# Information-Theoretic Analysis of a Family of Improper Discrete Constellations
## 2 Background and System Model
### Capacity Results for Gaussian Improper Signals
This means that we can always rotate the constellation \(V\) such that \(h_{1}\) is a real and positive parameter, whereas \(h_{2}\) is a complex value. With this restriction, \(h_{1}\) and \(h_{2}\) can be expressed as:  
\[h_{1} =\sqrt{\frac{1}{2}(1+\alpha)}, \tag{14}\] \[h_{2} =\sqrt{\frac{1}{2}(1-\alpha)}e^{i\phi}. \tag{15}\]  
Furthermore, the original proper \(M\)-QAM constellation \(X\) has a rotational symmetry of \(\pm n\pi/2\) for any integer \(n\), which means that \(\phi\) in (15) can be restricted to the interval \(\phi\in[0,\pi/2]\). Finally, the proposed family of improper constellations with circularity coefficient \(\kappa\) can be parameterized as:  
\[V=\sqrt{\frac{1}{2}\left(1+\alpha\right)}X+\sqrt{\frac{1}{2}\left(1-\alpha \right)}e^{i\phi}X^{*}, \tag{16}\]  
where \(\alpha=\sqrt{(1-\kappa^{2})}\), \(0\leq\kappa\leq 1\) and \(\phi\in[0,\pi/2]\). In what follows, to stress the dependency of the improper constellation on \(\kappa\), \(\phi\) and \(M\), we will denote this family of constellations as \(\mathcal{A}(\phi,\kappa,M)\). In terms of real variables, the transformation in (16) can be written as:  
\[ \begin{bmatrix}V_{I}\\ V_{Q}\end{bmatrix} =\underbrace{ \begin{bmatrix}\sqrt{0.5(1+\alpha)}+\sqrt{0.5(1- \alpha)}\cos(\phi)&\sqrt{0.5(1-\alpha)}\sin(\phi)\\ \sqrt{0.5(1-\alpha)}\sin(\phi)&\sqrt{0.5(1+\alpha)}-\sqrt{0.5(1-\alpha)}\cos( \phi)\end{bmatrix} }_{\mathbf{H}(\phi,\kappa)} \begin{bmatrix}X_{I}\\ X_{Q}\end{bmatrix} , \tag{17}\]  
where \(\mathbf{H}(\phi,\kappa)\) denotes the \(2\times 2\) transformation matrix. Observe that by setting \(\phi=0\) in (16), one obtains a rather naive rectangular QAM improper constellation, where its real and imaginary components are independent and have a power imbalance determined by the required circularity coefficient \(\kappa\). For any other \(\phi\
eq 0\) the constellation points are enclosed in a non-rectangular parallelogram. Figure 2 shows four improper constellations generated according to (16) for different values of \(\kappa\) and \(\phi\) when \(X\) is proper QPSK (\(M=4\)). For \(\kappa=1\) (maximally improper signal), in the right bottom panel, the points of the constellation lie on a line. Similarly, Figure 3 shows the constellation \(\mathcal{A}(\pi/2,0.7,256)\), where the shape of the constellation, a parallelogram, can be more clearly observed. We now provide an alternative geometric interpretation of the proposed improper constellations \(\mathcal{A}(\phi,\kappa,M)\). Consider the 2D integer lattice \(Z^{2}\subset\mathbb{R}^{2}\), consisting of all two-dimensional vectors \(\mathbf{z}=(a,b)^{T}\) with \(a\) and \(b\) integers. Then, the \(M\)-QAM constellation \(\frac{1}{2}[\pm 1,\pm 3,\ldots,\pm(\sqrt{M}-1)]^{2}\) can be obtained as the intersection of the translated lattice \(Z^{2}+(\frac{1}{2},\frac{1}{2})^{T}\) and the square region \(\mathcal{R}=[-\frac{\sqrt{M}}{2},\frac{\sqrt{M}}{2}]^{2}\subset\mathbb{R}^{2}\). If we apply the linear transformation \(\mathbf{H}(\phi,\kappa)\) to the lattice \(Z^{2}+(\frac{1}{2},\frac{1}{2})^{T}\), we obtain the new translated lattice \(\Lambda(\phi,\kappa)+\mathbf{H}(\phi,\kappa)(\frac{1}{2},\frac{1}{2})^{T}\), where:  
\[\Lambda(\phi,\kappa)=\{\lambda\in\mathbb{R}^{2}:\lambda=\mathbf{H}(\phi, \kappa)\mathbf{z},\ \forall\mathbf{z}\in Z^{2}\}. \tag{18}\]  
Reasoning as before, the unnormalized family of improper constellations \(\mathcal{A}(\phi,\kappa,M)\) can be obtained as the intersection of the translated lattice \(\Lambda(\phi,\kappa)+\mathbf{H}(\phi,\kappa)(\frac{1}{2},\frac{1}{2})^{T}\) and the parallelogram region \(\mathcal{P}\subset\mathbb{R}^{2}\) that results from the \(\mathbf{H}(\phi,\kappa)\)-transformation of a centered square with side length \(\sqrt{M}\), that is,  
\[\mathcal{A}(\phi,\kappa,M)=\left\{\Lambda(\phi,\kappa)+\mathbf{H}(\phi,\kappa )\left(\frac{1}{2},\frac{1}{2}\right)^{T}\right\}\cap\mathcal{P}. \tag{19}\]Figure 3: Improper constellation generated according to (16) for \(\kappa=0.7\) and \(\phi=\pi/2\) when \(X\) is proper 256-QAM (Quadrature Amplitude Modulation).