# Hybrid MU-MIMO Precoding Based on K-Means User Clustering
## 2 Precoding Schemes
### Linear Precoding
\begin{array}{l}\begin{bmatrix}\mathbf{y}_{1}\\ \mathbf{y}_{2}\\ \ldots\\ \mathbf{y}_{K}\end{bmatrix}&=\begin{bmatrix}\mathbf{H}_{1}&\mathbf{H}_{1}& \cdots&\mathbf{H}_{1}\\ \mathbf{H}_{2}&\mathbf{H}_{2}&\cdots&\mathbf{H}_{2}\\ \ldots&\ldots&\ldots&\ldots\\ \mathbf{H}_{K}&\mathbf{H}_{K}&\cdots&\mathbf{H}_{K}\end{bmatrix}\begin{bmatrix} \mathbf{W}_{1}\mathbf{x}_{1}\\ \mathbf{W}_{2}\mathbf{x}_{2}\\ \ldots\\ \mathbf{W}_{K}\mathbf{x}_{k}\end{bmatrix}+\begin{bmatrix}\mathbf{n}_{1}\\ \mathbf{n}_{2}\\ \ldots\\ \mathbf{n}_{K}\end{bmatrix}\\ =\begin{bmatrix}\mathbf{H}_{1}\mathbf{W}_{1}&\mathbf{H}_{1}\mathbf{W}_{2}& \cdots&\mathbf{H}_{1}\mathbf{W}_{K}\\ \mathbf{H}_{2}\mathbf{W}_{1}&\mathbf{H}_{2}\mathbf{W}_{2}&\cdots&\mathbf{H}_{2 }\mathbf{W}_{K}\\ \cdots&\ldots&\ldots&\ldots\\ \mathbf{H}_{K}\mathbf{W}_{1}&\mathbf{H}_{K}\mathbf{W}_{2}&\cdots&\mathbf{H}_{ K}\mathbf{W}_{K}\end{bmatrix}+\begin{bmatrix}\mathbf{n}_{1}\\ \mathbf{n}_{2}\\ \ldots\\ \mathbf{n}_{K}\end{bmatrix},\end{array}  \tag{4}\]  
where \(\mathbf{H}_{u}\mathbf{W}_{k,u\
eq k}\) is the effective channel corresponding to the signal sent to user \(k\) and received by user \(u\). This interferes with user \(k\) when \(\mathbf{H}_{u}\mathbf{W}_{k}\
eq\mathbf{0}_{N_{R}\times N_{R}}\). The effective channel matrix needs to be diagonalized in order to cancel the inter-user interference. \[\mathbf{H}_{u}\mathbf{W}_{k}=\mathbf{0}_{N_{R}\times N_{R}},\quad\forall u \
eq k. \tag{5}\]  
Figure 1: Linear precoding scheme [3]. To prevent any power increase after precoding, the BD precoder matrix has to be unitary. Assuming a matrix containing the channel gains of all users except \(u\),  
\[\widetilde{\mathbf{H}}_{u}=\left[\mathbf{H}_{1},\cdots,\mathbf{H}_{u-1},\mathbf{ H}_{u+1},\cdots,\mathbf{H}_{K}\right]^{T}, \tag{6}\]  
when \(N_{T}=K\times N_{R}\), the Singular-Value Decomposition (SVD) of \(\widetilde{\mathbf{H}}_{u}\) is:  
\[\widetilde{\mathbf{H}}_{u}=\widetilde{\mathbf{U}}_{u}\widetilde{\mathbf{\Lambda }}_{u}\left[\widetilde{\mathbf{V}}_{u}^{\text{non-zero}}\,\widetilde{\mathbf{V }}_{u}^{\text{zero}}\right]^{H}, \tag{7}\]  
where \(\widetilde{\mathbf{V}}_{u}^{\text{non-zero}}\in\mathcal{C}^{(K\times N_{R}-N_ {R})\times N_{T}}\) and \(\widetilde{\mathbf{V}}_{u}^{\text{zero}}\in\mathcal{C}^{N_{R}\times N_{T}}\) contain singular vectors corresponding to the non-zero and zero singular values, respectively. \[\widetilde{\mathbf{H}}_{u}\widetilde{\mathbf{V}}_{u}^{\text{zero}} =\widetilde{\mathbf{U}}_{u}\left[\widetilde{\mathbf{\Lambda}}_{u }^{\text{non-zero}}\,\mathbf{0}\right]\left[ \begin{array}{c}\left( \widetilde{\mathbf{V}}_{u}^{\text{non-zero}}\right)^{H}\\ \left(\widetilde{\mathbf{V}}_{u}^{\text{zero}}\right)^{H}\end{array} \right] \widetilde{\mathbf{V}}_{u}^{\text{zero}} \tag{8}\] \[=\widetilde{\mathbf{U}}_{u}\widetilde{\mathbf{\Lambda}}_{u}^{\text {zero}}\left(\widetilde{\mathbf{V}}_{u}^{\text{non-zero}}\right)^{H}\widetilde {\mathbf{V}}_{u}^{\text{zero}}\] \[=\widetilde{\mathbf{U}}_{u}\widetilde{\mathbf{\Lambda}}_{u}^{ \text{zero}}\mathbf{0}\] \[=\mathbf{0}.\]  
As \(\widetilde{\mathbf{V}}_{u}^{\text{zero}}\) lies in the null space of \(\widetilde{\mathbf{H}}_{u}\), the signal transmitted in the direction of \(\widetilde{\mathbf{V}}_{u}^{\text{zero}}\) is only received by user \(u\). Its corresponding precoder is \(\mathbf{W}_{u}=\widetilde{\mathbf{V}}_{u}^{\text{zero}}\).