# Downlink Training Design for FDD Massive MIMO Systems in the Presence of Colored Noise
## 6 Physical Channel Models  
Most of the research works on MIMO systems have considered the uncorrelated Rayleigh fading when modeling the channel covariance matrix. In this case, the elements of the covariance matrix are independent and uniformly distributed i.e., \(\mathbf{R}_{\rm h}=\mathbf{I}_{N}\) for all users. Therefore, in the uncorrelated Rayleigh fading model, all directions become equally important, since the energy is distributed in all directions [54]. However, the condition for the channel coefficients to be spatially uncorrelated is very strict. As mentioned earlier in Section 4, with a large number of BS antennas in MaMi systems, the channel coefficients may be highly correlated due to the spatially dependent of the antennas radiation patterns and near-field scattering environment. Furthermore, practical measurements have shown that MIMO channels are correlated [55; 56; 57]. Hence, spatial correlation models with different BS design typologies should be considered, in order to characterise a realistic performance assessment of the FDD MaMi systems. In the correlated Rayleigh fading, only few effective paths exist in the angular directions, while the rest of the directions could be ignored so as not to be used in the precoding. Exploiting the covariance correlation in the channel estimation is crucial for providing a feasible DTS design for the FDD MaMi systems. This is due to the fact that spatial correlation allows a rank deficient covariance matrices. As such, the rank of the channel covariance matrix can be increased sublinearly with \(N\), resulting in affordable DTS overhead for FDD operation mode with MaMi systems. In particular, spatial correlation allows channel training with a minimum duration, which is upper bounded by the rank of the correlation matrix, thus reducing the DTS overhead of MaMi systems. The degrees of spatial correlation in the channel covariance matrix depends on the statistical information and the array configurations. This paper considers both the ULA and UPA configurations, where the latter is modeled as a three-dimensional (3D) geometrical channel model. In what follows, we present the system geometries of the channel covariance matrix \(\mathbf{R}_{\rm h}\) in the one ring and Laplacian scattering channel models with UPA and ULA, respectively. For the UPA, the BS antennas consist of \(N=N_{\rm A}\times N_{\rm E}\) where each row has an \(N_{\rm A}\) and each column has \(N_{\rm E}\) antennas, where the subscript (A) denotes the azimuth direction and (E) stands for the elevation direction. Figure 2 shows a UPA based on the azimuth (horizontal) and elevation (vertical) directions. We consider a uniform distancesbetween the neighboring antennas of the BS array, i.e., \(d\) is considered for both azimuth and elevation directions, i.e., \(d_{\rm A}\), \(d_{\rm E}\), respectively. This implies that \(d_{\rm A}=d_{\rm E}=d\). Parameters \(\theta\) and \(\phi\) represent the azimuth and elevation angles, respectively, where \(\theta\) and \(\phi\in[0,\pi]\). The angular spread in the azimuth and elevation directions are given by \(\alpha_{\rm A}\) and \(\alpha_{\rm E}\), respectively. The spatial covariance matrix in the UPA, denoted by \(\mathbf{R}_{\rm h}\), can be well approximated using the Kronecker model as [22; 58]  
\[\mathbf{R}_{\rm h}=\mathbf{R}_{\rm h,A}\otimes\mathbf{R}_{\rm h,E}, \tag{46}\]  
where \(\otimes\) is the Kronecker operator and \(\mathbf{R}_{\rm h,A}\in\mathcal{C}^{N_{\rm A}\times N_{\rm A}}\) and \(\mathbf{R}_{\rm h,E}\in\mathcal{C}^{N_{\rm E}\times N_{\rm E}}\) are the covariance matrices in the azimuth and elevation directions, respectively.