## 3 Design and Analysis of 7-QAM Constellation
### SEP Expression of 7-QAM Constellation
The probability of correct decision for the (AFW) region could be written as  
\[P_{\mathrm{AFW}}=P_{\mathrm{rt}}\left(0,-1,\frac{\sigma}{d},\frac{\sigma}{ \sqrt{3}d/3}\right). \tag{11}\]  
In addition, according to the reference [28], the probability of correct decision for the (BCEF) region can be calculated by  
\[P_{\mathrm{BCEF}}=\left[1-2Q\left(\frac{d}{\sigma}\right)\right]\left[1-2Q \left(\frac{\sqrt{3}d}{3\sigma}\right)\right]. \tag{12}\]  
Figure 4: Schematic diagram for division of unconventional regions. (**a**) RT region; (**b**) decision region of signal point ‘1’. Combining Equations (11) and (12), and denoting the probability of correct decision for signal point '0' using \(P_{0}\), it could be written as  
\[P_{0} =P_{\text{BCEF}}+4P_{\text{AFW}}\] \[=\left[1-2Q\left(\frac{d}{\sigma}\right)\right]\left[1-2Q\left( \frac{\sqrt{3}d}{3\sigma}\right)\right]+4P_{\text{rt}}\left(0,-1,\frac{\sigma} {d},\frac{\sigma}{\sqrt{3}d/3}\right). \tag{13}\]  
As shown in Figure 3, the correct decision region for signal point '1' is (JFEK). Figure 4b shows the schematic diagram of the division for the (JFEK) region. Thus, the (JFEK) region can be expressed as (JUTL)+(RVEK)+(LTR)+(FUT)+(FEV). It can be known from Figure 4b that \(\overline{\text{VE}}=\overline{\text{UT}}=d\) and \(\overline{\text{VT}}=\frac{\sqrt{3}d}{3}\). The probabilities of correct decision for the (JUTL) region and the (RVEK) region can be written as  
\[P_{\text{JUTL}}=\frac{1}{2}\left[\frac{1}{2}-Q\left(\frac{d}{\sigma}\right) \right], \tag{14}\]  
\[P_{\text{REVK}}=\left[1-Q\left(\frac{\sqrt{3}d}{3\sigma}\right)\right]\left[ \frac{1}{2}-Q\left(\frac{d}{\sigma}\right)\right]. \tag{15}\]  
The angle of \(\angle\)LTR is \(\pi/3\). Thus, the probability of correct decision for the (LTR) region is  
\[P_{\text{LTR}}=\frac{1}{6}. \tag{16}\]  
For the (FUT) region and the (FEV) region, there are \(\overline{\text{VF}}=\overline{\text{UF}}=\frac{\sqrt{3}d}{3}\). Thus, the probabilities of correct decision for (FUT) and (FEV) regions are  
\[P_{\text{FUT}}=P_{\text{rt}}\left(1,0,\frac{\sigma}{d},\frac{\sigma}{\sqrt{3} d/3}\right), \tag{17}\]  
\[P_{\mathrm{AFW}}=P_{\mathrm{rt}}\left(0,-1,\frac{\sigma}{d},\frac{\sigma}{ \sqrt{3}d/3}\right). \tag{11}\]0  
Combining Equations (14) to (18), and denoting the correct decision probability for signal point '1' using \(P_{1}\), it could be expressed as  
\[P_{\mathrm{AFW}}=P_{\mathrm{rt}}\left(0,-1,\frac{\sigma}{d},\frac{\sigma}{ \sqrt{3}d/3}\right). \tag{11}\]1 \[P_{\mathrm{AFW}}=P_{\mathrm{rt}}\left(0,-1,\frac{\sigma}{d},\frac{\sigma}{ \sqrt{3}d/3}\right). \tag{11}\]2 \[P_{\mathrm{AFW}}=P_{\mathrm{rt}}\left(0,-1,\frac{\sigma}{d},\frac{\sigma}{ \sqrt{3}d/3}\right). \tag{11}\]3  
Finally, combining Equations (13) and (19), denoting the exact average SEP for the 7-QAM constellation using \(P_{\text{7-QAM}}\), it can be expressed as  
\[P_{\mathrm{AFW}}=P_{\mathrm{rt}}\left(0,-1,\frac{\sigma}{d},\frac{\sigma}{ \sqrt{3}d/3}\right). \tag{11}\]4Substituting the average symbol signal-to-noise ratio (SNR) into Equation (20), which is defined as \(\gamma_{s}=E_{s}/N_{0}=12d^{2}/(7\sigma^{2})\), the exact expression for the average SEP of 7-QAM over the AWGN channel is  
\[ \begin{split} P_{\text{7-QAM}}&=\frac{1}{7}\left(5Q \left(\frac{\sqrt{\gamma_{s}}}{6}\right)-10Q\left(\frac{\sqrt{\gamma_{s}}}{6} \right)Q\left(\frac{\sqrt{21\gamma_{s}}}{6}\right)+11Q\left(\frac{\sqrt{21 \gamma_{s}}}{6}\right)\right.\\ &\left.+\frac{1}{2}-6P_{\text{rt}}\left(1,0,\frac{6}{\sqrt{21 \gamma_{s}}},\frac{6}{\sqrt{\gamma_{s}}}\right)-10P_{\text{rt}}\left(0,-1, \frac{6}{\sqrt{21\gamma_{s}}},\frac{6}{\sqrt{\gamma_{s}}}\right)\right).\end{split}  \tag{21}\]