## 6 Results and Discussion
### Computational Complexity Evaluation  
The results from Figures 4-8 show plots of the overall computational complexity versus the number of BS antennas \(N\), comparing the proposed CGD training sequence design with the SMSE and SCMI training designs using the pilot length for BF and RZF precoding. The number of iterations each algorithm required to converge are provided in Tables 3-7, which are used to obtain the simulated results from Figures 4-8, respectively. The results are presented based on the scattering OR channel model, when the system parameters \(K\), \(N\), \(T_{\text{c}}\), SNR, angle of arrivals (AoAs), normalised antenna spacing \(D\), and angular spread \(\omega\) are used. The AoAs are distributed in the range \([-60^{\circ},60^{\circ}]\), i.e., \(\theta_{k}=\{-57.5^{\circ},-45^{\circ},-41.5^{\circ},-23^{\circ},-7.5^{\circ},7.5^{\circ},23.5^{\circ},41.5^{\circ},45^{\circ},57.5^{\circ}\}\), [30, 57]. The overall computational complexity of the proposed training optimization algorithm and the state-of-the-art SCMI [30] and SMSE [31] algorithms are obtained based on multiplying the number of iterations each algorithm needs to converge by the number of flops involved per iteration. The results show that the proposed CGD iterative algorithm for the DL training sequence optimization achieves a faster convergence rate in comparison with the state-of-the-art iterative algorithms. This is clearly indicated by the number of iterations given in Tables 3-7. The fast convergence speed provided by the proposed CGD algorithm arises from the use of matrix exponential search over the Riemannian manifold. Specifically, the proposed CGD iterative algorithm exhibits fewer iterations to converge due to the significant advantages of the almost periodic property of the matrix exponential search over the smooth Riemannian manifold. This signifies the feasibility of the proposed CGD algorithm to optimize the achievable sum rate of the FDD m-MIMO systems with limited coherence time in comparison with state-of-the-art iterative algorithms. Although the proposed CGD provides fast convergence speed, the results demonstrate that as the number of BS antennas \(N\) increases, the SCMI and proposed CGD iterative algorithms exhibit almost the same overall computational complexity. For example, the results in Figure 4 show that at \(N=350\), the proposed CGD iterative algorithm offers a comparable overall computational complexity to the SCMI algorithm, which implies that the SCMI iterative algorithm requires fewer flops at this BS array size. Nonetheless, the proposed CGD iterative algorithm still provides a larger achievable sum rate performance than the SCMI algorithm under a relatively weak correlated channels, see e.g., Figure 3. In addition, the SMSE algorithm grows asymptotically at about the same rate at large \(N\), since it exhibits the same order of complexity \(\mathcal{O}\left(N^{2}\right)\) as other algorithms. However, the \(N^{2}\) term for SMSE has a much higher multiplier, which gives the large gap above the SCMI and the proposed CGD algorithms. The results provided in this subsection indicate that the relative ratio between the three iterative algorithms for the training sequence optimization depends on the number of BS antennas \(N\) and the levels of spatial correlation. Noting that the non smooth curves associated with different iterative algorithms occur because the number of iterations each algorithm requires to converge are different for different values of \(N\). In addition, the results show that the performance of the SMSE based approach is decreased when \(N\) is increased as indicated by Figures 6b and 8a. This is attributed to the fact that the convergence rate of the SMSE algorithm is not propositional to the number of BS antenna \(N\). These results are linked to Tables 5 and 7, which clearly show that the convergence rates are not propositional to the number of BS antenna (N).