  Massive MIMO is a variant of multiuser MIMO where the number of base-station
antennas $M$ is very large (typically 100), and generally much larger than the
number of spatially multiplexed data streams (typically 10). Unfortunately, the
front-end A/D conversion necessary to drive hundreds of antennas, with a signal
bandwidth of the order of 10 to 100 MHz, requires very large sampling bit-rate
and power consumption.
  In order to reduce such implementation requirements, Hybrid Digital-Analog
architectures have been proposed. In particular, our work in this paper is
motivated by one of such schemes named Joint Spatial Division and Multiplexing
(JSDM), where the downlink precoder (resp., uplink linear receiver) is split
into the product of a baseband linear projection (digital) and an RF
reconfigurable beamforming network (analog), such that only a reduced number $m
\ll M$ of A/D converters and RF modulation/demodulation chains is needed. In
JSDM, users are grouped according to the similarity of their channel dominant
subspaces, and these groups are separated by the analog beamforming stage,
where the multiplexing gain in each group is achieved using the digital
precoder. Therefore, it is apparent that extracting the channel subspace
information of the $M$-dim channel vectors from snapshots of $m$-dim
projections, with $m \ll M$, plays a fundamental role in JSDM implementation.
  In this paper, we develop novel efficient algorithms that require sampling
only $m = O(2\sqrt{M})$ specific array elements according to a coprime sampling
scheme, and for a given $p \ll M$, return a $p$-dim beamformer that has a
performance comparable with the best p-dim beamformer that can be designed from
the full knowledge of the exact channel covariance matrix. We assess the
performance of our proposed estimators both analytically and empirically via
numerical simulations.
