# Information-Theoretic Analysis of a Family of Improper Discrete Constellations
## 2 Background and System Model
### Capacity Results for Gaussian Improper Signals  
We consider a complex AWGN channel:  
\[Y=g\sqrt{\mathrm{snr}}\;V+N, \tag{5}\]  
where \(N\sim\mathcal{CN}(0,1)\) is circular, \(V\) is the transmitted signal and the channel is a deterministic and known constant, which we take as \(g=1\) without loss of generality. The capacity (in bits per channel use) of this channel is [1]:  
\[C=\sup_{p_{V}(v)}\mathrm{I}(Y;V)=\sup_{p_{V}(v)}h(Y)-h(Y|V)=\sup_{p_{V}(v)}h(Y) -\log(\pi e), \tag{6}\]  
where the supremum is taken over all possible input distributions of \(V\) with \(E[|V|^{2}]=1\), and the logarithms have base two. It is well known that the supremum is achieved when \(V\sim\mathcal{CN}(0,1)\)[27]. In this paper, we consider the case where the input signal is unit-power improper with circularity coefficient \(\kappa\); then, the channel output is also improper Gaussian with variance \(E[|Y|^{2}]=1+\mathrm{snr}\) and circularity coefficient \(\kappa_{y}=\mathrm{rsnr}/(1+\mathrm{snr})\). By Theorem 1, the differential entropy of \(Y\) is:  
\[h(Y)=\log(\pi e(1+\mathrm{snr}))+\frac{1}{2}\log(1-\kappa_{y}^{2}), \tag{7}\]and the capacity is:  
\[C(\mathrm{snr},\kappa)=\log(1+\mathrm{snr})+\frac{1}{2}\log\left(1-\frac{\kappa^{ 2}}{(1+\mathrm{snr}^{-1})^{2}}\right)\ \ \mathrm{bits\ per\ complex\ dimension}. \tag{8}\]  
Observe that for \(0<\kappa\leq 1\), the second term in (8) is always negative, and therefore, there is a capacity loss with respect to proper Gaussian signaling. In Appendix A, we present a heuristic derivation of this formula based on a packing argument similar to [29]. **Remark 1**. : _As \(\mathrm{snr}\to\infty\), the term:_  
\[\frac{1}{2}\log\left(1-\frac{\kappa^{2}}{(1+\mathrm{snr}^{-1})^{2}}\right)\]  
_in (8) accounts for the loss in capacity due to enclosing the codewords in a multidimensional ellipsoid rather than in a hypersphere (as proper Gaussian signaling does) [30]._  
**3. A Family of Improper Constellations Generated by a Widely Linear Transformation**  
In this section, we propose a family of improper discrete constellations that are generated by a widely linear transformation of a standard \(M\)-QAM constellation, where \(M=4s^{2}\) with \(s\) a positive integer. To this end, let \(X=X_{I}+jX_{Q}\) be a symbol in a proper \(M\)-QAM constellation of unit power (\(E[|X|^{2}]=1\)). Then, we propose to generate the corresponding improper symbol as:  
\[V=V_{I}+jV_{Q}=h_{1}X+h_{2}X^{*}, \tag{9}\]  
where \(h_{1}=h_{1I}+jh_{1Q}\) and \(h_{2}=h_{2I}+jh_{2Q}\) are complex scalars such that \(|h_{1}|^{2}+|h_{2}|^{2}=1\) (to maintain unit transmit power). Alternatively, the widely linear precoding operation (9) can be expressed in terms of real variables as:  
\[ \begin{bmatrix}V_{I}\\ V_{Q}\end{bmatrix} = \begin{bmatrix}h_{1I}+h_{2I}&h_{2Q}-h_{1Q}\\ h_{1Q}+h_{2Q}&h_{1I}-h_{2I}\end{bmatrix}\begin{bmatrix}X_{I}\\ X_{Q}\end{bmatrix} . \tag{10}\]  
Under the constraint \(|h_{1}|^{2}+|h_{2}|^{2}=1\), the circularity coefficient of the generated symbol is:  
\[\kappa=\frac{|E[V^{2}]|}{E[|V|^{2}]}=2|h_{1}||h_{2}|. \tag{11}\]  
The question that now arises is: given \(\kappa\), what is the simplest parametrization of this family of improper constellations in terms of the complex variables \(h_{1}\) and \(h_{2}\)? All values of \(h_{1}\) and \(h_{2}\) producing a unit variance improper \(V\) with a given \(\kappa\) must satisfy:  
\[|h_{1}|^{2}|h_{2}|^{2} =\frac{\kappa^{2}}{4}, \tag{12}\] \[|h_{1}|^{2}+|h_{2}|^{2} =1.\]  
The nonlinear system of equations in (12) has a unique solution for the squared modulus \(|h_{1}|^{2}\) and \(|h_{2}|^{2}\), which can be written in terms of the variable \(\alpha=\sqrt{1-\kappa^{2}}\) as:  
\[|h_{1}|^{2} =\frac{1}{2}(1+\alpha), \tag{13}\] \[|h_{2}|^{2} =\frac{1}{2}(1-\alpha).\]Since for the AWGN channel considered in (5), the noise is Gaussian and proper (circular), it is clear that the mutual information does not change when the constellation is rotated by an arbitrary angle.