## 4 Multi-User MIMO System Model
### SLNR Precoding and MMSE Combining  
For notation simplicity, we omit the subcarrier index \(q\) in the following discussions. In order to analyze the performance of a multi-user multi-layer MIMO scheme under the 3 GPP 3-D channel model we split the processing at the transmitter and the receiver:  
1. at the transmitter, we adopt a Signal-to-Leakage-plus-Noise Ratio (_SLNR_) precoder [17], which is able to minimize the inter-user interference;
2. at each UT, we adopt a Minimum Mean Square Error (MMSE) combiner [42], which will minimize the intra-user inter-layer interference. Given a user \(k\), the interference caused to the other UTs can be defined as the leakage: \[\sum_{ \begin{subarray}{c}i=1\\ i\
eq k\end{subarray} }^{K}\|\mathbf{\tilde{H}}_{i}\mathbf{W}_{k}\|_{F}^{2}\] (68) this leads to the _SLNR_ expression the ratio between the signal power for user \(k\) and the noise plus the leakage: \[SLNR_{k}=\frac{\|\mathbf{\tilde{H}}_{k}\mathbf{W}_{k}\|_{F}^{2}}{N_{r}\sigma_ {k}^{2}+\sum_{ \begin{subarray}{c}i=1\\ i\
eq k\end{subarray} }^{K}\|\mathbf{\tilde{H}}_{i}\mathbf{W}_{k}\|_{F}^{2}}.\] (69) The _SLNR_ expression can be rewritten as: \[SLNR_{k}=\frac{\|\mathbf{\tilde{H}}_{k}\mathbf{W}_{k}\|_{F}^{2}}{N_{r}\sigma_ {k}^{2}+\|\mathbf{Z}_{k}\mathbf{W}_{k}\|_{F}^{2}}\] (70) where: \[\mathbf{Z}_{k}= \begin{bmatrix}\mathbf{\tilde{H}}_{1}\dots\mathbf{\tilde{H}}_{ k-1}\mathbf{\tilde{H}}_{k+1}\end{bmatrix} ^{T}\] (71) is an extended \(N_{r}\,K\times N_{t}\) channel matrix that only excludes user \(k\). The transmit precoding matrix targeted for user \(k\) that maximizes its _SLNR_ is given by \[\mathbf{\hat{W}}_{k}=\underset{\mathbf{W}_{k}\in\mathbb{C}^{N_{t}\times L}}{ \text{argmax}}\,SLNR_{k}=\underset{\mathbf{W}_{k}\in\mathbb{C}^{N_{t}\times L }}{\text{argmax}}\,\frac{\|\mathbf{\tilde{H}}_{k}\mathbf{W}_{k}\|_{F}^{2}}{N_{ r}\sigma_{k}^{2}+\|\mathbf{Z}_{k}\mathbf{W}_{k}\|_{F}^{2}}\] (72) it is shown in [20; 43] that the optimal precoder is linked to the solution of the generalized eigenvalue problem \[\left(\mathbf{\hat{H}}_{k}^{H}\mathbf{\hat{H}}_{k}\right)\mathbf{E}=\left(N_ {r}\sigma_{k}^{2}\mathbf{I}+\mathbf{Z}_{k}^{H}\mathbf{Z}_{k}\right)\mathbf{E}\bm {\Lambda}\] (73)  
Figure 6: Cell site with 30 users after sector assignment. 3-D view on the left, 2-D top-view on the right. where \(\mathbf{E}=\left[\mathbf{e}_{1},\ldots,\mathbf{e}_{N_{t}}\right]\) are the eigenvectors and \(\mathbf{\Lambda}=\text{diag}\left(\left[\lambda_{1},\ldots,\lambda_{N_{t}} \right]^{T}\right)\) the eigenvalues of \(\left(N_{r}\sigma_{k}^{2}\mathbf{I}+\mathbf{Z}_{k}^{H}\mathbf{Z}_{k}\right)^{- 1}\mathbf{\tilde{H}}_{k}^{H}\mathbf{\tilde{H}}_{k}\), respectively. Then the \(SLNR\) precoder is:  
\[\mathbf{W}_{k}\propto\left[\mathbf{e}_{1},\ldots,\mathbf{e}_{L}\right] \tag{74}\]  
i.e., \(\mathbf{W}_{k}\) is made by the \(L\) eigenvectors corresponding to the \(L\) largest eigenvalues \(\lambda_{1},\ldots,\lambda_{L}\). The design of the proper combiner at the UT, which will allow us to minimize the inter-layer interference, requires the definition of the equivalent channel matrix after precoding at the receiver \(k\):  
\[\mathbf{P}_{k}=\mathbf{\tilde{H}}_{k}\,\mathbf{W}_{k}. \tag{75}\]  
The designed \(L\times N_{r}\) combining matrix \(\mathbf{U}_{k}\) is a linear Minimum Mean Square Error (MMSE) receiver, defined as  
\[\mathbf{U}_{k}=\left(\mathbf{P}_{k}^{H}\,\mathbf{P}_{k}+\frac{1}{\rho} \mathbf{I}\right)^{-1}\mathbf{P}_{k}^{H} \tag{76}\]  
with  
\[\rho=\frac{P_{tot}}{Q\,K\,N_{r}\,\sigma^{2}}. \tag{77}\]  
The estimated signal vector \(\mathbf{\hat{s}}_{k}\) for UT \(k\) after MMSE combining becomes:  
\[\mathbf{\hat{s}}_{k}=\mathbf{U}_{k}\,\mathbf{\tilde{H}}_{k}\,\sum_{i=1}^{K} \mathbf{W}_{i}\,\mathbf{s}_{i}+\mathbf{U}_{k}\,\mathbf{n}_{k} \tag{78}\]  
By recalling the structure of precoding and combining matrices \(\mathbf{W}_{k}\) and \(\mathbf{U}_{k}\)  
\[\mathbf{W}_{k}=\left[\mathbf{w}_{k,1},\ldots,\mathbf{w}_{k,L}\right]\qquad \mathbf{U}_{k}= \begin{bmatrix}\mathbf{u}_{k,1}^{T}\\ \vdots\\ \mathbf{u}_{k,L}^{T}\end{bmatrix}