## 3 CSI Estimation Using DL Training Sequences  
To gain further insight into the problem of DL CSI estimation, this section investigates the training sequence design in an FDD m-MIMO system. To estimate the DL channel, the BS transmits a predetermined training sequence of length \(T_{\text{tr}}\) during the training-phase. The DL pilot sequences, collected by each user and returned to the BS, provide instantaneous channel realizations at the BS, which in turn implement regular BF or RZF precoding based on channel estimates. As such, the training sequences for channel estimation [41] are updated according to the dynamics of each \(\mathbf{R}_{k}\), which in turn is updated at every coherence time interval. To this end, the received signal during training-phase, \(\mathbf{y}_{k}\in\mathbb{C}^{T_{\text{tr}}}\), at the \(k\)-th user is given by:  
\[\mathbf{y}_{k}=\sqrt{\rho_{\text{tr}}}\mathbf{S}_{\text{tr}}^{\text{H}}\mathbf{ h}_{k}+\mathbf{z}_{k} \tag{4}\]  
where \(\mathbf{S}_{\text{tr}}\in\mathbb{C}^{N\times T_{\text{tr}}}\) is the training matrix. The receiver noise \(\mathbf{z}_{k}\in\mathbb{C}^{T_{\text{tr}}}\) exhibits a CSCG distribution \(\mathcal{CN}\big{(}\mathbf{0},\mathbf{I}_{T_{\text{tr}}}\big{)}\). Extensive research has been carried out to address the challenge of CSI estimation, see e.g., [11, 12, 13, 14, 15, 16, 17, 18]. In these investigations, a special scenario when the users exhibit common spatial correlation at the BS is considered. However, a more general scenario when the users exhibit distinct spatial correlation at the BS is needed. Besides, compressed sensing based approaches for CSI estimation have been used in [19, 20, 21, 22, 23, 24]. In these works, the sparsity structure on the virtual angular domain is used to reduce the training length required for the CSI estimation. However, the compressed sensing based approaches cannot be applied in practice due to the unknown sparsity nature of the channels that cannot be predicted in these approaches. Another area of research on CSI estimation is to exploit the hybrid two-stage precoding based approaches, see e.g., [25, 26, 27, 28]. In particular, the hybrid based approaches exploit correlation in the spatial domain, where the users within each group exhibit the same spatial correlation, and a linear superposition of each group correlation matrix is utilized to perform the first of two stages of precoding, thus forming a beam for each group. However, sophisticated scheduling and clustering algorithms of the user groups, and of the users inside each group, are required in these hybrid approaches. Hence, these works cannot be applied to predict the achievable sum rate performance in a more general single-stage scenario with \(K\) independent user covariance matrices. To address the challenge of CSI estimation in the more general single-stage precoding scenario when the users exhibit distinct correlated channels, several studies have been carried out, see, e.g., [30, 31, 42, 43]. Specifically, the training sequences in [30, 31, 42, 43] are designed by exploiting different iterative algorithms as a solution to an SCMI maximization criterion and an SMSE minimization criterion, respectively. However, the limited coherence time interval in practice implies that the CSI estimation should be determined more frequently, and thus, iterative-based solutions for the DL training sequence design may be infeasible. In addition, the algorithms used in the aforementioned optimizations exhibit slow convergence speed, and thus, deliver high latency and computational complexity. Hence, there is a crucial need to develop a lower complexity iterative algorithm solution for the FDD m-MIMO systems, which has the ability to optimize the DL training sequence, i.e., maximizes the achievable sum rate, while at the same time exhibits a fast convergence rate so that obtaining a fast CSI estimation when a limited coherence time is considered.