# Topological Interference Management for K-User Downlink Massive MIMO Relay Network Channel
## 4 Precoded Spatial-Multiplexing MIMO Systems
for fixed \(\mathbf{\hat{X}}^{[k],[R]}\) compute \(\mathbf{y}^{[2]}_{(2)},\mathbf{y}^{[3]}_{(3)}\) until recovered. ```  
**Algorithm 1** Multiuser MIMO Scheduling. The second key component is the multiuser MIMO scheduling Algorithm 1, which forces the active transmitters to be assigned during the time slots and provides the basic initialization necessary to recover the interference signals from the unintended receiver. In three time slots, the linear combination of the desired signal information is reconstructed from the unintended receiver. The third main component initializes the channel coefficients to identify the corresponding interference signals from the relay and the transmitters. These linear combinations of the desired signal information enable the reconstruction of the interference signals from the unintended receiver in which \(Z^{[k]}(t)\) is a realization of random variables for user \(K=\{1,2,3\}\). \(Z^{[k]}\sim\mathbb{C}N(0,1)\) impersonates the additive white Gaussian noise signal; the corresponding complex (static) channel coefficients [24,25] are \(h_{k}\)=\(\{R_{1},R_{2},R_{3},d_{1},d_{2},d_{3},c_{1},c_{2},c_{3}\}\), where \(R_{k},d_{k},c_{k}\) represent the relay signals, the directly transmitted signals, and the interference signals, respectively. In the Gaussian linear deterministic PIMAC models depicted in Figures 4-6, the corresponding transmitter binary vectors are \(x_{i}=\mathbb{F}^{q}_{2}\), where \(q=max\{n_{R_{1}},n_{R_{2}},n_{R_{3}},n_{d_{1}},n_{d_{2}},n_{d_{3}},n_{c_{1}},n_{ c_{2}},n_{c_{3}}\}\) and the integer of \(n_{k}\) represents the additive white Gaussian channel coefficients, \(n_{k}=\lfloor\log_{2}(P|h_{k}|^{2}\rfloor\). We assume that all of the nodes have global CSIR knowledge, where the noise, \(Z^{[k]}[t]\), is independent and identically distributed (i.i.d) over these time slots. **Over the first time slot**: Transmitters 1 and 2 are kept active because transmitter 3 is made silent, as shown in Figure 4. Under such conditions, receiver 1, \(\mathbb{F}^{[1,1]}[W_{1}]\) and receiver 2, \(F^{[2,2]}[W_{1}]\)  
Figure 4: (**a**) simple network topology over the first time slot and (**b**) block representation of the received signal. Figure 5: (**a**) simple network topology over the second time slot and (**b**) block representation of the received signal. Figure 6: (**a**) simple network topology over the third time slot and (**b**) block representation of the received signal. can acquire the superposition of the two desired symbols. Nevertheless, receiver 3 can eavesdrop on the linear combination of the unintended interference signal from the active transmitters:  
\[\mathbb{F}^{[1,1]}[W_{1}] = \mathbf{h}_{[c_{1}]}^{[1,1]}[1]\mathbf{u}_{1}^{[1]}+\mathbf{h}_{[c_{ 1}]}^{[1,1]}[1]\mathbf{u}_{2}^{[1]},\] \[\mathbb{F}^{[2,2]}[W_{1}] = \mathbf{h}_{[c_{2}]}^{[2,2]}[1]\mathbf{u}_{1}^{[1]}+\mathbf{h}_{[ c_{2}]}^{[2,2]}[1]\mathbf{u}_{2}^{[1]}. \tag{13}\]  
In order to deliver the same linear combination of undesired signals from receiver 3, the principal idea of the proposed beamforming scheme is to select the beamforming matrix from transmitters 1 and 2 with a shared relay:  
\[ \begin{bmatrix}\beta\mathbf{h}_{[R_{3}]}^{(1,R)T}[n]\\ \beta\mathbf{h}_{[R_{2}]}^{(2,R)T}[n]\end{bmatrix} \mathbf{V}^{(3)}[n]= \begin{bmatrix} \mathbf{h}_{[c_{1}]}^{(1,1)T}[n]\\ \mathbf{h}_{[c_{2}]}^{(2,2)T}[n]\end{bmatrix} .