## 3 Signal Spatial Reconstruction Method for Relay Cooperative Networks
### Extended to \(K\) Users  
In this section, we extend the above algorithm to a more general case where the number of users is \(K\). At this time, any two user pairs can send signals between them. Each user sends \(K-1\) private signals and one common signal through space separation technology. Therefore, the relay will receive a total of \((K-1)\times K+K=K^{2}\) signals from \(K\) users in the MAC phase. To separate private signals and common signals sent by each node, it must meet  
\[M\geq(K+1) \tag{26}\]  
In order to align the private signals and the common signal to the appropriate intersection subspace at the relay, the space precoding vectors \(\mathbf{v}_{p}^{[j,i]}\) and \(\mathbf{v}_{c,k}^{[i]}\) must be designed to satisfy \(span\left(\mathbf{H}^{[r,i]}\mathbf{v}_{p}^{[j,i]}\right)=span\left(\mathbf{ H}^{[r,i]}\mathbf{v}_{p}^{[i,j]}\right)\) and \(span\left(\mathbf{H}^{[r,i]}\mathbf{v}_{c,2}^{[i]}\right)=span\left(\mathbf{ H}^{[r,i+1]}\mathbf{v}_{c,1}^{[i+1]}\right)\), if  
\[2M\succ N+1 \tag{27}\]  
If the above conditions are satisfied, the intersection subspace composed of the column space of the equivalent channel matrix must exist. If each user sends \(K-1\) private signals, \(K(K-1)\) private signals will be transmitted simultaneously in the \(K\)-user network. For the \(K\)-user network to send common signals, we designed a novel transmission structure. User \(1\) and user \(K\) only send one common signal, while the other users send two copies of the common signal in two different directions. That is, \(\mathbf{x}_{c}^{[i]}=\mathbf{v}_{c,i}^{[i]}\mathbf{s}_{c}^{[i]}+\mathbf{v}_{c, 2}^{[i]}\mathbf{s}_{c}^{[i]}\), \(\mathbf{v}_{c,1}^{[1]}=\mathbf{v}_{c,2}^{[K]}=\mathbf{0}\). The principle for determining the transmission direction is that the second signal of the current user is spatially aligned with the first signal of the next user, \(span\left(\mathbf{H}^{[r,i]}\mathbf{v}_{c,2}^{[i]}\right)=span\left(\mathbf{ H}^{[r,i+1]}\mathbf{v}_{c,1}^{[i+1]}\right)\), \(i=1,\cdots,K-1\). The relay needs enough signal space to accommodate \(K(K-1)/2\) private signals and \(K-1\) common signals. Therefore, the number of relay antennas must meet  
\[N\geq K(K-1)/2+(K-1)= (K-1)(K+2)/2 \tag{28}\]  
while the relay obtains the \(K^{2}\) signals during the MAC phase. In the BC phase, in order to be correctly decoded by the target user, the interference of other signals must be suppressed. For private signals, we designed precoding vectors \(\mathbf{q}_{p}^{[i,j]}\) and \(\mathbf{q}_{p}^{[j,i]}\) to satisfy \(span\left(\mathbf{q}_{p}^{[i,j]}\mathbf{H}^{[r]}\right)=span\left(\mathbf{q} _{p}^{[j,i]}\mathbf{H}^{[r]}\right)\). For common signals, we designed precoding vectors \(\mathbf{q}_{c}^{[i,j]}\) and \(\mathbf{q}_{c}^{[j,i]}\) to satisfy \(span\left(\mathbf{q}_{c}^{[i,j]}\mathbf{H}^{[r,i]}\right)=span\left(\mathbf{ q}_{c}^{[i,j]}\mathbf{H}^{[r]}\right)\), the size of which is \((K\times(K-1))\times N\). The intersection subspace of the private signal and public signal of user \(i\) and user \(j\) is defined as \(\mathbf{z}_{p}^{\pi p(i,j)}\) and \(\mathbf{z}_{c}^{\pi c(i,j)}\), respectively. All the intersection subspaces together constitute the equivalent transmission channel. \[\mathbf{Z}=\left[\mathbf{z}_{p}^{\pi p(1,2)}\quad\ldots\quad\mathbf{z}_{p}^{ \pi p(K-1,K)}\mathbf{z}_{c}^{\pi c(1,2)}\quad\ldots\quad\mathbf{z}_{c}^{\pi c(K- 1,K)}\right]^{H}\cdot\mathbf{z}_{p}^{\pi p(i,j)}\]  
is defined as an equivalent matrix that does not contain \(\mathbf{z}_{p}^{\pi p(i,j)}\). Therefore, if  
\[N\geq(K\times(K-1)) \tag{29}\]the set of solutions for \(null\left( \begin{array}{c}\text{-}\pi p(i,j)\\ \mathbf{Z}_{p}\end{array} \right)\) exists. We can therefore conclude that \(\mathbf{T}_{p}^{\pi p(i,j)}\in null\left( \begin{array}{c}\text{-}\pi p(i,j)\\ \mathbf{Z}_{p}\end{array} \right)\).