  The capacity of the two-user Gaussian cognitive interference channel, a
variation of the classical interference channel where one of the transmitters
has knowledge of both messages, is known in several parameter regimes but
remains unknown in general. In this paper, we consider the following achievable
scheme: the cognitive transmitter pre-codes its message against the
interference created at its intended receiver by the primary user, and the
cognitive receiver only decodes its intended message, similar to the optimal
scheme for "weak interference"; the primary decoder decodes both messages,
similar to the optimal scheme for "very strong interference". Although the
cognitive message is pre-coded against the primary message, by decoding it, the
primary receiver obtains information about its own message, thereby improving
its rate. We show: (1) that this proposed scheme achieves capacity in what we
term the "primary decodes cognitive" regime, i.e., a subset of the "strong
interference" regime that is not included in the "very strong interference"
regime for which capacity was known; (2) that this scheme is within one
bit/s/Hz, or a factor two, of capacity for a much larger set of parameters,
thus improving the best known constant gap result; (3) we provide insights into
the trade-off between interference pre-coding at the cognitive encoder and
interference decoding at the primary receiver based on the analysis of the
approximate capacity results.
