to the empirical mean of $C_{mk}$. The small-scale fading gains are distributed as $$\alpha_{mk,cp} \sim \mathcal{CN}\left(0,\gamma_{mk,c}10^{-\operatorname{PL}(d_{mk})/10}\right),$$ where the cluster $c$ is assumed to contribute with a fraction of power given by $$\gamma_{mk,c}=\frac{N \gamma'_{mk,c}}{P_{mk}\sum_{j=1}^{C_{mk}} \gamma'_{mk,j}},$$ with $$\gamma'_{mk,j}=U_{mk,j}^{r_\tau-1} 10^{Z_{mk,j}/10},$$ $U_{mk,j} \sim \mathcal{U}[0,1]$, $Z_{mk,j} \sim \mathcal{N}(0,\zeta^2)$, and the constants $r_\tau$ and $\zeta^2$ being treated as model parameters (see [@Akdeniz14 Table I]).

Although the small-scale fading gains $\alpha_{mk,cp}$ are assumed to be static throughout the coherence interval and then change independently (i.e., block fading), the spatial covariance matrices $$\begin{split}
   {\boldsymbol}{R}_{mk}=&\mathbb{E}\left\{{\boldsymbol}{h}_{mk} {\boldsymbol}{h}_{mk}^H \right\} \\
              =&10^{-\operatorname{PL}(d_{mk})/10}\sum_{c=1}^{C_{mk}}\gamma_{mk,c} \\
               &\times \sum_{p=1}^{P_{mk}} {\boldsymbol}{a}\left(\theta_{mk,cp},\phi_{mk,cp}\right){\boldsymbol}{a}^H\left(\theta_{mk,cp},\phi_{mk,cp}\right),
\end{split}$$ are assumed to vary at a much smaller pace (i.e., ${\tau_L}_c \gg \tau_c$).

Large-scale training phase {#subsec:large-scale-training}
--------------------------

###  precoder/combiner design

In order to exploit the / channel reciprocity using the frame structure shown in Fig. \[fig:TDD\_frame\], it is assumed in this paper that the $N \times L_A$ matrix ${\boldsymbol}{W}_m^{RF}$, describing the effects of the active analog phase shifters at the $m$th , is common to the ( precoding phase) and ( combining phase). Furthermore, denoting by $\mathcal{K}_m=\left\{\kappa_{m 1}, \ldots, \kappa_{m L_A}\right\}$ the set of $L_A$ beamformed by the $m$th , it is assumed that ${\boldsymbol}{W}_m^{RF}$ is a function of only the spatial channel covariance 