# Information-Theoretic Analysis of a Family of Improper Discrete Constellations
## 4 Information-Theoretic Analysis
### Asymptotic Results at Low SNR  
The following theorem characterizes the mutual information at low snr up to the second order. **Theorem 2**: _Let \(V\) be a discrete improper random variable with circularity coefficient \(0\leq\kappa\leq 1\) taking values in \(\mathcal{A}(\phi,\kappa,M)\) and constructed as in (16) with \(X\) being a square proper \(M\)-QAM constellation. Then, the mutual information between \(Y\) and \(V\) in (5) admits the following second-order expansion around \(\mathrm{snr}=0\):_  
\[\mathrm{I}(\sqrt{\mathrm{snr}}\,V+N;V)=\mathrm{snr}-\frac{1}{2}(1+\kappa^{2}) \mathrm{snr}^{2}+O(\mathrm{snr}^{3}). \tag{21}\]  
From Expression (17), the random variable \(V=V_{I}+jV_{Q}\) can be expressed in terms of real variables as:  
\[\underbrace{ \begin{bmatrix}V_{I}\\ V_{Q}\end{bmatrix} }_{\mathbf{v}}=\underbrace{ \begin{bmatrix}\sqrt{0.5(1+\alpha) }+\sqrt{0.5(1-\alpha)}\cos(\phi)&\sqrt{0.5(1-\alpha)}\sin(\phi)\\ \sqrt{0.5(1-\alpha)}\sin(\phi)&\sqrt{0.5(1+\alpha)}-\sqrt{0.5(1-\alpha)}\cos( \phi)\end{bmatrix} }_{\mathbf{H}(\phi,\kappa)}\underbrace{ \begin{bmatrix}X_{I} \\ X_{Q}\end{bmatrix} }_{\mathbf{x}}, \tag{22}\]  
where \(\alpha=\sqrt{1-\kappa^{2}}\) and \(X_{I}+jX_{Q}\) is a proper uniformly-distributed \(M\)-QAM random variable. Then, the complex-valued AWGN channel reduces to the real-valued vector channel:  
\[\mathbf{y}=\mathbf{H}(\phi,\kappa)\mathbf{x}+\mathbf{n}, \tag{23}\]  
where \(\mathbf{y}=(Y_{I},Y_{Q})^{T}\) and \(\mathbf{n}=(N_{I},N_{Q})^{T}\), with \(N_{I}\) and \(N_{Q}\) being independent real Gaussian random variables with zero mean and variance \(\frac{1}{2}\). We exploit the known relationship between mmse and mutual information (MI) [31], namely,  
\[\frac{d}{d\,\mathrm{snr}}\mathrm{I}(\sqrt{\mathrm{snr}})=\mathrm{mmse}(\sqrt{ \mathrm{snr}}), \tag{24}\]  
and use a result in [31] showing that the mmse of a real-valued vector channel can be expanded around \(\mathrm{snr}=0\) up to the first order as (notice that the factor \(1/2\) in the expression is because, according to our formulation, \(E[\mathbf{x}\mathbf{x}^{T}]=\frac{1}{2}\mathbf{I}\)):  
\[\mathrm{mmse}(\mathrm{snr},\phi,\kappa)=\frac{1}{2}\mathrm{tr}(\mathbf{H}( \phi,\kappa)\mathbf{H}(\phi,\kappa)^{T})-\frac{1}{2}\mathrm{snr}\cdot\mathrm{ tr}((\mathbf{H}(\phi,\kappa)\mathbf{H}(\phi,\kappa)^{T})^{2})+O(\mathrm{snr}^{2}). \tag{25}\]  
Since the matrix \(\mathbf{H}(\phi,\kappa)\) is real and symmetric, it is always diagonalizable as:  
\[\mathbf{H}(\phi,\kappa)=\mathbf{Q}\Lambda\mathbf{Q}^{T},\]  
where \(\mathbf{Q}\) is an orthogonal matrix (\(\mathbf{Q}\mathbf{Q}^{T}=\mathbf{Q}^{T}\mathbf{Q}=\mathbf{I}\)) and:  
\[\Lambda=\mathrm{diag}\{\lambda_{1}(\phi,\kappa),\lambda_{2}(\phi,\kappa)\},\]  
with \(\lambda_{j}(\phi,\kappa),j=1,2\), being its real eigenvalues. It is easy to check that:  
\[\lambda_{1}(\phi,\kappa)=\sqrt{0.5(1+\alpha)}-\sqrt{0.5(1-\alpha)},\]  
\[\lambda_{2}(\phi,\kappa)=\sqrt{0.5(1+\alpha)}+\sqrt{0.5(1-\alpha)}.\]Note that these eigenvalues do not depend on \(\phi\). By the properties of the trace operator \(\operatorname{tr}(\mathbf{H}(\phi,\kappa)\mathbf{H}(\phi,\kappa)^{T})=\lambda_{1}^ {2}(\phi,\kappa)+\lambda_{2}^{2}(\phi,\kappa)=2\) and \(\operatorname{tr}((\mathbf{H}(\phi,\kappa)\mathbf{H}(\phi,\kappa)^{T})^{2})= \lambda_{1}^{4}(\phi,\kappa)+\lambda_{2}^{4}(\phi,\kappa)=2(1+\kappa^{2})\). Therefore, Expression (25) reduces to:  
\[\operatorname{mmse}(\operatorname{snr},\kappa)=1-(1+\kappa^{2})\operatorname{ snr}+O(\operatorname{snr}^{2}), \tag{26}\]  
and using the relationship (24), we obtain:  
\[\operatorname{I}(\sqrt{\operatorname{snr}}\,V+N;V)=\operatorname{snr}-\frac{1} {2}(1+\kappa^{2})\operatorname{snr}^{2}+O(\operatorname{snr}^{3}), \tag{27}\]  
as we wanted to prove. **Remark 2**. : _Theorem 2 is an extension to improper signaling (and up to the second-order term) of the well-known result that states that at low \(\operatorname{snr}\), the proper Gaussian capacity behaves linearly with \(\operatorname{snr}\)._  
**Remark 3**.