# Downlink Training Design for FDD Massive MIMO Systems in the Presence of Colored Noise
## 4 The Proposed DTS Design and Mean Square Error Analysis in the Presence of Colored Noise
Typically, the channel covariance matrix \(\mathbf{R}_{\text{h}}\) can be decomposed using the eigenvalue decomposition (EVD) as  
\[\mathbf{R}_{\text{h}}=\mathbf{U}_{\text{h}}\boldsymbol{\Lambda}_{\text{h}} \mathbf{U}_{\text{h}}^{\text{H}}, \tag{22}\]  
where matrix \(\mathbf{U}_{\text{h}}=[\mathbf{u}_{\text{h},1},\ldots,\mathbf{u}_{\text{h},N}] \in\mathbb{C}^{N\times N}\) is the eigenvectors and matrix \(\boldsymbol{\Lambda}_{\text{h}}\in\mathbb{R}^{N\times N}\) corresponds to the eigenvalues of \(\mathbf{R}_{\text{h}}\), which are ordered as \(\lambda_{\text{h},1}\geq\lambda_{\text{h},2}\geq\cdots\geq\lambda_{\text{h},N}\) (Matlab function (svd) provides the unitary matrix arranged in descending order). Using the EVD in (22), the MSE error cost function can be rewritten as  
\[C_{e}=\text{tr}\Big{(}\Big{(}\Big{(}\mathbf{U}_{\text{h}}\boldsymbol{\Lambda} _{\text{h}}\mathbf{U}_{\text{h}}^{\text{H}}\Big{)}^{-1}+\mathbf{S}\mathbf{R}_ {\text{n}}^{-1}\mathbf{S}^{\text{H}}\Big{)}^{-1}\Big{)}. \tag{23}\]  
It is straightforward to re-write (23) as  
\[C_{e}=\text{tr}\Big{(}\Big{(}\boldsymbol{\Lambda}_{\text{h}}^{-1}+\mathbf{U}_ {\text{h}}^{\text{H}}\mathbf{S}\mathbf{R}_{\text{n}}^{-1}\mathbf{S}^{\text{H} }\mathbf{U}_{\text{h}}\Big{)}^{-1}\Big{)}. \tag{24}\]  
Similar to channel covariance mtraix, the noise covariance matrix can be decomposed as  
\[\mathbf{R}_{\text{n}}=\mathbf{U}_{\text{n}}\boldsymbol{\Lambda}_{\text{n}} \mathbf{U}_{\text{n}}^{\text{H}}, \tag{25}\]  
where \(\mathbf{U}_{\text{n}}=[\mathbf{u}_{\text{h},1},\ldots,\mathbf{u}_{\text{n},T_ {\text{s}}}]\in\mathbb{C}^{T_{\text{s}}\times T_{\text{s}}}\) are the matrix eigenvectors and matrix \(\boldsymbol{\Lambda}_{\text{n}}\in\mathbb{R}^{T_{\text{s}}\times T_{\text{s}}}\) are the eigenvalues of \(\mathbf{R}_{\text{n}}\) ordered as \(\lambda_{\text{h},1}\geq\lambda_{\text{h},2}\geq\cdots\geq\lambda_{\text{h},T _{\text{s}}}\). To this end, the DTS framework in the presence of colored noise for an FDD MaMi system under consideration is provided in the following proposition. **Proposition 1**. : _A unique DTS matrix \(\mathbf{S}\in\mathbb{C}^{N\times T_{\text{s}}}\) for the training transmission phase in the FDD MaMi systems with colored noise is constructed from first \(T_{\text{s}}\) columns space eigenvectors of the transmit covariance matrix \(\mathbf{R}_{\text{h}}\), which correspond to \(T_{\text{s}}\) largest eigenvalues of \(\mathbf{R}_{\text{h}}\), multiplied by the columns space eigenvectors of the noise covariance matrix \(\mathbf{R}_{\text{n}}\) and scaled by the square root of the training power during the training phase, as expressed in (26)_  
\[\mathbf{S}=\sqrt{\rho_{\text{s}}}[\mathbf{U}_{\text{h},T_{\text{s}}}]_{1:T_{ \text{s}}}[\mathbf{U}_{\text{n}}], \tag{26}\]  
_where \(\mathbf{U}_{\text{h},T_{\text{s}}}\in\mathbb{C}^{N\times T_{\text{s}}}\) is a rectangular matrix consisting of \(T_{\text{s}}\) eigenvectors of \(\mathbf{R}_{\text{h}}\) that correspond to the largest \(T_{\text{s}}\) eigenvalues of \(\mathbf{R}_{\text{h}}\), i.e, \(\boldsymbol{\Lambda}_{\text{h},T_{\text{s}}}\in\mathbb{R}^{T_{\text{s}}\times T _{\text{s}}}\), and \(\mathbf{U}_{\text{n}}\in\mathbb{C}^{T_{\text{s}}\times T_{\text{s}}}\) is the eigenvectors of the noise covariance matrix \(\mathbf{R}_{\text{n}}\). This implies that the channel energy, which corresponds to the eigenvectors \([\mathbf{u}_{T_{\text{s}}+1},\ldots,\mathbf{u}_{N}]\), is not used in channel estimation process, which reduces the training overhead significantly. The obtained pilot design should satisfy the energy constraint \(E_{\text{s}}\) in the training-phase so that \(\text{tr}\big{(}\mathbf{S}^{\text{H}}\mathbf{S}\big{)}=\rho_{\text{s}}T_{\text {s}}\).