  On a single-input-single-out (SISO) interference channel (IC), conventional
non-cooperative strategies encourage players selfishly maximizing their
transmit data rates, neglecting the deficit of performance caused by and to
other players. In the case of proper complex Gaussian noise, the maximum
entropy theorem shows that the best-response strategy is to transmit with
proper signals (symmetric complex Gaussian symbols). However, such equilibrium
leads to degrees-of-freedom zero due to the saturation of interference.
  With improper signals (asymmetric complex Gaussian symbols), an extra freedom
of optimization is available. In this paper, we study the impact of improper
signaling on the 2-user SISO IC. We explore the achievable rate region with
non-cooperative strategies by computing a Nash equilibrium of a non-cooperative
game with improper signaling. Then, assuming cooperation between players, we
study the achievable rate region of improper signals. We propose the usage of
improper rank one signals for their simplicity and ease of implementation.
Despite their simplicity, rank one signals achieve close to optimal sum rate
compared to full rank improper signals. We characterize the Pareto boundary,
the outer-boundary of the achievable rate region, of improper rank one signals
with a single real-valued parameter; we compute the closed-form solution of the
Pareto boundary with the non-zero-forcing strategies, the maximum sum rate
point and the max-min fairness solution with zero-forcing strategies. Analysis
on the extreme SNR regimes shows that proper signals maximize the wide-band
slope of spectral efficiency whereas improper signals optimize the high-SNR
power offset.
