# The Role of Correlation in the Performance of Massive MIMO Systems
## 5 Physical Channel Model under Arbitrary Array Configurations
Interpolation techniques can be used [49] for obtaining the channel covariance matrices at the base station. Advanced signal processing and machine learning methods can also be exploited to achieve this purpose [50; 51; 52]. There are several approaches for modeling the spatially correlated m-MIMO channels. In the present study, we use a Laplacian physical channel model to design the channel covariance matrices of the UEs with different array geometries, i.e., ULA and URA configurations. Specifically, the Laplacian channel model is used to generate \(\mathbf{G}_{u}\), which represents the information of the physical structure of the channel statistics. This model represents a practical scenario since it captures an arbitrary array of geometries, where the power received by each antenna varies arbitrarily, so that it experiences an unequal contribution of each antenna for a given communication link. This channel model supports both the ULAand URA configurations. In the ULA configuration, the propagation signal happens in the azimuth direction only, while in URA configurations, the communication between the UEs and the base station happens in both azimuth and elevation directions. The physical channel model has some non-line-of-sight (NLoS) paths that correspond to the reflections from different scatterers, which produce a superposition of many paths that are jointly received. As shown in Figure 3, with the deployment of URA array topology, the parameters of the physical channel model include both azimuth and elevation angles so that the base station radiates the signals in three-dimensional (3D) space. The base station deploys the URA configuration in the y-z coordinate plane, and serves as a UE in the x-y coordinate plane. The UE channel is subject to an azimuth and elevation angle standard deviation. Each path has a corresponding angle of departure in the elevation direction \(\theta_{u,E}\) and angle of departure in the azimuth direction \(\theta_{u,A}\). In the horizontal ULA configuration, the signals arrive from directions within the x-y plane in the azimuth direction only along the y-axis. In the ULA configuration, the channel covariance matrix is designed in Toeplitz form and given as  
\[\left[\mathbf{G}_{u,A}\right]_{i,j}=\frac{1}{\sqrt{2}\alpha_{A}}\int_{-\pi+ \theta_{u,A}}^{\pi+\theta_{u,A}}e^{-\frac{\sqrt{2}}{\alpha_{A}}\left(x-\theta_ {u,A}\right)}e^{\frac{-2\pi d}{\lambda}\left(i-j\right)\sin(x)}\mathrm{d}x, \tag{21}\]  
where \(\alpha_{A}\) is the azimuth standard deviation, also known as the angular spread (AS) in the azimuth direction. In the URA topology, the channel covariance matrix in the azimuth direction can be written as  
\[\left[\mathbf{G}_{u,A}\right]_{i,j}=\frac{1}{\sqrt{2}\alpha_{A}}\int_{-\pi+ \theta_{u,A}}^{\pi+\theta_{u,A}}e^{-\frac{\sqrt{2}}{\alpha_{A}}\left(x-\theta _{u,A}\right)}e^{\frac{-2\pi d}{\lambda}\left(i-j\right)\sin(x)}\mathrm{d}x, \tag{22}\]  
where \(\alpha_{A}\) is the azimuth standard deviation, also known as the angular spread (AS) in the azimuth direction. In the URA, the channel covariance matrix in the elevation direction is given as  
\[\left[\mathbf{G}_{u,E}\right]_{i,j}=\frac{1}{\sqrt{2}\alpha_{E}}\int_{-\pi+ \theta_{u,E}}^{\pi+\theta_{u,E}}e^{-\frac{\sqrt{2}}{\alpha_{E}}\left(x-\theta _{u,E}\right)}e^{\frac{-2\pi d}{\lambda}\left(i-j\right)\sin(x)}\mathrm{d}x, \tag{23}\]  
Figure 3: Base station deploys a URA topology, the channel model shows that the propagation link from the UEs to the base station with the azimuth and elevation angles in the coordinate plane. where \(\alpha_{E}\) is the elevation standard deviation, also known as the angular spread (AS) in the elevation direction.