# Topological Interference Management for K-User Downlink Massive MIMO Relay Network Channel
## 3 Three-User MIMO Relay-Aided Broadcast Channel  
In this section, we propose a simple three-user network topology with a shared relay, as shown in Figure 2. We summarize the important steps in the proposed optimization framework scheme shown in Figure 3. We have considered \(K=3\) users, where the transmitter, \(N_{tx,k}\) and the relay, \(N_{R}\), have massive antennas and the receiver, \(N_{rx,k}\), has a single antenna [23]. We primarily focus on scenarios in which the CSIT is not known at the receiver-side, where it has access to limited CSIR knowledge only:  
\[\mathbf{\hat{X}}^{[k]}[n]=\sum_{k=1}^{3}\mathbf{h}_{n}^{[k]}[k] \mathbf{u}_{n}^{[k]}. \tag{8}\]  
Based on this space-time relay beamforming method, the relay forwards the received signals, while all of the three transmitters remain silent. More specifically, the space-time relay beamforming matrix can carry only two symbols for the receiver, \(K\in\{1,2,3\}\), which can be denoted by \(\mathbf{V}^{[k]}=[\mathbf{V}_{k}^{[1]},\mathbf{V}_{k}^{[2]},\mathbf{V}_{k}^{ [3]}]\in\mathbb{C}^{3\times 3}\) and the transmitter signal vector at the relay is given by  
\[\mathbf{\hat{X}}^{[k]}[n] = \beta(\mathbf{V}_{1}^{[1]}\hat{\mathbf{u}}_{1}^{[1]}+\mathbf{V}_ {2}^{[1]}\hat{\mathbf{u}}_{2}^{[1]})+\ \beta(\mathbf{V}_{1}^{[2]}\hat{\mathbf{u}}_{2}^{[2]}+\mathbf{V}_{2}^{[2]}\hat {\mathbf{u}}_{2}^{[2]})+\ \beta(\mathbf{V}_{1}^{[3]}\hat{\mathbf{u}}_{1}^{[3]}+ \mathbf{V}_{2}^{[3]}\hat{\mathbf{u}}_{2}^{[3]}), \tag{9}\] \[= \beta\mathbf{V}_{1}^{[1]}\mathbf{u}_{1}^{[1]}+\beta\mathbf{V}_{2} ^{[1]}\mathbf{u}_{2}^{[1]}\ +\ \beta\mathbf{V}_{1}^{[2]}\mathbf{u}_{1}^{[2]}+\beta \mathbf{V}_{2}^{[2]}\mathbf{u}_{2}^{[2]}+\ \beta\mathbf{V}_{1}^{[3]}\mathbf{u}_{1}^{[3]}+ \beta\mathbf{V}_{2}^{[3]}\mathbf{u}_{2}^{[3]},\]  
where \(\beta\) is the power normalization factor. \(\hat{\mathbf{x}}^{[k]}[n]\) and \(\hat{\mathbf{x}}^{[k]}[n]\) are the original symbol vectors for the transmitter and relay, respectively. Although the pre-equalization can be characterized by a pre-equalizer weight matrix, \(\mathbf{W}\in\mathbb{C}^{N_{tx,k}\times N_{tx,k}}\) and the precoding symbol vector, \(x\), in terms of \(\mathbf{W}\), can be defined as \(\mathrm{x}=W\hat{\mathrm{x}}\). We provide a refined analysis of the corresponding weight matrix, described under the MMSE pre-equalization.  
The total Gaussian point-to-point interfering with the multiple access relay channel (PIMAC) power constraints after pre-equalization can be expressed as follows  
\[\mathbb{E}\left[tr\left(\mathbf{x}^{[R]}(n)\mathbf{x}^{[R]^{ \dagger}}(n)\right)\right]\leq P, \tag{10}\]  
where \(\dagger\) represents the conjugate transpose and the power normalization factor \(\beta\) is expressed as  
\[\beta=\ \sqrt{\frac{P}{\mathbb{E}\{Tr(\mathbf{H}\hat{a}\hat{a}^{ \dagger}\mathbf{H}^{-1})\}}}. \tag{11}\]Furthermore, the MMSE pre-equalization weight matrix is selected as  
\[\text{W}_{\text{MMSE}} = \beta\times\,argmin\,\text{E}\left[\parallel\beta^{-1}(\text{HW} \hat{\mathbf{x}})-\hat{\mathbf{x}}\parallel^{2}\right], \tag{12}\] \[= \beta\times\text{H}^{\text{H}}\left(\text{HH}^{\text{H}}+\frac{ \sigma_{x}^{2}}{\sigma_{x}^{2}}\text{I}\right)^{-1}.\]