## 3 CSI Estimation Using DL Training Sequences
This paper proposes a computationally efficient CGD iterative algorithm, described in detail in Section 4, to optimize \(\mathbf{S}_{\text{tr}}\) for pre-selected training length \(T_{\text{tr}}\), and training power \(\rho_{\text{tr}}\). The proposed algorithm aims to achieve a robust sum rate performance when the users exhibit different correlations while at the same time reducing the computational complexity. Let \(f(\mathbf{S}_{\text{tr}})\in\mathbb{R}\) denote the sum MSE of the channel estimate cost function, which needs to be minimized with respect to the DL training matrix \(\mathbf{S}_{\text{tr}}\), as  
\[f(\mathbf{S}_{\text{tr}})=\sum_{k=1}^{K}\text{tr}(\mathbf{R}_{k})-\sum_{k=1}^{K }\text{tr}(\mathbf{\Phi}_{k}) \tag{5}\]  
where \(\mathbf{\Phi}_{k}=\mathbf{R}_{k}\mathbf{S}_{\text{tr}}\big{(}\mathbf{S}_{\text {tr}}^{\text{H}}\mathbf{R}_{k}\mathbf{S}_{\text{tr}}+\frac{1}{\rho_{\text{tr}}} \mathbf{I}_{\text{tr}}\big{)}^{-1}\mathbf{S}_{\text{tr}}^{\text{H}}\mathbf{R}_{k}\), which denotes the \(k\)-th user's covariance matrix with minimum mean square error (MMSE) channel estimation. To this end, the formulation of the CSI estimation optimization problem is provided. Minimizing the sum MSE over the training sequence \(\mathbf{S}_{\text{tr}}\) for a given training-phase duration \(T_{\text{tr}}\) in the FDD m-MIMO system equates to the optimization problem defined in (6). \[ \begin{split}\underset{\mathbf{S}_{\text{tr}}}{\text{minimize}} \ f(\mathbf{S}_{\text{tr}})\\ \text{subject to}\ \mathbf{S}_{\text{tr}}^{\text{H}}\mathbf{S}_{ \text{tr}}=\mathbf{I}_{\text{tr}}\end{split}  \tag{6}\]  
The sum MSE cost function, which corresponds to a function of the subspace that is spanned by the pilot matrix \(\mathbf{S}_{\text{tr}}\), is invariant to the unitary rotation, i.e., \(f(\mathbf{S}_{\text{tr}}\mathbf{U})=f(\mathbf{S}_{\text{tr}})\). This property allows the CGD algorithm over the Riemannian manifold to be effectively used to solve the optimization problem in (6). To apply the CGD optimization algorithm on a Riemannian manifold, the partial derivative of the sum MSE cost function in (5) needs to be determined with respect to \(\mathbf{S}_{\text{tr}}\). To this end, the partial derivative \(\mathbf{\Gamma}\in\mathbb{C}^{N\times T_{\text{tr}}}\) is given by [31]:  
\[ \begin{split}\mathbf{\Gamma}=2\sum_{k=1}^{K}\bigg{(}-\mathbf{R}_ {k}^{2}\mathbf{S}_{\text{tr}}\big{(}\mathbf{S}_{\text{tr}}^{\text{H}}\mathbf{ R}_{k}\mathbf{S}_{\text{tr}}+\frac{1}{\rho_{\text{tr}}}\mathbf{I}_{\text{tr}} \big{)}^{-1}+\mathbf{R}_{k}\mathbf{S}_{\text{tr}}\times\\ \Big{(}\mathbf{S}_{\text{tr}}^{\text{H}}\mathbf{R}_{k}\mathbf{S}_ {\text{tr}}+\frac{1}{\rho_{\text{tr}}}\mathbf{I}_{\text{tr}}\Big{)}^{-1} \mathbf{S}_{\text{tr}}^{\text{H}}\mathbf{R}_{k}^{2}\mathbf{S}_{\text{tr}} \Big{(}\mathbf{S}_{\text{tr}}^{\text{H}}\mathbf{R}_{k}\mathbf{S}_{\text{tr}}+ \frac{1}{\rho_{\text{tr}}}\mathbf{I}_{\text{tr}}\Big{)}^{-1}\bigg{)}\end{split}  \tag{7}\]  
where the right hand side in (7) is obtained by differentiating the cost function in (6) with respect to pilot matrix, (i.e., \(\partial f(\mathbf{S}_{\text{tr}})/\partial\mathbf{S}_{\text{tr}}^{*}\)).