# Information-Theoretic Analysis of a Family of Improper Discrete Constellations
## 4 Information-Theoretic Analysis  
### Introduction  
To assess the performance of the proposed family of improper constellations \(\mathcal{A}(\phi,\kappa,M)\) in coded communication systems, one has to check whether the maximum reliable communication rates achievable by channel encoders designed with coded symbols in these constellations are close to the rates dictated by the improper Gaussian capacity (8). To that end, let us define the mutual information envelope, denoted by \(I^{*}(\mathrm{snr},\kappa)\), as follows. Given \(\kappa\) and \(M\), we obtain for each snr the optimal phase \(\hat{\phi}(\kappa,\mathrm{snr},M)\) that maximizes the mutual information \(I(Y,V)\). By repeating this optimization process for each \(M\), a family of mutual information curves is obtained. \(I^{*}(\mathrm{snr},\kappa)\) is now obtained as the envelope of these curves, that is,  
\[I^{*}(\mathrm{snr},\kappa)=\max_{M}\{I(Y;V):V\in\mathcal{A}(\hat{\phi}(\kappa, \mathrm{snr},M),\kappa,M)\}.\]  
The closer \(I^{*}(\mathrm{snr},\kappa)\) is to the improper Gaussian capacity \(C(\kappa,\mathrm{snr})\), the better the proposed family of constellations will be. A way of quantifying performance is to obtain the signal-to-noise ratio gap, \(\Delta(\mathrm{snr},\kappa)\):  
\[C(\kappa,\mathrm{snr})=I^{*}(\mathrm{snr}+\Delta(\mathrm{snr},\kappa),\kappa).\]  
Although the analytical derivation of \(\Delta(\mathrm{snr},\kappa)\) looks intractable, useful insights into the behavior of \(\Delta(\mathrm{snr},\kappa)\) versus snr can be gained by analyzing in the high- and low-snr regimes the family of mutual information values, \(\{I(Y;V):V\in\mathcal{A}(\hat{\phi}(\kappa,\mathrm{snr},M),\kappa,M),\text{ for all }M\text{ and snr values}\}\). This is studied in the following sections.  
Section 4.2 shows that at low snr, the behavior of \(C(\kappa,\mathrm{snr})\) and \(I^{*}(\mathrm{snr},\kappa)\) is the same up to the second order in snr. In Section 4.3, we show that at high snr, the optimal phase \(\hat{\phi}(\kappa,\mathrm{snr},M)\) that maximizes the mutual information \(I(Y;V)\) with \(V\in\mathcal{A}(\phi,\kappa,M)\) does not depend on \(M\) and is given by \(\phi^{*}(\kappa)\), i.e., the phase that maximizes the minimum distance of the lattice \(\Lambda(\phi,\kappa)\) (refer to Proposition 1). Observe that maximizing the mutual information in this context is equivalent to minimizing the BER for uncoded systems.  
In Section 4.4, we compute the shaping loss of the proposed family of constellations \(\mathcal{A}(\phi,\kappa,M)\) with respect to improper Gaussian signaling. It is shown that this loss is 1.53 dB, for all \(\phi\) and \(\kappa\). Although this result can easily be checked for rectangular QAM improper constellations (when \(\phi=0\)), it has to be proven for any other value of \(\phi\in(0,\pi/2]\), which generates non-rectangular constellations.  
Section 5 shows by computer simulation that the gap, \(\Delta(\mathrm{snr},\kappa)\), increases monotonically with the snr up to a saturation value of 1.53 dB. Since at low snr, the value of \(\Delta(\mathrm{snr},\kappa)\) is negligible, we conclude that the only loss in rate is due to shaping. In this sense, the proposed family of constellations can be viewed as the improper counterpart of the standard proper \(M\)-QAM constellations.