# Downlink Training Design for FDD Massive MIMO Systems in the Presence of Colored Noise
## 1 Introduction
### Paper Contributions and Findings  
In this paper, we address the challenge of DTS design and CSI estimation in the FDD operation in MaMi systems with single-stage precoding and in the presence of spatially colored noise and with limited coherence time. A computational feasible solution for the DTS design and CSI estimation with colored noise environments is proposed by exploring the statistical information denoted by the channel and noise covariance matrices. We analyse the DL CSI estimation with minimum-mean square-error (MMSE) filter and derive a new analytical solution for the MSE, which incorporates a practical impairment of colored noise. Additionally, an asymptotic closed form analytical solution for the achievable sum rate of the regularized zero forcing beamforming (RZFBF) precoder is developed by exploiting a random matrix theory method. This explicit analytical solution allows the achievable sum rate performance of an FDD operation in MaMi system to be straightforwardly obtained without resorting to a computationally demanding exhaustive search. Comparisons between the MSE and sum rate performances of the proposed DTS design and the state-of-the-art DTS designs based on the white noise environments are carried out by considering the one ring (OR) [24; 32; 33] and Laplacian [26; 34] channel models with uniform linear array (ULA) and uniform planar array (UPA) configurations. The results show that the proposed DTS design improves the MSE performance significantly compared with the conventional training designs based on white noise. This improvement results in maximizing the DL achievable sum rate of the FDD MaMi systems considerably, without increasing in the system overhead and design complexity. In particular, this paper shows that a feasible DTS design with minimal overheads can be achieved by jointly considering the eigen-decompositions of the channel and noise covariance matrices. Although the results in this paper are presented for practical antennas sizes, i.e., \(N=128\), we believe that the same trends can also be observed for the other antenna arrays sizes. Finally, the results show that the analyses of the MSE and achievable sum are developed in this paper, and greatly agree with the simulations, which underpin the contributions of this work.  
_Paper organization_: In Section 2, we describe the system model and introduce the achievable sum rate of the RZFBF precoding. Section 3 explains the DL channel estimation process based on the DTS design and formalized the optimization problem of the FDD operation in MaMi systems in the presence of colored noise. In Section 4, the proposed DTS design for CSI estimation in the presence of colored noise is developed and the analysis of the MSE is derived. In Section 5, we adopt a random matrix theory to provide an asymptotic analysis for the signal-to-interference-plus-noise ratio (SINR) and the achievable sum rate of the RZFBF precoding. In Section 6, the system geometries of the channel covariance matrices in the ULA and UPA are presented. In Section 7, we characterize the system performance using the proposed training framework by providing some simulation results. Finally, we conclude the paper in Section 8.  
_Notation_: We use an upper boldface symbol for a matrix and lower boldface symbol for a vector. We use the term \(\mathbb{E}[\cdot]\) for the expectation operator. The following mathematical operations, trace, transpose, Hermitian transpose, inverse and absolute, are denoted by \(\text{tr}(\cdot)\), \((\cdot)^{\text{T}}\), \((\cdot)^{\text{H}}\), \((\cdot)^{-1}\), and \(|\cdot|\), respectively.