## 3 Design and Analysis of 7-QAM Constellation
### SEP Expression of 7-QAM Constellation  
The formula for calculating the SEP of a constellation helps to analyze its anti-noise performance. In this part, we derive the exact closed-form expression for the SEP of the new constellation over the AWGN channel for further theoretical analysis to its error performance. The decision region for each signal point of the 7-QAM constellation is shown in Figure 3, and DL1, DL2 (Q axis), and DL3 are decision boundaries. Obviously, according to the decision region, the signal points could be divided into two classes, interior point and points on the outer circle. As shown in Figure 3, the correct decision region for signal point '0' is the regular hexagon (ABCDEF). It is decomposed into five regions, which can be expressed as (ABW) + (AFW) + (BCEF) + (DCS) + (DES). In addition, the probabilities of correct decision for the four right triangle (RT) regions are equal. In order to express the probability of correct decision for the RT region (AFW) in terms of the Gaussian Q-function \(Q\left(x\right)\), which is the probability that the random variable will obtain a value larger than \(x\) and is defined as \(Q\left(x\right)=\frac{1}{\sqrt{2\pi}}\int_{x}^{\infty}e^{-t^{2}/2}dt\), the RT region with right angle edge lengths of \(l_{1}\) and \(l_{2}\) can be divided into an infinite number of rectangles, as shown in Figure 4a. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Constellation** & \(\mathbf{E_{s}}\) & \(\mathbf{E_{max}}\) & **PAPR** \\ \hline
7-QAM & \(24d^{2}/7\) & \(4d^{2}\) & 1.1667 \\
8-PSK & \(d^{2}/\sin^{2}\left(\pi/8\right)\) & \(d^{2}/\sin^{2}\left(\pi/8\right)\) & 1 \\
8-QAM & \(\left(3+\sqrt{3}\right)d^{2}\) & \(\left(4+2\sqrt{3}\right)d^{2}\) & 1.5774 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: PAPRs of three constellations. Figure 3: The decision regions of 7-QAM constellation points. There are \(2^{n-1}\) rectangles with length of \(2^{-n}l_{1}\) and width of \(2^{-n}l_{2}\) in the RT region, and each rectangle is denoted by  
\[R_{n}^{k}=\left\{\left(k2^{1-n}\right)l_{1}\leq x<\left(\left(1+2k\right)2^{-n} \right)l_{1},\,\left(1-\left(1+k\right)2^{1-n}\right)l_{2}\leq y<\left(1- \left(1+2k\right)2^{-n}\right)l_{2}\right\}, \tag{8}\]  
where \(n=1,2,\ldots\) and \(k=0,1,2,\ldots,2^{n-1}-1\). If the transmitted signal is with a coordinate of \(\left(S_{\mathrm{I}},S_{\mathrm{Q}}\right)\), which is determined by using two right-angled sides of the RT region as coordinate axes, the probability that the received signal falls in the region \(R_{n}^{k}\) can be written as  
\[ \begin{split} P_{n}^{k}=&\left|Q\left(\frac{\frac{S _{\mathrm{I}}}{l_{1}}-k2^{1-n}}{\sigma/l_{1}}\right)\,-\,Q\left(\frac{\frac{S_ {\mathrm{I}}}{l_{1}}-\left(1+2k\right)2^{-n}}{\sigma/l_{1}}\right)\right|\\ &\times\left|Q\left(\frac{\frac{S_{\mathrm{Q}}}{l_{2}}-\left(1- \left(1+k\right)2^{1-n}\right)}{\sigma/l_{2}}\right)\,-\,Q\left(\frac{\frac{S _{\mathrm{Q}}}{l_{2}}-\left(1-\left(1+2k\right)2^{-n}\right)}{\sigma/l_{2}} \right)\right|,\end{split}  \tag{9}\]  
which is obtained from [27]. The probability that the received signal falls in the RT region can be obtained through summing all the probabilities \(P_{n}^{k}\), namely  
\[P_{\mathrm{rt}}\left(\frac{S_{\mathrm{I}}}{l_{1}},\frac{S_{\mathrm{Q}}}{l_{2}},\frac{\sigma}{l_{1}},\frac{\sigma}{l_{2}}\right)=\sum_{n=1}^{\infty}\sum_{k=0 }^{2^{n-1}-1}P_{n}^{k}. \tag{10}\]  
The probability that the received signal falls in the residual area of the RT region after division \(n\) times decreases exponentially with the increase of \(n\), so Equation (10) can be made arbitrarily accurate [27]. The distance between points 'F' and 'W' is \(d\) and that between points 'A' and 'W' is \(\frac{\sqrt{3}d}{3}\), where \(d\) is half of the minimum distance between two adjacent signal points.