# Hybrid MU-MIMO Precoding Based on K-Means User Clustering
## 3 Spatial Compatibility and Channel Modeling
### Correlation Model  
The authors in [17] mentioned that the channel separation metric should also be a function of the spatial correlation between \(\mathbf{h}_{k}\) and \(\mathbf{h}_{i}\) and considered the inner correlation of each multi-antenna user. However, the precoding performance was primarily affected by the correlation between transmit antennas, whereas receive antenna correlation had an insignificant impact on the precoding design [18]. In a MIMO context, a spatially-correlated random channel vector \(\mathbf{h}_{k}\) can be created using Karhunen-Loeve expansion [15]:  
\[\mathbf{h}_{k}=\mathbf{R}_{k}\widehat{\mathbf{e}}_{k}=\mathbf{U}_{k}\mathbf{D }_{k}^{\frac{1}{2}}\mathbf{U}_{k}^{H}\widehat{\mathbf{e}}_{k}\sim\mathbf{U}_ {k}\mathbf{D}_{k}^{\frac{1}{2}}\mathbf{e}_{k}, \tag{13}\]  
where \(\widehat{\mathbf{e}}_{k}\sim\mathcal{N}_{\mathbb{C}}\left(\mathbf{0}_{N_{T}}, \mathbf{I}_{N_{T}}\right)\) and \(\mathbf{e}_{k}\sim\mathcal{N}_{\mathbb{C}}\left(\mathbf{0}_{r},\mathbf{I}_{r}\right)\). The eigenvalue decomposition of \(\mathbf{R}_{k}\in\mathbb{C}^{N_{T}\times N_{T}}\) is \(\mathbf{R}_{k}=\mathbf{U}_{k}\mathbf{D}_{k}\mathbf{U}_{k}^{H}\), where \(\mathbf{D}_{k}\in\mathbb{R}^{r\times r}\) is a diagonal matrix containing \(r=\text{rank}(\mathbf{R}_{k})\) positive non-zero eigenvalues of \(\mathbf{R}_{k}\) and \(\mathbf{U}_{k}\in\mathbb{C}^{N_{T}\times r}\) contains the associated eigenvectors. The last part means that the distributions of \(\mathbf{h}_{k}\) of \(\mathbf{U}_{k}\mathbf{D}_{k}^{\frac{1}{2}}\mathbf{e}_{k}\) are identical. It can be shown that the spatially-correlated Rayleigh channel corresponding to user \(k\) is a complex normal random vector, with zero mean and spatial correlation matrix \(\mathbb{E}\left\{\mathbf{h}_{k}\mathbf{h}_{k}^{H}\right\}=\mathbf{R}_{k}\), \(\mathbf{h}_{k}\sim\mathcal{N}_{\mathbb{C}}\left(\mathbf{0}_{N_{T}},\mathbf{R} _{k}\right)\)[15; 19]. The normalized trace \(\gamma=\frac{1}{N_{T}}\text{tr}\left(\mathbf{R}_{k}\right)\) captures the average channel gain from one of the antennas at the base station to the UE and includes the macroscopic large-scale fading. The Rayleigh distribution of \(\mathbf{h}_{k}\) is motivated by the presence of small-scale fading due to multipath propagation. When the channel gains between different antennas are uncorrelated, the channel model can be referred to as uncorrelated Rayleigh fading. \(\gamma\) can be modeled as:  
\[\gamma=\gamma_{0}-10\alpha\log_{10}\left(\frac{D}{D_{0}}\right)+F, \tag{14}\]where \(D\) is the distance between the transmitter and receiver, \(\alpha\) is the path loss exponent, which determines how fast the signal attenuates with distances, and \(\gamma_{0}\) determines the median channel gain at a reference distance \(D_{0}\). In theoretical studies, those two parameters can be computed with one of the many established propagation models. The only non-deterministic terms is \(F\sim\mathcal{N}(0,\sigma_{sf}^{2})\) and represents the shadow fading that creates log-normal random variations around the mean value, \(\gamma_{0}-10\alpha\log_{10}\left(\frac{D}{D_{0}}\right)\). The shadow fading can be viewed as a model of blockage from large obstacles. The variance \(\sigma_{sf}^{2}\) determines how large the variations are. Realistic performance assessment of MIMO systems requires the use of a channel model that reflects the main characteristics of large antenna arrays such as the array geometry, the correlation between the channel responses of different antennas and the physical location and orientation of BSs and UEs. In [15], a correlation-based model was proposed, where the channel responses were all Gaussian distributed with zero mean and entirely defined through the correlation matrices. Having an intuitive structure, the correlation model allowed the subspaces of the correlation matrices to be parametrized by the azimuth angles of the UEs. This makes it easy to determine if two UEs are spatially separable by comparing their respective angles.