  Sharing the spectrum with in-band full-duplex (FD) primary users (PU) is a
challenging and interesting problem in the underlay cognitive radio (CR)
systems. The self-interference introduced at the primary network may
dramatically impede the secondary user (SU) opportunity to access the spectrum.
To tackle this problem, we use the so-called improper Gaussian signaling.
Particularly, we assume a system with a SU pair working in a half-duplex mode
that uses improper Gaussian signaling while the FD PU pair implements the
regular proper Gaussian signaling. First, we derive a closed form expression
and an upper bound for the SU and PU outage probabilities, respectively.
Second, we optimize the SU signal parameters to minimize its outage probability
while maintaining the required PU quality-of-service based on the average
channel state information (CSI). Moreover, we provide the conditions to reap
merits from employing improper Gaussian signaling at the SU. Third, we design
the SU signal parameters based on perfect knowledge of its direct link
instantaneous CSI and investigate all benefits that can be achieved at both the
SU and PU. Finally, we provide some numerical results that demonstrate the
advantages of using improper Gaussian signaling to access the spectrum of the
FD PU.
