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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, the authors 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 co-variances only, and an additional term, which is a function of both the users' co-variances and pseudo-co-variances. 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.
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