  This paper provides a complete characterization of the boundary of an
achievable rate region, called the Pareto boundary, of the single-antenna Z
interference channel (Z-IC), when interference is treated as noise and users
transmit complex Gaussian signals that are allowed to be improper. By
considering the augmented complex formulation, we derive a necessary and
sufficient condition for improper signaling to be optimal. This condition is
stated as a threshold on the interference channel coefficient, which is a
function of the interfered user rate and which allows insightful
interpretations into the behavior of the achievable rates in terms of the
circularity coefficient (i.e., degree of impropriety). Furthermore, the optimal
circularity coefficient is provided in closed form. The simplicity of the
obtained characterization permits interesting insights into when and how
improper signaling outperforms proper signaling in the single-antenna Z-IC. We
also provide an in-depth discussion on the optimal strategies and the
properties of the Pareto boundary.
