  It has been shown that improper Gaussian signaling (IGS) can improve the
performance of wireless interference-limited systems when perfect channel state
information (CSI) is available. In this paper, we investigate the robustness of
IGS against imperfect CSI on the transmitter side in a two-user single-input
single-output (SISO) interference channel (IC) as well as in a SISO Z-IC, when
interference is treated as noise. We assume that the true channel coefficients
belong to a known region around the channel estimates, which we call the
uncertainty region. Following a worst-case robustness approach, we study the
rate-region boundary of the IC for the worst channel in the uncertainty region.
For the two-user IC, we derive a robust design in closed-form, which is
independent of the phase of the channels by allowing only one of the users to
transmit IGS. For the Z-IC, we provide a closed-form design for the
transmission parameters by considering an enlarged uncertainty region and
allowing both users to employ IGS. In both cases, the IGS-based designs are
ensured to perform no worse than proper Gaussian signaling. Furthermore, we
show, through numerical examples, that the proposed robust designs
significantly outperform non-robust solutions.
