  A framework for energy-efficient resource allocation in a single-user,
amplify-and-forward relay-assisted MIMO system is devised in this paper.
Previous results in this area have focused on rate maximization or sum power
minimization problems, whereas fewer results are available when bits/Joule
energy efficiency (EE) optimization is the goal. The performance metric to
optimize is the ratio between the system's achievable rate and the total
consumed power. The optimization is carried out with respect to the source and
relay precoding matrices, subject to QoS and power constraints. Such a
challenging non-convex problem is tackled by means of fractional programming
and and alternating maximization algorithms, for various CSI assumptions at the
source and relay. In particular the scenarios of perfect CSI and those of
statistical CSI for either the source-relay or the relay-destination channel
are addressed. Moreover, sufficient conditions for beamforming optimality are
derived, which is useful in simplifying the system design. Numerical results
are provided to corroborate the validity of the theoretical findings.
