  The capacity of the Gaussian cognitive interference channel, a variation of
the classical two-user interference channel where one of the transmitters
(referred to as cognitive) has knowledge of both messages, is known in several
parameter regimes but remains unknown in general. In this paper we provide a
comparative overview of this channel model as we proceed through our
contributions: we present a new outer bound based on the idea of a broadcast
channel with degraded message sets, and another series of outer bounds obtained
by transforming the cognitive channel into channels with known capacity. We
specialize the largest known inner bound derived for the discrete memoryless
channel to the Gaussian noise channel and present several simplified schemes
evaluated for Gaussian inputs in closed form which we use to prove a number of
results. These include a new set of capacity results for the a) "primary
decodes cognitive" regime, a subset of the "strong interference" regime that is
not included in the "very strong interference" regime for which capacity was
known, and for the b) "S-channel" in which the primary transmitter does not
interfere with the cognitive receiver. Next, for a general Gaussian cognitive
interference channel, we determine the capacity to within one bit/s/Hz and to
within a factor two regardless of channel parameters, thus establishing rate
performance guarantees at high and low SNR, respectively. We also show how
different simplified transmission schemes achieve a constant gap between inner
and outer bound for specific channels. Finally, we numerically evaluate and
compare the various simplified achievable rate regions and outer bounds in
parameter regimes where capacity is unknown, leading to further insight on the
capacity region of the Gaussian cognitive interference channel.
