# Downlink Training Design for FDD Massive MIMO Systems in the Presence of Colored Noise
## 3 Problem Formulation of Channel Estimation with Colored Noise
Since the DL channel \(\mathbf{h}_{k}\) exhibits a CSCG distribution and the statistics \(\mathbf{R}_{\mathrm{h}}\) are known, the probability density function (PDF) is given as  
\[f_{\mathrm{h}_{k}}(\mathbf{h}_{k})=\frac{\exp\{-\mathbf{h}_{k}^{\mathrm{H}} \mathbf{R}_{\mathrm{h}}^{-1}\mathbf{h}_{k}\}}{\det(\pi\mathbf{R}_{\mathrm{h}})}. \tag{9}\]  
Similarly, the PDF of the noise vector with noise temporal correlation matrix is given as  
\[f_{\mathrm{h}_{k}}(\mathbf{n}_{k})=\frac{\exp\{-\mathbf{n}_{k}^{\mathrm{H}} \mathbf{R}_{\mathrm{h}}^{  
Since the channel \(\mathbf{h}_{k}\) and noise \(\mathbf{n}_{k}\) vectors are independent Gaussian distributions, by the definition of the posterior probability density function, we have  
\[f_{\mathbf{h}_{k}|y_{k}}\big{(}\mathbf{h}_{k}|y_{k}\big{)} =\frac{f_{\mathbf{h}_{k},y_{k}}\big{(}\mathbf{h}_{k},\mathbf{y}_{k }\big{)}}{f_{\mathbf{y}_{k}}\big{(}\mathbf{y}_{k}\big{)}}\] \[=\frac{f_{\mathbf{h}_{k}}\big{(}\mathbf{h}_{k}\big{)}f_{\mathbf{h} _{k}|y_{k}}\big{(}\mathbf{y}_{k}|\mathbf{h}_{k}\big{)}}{f_{\mathbf{y}_{k}} \big{(}\mathbf{y}_{k}\big{)}}, \tag{13}\]  
substituting (9)-(12) into (13), yields  
(14)  
The expression of the PDF in (14) can be expressed in a canonical Hermitian with Gaussian distribution [48], and thus, the conditional channel covariance matrix of \(\mathbf{h}_{k}\) given \(\mathbf{y}_{k}\) can be written as  
\[\mathbf{R}_{\mathbf{h}_{k}|\mathbf{y}_{k}}=\Big{(}\mathbf{R}_{\mathbf{h}}- \mathbf{\Phi}_{\hat{\mathbf{h}}}\Big{)}, \tag{15}\]  
where \(\mathbf{\Phi}_{\hat{\mathbf{h}}}\) is the covariance of the MMSE channel estimation that is given by  
\[\mathbf{\Phi}_{\hat{\mathbf{h}}}=\mathbf{R}_{\mathbf{h}}\mathbf{S}\big{(} \mathbf{S}^{\mathrm{H}}\mathbf{R}_{\mathbf{h}}\mathbf{S}+\mathbf{R}_{\mathbf{ n}}\big{)}^{-1}\mathbf{S}^{\mathrm{H}}\mathbf{R}_{\mathbf{h}}, \tag{16}\]  
where the expression in (15) is minimised by maximising the expression in (16), which represents the covariance matrix with MMSE channel estimate that is required to be as similar to \(\mathbf{R}_{\mathbf{h}}\) as possible. **Lemma 1**. : _For a positive definite matrices \(\mathbf{A},\mathbf{U},\mathbf{C},\mathbf{V}\), using the Woodbury matrix identity, the following holds:_  
\[(\mathbf{A}+\mathbf{UCV})^{-1}=\mathbf{A}^{-1}-\mathbf{A}^{-1}\mathbf{U}( \mathbf{C}^{-1}+\mathbf{VA}^{-1}\mathbf{U})^{-1}\mathbf{VA}^{-1}. \tag{17}\]  
Using Lemma 1, the conditional covariance matrix of \(\mathbf{h}_{k}\) given \(\mathbf{y}_{k}\) can be expressed as  
\[\mathbf{R}_{\mathbf{h}_{k}|\mathbf{y}_{k}}=\Big{(}\mathbf{R}_{\mathbf{h}}^{-1 }+\mathbf{S}\mathbf{R}_{\mathbf{n}}^{-1}\mathbf{S}^{\mathrm{H}}\Big{)}^{-1}. \tag{18}\]  
Accordingly, the MSE cost function \(C_{e}\in\mathbb{R}\) with MMSE channel estimates can be written as  
\[C_{e}=\text{tr}\big{(}\mathbf{R}_{\mathbf{h}_{k}|\mathbf{y}_{k}}\big{)}. \tag{19}\]  
Clearly, the MSE performance depends on the structure of the training matrix \(\mathbf{S}\), which explicitly depends on the pilot energy allocated during the training phase, i.e., \((\rho_{\mathrm{s}}T_{\mathrm{s}})\), and on the structure of the channel and noise statistics. Specifically, the MSE performance depends on the eigenstructure of the term \(\Big{(}\mathbf{R}_{\mathbf{h}}^{-1}+\mathbf{S}\mathbf{R}_{\mathbf{n}}^{-1} \mathbf{S}^{\mathrm{H}}\Big{)}^{-1}\), i.e., the energy that corresponds to the eigenvalues distribution over the diagonal of this term. This observation motivates the use of the structures of the channel covariance matrix \(\mathbf{R}_{\mathbf{h}}\) and noise covariance matrix \(\mathbf{R}_{\mathbf{n}}\) in optimizing the DTS, and thus performance, of the FDD MaMi systems. More details about the structure of the DL training matrix \(\mathbf{S}\) are provided in the next section.