# 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  
The DL sum rate of RZFBF precoding in an FDD MaMi relies on the channel statistic and noise statistic and the DL channel estimation. In non-reciprocal channels, such as FDD based systems, the BS would need to transmit DL training sequence to the users, and each user will then quantize the received signal and transmit it back to the BS to estimate the DL channels and complete the precoder design. However, in some cases, each user could estimate its own DL channel and send the quantized channel estimates back to the BS. Both approaches are suitable with the poposed design in the paper. This section derives the optimum training sequence that needs to be used to design the precoding at the BS in order to achieve a maximum achievable sum rate in an FDD MaMi system. In what follows, we provide the proposed DTS framework for an FDD MaMi system in the presence of colored noise. The proposed training design aims to improve the quality of the channel estimation, and thus, achieves a robust sum rate performance with a reduced training overhead. Due to the use of hundreds of antennas at the BS, which are grouped together and packed into a limited physical area, the MaMi channels may be highly correlated. To this end, the channel covariance matrix would exhibit a large eigenvalue spread so that it can be partitioned into a small dimensions, which is effectively less than \(N\). Moreover, a large portion of the energy in the channel can be focused into a few directions (eigenmodes). In what follows, the basic concept of majorization theory is introduced in order to understand the fundamental impact of spatial correlation on the MSE of channel estimate. Definition 1: Majorization [49]: Let \(\mathbf{x}=[\mathbf{x}_{1},\ldots,\mathbf{x}_{N}]^{\text{T}}\) and \(\mathbf{z}=[\mathbf{z}_{1},\ldots,\mathbf{z}_{N}]^{\text{T}}\) be as two positive real-valued vectors, containing the elements of \(\mathbf{x}\) and \(\mathbf{z}\), respectively, which are arranged in descending order. Vector \(\mathbf{x}\) majorizes \(\mathbf{z}\)\((\mathbf{x}\succeq\mathbf{z})\) if  
\[\sum_{i=1}^{n}\mathbf{x}\succeq\sum_{i=1}^{n}\mathbf{z}, \tag{21}\]  
for all \(1\leq n<N\). If vectors \(\mathbf{x}\) and \(\mathbf{z}\) contain eigenvalues of the channel covariance matrices, then by the majorization property, \(\mathbf{z}\) is less spread out than \(\mathbf{x}\) and vector \(\mathbf{x}\) is more spatially correlated than vector \(\mathbf{z}\). Similarly, for the MSE cost function under consideration, \(C_{\text{r}}(\mathbf{x})\leq C_{\text{r}}(\mathbf{z})\), which implies that the MSE of the stronger eigenvalues are less than the MSE of the weakest eigenvalues. As such, high spatial correlation reduces the estimation error and the eigendirections of \(\mathbf{R}_{\text{h}}\), with large eigenvalues have a smaller estimation error variance than the eigendirections with smaller eigenvalues. Specifically, based on the majorization theory, the MSE performance decreases with increasing spatial channel correlations. As such, the minimum number of DTS length can be chosen to be significantly less than \(N\) in correlated channel scenarios. This is particularly important for the FDD MaMi systems where the MSE needs to be minimised and the sum rate requires being maximized using as little DTS length as possible. We design the structure of the DTS matrix \(\mathbf{S}\) with the single stage precoding, based on the second order channel and noise statistics, which are denoted by the covariance matrix and the noise covariance matrix. In particular, the effective eigenvectors denoted by the largest eigenvalues of the channel covariance matrix and noise covariance matrix can be exploited in the pilot design, in order to minimize the MSE while reducing the DTS overhead. This would allow the DL achievable sum rate of the FDDMaMi communications systems to be maximized.