  Motivated by recent developments in wireless power transfer, we study
communication with a remotely powered transmitter. We propose an
information-theoretic model where a charger can dynamically decide on how much
power to transfer to the transmitter based on its side information regarding
the communication, while the transmitter needs to dynamically adapt its coding
strategy to its instantaneous energy state, which in turn depends on the
actions previously taken by the charger. We characterize the capacity as an
$n$-letter mutual information rate under various levels of side information
available at the charger. When the charger is finely tunable to different
energy levels, referred to as a "precision charger", we show that these
expressions reduce to single-letter form and there is a simple and intuitive
joint charging and coding scheme achieving capacity. The precision charger
scenario is motivated by the observation that in practice the transferred
energy can be controlled by simply changing the amplitude of the beamformed
signal. When the charger does not have sufficient precision, for example when
it is restricted to use a few discrete energy levels, we show that the
computation of the $n$-letter capacity can be cast as a Markov decision process
if the channel is noiseless. This allows us to numerically compute the capacity
for specific cases and obtain insights on the corresponding optimal policy, or
even to obtain closed form analytical solutions by solving the corresponding
Bellman equations, as we demonstrate through examples. Our findings provide
some surprising insights on how side information at the charger can be used to
increase the overall capacity of the system.
