Improper Signaling Versus Time-Sharing in the MISO Z-Interference Channel

In the two-user Gaussian interference channel (IFC) with Gaussian inputs and treating interference as noise (TIN), it depends on the assumptions about time-sharing whether improper signals (asymmetric complex signals) can bring benefits over proper signals. If no time-sharing is allowed (pure strategies) or the convex hull formulation is used (only data rates can be averaged), improper signaling can lead to a larger rate region than optimal proper signaling. However, when coded time-sharing is allowed, i.e., both the data rates and the transmit powers can be averaged, the whole rate region can be achieved by proper signaling. Here, we show that the same is true in the multiple-input/single-output (MISO) Z-interference channel (ZIFC). We provide an analytical proof for the optimality of proper signaling in the case of coded time-sharing, and we numerically demonstrate gains by improper signaling in the cases without coded time-sharing.