# Information-Theoretic Analysis of a Family of Improper Discrete Constellations
## 5 Numerical Results
### Capacity Curves  
Figure 7 plots the mutual information achieved by improper \(M\)-QAM signals with \(\kappa=0.95\) when the optimum \(\phi^{*}=0.9538\) at high snr is used (black solid lines), in comparison to the case when we use fixed \(\phi=\pi/2\) (dashed blue lines) or \(\phi=0\) (red dashed lines). We consider three increasing values of the constellation size: \(M=16\), 64 and 144. The constellation with \(\phi=0\) is a rectangular constellation with uncorrelated real and imaginary parts, but with a power imbalance to achieve the desired circularity coefficient. We observe that this naive family \(\mathcal{A}(\phi=0,\kappa,M)\) of constellations attains lower MI than using the optimum \(\phi^{*}=0.9538\) given by Theorem 4. This rectangular constellation is also outperformed by the constellation generated with \(\phi=\pi/2\). The figure also depicts the Gaussian capacity curve with improper signaling and the envelope of the capacity curves with \(M\)-QAM constellations for \(M\) sufficiently large. The distance between both curves is the shaping gain, which as proven in Theorem 5 is 1.53 dB. It should be mentioned that, since the shaping loss does not depend on \(\phi\), the mutual information envelope \(I^{*}(\mathrm{snr},\kappa)\) obtained from the set of constellations with \(\phi=0\), \(\phi=\pi/2\) or with the optimum \(\phi^{*}\) are all the same as \(M\) increases. However, for a given snr and \(\kappa\), the minimum \(M\) that makes \(I(Y;V)\approx I^{*}(\mathrm{snr},\kappa)\), will be smaller when \(\phi=\phi^{*}(\kappa)\) than when \(\phi=0\). That is, \(\phi=0\) needs a larger constellation expansion ratio [30] to achieve the same rate. Figure 6: \(\phi_{opt}\) versus snr in dBs for different values of \(\kappa\). The situation is different for maximally improper signals (\(\kappa=1\)) (i.e., rectilinear constellations). In this case, using \(\phi=\pi/2\) makes some signal points of the original proper constellation collapse into the same symbol, thus losing rate at high snr. This effect becomes evident in Figure 8, which plots the mutual information achieved by the proposed improper constellation with \(\kappa=1\), and \(M=4\) and \(M=16\) signal points, for \(\phi=\pi/2\) and for the optimal \(\phi^{*}=0.9273\) at high snr. From a practical point of view, we advocate the use of the optimal \(\phi^{*}\) given in (30), which maximizes the minimum distance between constellation points. Finally, Figure 9 depicts the capacity curves for proper and improper (with \(\kappa=0.95\)) \(M\)-QAM discrete constellations. For the improper \(M\)-QAM, the optimal \(\phi^{*}=0.9538\) at high snr is used. The proper and improper Gaussian capacity is also depicted for comparison. From these results, the following conclusions are drawn:  
1. For low-rate transmissions (\(M=4\) or \(M=16\)), there is no rate loss with respect to the improper Gaussian capacity when using coded symbols uniformly drawn from the proposed family of improper constellations. Therefore, like in the proper case, the proposed improper QPSK and 16-QAM constellations are optimal in the wideband regime [32]. 2. For high-rate transmissions (\(M>16\)), the proposed family of improper constellations incurs a shaping loss of up to 1.53 dB. As in the case of proper \(M\)-QAM signaling, this snr loss can be reduced by designing codes with some shaping mechanism to make the coded symbols in \(\mathcal{A}(\phi,\kappa,M)\) more Gaussian-like distributed. 3. The optimal \(\phi^{*}\), derived at high snr, can be safely used without any observable degradation in \(\Delta(\mathrm{snr},\kappa)\), independently of the value of \(M\) and snr. Therefore, we arrive at the main conclusion of the paper, namely that the proposed family of improper constellations is a good candidate to be used in communication systems with improper signaling. In fact, it can be viewed as the improper counterpart of the standard proper \(M\)-QAM constellations.