# Information-Theoretic Analysis of a Family of Improper Discrete Constellations
## 2 Background and System Model
### A Motivating Example  
In this subsection, we present a motivating example for the study conducted in this paper. Let us consider an underlay cognitive radio (UCR) system transmitting Gaussian codewords in which one single-antenna secondary user (SU) wishes to access the channel in the presence of a single-antenna primary user (PU) as shown in Figure 1, where the top and bottom links represent the PU and the SU, respectively. Without loss of generality, we consider the standard or canonical form of this scenario for which the direct channels are normalized to one, and the noise at both receivers is Gaussian and proper with zero mean and unit variance. This scenario is known in the information theory literature as the Z (or one-sided) interference channel [28]. In the UCR paradigm, the unlicensed SU is allowed to coexist with the PU as long as they ensure a minimum transmission rate \(\tilde{R}_{PU}\geq\xi\log(1+p)\) for the PU, where \(\xi\in[0,1]\) is the loading factor. That is, the transmit power and signaling scheme of the SU must be selected to control the interference level at the primary receiver such that the PU's rate requirement is guaranteed. Since the PU is typically unaware of the SU, it is also assumed that the PU always transmits proper Gaussian signals, whereas the SU may transmit either proper or improper Gaussian signals depending on which signaling scheme  
Figure 1: Underlay cognitive radio scenario with a secondary user that may transmit improper Gaussian signals. performs better. This problem has been analyzed in [7], where it has been proven that the SU improves its rate by transmitting Gaussian improper signals if and only if (cf. Theorem 1 in [7]):  
\[a\geq\beta=1-\frac{p}{(1+p)^{2\xi}-1}, \tag{4}\]  
where \(p\) is the power transmitted by the PU. Furthermore, Theorem 1 in [7] also gives us the optimal circularity coefficient \(\kappa_{s}\) that must be used by the SU depending on its power budget. For example, let us suppose that the cross-link channel is \(a=4\); the power budgets for the PU and SU are 100 and 20, respectively; and the loading factor is \(\xi=0.4\). With these parameters \(\beta=-1.56\) and, since \(a\geq\beta\), improper signaling with \(\kappa_{s}=0.96\) (close to maximally improper) improves the rate of the SU. Specifically, the rate achieved by the SU using improper Gaussian codewords is \(R_{SU}=3.09\) b/s/Hz, whereas using proper Gaussian codewords, the rate would be only \(R_{SU}=2.44\) b/s/Hz. As we have already argued, practical constraints dictate the use of discrete constellations instead of Gaussian signals. Nevertheless, the previous analysis with Gaussian signals is still valuable as an upper bound in the sense that any finite constellation with the same circularity coefficient will have lower differential entropy than a Gaussian and, consequently, will be less harmful for the PU. Therefore, in our example, for any improper discrete constellation, we might transmit with \(\kappa_{s}=0.96\), and the rate requirement of the PU would still be satisfied. When the PU transmits Gaussian proper signals and the interference at the SU is treated as noise, the SU link is an AWGN channel over which we have to send improper digital signals with a given \(\kappa_{s}\). In a more realistic scenario, the PU would also transmit a discrete non-Gaussian constellation. In this case, the interference power tolerated by the PU could potentially vary with respect to Gaussian signaling, and the SU would receive non-Gaussian interference, as well. While all Gaussian signals with a given circularity coefficient are identical in terms of performance, this is not the case with our family of discrete constellations. As we will see shortly, with discrete improper constellations, we have one degree of freedom to optimize the SU performance. In this work, we study a family of improper constellations and design its parameters to maximize the mutual information over the SU link modeled as an AWGN channel.