  This paper is on beamforming in wireless relay networks with perfect channel
information at relays, the receiver, and the transmitter if there is a direct
link between the transmitter and receiver. It is assumed that every node in the
network has its own power constraint. A two-step amplify-and-forward protocol
is used, in which the transmitter and relays not only use match filters to form
a beam at the receiver but also adaptively adjust their transmit powers
according to the channel strength information. For a network with any number of
relays and no direct link, the optimal power control is solved analytically.
The complexity of finding the exact solution is linear in the number of relays.
Our results show that the transmitter should always use its maximal power and
the optimal power used at a relay is not a binary function. It can take any
value between zero and its maximum transmit power. Also, this value depends on
the quality of all other channels in addition to the relay's own channels.
Despite this coupling fact, distributive strategies are proposed in which, with
the aid of a low-rate broadcast from the receiver, a relay needs only its own
channel information to implement the optimal power control. Simulated
performance shows that network beamforming achieves the maximal diversity and
outperforms other existing schemes. Then, beamforming in networks with a direct
link are considered. We show that when the direct link exists during the first
step only, the optimal power control is the same as that of networks with no
direct link. For networks with a direct link during the second step, recursive
numerical algorithms are proposed to solve the power control problem.
Simulation shows that by adjusting the transmitter and relays' powers
adaptively, network performance is significantly improved.
