  This paper studies the achievable rates of Gaussian interference channels
with additive white Gaussian noise (AWGN), when improper or circularly
asymmetric complex Gaussian signaling is applied. For the Gaussian
multiple-input multiple-output interference channel (MIMO-IC) with the
interference treated as Gaussian noise, we show that the user's achievable rate
can be expressed as a summation of the rate achievable by the conventional
proper or circularly symmetric complex Gaussian signaling in terms of the
users' transmit covariance matrices, and an additional term, which is a
function of both the users' transmit covariance and pseudo-covariance matrices.
The additional degrees of freedom in the pseudo-covariance matrix, which is
conventionally set to be zero for the case of proper Gaussian signaling,
provide an opportunity to further improve the achievable rates of Gaussian
MIMO-ICs by employing improper Gaussian signaling. To this end, this paper
proposes widely linear precoding, which efficiently maps proper
information-bearing signals to improper transmitted signals at each transmitter
for any given pair of transmit covariance and pseudo-covariance matrices. In
particular, for the case of two-user Gaussian single-input single-output
interference channel (SISO-IC), we propose a joint covariance and
pseudo-covariance optimization algorithm with improper Gaussian signaling to
achieve the Pareto-optimal rates. By utilizing the separable structure of the
achievable rate expression, an alternative algorithm with separate covariance
and pseudo-covariance optimization is also proposed, which guarantees the rate
improvement over conventional proper Gaussian signaling.
