  We consider an energy harvesting channel, in which the transmitter is powered
by an exogenous stochastic energy harvesting process $E_t$, such that $0\leq
E_t\leq\bar{E}$, which can be stored in a battery of finite size $\bar{B}$. We
provide a simple and insightful formula for the approximate capacity of this
channel with bounded guarantee on the approximation gap independent of system
parameters. This approximate characterization of the capacity identifies two
qualitatively different operating regimes for this channel: in the large
battery regime, when $\bar{B}\geq \bar{E}$, the capacity is approximately equal
to that of an AWGN channel with an average power constraint equal to the
average energy harvesting rate, i.e. it depends only on the mean of $E_t$ and
is (almost) independent of the distribution of $E_t$ and the exact value of
$\bar{B}$. In particular, this suggests that a battery size
$\bar{B}\approx\bar{E}$ is approximately sufficient to extract the infinite
battery capacity of the system. In the small battery regime, when
$\bar{B}<\bar{E}$, we clarify the dependence of the capacity on the
distribution of $E_t$ and the value of $\bar{B}$. There are three steps to
proving this result which can be of interest in their own right: 1) we
characterize the capacity of this channel as an $n$-letter mutual information
rate under various assumptions on the availability of energy arrival
information; 2) we characterize the approximately optimal online power control
policy that maximizes the long-term average throughput of the system; 3) we
show that the information-theoretic capacity of this channel is equal, within a
constant gap, to its long-term average throughput. This last result provides a
connection between the information- and communication-theoretic formulations of
the energy-harvesting communication problem that have been so far studied in
isolation.
