# Downlink Training Design for FDD Massive MIMO Systems in the Presence of Colored Noise
## 7 Results and Findings  
This section provides several simulation and theoretical results, which characterize the FDD performance in MaMi, based on the normalized MSE and the achievable sum rate of RZFBF precoding in the FDD MaMi systems. To this end, the performance of the proposed DTS and CSI estimation of FDD operation mode in MaMi systems with colored noise and limited coherence time is evaluated. The simulation parameters that are used in the FDD MaMi evaluation are provided in Table 1. We consider a dense urban scenario where the uses are located within the range of 200 m from the BS. The coherence block length is given as \(\tau=100\) symbols and \(K=8\) users. The one ring and Laplacian channel models parameters are selected with angular spread \(\alpha_{\mathrm{A}}=\{5^{\circ},15^{\circ}\}\), which imply relatively high spatial channel correlation and weak spatial channel correlation, respectively. Parameter \(\sigma_{\mathrm{d}}\) is given by \(\sigma_{\mathrm{d}}=1/\mathrm{tr}(\mathbf{R}_{\mathrm{n}})\rho_{\mathrm{d}}\), where \(\mathbf{R}_{\mathrm{n}}\) denotes the colored noise covariance matrix, which is given in [59] as \([\mathbf{R}_{\mathrm{n}}]_{i,j}=p^{|j-i|}\) with \(0\leq p<1\). For relatively high correlated colored noise, we consider \(p>0.9\). Each simulated point in our figures is obtained by averaging over \(10^{5}\) independent simulation runs. In practice, the channel covariance matrix based on the one ring and Laplacian channel models will not have singular values that are exactly equal zero. As such, programs such as MATLAB might over estimate the effective rank of channel covariance matrix. To overcome this issue, this paper uses a threshold-based pilot selection criteria to provide a feasible DTS length with a reduced overhead for the FDD MaMi systems. In particular, only the effective eigenvalues of the transmit covariance matrix are selected, in order to reduce the overhead of the FDD MaMi systems. To this end, the effective eigenvalues of the transmit covariance matrix that are above a threshold of 0.001, which can be considered as approximately equal zero, are selected for the training transmission phase, while the rest of the eigenvalues are ignored and not used in the training transmission phase, and thus, not used in precoding design and data transmission phase. It is also worth noting that, orthogonal DTS and/or random DTS are ineffective with the FDD MaMi systems, because such sequences would span all the dimensional space. Specifically, the aforementioned sequences do not capture the dominant effective channel dimensions where the energy is concentrated, and thus wasting the resources in the training-phase, which will not be used for broadcasting useful information to the users. Performance Evaluation of the Proposed DTS Design and CSI Estimation in the Presence of Colored NoiseIn what follows, the performance of the proposed DTS design for the FDD MaMi systems is evaluated and compared with the conventional DTS designs, which are proposed based on idealistic assumption of white noise. The performance evaluation is carried out based on the normalized MSE and the achievable sum rate maximization. \begin{table}
\begin{tabular}{l l l} \hline \hline
**Parameters** & **Symbol** & **Value** \\ \hline Number of BS antennas & \(N\) & 128 \\ Number of users & \(K\) & 8 \\ Azimuth angular spread & \(\alpha_{\mathrm{A}}\) & \(\{5^{\circ},15^{\circ}\}\) \\ Antenna element separation & \(d_{\mathrm{A}}=d_{\mathrm{E}}=d\) & \(\lambda/2\) \\ Coherence time & \(T_{\mathrm{c}}\) & 1 ms \\ Coherence frequency & \(B_{\mathrm{c}}\) & 100 kHz \\ Coherence block length & \(\tau\) & 100 symbols \\ Training power & \(\rho_{\mathrm{s}}\) & \(-20\) dB–10 dB \\ BS height & \(h\) & 30 m \\ Ring radius & \(s\) & 30 m \\ Users distance & \(D\) & 200 m \\ Azimuth angle & \(\theta\) & \(\pi/6\) \\ BS configurations & BS & \{ULA, UPA\} \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation parameters.