# Hybrid MU-MIMO Precoding Based on K-Means User Clustering
## 3 Spatial Compatibility and Channel Modeling
### Correlation Model
Since all the multipath components are assumed to arrive with the same delay, the model is a frequency flat-fading channel model. This limitation is less significant, as we focus only on communication over a coherence block, where the channel is assumed to be constant. For these reasons, in the context of user clustering for MU-MIMO, the model was employed in several publications like [13; 19]. To model the small-scale fading, each path from the antenna to the user is assumed to represent a superposition of \(N_{p}\) multipaths (\(N_{p}\) is a large number). It is assumed that the macro base station is elevated and that the scatterers only surround the user, and each multipath is a plane wave, leaving from the uniform linear spaced array at an angle \(\bar{\varphi}_{n}\). The equivalent channel of each user becomes:  
\[\mathbf{h}=\sum_{n=1}^{N_{p}}\mathbf{a}_{n}, \tag{15}\]  
where the channel vector of the \(n\)th path towards the \(N_{T}\) antenna array is:  
\[\mathbf{a}_{n}=g_{n}\left[1\,e^{2\pi jd\sin(\bar{\varphi}_{n})}\,\ldots\,e^{2 \pi jd(N_{T}-1)\sin(\bar{\varphi}_{n})}\right]^{T}, \tag{16}\]  
where \(\mathbf{X}^{T}\) denotes the transpose of an array \(\mathbf{X}\), \(g_{n}\) is a complex number that models the gain and phase of multipath \(n\), and \(d\) is the array antenna spacing (measured in wavelengths). It is assumed that the angles \(\bar{\varphi}_{n}\) are random variables with the Probability Density Function (PDF) \(f(\bar{\varphi})\) and \(g_{n}\) are zero mean random variables with variance \(\mathbb{E}\left\{|g_{n}|^{2}\right\}\). The variance is the \(n\)th multipath component average gain, and the total gain of the multipaths is \(\gamma=\sum_{n=1}^{N_{p}}\mathbb{E}\left\{|g_{n}|^{2}\right\}\). Using the multidimensional central limit theorem, \(\mathbf{h}\rightarrow\mathcal{N}_{\mathbb{C}}(\mathbf{0}_{N_{T}},\mathbf{R})\) when \(N_{p}\rightarrow\infty\), where the convergence is in its distribution and has a spatial correlation matrix \(\mathbf{R}=\mathbb{E}\left\{\sum_{n}\mathbf{a}_{n}\mathbf{a}_{n}^{H}\right\}\). This is the reason why the channel model is a correlated Rayleigh fading model. The element \(r_{l,m}\) of matrix \(\mathbf{R}\) is defined as:  
\[ \begin{array}{ll}r_{l,m}&=\sum_{n=1}^{N_{p}}\mathbb{E}\left\{|g_{n}|^{2} \right\}\mathbb{E}\left\{e^{2\pi jd(l-1)\sin(\bar{\varphi})}e^{-2\pi jd(m-1) \sin(\bar{\varphi})}\right\}\\ &=\gamma\int e^{2\pi jd(l-m)\sin(\bar{\varphi})}f(\bar{\varphi})d\bar{\varphi }\end{array}  \tag{17}\]  
As \(r_{l,m}\) depends on the \(l-m\) difference and not on each individual value, \(\mathbf{R}\) is a Toeplitz matrix. It is assumed that the multipath components are originating from a scattering cluster around the user [due to the lack of scatterers around the Base Station (BS)], located at an angle \(\bar{\varphi}=\varphi+\delta\). \(\varphi\) is a deterministic nominal angle, and \(\delta\) is a random variable with a standard deviation from the nominal angle given by \(\sigma_{\varphi}\). Depending on how \(\delta\) is modeled, several correlation models can be found in the literature. The standard deviation \(\sigma_{\varphi}>0\) is measured in radians and is called the Angular Standard Deviation (ASD). A usual value for urban environments is \(\sigma_{\varphi}>0\) (\(10^{\circ}\)), but smaller values are expected in rural areas or hills.