## 7 Conclusions  
This paper explained that optimizing the DL training sequences for CSI estimation in a general scenario with single-stage precoding and \(K\) independent channel correlation matrices is very challenging. To overcome this issue, this paper investigated a geometrical optimization approach, which utilizes a computationally efficient Riemannian manifold, with an aim to maximize the achievable sum rate performance, while improving the convergence rate to achieve a fast CSI estimation for an FDD m-MIMO system. To this end, a new computationally efficient gradient-based iterative algorithm, which uses the matrix exponential search on the Riemannian manifold, is proposed to optimize the DL training sequence in an FDD m-MIMO system, while delivering a lower overall computational complexity owing to a faster convergence rate. The sum rate performance of the proposed CGD algorithm is also compared with the rates achieved by the best known state-of-the-art iterative algorithms. Additionally, the computational complexity is analyzed and compared with the state-of-the-art SCMI and SMSE iterative algorithms. The number of convergence iterations of the proposed CGD training design is also provided. Consequently, the overall computational complexity of the proposed CGD training design is compared with the state-of-the-art iterative algorithms. Importantly, the results demonstrated that, under relatively weakly correlated channels, over a 5 bit/s/Hz improvement in the achievable sum rate is obtained using the proposed CGD iterative algorithm in comparison to the SCMI algorithm. This achievement signifies the advantages of the proposed approach for the training sequence optimization of FDD m-MIMO systems compared with the SCMI approach. The results indicated that the proposed CGD algorithm is able to achieve the best sum rate performance owing to a faster convergence rate. This is attributed to the efficient matrix exponential search on the Riemannian manifold, which allows the algorithm to converge faster with fewer iterations. The results showed that the proposed CGD algorithm maximized the achieved sum rate of FDD m-MIMO systems with a reduced overall computational complexity. \begin{table}
\begin{tabular}{l l} \hline \hline
**Algorithms** & **Fundamental Characteristics** \\ \hline \multirow{6}{*}{SCMI [30]} & \(\bullet\) & Training sequences optimization based on a sum conditional mutual information (SCMI) maximization problem. \\  & \(\bullet\) & Uses gradient with Karush-Kuhn-Tucker (KKT) conditions of the Lagrangian dual problem based approach. \\  & \(\bullet\) & Requires high number of iterations to converges. \\  & \(\bullet\) & The complexity is still high in comparison to the proposed CGD. \\ \hline \multirow{6}{*}{SMSE [31]} & \(\bullet\) & Training sequences optimization based on a sum conditional mutual information (SCMI) maximization problem. \\  & \(\bullet\) & Uses a Grassmannian manifold based approach. \\  & \(\bullet\) & Requires a highest number of iterations to converges. \\  & \(\bullet\) & Delivers a highest overall computational complexity. \\ \hline \multirow{6}{*}{Proposed CGD} & \(\bullet\) & Training sequences optimization based on a sum achievable sum rate maximization problem. \\  & \(\bullet\) & Uses a Riemannian manifold based approach. \\ \cline{1-1}  & \(\bullet\) & Requires a lowest number of iterations to converges. \\ \cline{1-1}  & \(\bullet\) & Delivers a lower overall computational complexity. \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Comparison between the methods to compare the fundamental characteristics of the proposed method with the state-of-the-art-methods. **Author Contributions:** Conceptualization, M.A. ; methodology, M.A. and M.A.N. ; software, S.H.A. and M.A. ; validation, M.A.N., M.A. and B.M.M. ; investigation, M.A.N. and B.M.M. ; writing--original draft preparation, M.A. and B.M.M. ; writing--review and editing, S.H.A. and M.A.N. and B.M.M. ; supervision, M.A. All authors have read and agreed to the published version of the manuscript.