  We analyze coding for energy efficiency in relay channels at a fixed source
rate. We first propose a half-duplex decode-forward coding scheme for the
Gaussian relay channel. We then derive three optimal sets of power allocation,
which respectively minimize the network, the relay and the source energy
consumption. These optimal power allocations are given in closed-form, which
have so far remained implicit for maximum-rate schemes. Moreover, analysis
shows that minimizing the network energy consumption at a given rate is not
equivalent to maximizing the rate given energy, since it only covers part of
all rates achievable by decode-forward. We thus combine the optimized schemes
for network and relay energy consumptions into a generalized one, which then
covers all achievable rates. This generalized scheme is not only energy-optimal
for the desired source rate but also rate-optimal for the consumed energy. The
results also give a detailed understanding of the power consumption regimes and
allow a comprehensive description of the optimal message coding and resource
allocation for each desired source rate and channel realization. Finally, we
simulate the proposed schemes in a realistic environment, considering path-loss
and shadowing as modelled in the 3GPP standard. Significant energy gain can be
obtained over both direct and two-hop transmissions, particularly when the
source is far from relay and destination.
