# The Role of Correlation in the Performance of Massive MIMO Systems
## 5 Physical Channel Model under Arbitrary Array Configurations  
In practice, the channels intended for different UEs are subject to spatial correlations due to different near-field scattering environments. Measurements of propagation environments have revealed that the elements of the channel are correlated [20, 21, 22]. Moreprecisely, the realistic propagation environment produces more multipath components toward the serving base station from the more dominated spatial directions than from others. Therefore, to obtain a realistic propagation environment of an m-MIMO system, factors such as correlations between antennas and different scattering around the UEs should be considered. In an m-MIMO system, the antenna spacing is sufficiently small due to the large size of the array, which leads to strong correlations between the adjusted elements. This also implies that the degrees of correlation would depend on the antenna configurations and the topology of the array deployed by the base station, i.e., URA and ULA configurations. This would allow for a variation in the polarization and antenna patterns. Physically, the spatial covariance matrices are represented by the distance between the antenna elements, angle of departure, and angular spread (AS) [39]. A denser antenna array with a massive number of base station antennas may enhance the spatial resolution and improve the channel estimation accuracy. However, if the interference between the elements of the channels is very high, this might affect the received signal considerably and, thus, result in degrading the SE performance. The eigenvalue distribution of the channel covariance matrix at the base station is a metric for measuring the degrees of the correlation in the channel. For example, an identical eigenvalue distribution for all UEs is denoted by a very weak correlation or no correlation, while high channel correlations correspond to a small portion of eigenvalues that are dominated and the rest of the eigenvalues are closed to zero. In addition, strong correlations account for large eigenvalue variations. The channel covariance matrices are subject to change on a time scale that is much slower than the coherence time. The channel gains between the base station and the UEs within the same local area can be represented by a certain correlation that stays fixed during the coherence time. This is because the channel may have the same statistics that depend on the power azimuth around the base station antenna array [40]. The eigenstructure of the channel covariance matrix \(\mathbf{G}_{u}\) determines the spatial correlation properties of the channel vector intended for the specific \(u\)-th UE with a channel \(\mathbf{h}_{u}\), which represents the geometry of the propagation paths [41]. The eigenvalues and the corresponding eigenvectors imply in which spatial directions the signal components are statistically dominated. In contrast to the conventional MIMO systems, the typical assumption in the m-MIMO literature is to have a single antenna per UE and the UEs are separable so that there is no correlations between the UEs. Extensions to multi-antenna at the UE are possible and could be considered in future work. The covariance matrix information of spatial channel characteristics is essential for channel estimation, especially when the MMSE estimator is used. In addition, the covariance matrix information can also play an essential role in resource allocation. Methods for estimating a large dimensional covariance matrix by using a small number of observations are also possible, which can be achieved by a regularization of the sample covariance matrix [42; 43]. The channel covariance matrix information can be known by the base station through the uplink channel covariance matrix [44]. Estimating the channel covariance matrix is also possible by utilizing subspace approaches, thereby avoiding the need for an instantaneous estimation of the channel [45]. An example of estimating the base station covariance matrices is provided in [46; 47; 48].