## 3 Design and Analysis of 7-QAM Constellation
### Design of 7-QAM Constellation  
In order to reduce the influence of non-linear distortions on the signal points and simultaneously obtain better anti-noise performance, the new constellation is designed based on the hexagonal grid signal sets of seven signals given in [14]. In addition, the configuration of new constellation is \(1+6\). Specifically, as shown in Figure 2a, one signal point is on the origin point, and the other six signal points are evenly arranged on the outer circle with the phase difference of \(\pi/3\). The vector expression of 7-QAM signals can be written as  
\[\left\{\begin{array}{ll}x_{k}=[0,0],&k=0,\\ \\ x_{k}=\Big{[}R_{1}\cos\frac{2\pi(k-1)}{6},R_{1}\sin\frac{2\pi(k-1)}{6}\Big{]},& 1\leq k\leq 6,\end{array}\right. \tag{5}\]  
where \(R_{1}\) is the radius of the 7-QAM constellation.  
The constellation diagrams of the conventional 8-PSK and 8-QAM are also demonstrated in Figure 2. The signal points are equiprobable, so that, for the normalization of power, the radiuses of these constellations are \(R_{1}\approx 1.0801\), \(R_{2}=1\), \(R_{3}\approx 0.6501\) and \({R_{3}}^{\prime}\approx 1.2559\).  
Table 1 shows the comparison for parameters of the three constellations presented in Figure 2, and the minimum phase difference \(\theta_{\mathrm{min}}\) of 7-QAM is the value except the signal point '0'. It can be known from Table 1 that the minimum Euclidean distance \(d_{\mathrm{min}}\) of the 7-QAM constellation is 3 dB higher than that of the 8-PSK constellation and is 1.4 dB higher than that of 8-QAM constellation, which reflects the fact that the anti-noise performance of 7-QAM is much better than that of 8-PSK and 8-QAM.  
Figure 1: The non-linear channel model.  
More serious non-linear distortion would be introduced as applying the constellation with higher PAPR to the satellite communication system. For the practical consideration, the PAPR of 7-QAM constellation is calculated. The PAPR of the constellation can be calculated by  
\[PAPR=\frac{E_{\text{max}}}{E_{\text{s}}}, \tag{6}\]  
where \(E_{\text{s}}\) is the average energy per symbol of the constellation and \(E_{\text{max}}\) is the maximum energy of the modulated signal. Specifically, in the case that half of the minimum distance between adjacent symbols is \(d\), the average energy per symbol of the 7-QAM constellation, denoted as \(E_{7\text{-QAM}}\), can be calculated as  
\[\begin{split} E_{7\text{-QAM}}&=\frac{1}{7}\left[ \sum_{k=1}^{7}\left(\{\text{Re}\left(S_{k}\right)\}^{2}+\{\text{Im}\left(S_{k} \right)\}^{2}\right)\right]\\ &=\frac{1}{7}\left[6\times(2d)^{2}\right]=\frac{24d^{2}}{7},\end{split} \tag{7}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Constellation** & \(\mathbf{\theta_{\text{min}}}\) & \(\mathbf{d_{\text{min}}}\) \\ \hline
7-QAM & \(\pi/3\) & \(2R_{1}\sin\frac{\pi}{4}\approx 1.0801\) \\
8-PSK & \(\pi/4\) & \(2R_{2}\sin\frac{\pi}{4}\approx 0.7654\) \\
8-QAM & \(\pi/4\) & \(2R_{3}\sin\frac{\pi}{4}\approx 0.9194\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters of three constellations.  
Figure 2: Constellation diagrams. (**a**) 7-QAM; (**b**) 8-PSK; (**c**) 8-QAM.  
where \(S_{k}\) represents the \(k\)-th signal point of the 7-QAM constellation, \(\text{Re}\left(S_{k}\right)\) and \(\text{Im}\left(S_{k}\right)\) mean the real and imaginary values of the signal point \(S_{k}\), respectively. In addition, the maximum energy of 7-QAM signal is \(4d^{2}\). Thereby, the PAPR of the 7-QAM constellation is 1.1667. With the same method, the PAPRs of the 8-PSK and 8-QAM constellations are also calculated for comparison, and the calculation results are presented in Table 2.  
It can be known that the PAPR of 7-QAM constellation is 1.31 dB lower than that of 8-QAM constellation, and is 0.67 dB higher than that of 8-PSK constellation.