###### Abstract
However, in practice, users could exhibit distinct spatial correlation patterns, which arise due to independent propagation conditions and scattering geometries; hence, the optimization framework of the training sequences developed in [11, 12, 13, 14, 15, 16, 17, 18] does not hold in general multiuser scenarios with \(K\) distinct spatial correlations. Another line of research has focused on the channel estimation by using compressed sensing (CS) based techniques [19, 20, 21, 22, 23, 24]. However, due to the unknown sparse nature of the channels, CS based approaches cannot be applied in practice to predict the DL achievable sum rate of an FDD m-MIMO system. Another development for CSI estimation is to use a hybrid two-stage precoding approaches, see e.g., [25, 26, 27, 28]. However, the aforementioned research works do not address the single-stage precoding scenario with \(K\) independent user covariance matrices. To address the challenge of having distinct correlated channels and to find a more realistic solution with single-stage precoding scheme, several studies have investigated the design of training sequences by utilizing different iterative algorithms. For example, in [29] the DL training sequences are optimized iteratively based on the average sum rate loss due to CSI estimation with zero forcing (ZF) precoding. In refs. [30, 31] the DL training sequences are optimized iteratively as a solution to a sum conditional mutual information (SCMI) maximization problem and a sum mean square error (SMSE) minimization problem, respectively. While the investigations in refs. [30, 31] have optimized the DL training sequences iteratively, the algorithms used in the optimization exhibit slow convergence speed, and thus, deliver high latency and computational complexity. However, in real-time systems, the CSI estimation must be obtained more frequently with an acceptable latency from channel estimation to precoding, especially when the channel exhibits a shorter coherence time. Therefore, there is an essential requirement for developing a new efficient iterative algorithm for the FDD m-MIMO systems that optimizes the DL training sequence, i.e., maximizes the sum rate, while at the same time exhibits the fast convergence rate so as to obtain the CSI estimation more quickly over a limited coherence time. Note that the design principles in refs. [30, 31] are more general compared with [29] since they do not enforce a specific precoding structure and do not consider a heuristic approximate upper bound of the sum rate loss for CSI estimation. This makes the approaches in refs. [30, 31] more rigorous than the approach in [29] for DL sequences optimization. This paper addresses the challenge of CSI DL channel estimation in the FDD m-MIMO systems with limited coherence time considering the users have different correlations. We propose a conjugate gradient descent (CGD) optimization algorithm over the Riemannian manifold for optimizing the DL training sequences. The proposed algorithm is essentially required to speed up the convergence rate by lessening the required number of iterations for achieving fast CSI estimation and hence reduces the overall computational complexity. This proposed algorithm is also very useful for massive communication systems and networks. In this paper, a random matrix theory (RMT) method [32, 33] is used to validate our Monte-Carlo simulations. This allows the achievement of a straightforward methodology without resorting to a computationally demanding exhaustive search. In addition, the computational complexity analyses of the proposed CGD algorithm and the state-of-the-art SCMI [30] and SMSE [31] iterative algorithms are provided. Comparisons are presented for the sum rate and overall complexity performances between the proposed CGD algorithm and existing SCMI [30] and SMSE [31] iterative algorithms for DL sequence optimization in both eigenbeamforming (BF) and the regularized ZF (RZF) precoding. The aforementioned algorithms are considered the best known state-of-the-art iterative algorithms.