# Information-Theoretic Analysis of a Family of Improper Discrete Constellations
## 4 Information-Theoretic Analysis
### Asymptotic Results at High SNR  
The optimum \(\hat{\phi}\) that maximizes \(\operatorname{I}(\sqrt{\operatorname{snr}}\,V+N;V)\) depends in general on \(M\), \(\kappa\) and the \(\operatorname{snr}\). The analytical derivation of \(\hat{\phi}(\operatorname{snr},\kappa,M)\) for all \(\operatorname{snr}\) appears to be intractable in general. However, at high \(\operatorname{snr}\), the optimal phase depends only on \(\kappa\) and is given by the following theorem.  
**Theorem 4**.: _At high \(\operatorname{snr}\), the optimal \(\hat{\phi}\) that maximizes \(\operatorname{I}(\sqrt{\operatorname{snr}}\,V+N;V)\) over \(0\leq\phi\leq\frac{\pi}{2}\) is equal to \(\phi^{*}(\kappa)\) in Expression (20) and is given by:_  
\[\lim_{\operatorname{snr}\to\infty}\hat{\phi}(\operatorname{snr},\kappa,M)=\phi ^{*}(\kappa)=\begin{cases}\pi/2,&0<\kappa\leq 0.5,\\ \phi^{*},&0.5<\kappa\leq 1,\end{cases} \tag{30}\]  
_where \(\phi^{*}\) is the solution of the equation \(2\sin(\phi)-\cos(\phi)=1/\kappa\)._Proof.: In order to proceed, we use the fact [26] that for any discrete constellation, the mutual information, the minimum mean-squared error (mmse) and the symbol error probability all have an asymptotic behavior proportional to \(Q\left(\sqrt{\text{s}\text{n}}d_{min}/2\right)\), where \(Q(\cdot)\) denotes the Gaussian Q-function and \(d_{min}\) is the minimum Euclidean distance of the constellation.  
In Proposition 1, we showed that the value of \(\phi^{*}(\kappa)\) that maximizes the minimum distance of the family of constellations \(\mathcal{A}(\phi,\kappa,M)\) is given by:  
\[\phi^{*}(\kappa)=\arg\max_{\phi\in[0,\pi/2]}d_{min}(\Lambda(\phi,\kappa)) \tag{31}\]  
where \(\Lambda(\phi,\kappa)\) is the lattice that results from the \(\mathbf{H}(\phi,\kappa)\)-transformation of the integer lattice \(Z^{2}\) (refer to (18)). From (18) and exploiting the fact that \(0\leq\phi\leq\pi/2\), it can be computed that:  
\[d_{min}^{2}=\min\{\left(1-\kappa\cos(\phi)\right),2\left(1-\kappa\sin(\phi) \right)\},\]  
which can be obtained from the real representation given by (17) when the signal points of the original proper constellation belong to a regular square lattice. It is easy to check that \(0\leq\kappa\leq 0.5\Rightarrow\left(1-\kappa\cos(\phi)\right)\leq 2\left(1- \kappa\sin(\phi)\right)\). This, in turn, means that the optimal solution of (31) when \(0\leq\kappa\leq 0.5\) is \(\phi^{*}(\kappa)=\pi/2\). On the other hand, when \(\kappa>0.5\), the optimal phase satisfies the nonlinear equation:  
\[\left(1-\kappa\cos(\phi)\right)=2\left(1-\kappa\sin(\phi)\right)\Leftrightarrow 2 \sin(\phi)-\cos(\phi)=\frac{1}{\kappa}, \tag{32}\]  
which proves the theorem.  
**Remark 5**.: _For \(\kappa=1\) (maximally improper, rectilinear, constellations), the optimal phase in Equation (30) is \(\phi^{*}\approx 0.9273\). This value results in a rectilinear constellation with equidistant points, which would be equivalent to a rotated M-ary pulse amplitude modulation (PAM). As an example, starting from a standard QPSK constellation and using \(\kappa=1\) and \(\phi^{*}\approx 0.9273\) in (16), we obtain a unit-norm rectilinear constellation with equidistant signal points:_  
\[V=\left[1.2+0.6i\quad 0.4+0.2i\quad-0.4-0.2i\quad-1.2-0.6i\right].\]