  In this paper, we study the performance of the downlink of a cellular network
with automatic repeat-request (ARQ) and a half duplex decode-and-forward shared
relay. In this system, two multiple-input-multiple-output (MIMO) base stations
serve two single antenna users. A MIMO shared relay retransmits the lost
packets to the target users. First, we study the system with direct
retransmission from the base station and derive a closed form expression for
the outage probability of the system.We show that the direct retransmission can
overcome the fading, however, it cannot overcome the interference. After that,
we invoke the shared relay and design the relay beamforming matrices such that
the signal-to-interference-and-noise ratio (SINR) is improved at the users
subject to power constraints on the relay. In the case when the transmission of
only one user fails, we derive a closed form solution for the relay
beamformers. On the other hand when both transmissions fail, we pose the
beamforming problem as a sequence of non-convex feasibility problems. We use
semidefinite relaxation (SDR) to convert each feasibility problem into a convex
optimization problem. We ensure a rank one solution, and hence, there is no
loss of optimality in SDR. Simulation results are presented showing the
superior performance of the proposed relay beamforming strategy compared to
direct ARQ system in terms of the outage probability.
