  This work studies the net sum-rate performance of a distributed
reconfigurable intelligent surfaces (RISs)-assisted multi-user
multiple-input-single-output (MISO) downlink communication system under
imperfect instantaneous-channel state information (I-CSI) to implement
precoding at the base station (BS) and statistical-CSI (S-CSI) to design the
RISs phase-shifts. Two channel estimation (CE) protocols are considered for
I-CSI acquisition: (i) a full CE protocol that estimates all direct and
RISs-assisted channels over multiple training sub-phases, and (ii) a
low-overhead direct estimation (DE) protocol that estimates the end-to-end
channel in a single sub-phase. We derive the deterministic equivalents of
signal-to-interference-plus-noise ratio (SINR) and ergodic net sum-rate under
Rayleigh and Rician fading and both CE protocols, for given RISs phase-shifts,
which are then optimized based on S-CSI. Simulation results reveal that the
low-complexity DE protocol yields better net sum-rate than the full CE protocol
when used to obtain CSI for precoding. A benchmark full I-CSI based RISs design
is also outlined and shown to yield higher SINR but lower net sum-rate than the
S-CSI based RISs design due to the large overhead associated with full I-CSI
acquisition. Therefore the proposed DE-S-CSI based design for precoding and
reflect beamforming achieves high net sum-rate with low complexity, overhead
and power consumption.
