  In this letter we study the potential benefits of improper signaling for a
secondary user (SU) in underlay cognitive radio networks. We consider a basic
yet illustrative scenario in which the primary user (PU) always transmit proper
Gaussian signals and has a minimum rate constraint. After parameterizing the SU
transmit signal in terms of its power and circularity coefficient (which
measures the degree of impropriety), we prove that the SU improves its rate by
transmitting improper signals only when the ratio of the squared modulus
between the SU-PU interference link and the SU direct link exceeds a given
threshold. As a by-product of this analysis, we obtain the optimal circularity
coefficient that must be used by the SU depending on its power budget. Some
simulation results show that the SU benefits from the transmission of improper
signals especially when the PU is not highly loaded.
