Inner and Outer Bounds for the Gaussian Cognitive Interference Channel and New Capacity Results

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. This paper provides a comparative overview of this channel model as it proceeds through the following contributions. First, several outer bounds are presented: a) a new outer bound based on the idea of a broadcast channel with degraded message sets, and b) an outer bound obtained by transforming the channel into channels with known capacity. Next, a compact Fourier-Motzkin eliminated version of the largest known inner bound derived for the discrete memoryless cognitive interference channel is presented and specialized to the Gaussian noise case, where several simplified schemes with jointly Gaussian input are evaluated in closed form and later used to prove a number of results. These include a new set of capacity results for: a) the "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 b) the "S-channel in strong interference" in which the primary transmitter does not interfere with the cognitive receiver and the primary receiver experiences strong interference. Next, for a general Gaussian channel the capacity is determined to within one bit/s/Hz and to within a factor two regardless of the channel parameters, thus establishing rate performance guarantees at high and low SNR, respectively. The paper concludes with numerical evaluations and comparisons of the various simplified achievable rate regions and outer bounds in parameter regimes where capacity is unknown, leading to further insight on the capacity region.