# Topological Interference Management for K-User Downlink Massive MIMO Relay Network Channel
## 4 Precoded Spatial-Multiplexing MIMO Systems  
In this section, we focus on a precoded spatial-multiplexing MIMO system to transmit the independent and separately encoded desired signals using multiple antennas at the transmitters and relay, where the common relay shares limited CSIR knowledge with the receivers, as shown in Figure 2. The three key components in relay aided three-user MIMO broadcast relay channels are as follows. The first key component initializes the three time slots in which each user sends two independent intended symbols, \(\text{u}_{1}^{(K)}\) and \(\text{u}_{2}^{(K)}\), using two antennas. Over each of the three time slots, the corresponding network topology of the active transmitters and the block representation of the received signals are illustrated in Figures 4-6, respectively. In time slot-1, the user-1 transmitter signal is \(\mathbf{X}^{[1]}[1]=[\text{u}_{1}^{(1)},\text{u}_{2}^{(1)}]^{T}\), where \(\text{u}_{1}^{(1)}\) and \(\text{u}_{2}^{(1)}\) are the respective intended symbols. In time slot-2, the user-2 transmitter signal is \(\mathbf{X}^{[2]}[2]=[\text{u}_{1}^{(2)},\text{u}_{2}^{(2)}]^{T}\), where \(\text{u}_{1}^{(2)}\) and \(\text{u}_{2}^{(2)}\) are the respective intended symbols. In time slot-3, the user-3 transmitter signal is \(\mathbf{X}^{[3]}[3]=[\text{u}_{1}^{(3)},\text{u}_{2}^{(3)}]^{T}\), where \(\text{u}_{1}^{(3)}\) and \(\text{u}_{2}^{(3)}\) are  
Figure 3: Proposed optimization framework. Figure 2: (a) simple three-user network topology with a relay and (b) side information conflict diagram. the corresponding intended symbols. The transmitter corresponding to each of the above-mentioned time slots, \(\{K=1,2,3\}\), sends two independent symbols for sharing the information with the intended receiver [1]. Thus, three transmitters belonging to three time slots each send a total of six independent symbols. The intended symbols are indicated by \(F^{[k,l]}[W_{n}]\), which express the linear combination of the transmitting signals from user \(K\). **Step : 1. Initialize**:  
\(K\in\{1,2,...,k\}\); \(\mathbf{V}^{[k]}=[\mathbf{V}^{[1]}_{k},\mathbf{V}^{[2]}_{k},...,\mathbf{V}^{[ k]}_{k}]\);  
Transmitter & relay signal: \(\mathbf{\hat{X}}^{[k]}[n]=\sum_{k=1}^{k}\mathbf{h}^{[k]}_{c_{n}}[k]\mathbf{u}^{ [k]}_{n}\); \(\mathbf{\hat{X}}^{[k]}[n]=\beta(\mathbf{V}^{[1]}_{k}\hat{\mathbf{u}}^{[k]}_{1} +\mathbf{V}^{[1]}_{k}\hat{\mathbf{u}}^{[k]}_{2})\); \(\beta=\sqrt{\frac{P}{\mathbf{E}\left(\gamma(\mathbf{H}\mathbf{0}^{k}\mathbf{0}^ {t}\mathbf{H}^{-1})\right)}}\). **Step : 2. Repeat**:  
```
1:for\(k=1,2,3\)do
2: Compute the transmitter&relay signals at the unintended receiver: \[\mathbb{F}^{[3,3]}[W_{3}]=\mathbf{h}^{[3]}_{[c_{3}]}[k]\mathbf{u}^{[k]}_{1}+ \mathbf{h}^{[k]}_{[c_{3}]}[k]\mathbf{u}^{[k]}_{2},\] \[\mathbb{F}^{[2,2]}[W_{2}]=\mathbf{h}^{[2]}_{[c_{2}]}[k]\mathbf{u}^{[k]}_{1}+ \mathbf{h}^{[k]}_{[c_{2}]}[k]\mathbf{u}^{[k]}_{2},\] \[ \begin{bmatrix}\beta\mathbf{h}^{(3,R)}_{[R_{3}]}T[1]\\ \beta\mathbf{h}^{(2,R)}_{[R_{2}]}T[1]\end{bmatrix} \mathbf{V}^{(k)}[n],\]
3: User-1 reconstructed signal from the transmitter and relay. Decoding: \(\mathbf{y}^{[1]}_{(1)}=\mathbf{y}^{[1]}_{(3)}+\mathbf{y}^{[1]}_{(2)}\Longrightarrow\) \[=(\mathbf{h}^{[1,R]}_{[R_{1}]}(3)+\mathbf{h}^{[1,R]}_{[R_{2}]}(3))\mathbf{X}^ {[R]}[3]+(\mathbf{h}^{[1,R]}_{[R_{3}]}(2)+\mathbf{h}^{[1,R]}_{[R_{1}]}(2)) \mathbf{X}^{[R]}[2].\] \[=-\{\mathbf{h}^{[1,1]^{\prime}}_{[c_{2}]}(2)+\mathbf{h}^{[1,1]}_{[c_{3}]}(3) \}\mathbf{u}^{[1]}_{1}+\{\mathbf{h}^{[1,1]^{\prime}}_{[c_{1}]}(1)+\mathbf{h}^ {[1,1]}_{[c_{1}]}(1)\}\}\mathbf{u}^{[1]}_{2}.\]
4: Achievable Sum-rate for K-user MIMO relay channel: \[\mathrm{R}_{K}\leq\log\left(1+\frac{\parallel\mathbf{h}_{j,j}\parallel^{2}P+ \parallel\mathbf{h}_{R_{j}}\parallel^{2}\left(\frac{P_{R}}{2}\right)}{\parallel \mathbf{h}_{j,i}\parallel^{2}P+1}+\frac{2\pi\parallel\mathbf{h}_{j,j} \parallel\parallel\mathbf{h}_{R_{j}}\parallel\sqrt{P(\frac{P_{R}}{2})}}{ \parallel\mathbf{h}_{j,i}\parallel^{2}P+1}\right)\]
5:endfor.