Improper Gaussian signaling (IGS) scheme has been recently shown to provide performance improvements in interference-limited networks as opposed to the conventional proper Gaussian signaling (PGS) scheme. In this paper, we implement the IGS scheme in an overlay cognitive radio system, where the secondary transmitter broadcasts a mixture of two different signals. The first signal is selected from the PGS scheme to match the primary message transmission. On the other hand, the second signal is chosen to be from the IGS scheme in order to reduce the interference effect on the primary receiver. We then optimally design the overlay cognitive radio to maximize the secondary link achievable rate while satisfying the primary network quality of service requirements. In particular, we consider full and partial channel knowledge scenarios and derive the feasibility conditions of operating the overlay cognitive radio systems. Moreover, we derive the superiority conditions of the IGS schemes over the PGS schemes supported with closed-form expressions for the corresponding power distribution and the circularity coefficient and parameters. Simulation results are provided to support our theoretical derivations.