## 3 CSI Estimation Process  
As previously noted, the accuracy of DL CSI estimation is crucial for the performance of the DL precoding scheme. Therefore, it is essential to design a feasible TS for CSI estimation in m-MIMO systems with FDD transmission mode. This section presents the TS design, which depends on the received training signal used for CSI estimation conditioned on the channel covariance matrix (CCM), as subsequently explained. To estimate the DL CSI, the BS transmits a TS matrix of length \(T_{\mathrm{tr}}\) during the training period. Thus, the  
Figure 1: DL TS and data transmission in m-MIMO systems with an FDD transmission mode.  
received training signal per user, denoted as \(\mathbf{y}k\in\mathbb{C}^{\mathrm{Ttr}}\) during the training period, can be expressed as  
\[\mathbf{y}_{k}=\mathbf{S}_{\mathrm{tr}}^{\mathrm{H}}\mathbf{h}_{k}+\mathbf{z}_{ k}, \tag{5}\]  
where \(\mathbf{S}_{\mathrm{tr}}\in\mathbb{C}^{N\times T_{\mathrm{tr}}}\) denotes the training matrix that satisfies \(\mathrm{tr}\big{(}\mathbf{S}_{\mathrm{tr}}^{\mathrm{H}}\mathbf{S}_{\mathrm{tr }}\big{)}=\rho_{\mathrm{tr}}T_{\mathrm{tr}}\), where \(\rho_{\mathrm{tr}}\) represents the transmitted power during the training stage. Parameter \(\mathbf{z}_{k}\in\mathbb{C}^{T_{\mathrm{tr}}}\) denotes the receiver noise that is modeled with iid CSCG with \(\mathcal{CN}\big{(}\mathbf{0},\mathbf{I}_{\mathrm{tr}}\big{)}\). This paper assumes equal training and data power allocation, i.e., \(\rho_{\mathrm{tr}}=\rho_{\mathrm{d}}\). However, optimizing the power allocation between \(\rho_{\mathrm{tr}}\) and \(\rho_{\mathrm{d}}\) might be considered in future.  
Linear filters can be utilized to optimize channel estimation performance, given that the channel vector \(\mathbf{h}_{k}\) follows a CSCG distribution with known statistics at the BS. Linear filters can be utilized to enhance channel estimation performance by taking advantage of the statistics of the channel vector \(\mathbf{h}_{k}\), which is denoted by CSCG with known properties at the BS. In order to achieve the best possible channel estimation performance in the DL with \(T_{\mathrm{tr}}<N\), Bayesian estimation is employed. This is done by utilizing an MMSE filter that takes into account both channel and noise statistics. As such, CSI estimate \(\hat{\mathbf{h}}_{k}\sim\mathcal{CN}(\mathbf{0},\mathbf{\Phi}_{k})\) can be expressed as  
\[\hat{\mathbf{h}}_{k}=\mathbf{R}_{k}\mathbf{S}_{\mathrm{tr}}\big{(}\mathbf{S}_ {\mathrm{tr}}^{\mathrm{H}}\mathbf{R}_{k}\mathbf{S}_{\mathrm{tr}}+\mathbf{I}_{ T_{\mathrm{tr}}}\big{)}^{-1}\mathbf{y}_{k}, \tag{6}\]  
where \(\mathbf{y}_{k}\) denotes the per user-received signal during the training stage. As such, CSI estimation error might be written as  
\[\mathbf{e}_{k}=\mathbf{h}_{k}-\hat{\mathbf{h}}_{k}. \tag{7}\]  
The covariance matrix of the CSI estimation error, which is denoted by \(\mathbf{C}_{e,k}\in\mathbb{C}^{N\times N}\), can be expressed as  
\[\mathbf{C}_{e,k} =\mathbb{E}\left\{\,\mathbf{e}_{k}\mathbf{e}_{k}^{\mathrm{H}}\right\} \tag{8}\] \[=\mathbb{E}\Big{\{}\big{(}\mathbf{h}_{k}-\hat{\mathbf{h}}_{k} \big{)}\big{(}\mathbf{h}_{k}-\hat{\mathbf{h}}_{k}\big{)}^{\mathrm{H}}\Big{\}},\] (9) \[=\Big{(}\mathbf{R}_{k}-\mathbf{R}_{k}\mathbf{S}_{\mathrm{tr}} \big{(}\mathbf{S}_{\mathrm{tr}}^{\mathrm{H}}\mathbf{R}_{k}\mathbf{S}_{\mathrm{ tr}}+\mathbf{I}_{T_{\mathrm{tr}}}\big{)}^{-1}\mathbf{S}_{\mathrm{tr}}^{\mathrm{H}} \mathbf{R}_{k}\Big{)}. \tag{10}\]  
The mean square error can be written as  
\[\mathrm{MSE}_{k}=\mathrm{tr}(\mathbf{C}_{e,k}). \tag{11}\]  
Let \(\mathbf{R}_{\mathrm{tot}}=\sum_{k=1}^{K}\mathbf{R}_{k}\) be the superposition of different user channel covariance matrices. Using the Woodbury matrix identity, the error covariance matrix can be written as  
\[\mathbf{C}_{e}=\Big{(}\mathbf{R}_{\mathrm{tot}}^{-1}+\mathbf{S}_{\mathrm{tr}} \mathbf{S}_{\mathrm{tr}}^{\mathrm{H}}\Big{)} \tag{12}\]