## 5 Channel Correlation Models  
When modeling the CCM in MIMO systems, many studies have assumed Rayleigh fading is uncorrelated. This means that the elements of the matrix are uniformly distributed and independent for all users [65]. As a result, all directions are equally important in this model since the energy is broadcast evenly over all directions. The assumption of spatially uncorrelated channel coefficients is quite strict. In real-world scenarios, m-MIMO systems with a large \(N\) may exhibit significant correlations among the channel coefficients. Moreover, empirical measurements have provided evidence that MIMO channels do exhibit correlation in practice [66; 67; 68]. Therefore, it is important to consider correlation models. When dealing with correlated Rayleigh fading, only a small number of effective paths are present in the angular directions, which means that the remaining directions can be disregarded. The level of spatial correlation present in the CCM is determined by both the statistical information available and the specific configurations of the antenna array. Note that the radiation pattern of an antenna array can be calculated by taking into account the phase and amplitude of the signals at each antenna element, as well as the spacing and geometry of the elements. The pattern can be represented graphically in two or three dimensions. In general, the radiation pattern of an antenna array depends on the array configuration, which includes the number of elements, their spacing, and their relative phase and amplitude. In this paper, we use a uniform rectangular array in which the radiation pattern is distributed in both the horizontal and vertical directions. We will now describe the geometric characteristics of the CCM \(\mathbf{R}_{\text{k}}\) for the Laplacian and the OR model with UPA. The attributes of the UPA are established by the angles of arrival (AoA) in the horizontal \(\theta_{k,\text{H}}\) and vertical \(\theta_{k,\text{V}}\) directions. We assume the angular spread in the horizontal direction \(\omega_{k,\text{H}}\) and vertical direction \(\omega_{k,\text{V}}\) and the normalized antenna spacing \(D_{H}\) and \(D_{V}\) (measured in wavelengths). We also assume that the BS array has uniform spacing between its antenna elements, denoted by \(D\), which is equal in both the horizontal and vertical directions, i.e., \(D_{H}\) and \(D_{V}\). The number of antennas arrays is denoted by \(N\), which is equal to the product of the number of antenna arrays in the horizontal (azimuth), denoted by \(N_{H}\), and the number of antenna arrays in the vertical (elevation), denoted by \(N_{V}\). A uniform spacing between antennas is considered so that the same distance \(d\) is assumed for both azimuth and elevation directions denoted as \(d_{k,\text{H}}\) and \(d_{k,\text{V}}\), respectively. Hence, \(d_{k,\text{H}}=d_{k,\text{V}}=d_{k}\). Based on this assumption, the CCMs in the horizontal and vertical directions are denoted as \(\mathbf{R}_{k,\text{H}}\in\mathcal{C}^{\text{NH}\times\text{NH}}\) and \(\mathbf{R}_{k,\text{V}}\in\mathcal{C}^{\text{NV}\times\text{NV}}\)respectively. Thus, the elements of \(\mathbf{R}_{k,\mathrm{H}}\) and \(\mathbf{R}_{k,\mathrm{V}}\) can be expressed in Toeplitz form as \((m,n)\)th element [28]. \[[\mathbf{R}_{k,\mathrm{H}}]_{m,n}=\frac{1}{2\omega_{\mathrm{H}}}\int_{-\omega_{ \mathrm{H}}+\theta_{k,\mathrm{H}}}^{\omega_{\mathrm{H}}+\theta_{k,\mathrm{H}} }e^{-\left|2\pi\,D\,(m-n)\sin(\Delta)\right.}d\Delta. \tag{20}\]  
\[[\mathbf{R}_{k,\mathrm{V}}]_{m,n}=\frac{1}{2\omega_{\mathrm{V}}}\int_{-\omega_{ \mathrm{V}}+\theta_{k,\mathrm{V}}}^{\omega_{\mathrm{V}}+\theta_{k,\mathrm{V}} }e^{-\left|2\pi\,D\,(m-n)\sin(\Delta)\right.}d\Delta, \tag{21}\]  
The computation of the integration in equations (20) and (21) is done through numerical methods. The vertical angular spread is denoted by the parameter \(\omega_{\mathrm{V}}\) and its value is obtained according to reference [28].