# Downlink Training Design for FDD Massive MIMO Systems in the Presence of Colored Noise
## 2 System Model
### Downlink Achievable Sum Rate  
In general, the user does not know the exact channel vector and precoding matrix, but instead estimates their average effect through \(\sqrt{\xi}\mathbb{E}\big{[}\mathbf{h}_{k}^{\mathrm{H}}\mathbf{w}_{k}\big{]}\), as considered in [9]. As such, the DL received signal at the \(k\)-th user can be expressed as  
\[y_{k}=\sqrt{\rho_{\mathrm{d}}\xi}\mathbb{E}\big{[}\mathbf{h}_{k} ^{\mathrm{H}}\mathbf{w}_{k}\big{]}x_{k}+\sqrt{\rho_{\mathrm{d}}\xi}\big{(} \mathbf{h}_{k}^{\mathrm{H}}\mathbf{w}_{k}-\mathbb{E}\big{[}\mathbf{h}_{k}^{ \mathrm{H}}\mathbf{w}_{k}\big{]}\big{)}x_{k}\] \[+\sqrt{\rho_{\mathrm{d}}\xi}\mathbf{h}_{k}^{\mathrm{H}}\sum_{i\
eq k }^{K}\mathbf{w}_{i}x_{i}+n_{k}. \tag{3}\]  
The first term in the DL received signal in (3) represents the desired signal at the \(k\)-th user. The second term in (3) is the error caused by the imperfect channel knowledge at the user. The third term in (3) represents the residual multiuser interference after precoding. In general, the interference and error introduced by channel estimation terms are neither Gaussian nor independent of the desired information signal. Thus, an ergodic sum rate lower bound is determined by considering that both terms are independent of the desired signal, i.e., a worst-case scenario [9; 16]. To this end, the achievable sum rate in the DL of the FDD MaMi systems is given as  
\[\mathrm{Rate}=\left(1-\frac{T_{\mathrm{s}}}{\tau}\right)\cdot K\cdot\log_{2} \big{(}1+\gamma_{\mathrm{h}}\big{)}\quad[\mathrm{bit/s/Hz}], \tag{4}\]  
where \(T_{\mathrm{s}}\) denotes the DTS length in symbol, \(\tau\) represents the coherence block length in symbol and \(\gamma_{\mathrm{h}}\) is the \(k\)-th user (\(\mathrm{SINR}\)), which is provided by [9; 10]  
\[\gamma_{\mathrm{h}}=\frac{\xi\big{|}\mathbb{E}\big{[}\mathbf{h}_{k}^{\mathrm{ H}}\mathbf{w}_{k}\big{]}\big{|}^{2}}{\sigma_{\mathrm{d}}+\xi\mathbb{E}\big{[} \big{|}\,\mathbf{h}_{k}^{\mathrm{H}}\mathbf{w}_{k}-\mathbb{E}\big{[}\mathbf{ h}_{k}^{\mathrm{H}}\mathbf{w}_{k}\big{]}\,\big{|}^{2}\big{]}+\xi\sum_{i\
eq k }^{K}\mathbb{E}\big{[}\big{|}\,\mathbf{h}_{k}^{\mathrm{H}}\mathbf{w}_{i}\, \big{|}^{2}\big{]}}. \tag{5}\]  
The SINR expression in (5) can be re-written as  
\[\gamma_{\mathrm{h}}=\frac{\xi\big{|}\mathbb{E}\big{[}\mathbf{h}_{k}^{\mathrm{ H}}\mathbf{w}_{k}\big{]}\big{|}^{2}}{\sigma_{\mathrm{d}}+\xi\sum_{l=1}^{K} \mathbb{E}\big{[}\big{|}\,\mathbf{h}_{k}^{\mathrm{H}}\mathbf{w}_{l}\,\big{|}^ {2}\big{]}-\big{|}\,\mathbb{E}\big{[}\mathbf{h}_{k}^{\mathrm{H}}\mathbf{w}_{k }\big{]}\,\big{|}^{2}}, \tag{6}\]  
where the term \(\sigma_{\mathrm{d}}\) is the noise variance. The expectations in (6) are calculated based on different channel variations. These different channel realizations which are determined using extensive Monte Carlo simulations. However, this averaging process is very computationally demanding, because the different expectations in (6) are determined for different choices of \(\sigma_{\mathrm{d}}\), \(\sigma_{\mathrm{p}}\), \(T_{\mathrm{s}}\), \(\tau\) and \(N\) values using extensive Monte Carlo simulations. In this paper, a feasible and computationally effective solution for the SINR and the achievable sum rate of the RZFBF precoder in an FDD MaMi system is obtained in Section 5 by exploiting the asymptotic random matrix theory method.