  This paper studies the achievable rate region of the K-user Gaussian
multiple-input single-output interference channel (MISO-IC) with the
interference treated as noise, when improper or circularly asymmetric complex
Gaussian signaling is applied. The transmit optimization with improper Gaussian
signaling involves not only the signal covariance matrix as in the conventional
proper or circularly symmetric Gaussian signaling, but also the signal
pseudo-covariance matrix, which is conventionally set to zero in proper
Gaussian signaling. By exploiting the separable rate expression with improper
Gaussian signaling, we propose a separate transmit covariance and
pseudo-covariance optimization algorithm, which is guaranteed to improve the
users' achievable rates over the conventional proper Gaussian signaling. In
particular, for the pseudo-covariance optimization, we establish the optimality
of rank-1 pseudo-covariance matrices, given the optimal rank-1 transmit
covariance matrices for achieving the Pareto boundary of the rate region. Based
on this result, we are able to greatly reduce the number of variables in the
pseudo-covariance optimization problem and thereby develop an efficient
solution by applying the celebrated semidefinite relaxation (SDR) technique.
Finally, we extend the result to the Gaussian MISO broadcast channel (MISO-BC)
with improper Gaussian signaling or so-called widely linear transmit precoding.
