## 3 Performance Analysis
 \tag{14}\]According to (14)-(16), the error rate of the ground-HAP link's HQAM can be determined as:  
\[P_{g-h(HQAM)}=\sum\limits_{m=1}^{n}H_{m}K_{NN}Q\Big{(}\sqrt{\delta a_{m}}\Big{)} \frac{\alpha_{1}\beta_{1}}{2\eta_{1}\sqrt{\gamma}}\bigg{(}\frac{\sqrt{a_{m}}}{ \sqrt{\gamma}<l>\eta_{1}}\bigg{)}^{\beta_{1}-1}e^{a_{m}-[-(\frac{\sqrt{a_{m}}}{ \sqrt{l<l>\eta_{1}}})]^{\beta_{1}}}\left(1-e^{-(\frac{\sqrt{a_{m}}}{\sqrt{l<l> \eta_{1}}})^{\beta_{1}}}\right)^{\alpha_{1}-1} \tag{17}\]  
Similarly, it can be concluded that:  
\[P_{g-h(XQAM)}=\sum\limits_{m=1}^{n}H_{m}K_{CN}Q(\tau\sqrt{a_{m}})\frac{\alpha_ {1}\beta_{1}}{2\eta_{1}\sqrt{\gamma}}\bigg{(}\frac{\sqrt{a_{m}}}{\sqrt{\gamma }<l>\eta_{1}}\bigg{)}^{\beta_{1}-1}e^{a_{m}-[-(\frac{\sqrt{a_{m}}}{\sqrt{l<l> \eta_{1}}})]^{\beta_{1}}}\left(1-e^{-(\frac{\sqrt{a_{m}}}{\sqrt{l<l>\eta_{1}} })^{\beta_{1}}}\right)^{\alpha_{1}-1} \tag{18}\]  
\[P_{h-s(HQAM)}=\sum\limits_{m=1}^{n}H_{m}K_{NN}Q\Big{(}\sqrt{\delta a_{m}}\Big{)} \frac{\alpha_{2}\beta_{2}}{2\eta_{2}\sqrt{\gamma}}\bigg{(}\frac{\sqrt{a_{m}}}{ \sqrt{\gamma}<l>\eta_{2}}\bigg{)}^{\beta_{2}-1}e^{a_{m}-[-(\frac{\sqrt{a_{m}} }{\sqrt{l<l>\eta_{2}}})]^{\beta_{2}}}\left(1-e^{-(\frac{\sqrt{a_{m}}}{\sqrt{l <l>\eta_{2}}})^{\beta_{2}}}\right)^{\alpha_{2}-1} \tag{19}\]  
\[P_{h-s(XQAM)}=\sum\limits_{m=1}^{n}H_{m}K_{CN}Q(\tau\sqrt{a_{m}})\frac{\alpha_ {2}\beta_{2}}{2\eta_{2}\sqrt{\gamma}}\bigg{(}\frac{\sqrt{a_{m}}}{\sqrt{\gamma }<l>\eta_{2}}\bigg{)}^{\beta_{2}-1}e^{a_{m}-[-(\frac{\sqrt{a_{m}}}{\sqrt{l<l> \eta_{2}}})]^{\beta_{2}}}\left(1-e^{-(\frac{\sqrt{a_{m}}}{\sqrt{l<l>\eta_{2}} })^{\beta_{2}}}\right)^{\alpha_{2}-1} \tag{20}\]  
where \(K_{CN}=\big{(}4-\frac{2}{M}-\frac{2}{N}\big{)},\alpha_{1},\beta_{1},\eta_{1}\) and \(\alpha_{2},\beta_{2},\eta_{2}\) are the parameters of the ground-HAP and HAP-satellite links, respectively. Finally, the average bit error rate closed expression of the system is obtained by bringing Equations (17)-(20) into Equation (13):  
\[\begin{split} P_{BER(HQAM)}=\sum\limits_{m=1}^{n}H_{m}\frac{K_{ NN}Q\big{(}\sqrt{\delta a_{m}}\big{)}}{2\sqrt{\gamma}}\left( \begin{array}{c} \frac{\alpha_{1}\beta_{1}}{\eta_{1}}\big{(}\frac{\sqrt{a_{m}}}{\sqrt{l<l>\eta _{1}}}\big{)}^{\beta_{1}-1}e^{a_{m}-[-(\frac{\sqrt{a_{m}}}{\sqrt{l<l>\eta_{1 }}})]^{\beta_{1}}}\left(1-e^{-(\frac{\sqrt{a_{m}}}{\sqrt{l<l>\eta_{1}}})^{ \beta_{1}}}\right)^{\alpha_{1}-1}+\\ \frac{\alpha_{2}\beta_{2}}{\eta_{2}}\left(\frac{\sqrt{a_{m}}}{\sqrt{l<l>\eta _{2}}}\right)^{\beta_{2}-1}e^{a_{m}-[-(\frac{\sqrt{a_{m}}}{\sqrt{l<l>\eta_{2} }})]^{\beta_{2}}}\times\left(1-e^{-(\frac{\sqrt{a_{m}}}{\sqrt{l<l>\eta_{2}}})^{ \beta_{2}}}\right)^{\alpha_{2}-1}\end{array} \right)\\ -\bigg{(}\sum\limits_{m=1}^{n}H_{m}\frac{K_{NN}Q\big{(}\sqrt{\delta a_{m}}\big{)}}{ \sqrt{\gamma}}\bigg{)}^{2}\left( \begin{array}{c}\frac{\alpha_{1}\beta_{1}}{ \eta_{1}}\big{(}\frac{\sqrt{a_{m}}}{\sqrt{l<l>\eta_{1}}}\big{)}^{\beta_{1}-1}e ^{a_{m}-[-(\frac{\sqrt{a_{m}}}{\sqrt{l<l>\eta_{1}}})]^{\beta_{1}}}\left(1-e^{ -(\frac{\sqrt{a_{m}}}{\sqrt{l<l>\eta_{1}}})^{\beta_{1}}}\right)^{\alpha_{1}-1 }\times\\ \frac{\alpha_{2}\beta_{2}}{\eta_{2}}\left(\frac{\sqrt{a_{m}}}{\sqrt{l<l>\eta _{2}}}\right)^{\beta_{2}-1}e^{a_{m}-[-(\frac{\sqrt{a_{m}}}{\sqrt{l<l>\eta_{2} }})]^{\beta_{2}}}\times\left(1-e^{-(\frac{\sqrt{a_{m}}}{\sqrt{l<l>\eta_{2}}}) ^{\beta_{2}}}\right)^{\alpha_{2}-1}\end{array} \right)\\ P_{BER(XQAM)}=\sum\limits_{m=1}^{n}H_{m}\frac{K_{CN}Q\big{(}\sqrt{\delta a_{m}}\big{)}}{ \sqrt{\gamma}}\left( \begin{array}{c}\frac{\alpha_{1}\beta_{1}}{\eta_{1}} \big{(}\frac{\sqrt{a_{m}}}{\sqrt{l<l>\eta_{1}}}\big{)}^{\beta_{1}-1}e^{a_{m}-[ -(\frac{\sqrt{a_{m}}}{\sqrt{l<l>\eta_{1}}})]^{\beta_{1}}}\left(1-e^{-(\frac{ \sqrt{a_{m}}}{\sqrt{l<l>\eta_{1}}})^{\beta_{1}}}\right)^{\alpha_{1}-1}+\\ \frac{\alpha_{2}\beta_{2}}{\eta_{2}}\left(\frac{\sqrt{a_{m}}}{\sqrt{l<l>\eta_{2} }}\big{)}^{\beta_{2}-1}e^{a_{m}-[-(\frac{\sqrt{a_{m}}}{\sqrt{l<l>\eta_{2}}})]^{ \beta_{2}}}\left(1-e^{-(\frac{\sqrt{a_{m}}}{\sqrt{l<l>\eta_{2}}})^{\beta_{2}}} \right)^{\alpha_{2}-1}\end{array}