matrices $\left\{{\boldsymbol}{R}_{mk}\right\}_{k\in\mathcal{K}_m}$, known at the $m$th through spatial channel covariance estimation for hybrid analog-digital precoding architectures (see e.g. [@Adhikary14; @Mendez15; @Park16; @Park17]).

Using eigen-decomposition, the covariance matrix of the propagation channel linking k and $m$ can be expressed as ${\boldsymbol}{R}_{mk}={\boldsymbol}{U}_{mk}{\boldsymbol}{\Lambda}_{mk}{\boldsymbol}{U}_{mk}^H$, where ${\boldsymbol}{\Lambda}_{mk}=\operatorname{diag}\left(\left[\lambda_{mk,1}\,\ldots\,\lambda_{mk,r_{mk}}\right]\right)$ contains the $r_{mk}$ non-null eigenvalues of ${\boldsymbol}{R}_{mk}$, and ${\boldsymbol}{U}_{mk}$ is the $N \times r_{mk}$ matrix of the corresponding eigenvectors. Hence, assuming the use of (constrained) statistical eigen beamforming [@Park17TSC; @Mai18], the analog precoder/combiner can be designed as $$\begin{split}
   {\boldsymbol}{W}_m^{RF}=&\begin{bmatrix}
                    {\boldsymbol}{w}_{m \kappa_{m 1}}^{RF} & \ldots & {\boldsymbol}{w}_{m \kappa_{m L_A}}^{RF}
                 \end{bmatrix}\\
                =&\begin{bmatrix}
                    e^{-j\angle{\boldsymbol}{u}_{m \kappa_{m 1},\max}} & \ldots & e^{-j\angle{\boldsymbol}{u}_{m \kappa_{m L_A},\max}}
                 \end{bmatrix},
\end{split}$$ where ${\boldsymbol}{u}_{mk,\max}$ is the dominant eigenvector of ${\boldsymbol}{R}_{mk}$ associated to the maximum eigenvalue $\lambda_{mk,\max}$, and the function $\angle{\boldsymbol}{x}$ returns the phase angles, in radians, for each element of the complex vector ${\boldsymbol}{x}$. Note that using the precoding/combining matrix, the equivalent channel vector between k and $m$, including the precoding/decoding matrix, is defined as $${\boldsymbol}{g}_{mk} = {{\boldsymbol}{W}_m^{RF}}^T {\boldsymbol}{h}_{mk}\ \in\ \mathbb{C}^{L_A \times 1},$$ whose dimension is much less than the number of antennas of the massive array used at the $m$th , thus largely simplifying the small-scale training phase.

### Selection of to beamform from each 

As previously stated, 