## 4 Proposed Low Complexity DL TS Design Based on a Waterfilling Algorithm  
In this section, we discuss the proposed TS framework based on a waterfilling power allocation algorithm. The structure of TS matrix \(\mathbf{S}_{\text{tr}}\) is designed by considering the \(K\) different CCMs where the CCMs of all users are combined together. As such, the effective eigenvectors of the superposition CCMs and the corresponding eigenvalues are used in the pilot design. The superposition CCM \(\mathbf{R}_{\text{tot}}\) can be expressed by considering the eigenvalue decomposition (EVD) as  
\[\mathbf{R}_{\text{tot}}=\mathbf{U}_{\text{tot}}\mathbf{\Lambda}_{\text{tot}} \mathbf{U}_{\text{tot}}^{\text{H}}, \tag{14}\]  
where unitary matrix \(\mathbf{U}_{\text{tot}}=[\mathbf{u}_{\text{tot},1},\ldots,\mathbf{u}_{\text{ tot},N}]\in\mathbb{C}^{N\times N}\) represents the eigenvectors of CCM and \(\mathbf{\Lambda}_{\text{tot}}\in\mathbb{R}^{N\times N}\) denotes the eigenvalues of CCM \(\mathbf{R}_{\text{tot}}\), which are arranged in descending order so that \(\lambda_{\text{tot},1}\geq\lambda_{\text{tot},2}\geq\cdots\geq\lambda_{\text{ tot},N}\). Using the eigenvalue decomposition in (14), the MSE error can be expressed as  
\[\mathbf{C}_{e}=\text{tr}\Big{(}\mathbf{U}_{\text{tot}}\mathbf{\Lambda}_{\text {tot}}^{-1}\mathbf{U}_{\text{tot}}^{\text{H}}+\mathbf{S}_{\text{tr}}\mathbf{ S}_{\text{tr}}^{\text{H}}\Big{)}. \tag{15}\]  
Expression (15) can be written as  
\[\mathbf{C}_{e}=\text{tr}\Big{(}\mathbf{\Lambda}_{\text{tot}}^{-1}+\mathbf{U}_ {\text{tot}}^{\text{H}}\mathbf{S}_{\text{tr}}\mathbf{S}_{\text{tr}}^{\text{H }}\mathbf{U}_{\text{tot}}\Big{)}. \tag{16}\]  
**Lemma 1**. : _The following inequality holds for positive definite matrix \(\mathbf{A}\):_  
\[\text{tr}\big{(}\mathbf{A}^{-1}\big{)}\geq\sum_{i=1}^{m}(a_{i,i})^{-1}, \tag{17}\]  
_where \(a_{i,i}\) is a diagonal element of the matrix \(\mathbf{A}\). The equality in (17) holds if \(\mathbf{A}\) is a diagonal matrix._  
Proof. : A proof of Lemma 1 can be found in [60]. Based on Lemma 1 and majorization theory, the MSE in (16) is minimized if \(\mathbf{U}_{\text{tot}}^{\text{H}}\mathbf{S}_{\text{tr}}\mathbf{S}_{\text{tr}} ^{\text{H}}\mathbf{U}_{\text{tot}}\) is a diagonal matrix with all non-negative elements. Consider the EVD of \(\mathbf{S}_{\text{tr}}\mathbf{S}_{\text{tr}}^{\text{H}}\), i.e., \(\mathbf{S}_{\text{tr}}\mathbf{S}_{\text{tr}}^{\text{H}}=\mathbf{U}\mathbf{ \Sigma}\mathbf{U}^{\text{H}}\). To this end, the right hand side in (16) can be written as \(\mathbf{U}_{\text{tot}}^{\text{H}}\mathbf{U}\mathbf{\Sigma}\mathbf{U}^{\text{H }}\mathbf{U}_{\text{tot}}\), which can be represented by a diagonal matrix \(\mathbf{\Sigma}\) if we choose \(\mathbf{U}=\mathbf{U}_{\text{tot}}\), i.e., \(\mathbf{\Sigma}=\mathbf{U}_{\text{tot}}\). The diagonal matrix \(\mathbf{\Sigma}\succeq\mathbf{0}_{\text{Tr}}\) with an optimum power assignment and under the transmit power constraint can be found through the waterfilling algorithm; see, e.g., [27; 61; 62; 63; 64]. As such, the TS framework is provided next. **Proposition 1**. : _A new DL TS \(\mathbf{S}_{\text{tr}}\in\mathbb{C}^{N\times\text{Tr}}\) for \(m\)-MIMO systems with FDD transmission mode considering different spatial correlation is designed based on the first \(T_{\text{tr}}\) eigenvectors of the superposition CCMs \(\mathbf{R}_{\text{tot}}\), denoting the largest eigenvalues, scaled by an optimized power allocation that is obtained using the waterfilling solution as expressed in (18)_  
\[\mathbf{S}_{\text{tr}}=\sqrt{\text{T}_{\text{tr}}}[\mathbf{U}_{\text{tot}}][ \mathbf{\Sigma}_{\text{tot}}^{1/2}], \tag{18}\]  
_where \(\mathbf{U}_{\text{tot}}\in\mathbb{C}^{N\times\text{Tr}_{\text{tr}}}\) is a rectangular matrix consisting of \(T_{\text{tr}}\) eigenvectors of \(\mathbf{R}_{\text{tot}}\) associated with the first \(T_{\text{tr}}\) largest eigenvalues with an optimized power assignment.