  This paper considers an unlicensed multiple-access channel (MAC) that
coexists with a licensed point-to-point user, following the underlay cognitive
radio paradigm. We assume that every transceiver except the secondary base
station has one antenna, and that the primary user (PU) is protected by a
minimum rate constraint. In contrast to the conventional assumption of proper
Gaussian signaling, we allow the secondary users (SUs) to transmit improper
Gaussian signals, which are correlated with their complex conjugate. When the
secondary base station performs zero-forcing, we show that improper signaling
is optimal if the sum of the interference channel gains (in an equivalent
canonical model) is above a certain threshold. Additionally, we derive an
efficient algorithm to compute the transmission parameters that attain the rate
region boundary for this scenario. The proposed algorithm exploits a
single-user representation of the secondary MAC along with new results on the
optimality of improper signaling in the single-user case when the PU is
corrupted by improper noise.
