  The problem of energy-efficient precoding is investigated when the terminals
in the system are equipped with multiple antennas. Considering static and
fast-fading multiple-input multiple-output (MIMO) channels, the
energy-efficiency is defined as the transmission rate to power ratio and shown
to be maximized at low transmit power. The most interesting case is the one of
slow fading MIMO channels. For this type of channels, the optimal precoding
scheme is generally not trivial. Furthermore, using all the available transmit
power is not always optimal in the sense of energy-efficiency (which, in this
case, corresponds to the communication-theoretic definition of the
goodput-to-power (GPR) ratio). Finding the optimal precoding matrices is shown
to be a new open problem and is solved in several special cases: 1. when there
is only one receive antenna; 2. in the low or high signal-to-noise ratio
regime; 3. when uniform power allocation and the regime of large numbers of
antennas are assumed. A complete numerical analysis is provided to illustrate
the derived results and stated conjectures. In particular, the impact of the
number of antennas on the energy-efficiency is assessed and shown to be
significant.
