  This paper studies the achievable rate region of the two-user
single-input-single-output (SISO) Gaussian interference channel, when the
improper Gaussian signaling is applied. Under the assumption that the
interference is treated as additive Gaussian noise, we show that the user's
achievable rate can be expressed as a summation of the rate achievable by the
conventional proper Gaussian signaling, which depends on the users' input
covariances only, and an additional term, which is a function of both the
users' covariances and pseudo-covariances. The additional degree of freedom
given by the pseudo-covariance, which is conventionally set to be zero for the
case of proper Gaussian signaling, provides an opportunity to improve the
achievable rate by employing the improper Gaussian signaling. Since finding the
optimal solution for the joint covariance and pseudo-covariance optimization is
difficult, we propose a sub-optimal but efficient algorithm by separately
optimizing these two sets of parameters. Numerical results show that the
proposed algorithm provides a close-to-optimal performance as compared to the
exhaustive search method, and significantly outperforms the optimal proper
Gaussian signaling and other existing improper Gaussian signaling schemes.
