# 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
Substituting the DTS design in (26) into (24) with some straightforward algebra yields, a novel analytical form for the MSE of the MMSE channel estimate with colored noise as_  
\[C_{e}=\text{tr}\Big{(}\Big{(}\boldsymbol{\Lambda}_{\text{h},T_{\text{s}}}^{-1}+ \rho_{\text{s}}\big{(}\mathbf{U}_{\text{n}}\big{)}^{2}\boldsymbol{\Lambda}_{ \text{n}}^{-1}\big{(}\mathbf{U}_{\text{n}}^{\text{H}}\big{)}^{2}\Big{)}^{-1} \Big{)}. \tag{27}\]  
For a spacial case of white noise, so that \(\mathbf{R}_{\text{n}}=\mathbf{I}_{T_{\text{s}}}\), the MSE expression in (27) can be simplified to  
\[C_{e}=\text{tr}\Big{(}\Big{(}\boldsymbol{\Lambda}_{\text{h},T_{\text{s}}}^{-1}+ \rho_{\text{s}}\mathbf{I}_{T_{\text{s}}}\Big{)}^{-1}\Big{)}. \tag{28}\]  
Using the trace property, expression in (28) can be written as  
\[C_{e}=\sum_{t=1}^{T_{\text{s}}}\bigg{(}\frac{1}{\boldsymbol{\lambda}_{\text{h},t }}+\rho_{\text{s}}\bigg{)}^{-1}. \tag{29}\]Straightforward algebra simplifies (29) to  
\[C_{e}=\sum_{t=1}^{T_{\rm s}}\frac{\lambda_{\rm h,f}}{\rho_{\rm s}\lambda_{\rm h, f}+1}. \tag{30}\]  
Substituting the DTS design in (26) into (16) simplifies the trace of the MMSE estimation covariance matrix to  
\[\mathbf{R}_{\text{h}}=\mathbf{U}_{\text{h}}\boldsymbol{\Lambda}_{\text{h}} \mathbf{U}_{\text{h}}^{\text{H}}, \tag{22}\]0  
and for the spacial scenario of weight noise to  
\[\mathbf{R}_{\text{h}}=\mathbf{U}_{\text{h}}\boldsymbol{\Lambda}_{\text{h}} \mathbf{U}_{\text{h}}^{\text{H}}, \tag{22}\]1  
which can be further simplified to  
\[\mathbf{R}_{\text{h}}=\mathbf{U}_{\text{h}}\boldsymbol{\Lambda}_{\text{h}} \mathbf{U}_{\text{h}}^{\text{H}}, \tag{22}\]2  
Clearly, increasing the transmit power during the training phase, i.e., \(\rho_{\rm s}\) reduces the MSE. **Remark 1**. : _Typically, increasing \(T_{\rm s}\) allows for more \(DTS\) energy to be collected, which results in reducing the MSE, but it will reduce the achievable sum rate due to a shorter useful information transmission. Hence, it is essential to conduct a suitable system analysis in order to determine a feasible \(DTS\) length based on general statistics. This is because the loss in the MSE with shorter pilot length might be minor compared with the gain obtained by carrying out more data transmission. The proposed approach suggests that for a given training power, only the eigenvectors that correspond to the effective eigenvalues of \(\mathbf{R}_{\rm h}\), which are deemed as essential, should be exploited in the pilot design. The intuition behind this pilot framework is to take advantage of the strong subspaces (eigendirections) of \(\mathbf{R}_{\rm h}\) to send the common pilot signal to the users. The minimum number of DTS symbols is determined by the number of dominant eigenvalues of the spatial correlation matrix. The pilot selection criteria based on the effective eigendirection will be discussed later in Section 7._  
**Remark 2**. : _As discussed earlier, the vast majority of the previous studies on the DTS design and CSI estimation in the FDD MaMi systems have assumed an ideal scenario of wight noise environment. Specifically, the impact of colored noise with imperfect CSI estimation on the FDD MaMi systems has not yet been characterized in prior works. In summary, unlike the state-of-the-art research in the FDD MaMi systems where the DL channel estimation is considered by assuming an uncorrelated white noise environment, we focus on DTS design in the colored environments. To the best of our knowledge, this research study is the first work that takes into consideration the noise covariance in pilot design of the FDD MaMi systems. This paper also focuses on investigating the achievable DL sum rate of the proposed DTS design in the limited coherence time. Our approach benefits from the statistical information of the channel covariance matrix and the noise covariance matrix, resulting in a feasible DTS solution during the DL channel estimation, even when the channel coherence time is short._