  The performance of full-duplex (FD) relay systems can be greatly impacted by
the self-interference (SI) at relays. By exploiting multi-antenna in FD relay
systems, the spectral efficiency of FD relay systems can be enhanced through
spatial SI mitigation. This paper studies joint source transmit beamforming and
relay processing to achieve rate maximization for FD MIMO amplify-and-forward
(AF) relay systems with consideration of relay processing delay. The problem is
difficult to solve due mainly to the SI constraint induced by the relay
processing delay. In this paper, we first present a sufficient condition under
which the relay amplification matrix has rank one structure. Then, for the case
of rank one amplification matrix, the rate maximization problem is equivalently
simplified into an unconstrained problem which can be locally solved using
gradient ascent method. Next, we propose a penalty-based algorithmic framework,
called P-BSUM, for a class of constrained optimization problems which have
difficult equality constraints in addition to some convex constraints. By
rewriting the rate maximization problem with a set of auxiliary variables, we
apply the P-BSUM algorithm to the rate maximization problem in the general
case. Finally, numerical results validate the efficiency of the proposed
algorithms and show that the joint source-relay design approach under the rank
one assumption could be strictly suboptimal as compared to the P-BSUM-based
joint source-relay design approach.
