  This paper deals with joint source-relay-destination beamforming (BF) design
for an amplify-and-forward (AF) relay network. Considering the channel state
information (CSI) from the relay to the destination is imperfect, we first aim
to maximize the worst case received SNR under per-antenna power constraints.
The associated optimization problem is then solved in two steps. In the first
step, by revealing the rank-one property of the optimal relay BF matrix, we
establish the semi-closed form solution of the joint optimal BF design that
only depends on a vector variable. Based on this result, in the second step, we
propose a low-complexity iterative algorithm to obtain the remaining unknown
variable. We also study the problem for minimizing the maximum per-antenna
power at the relay while ensuring a received signal-to-noise ratio (SNR)
target, and show that it reduces to the SNR maximization problem. Thus the same
methods can be applied to solve it. The differences between our result and the
existing related work are also discussed in details. In particular, we show
that in the perfect CSI case, our algorithm has the same performance but with
much lower cost of computational complexity than the existing method. Finally,
in the simulation part, we investigate the impact of imperfect CSI on the
system performance to verify our analysis.
