cdot\mathbf{v}^{[3,1]}=\mathbf{H}^{[r,3]}\cdot\mathbf{v}^{[ 1,3]}\\ \mathbf{H}^{[r,2]}\cdot\mathbf{v}^{[3,2]}=\mathbf{H}^{[r,3]}\cdot\mathbf{v}^{[ 2,3]}\end{array} \right.\), i.e.,  
\[\left[ \begin{array}{l}\mathbf{v}^{[2,1]}\\ \mathbf{v}^{[1,2]}\end{array} \right]=null\Big{[}\mathbf{H}^{[r,1]}\quad- \mathbf{H}^{[r,2]}\Big{]} \tag{18}\]  
where \(\left[\mathbf{H}^{[r,1]}\quad-\mathbf{H}^{[r,2]}\right]\) is a matrix of \(3\times 6\) in dimensions. If these conditions are met, \(null\Big{[}\mathbf{H}^{[r,1]}\quad-\mathbf{H}^{[r,2]}\Big{]}\) has a redundancy of \(6-3=3\) dimensions. Each direction is regarded by a vector of \(6\times 1\) dimensions. In other words, there are three groups of selectable vectors in this group of solved precoding vectors. Therefore, the corresponding aligned spatial direction of this group of precoding vectors formed on the relay node also has three directions to choose from. On the other hand, the first determined spatial direction will affect the precoding vectors of the other two spatial directions. If the first determined spatial direction is \(\mathbf{U}_{12}\), its corresponding precoding vectors \(\mathbf{v}^{[2,1]},\mathbf{v}^{[1,2]}\) are also determined. According to orthogonality conditions, another two groups of precoding vectors can be obtained:  
\[\left\{ \begin{array}{l}\mathbf{v}^{[1,3]}=null(\mathbf{v}^{[1,2]}{}^{H} \cdot\mathbf{H}^{[r,2]}{}^{H}\cdot\mathbf{H}^{[r,3]})\\ \mathbf{v}^{[2,3]}=null(\mathbf{v}^{[1,2]}{}^{H}\cdot\mathbf{H}^{[r,2]}{}^{H} \cdot\mathbf{H}^{[r,3]})\end{array} \right. \tag{19}\]  
It can be seen from the above formula that \(\mathbf{v}^{[2,3]}\) and \(\mathbf{v}^{[1,3]}\) are in the same null space. In the above formula, \(null(\mathbf{v}^{[1,2]}{}^{H}\cdot\mathbf{H}^{[r,2]}{}^{H}\cdot\mathbf{H}^{[r, 3]})\) has \(3-1=2\) directional dimensions. Each direction can be regarded by a vector of \(3\times 1\) dimensions. Similarly, the orthogonality condition between \(\mathbf{U}_{12}\) and \(\mathbf{U}_{13}\) must be satisfied when determining \(\mathbf{v}^{[2,3]}\) and \(\mathbf{v}^{[1,3]}\). According to the above analysis, \(\mathbf{v}^{[1,3]}\) is represented by \(\mathbf{v}^{[1,3]}=a_{1}\cdot\mathbf{x}+b_{1}\cdot\mathbf{y}\), \(\mathbf{v}^{[2,3]}\) is represented by \(\mathbf{v}^{[2,3]}=a_{2}\cdot\mathbf{x}+b_{2}\cdot\mathbf{y}\). \(\mathbf{x}\) and \(\mathbf{y}\) are a group of \(3\times 1\)-dimensional bases in null space. After \(a_{1}\) and \(b_{1}\) are arbitrarily selected, \(\mathbf{v}^{[1,3]}\), \(\mathbf{v}^{[3,1]}\) and \(\mathbf{U}_{13}\) are determined according to the signal space alignment condition of \(\left[ \begin{array}{l}\mathbf{v}^{[3,1]}\\ \mathbf{v}^{[1,3]}\end{array} \right]=null\Big{[}\mathbf{H}^{[r,1]}\quad- \mathbf{H}^{[r,3]}\Big{]}\). Therefore, the second spatial direction can be determined. Figure 3: Signal Space Alignment of Relay Nodes. When solving the third spatial direction, the orthogonality condition in Equation (17) can give:  
\[\left[ \begin{matrix}a_{1}&b_{1}\end{matrix} \right]\cdot\left[ \begin{matrix} \mathbf{x}&\mathbf{y}\end{matrix} \right]^{H}\cdot\left[\mathbf{H}^{[r,3]^{H}} \right]\cdot\left[\mathbf{H}^{[r,3]}\right]\cdot\left[ \begin{matrix}\mathbf{x} &\mathbf{y}\end{matrix} \right]\cdot\left[a_{2}&b_{2}\right]^{T}=0 \tag{20}\]  
After the above equations \(a_{1}\) and \(b_{1}\) are determined, only the unknowns \(a_{2}\) and \(b_{2}\) are not determined. Therefore, after the precoding vector \(\mathbf{v}^{[1,3]}\) is determined, \(\mathbf{v}^{[2,3]}\) can be obtained from the alignment condition:  
\[\left[ \begin{matrix}\mathbf{v}^{[3,2]}\\ \mathbf{v}^{[2,3]}\end{matrix} \right]=null\left[\mathbf{H}^{[r,2]}&-\mathbf{ H}^{[r,3]}\right] \tag{21}\]  
After solving \(\mathbf{v}^{[2,3]}\) by Equation (21), the third spatial direction \(\mathbf{U}_{23}\) can also be uniquely determined according to the corresponding channel matrix.