### Appendix A.2  
According to (07.34.21.0011.01) in [47], which is denoted as  
\[\int_{0}^{\infty}\tau^{a-1}\hat{\mathrm{G}}_{u,v}^{s,t}\left( \tau w\left| \begin{array}{l}c_{1},\ldots,c_{t},c_{t+1},\ldots,c_{u}\\ d_{1},\ldots,d_{s},d_{s+1},\ldots,d_{v}\end{array} \right.\right)\hat{\mathrm{G }}_{p,q}^{m,n}\left(\tau z\left| \begin{array}{l}a_{1},\ldots,a_{n},a_{n+1}, \ldots,a_{p}\\ b_{1},\ldots,b_{m},b_{m+1},\ldots,b_{q}\end{array} \right.\right)d\tau==\] \[w^{-a}\hat{\mathrm{G}}_{v+p,n+q}^{m+t,n+s}\left(\frac{z}{w} \left| \begin{array}{l}a_{1},\ldots,a_{n},1-\alpha-d_{1},\ldots,1-\alpha-d_{s },1-\alpha-d_{s+1},\ldots,1-\alpha-d_{v},a_{n+1},\ldots,a_{p}\\ b_{1},\ldots,b_{m},1-\alpha-c_{1},\ldots,1-\alpha-c_{t},1-\alpha-c_{t+1}, \ldots,1-\alpha-c_{u},b_{m+1},\ldots,b_{q}\end{array} \right.\right). \tag{10}\]  
where parameters \(s=0\), \(t=2\), \(u=2\), \(v=0\), \(m=3\), \(n=0\), \(p=1\), and \(q=1\). Consequently, (14) is derived as (15). According to (17) and (24), the Webb based PPM SER derivation process is as follows:\[E_{\text{W}}= \int_{0}^{\infty}f_{h}(h)\Bigg{(}1-\int_{-\infty}^{\infty}p_{w}(i_{ x}|h\bar{n}_{s}+\bar{n}_{b})di_{x}\bigg{(}\int_{-\infty}^{i_{x}}p_{w}(i_{y}|\bar{n}_{ b})di_{y}\bigg{)}^{Q-1}\Bigg{)}dh\] \[= \int_{0}^{\infty}f_{h}(h)\Bigg{\{} \begin{array}{l}1-\int_{- \infty}^{\infty}\sum_{m=0}^{\infty}\left(\frac{\left(1+\frac{m-C(h_{m}+\bar{n}_ {b})}{c_{f}x_{b}+\bar{n}_{b}}\right)^{2/3}}{2\pi\
u_{m}\sqrt{(i\bar{n}_{h}+\bar{ n}_{b})C^{2}\bar{n}_{a}}}\exp\left(-\frac{\frac{(m-C(h_{m}+\bar{n}_{b}))^{2}}{2( \bar{n}_{b}+\bar{n}_{b})C^{2}\bar{n}_{a}}}{1+\frac{m-C(h_{m}+\bar{n}_{b})}{c_{ f}(h_{m}+\bar{n}_{b})}}-\frac{(i_{x}-\bar{n}_{m})^{2}}{2\sigma_{m}^{2}} \right)\right)di_{x}\\ \cdot\left(\int_{-\infty}^{i_{x}}\sum_{m=0}^{\infty}\left(\frac{\left(1+\frac {m-\bar{n}_{b}}{c_{f}x_{b}}\right)^{2/3}}{2\pi\
u_{m}\sqrt{h_{b}}G^{2}\bar{n}_ {a}}\exp\left(-\frac{(m-\bar{n}_{b})^{2}}{2\pi G^{2}\bar{n}_{a}\left(1+\frac{m -\bar{n}_{b}}{c_{f}\bar{n}_{b}}\right)}-\frac{\left(i_{y}-\bar{n}_{m}\right)^{ 2}}{2\sigma_{m}^{2}}\right)\right)di_{y}\right)^{Q-1}\\ =\int_{0}^{\infty}f_{h}(h)\left(1-\text{InS}\left(i_{x},\frac{\frac{1}{c_{2} \sqrt{x}}\Big{(}1+\frac{c_{x}}{c_{f}x}\Big{)}^{2/3}}{\exp\left(\frac{c_{x}^{2 }}{c_{1}\bar{n}+\frac{c_{x}\times 2\bar{n}_{b}}}+\frac{\left(i_{x}-\bar{n}_{m} \right)^{2}}{2\sigma_{m}^{2}}\right)}\right)\text{InS}^{Q-1}\left(i_{y},\frac{ \frac{1}{c_{2}\sqrt{h_{b}}}\Big{(}1+\frac{c_{y}}{c_{f}\bar{n}_{b}}\Big{)}^{2/ 3}}{\exp\left(\frac{c_{x}^{2}}{c_{1}\bar{n}+\frac{c_{x}\times 2\bar{n}_{b}}}+ \frac{\left(i_{y}-\bar{n}_{m}\right)^{2}}{2\sigma_{m}^{2}}\right)\right)} \right)dh.\end{array}  \tag{10}\]  
For Gaussian model estimated PPM SER, the deriving process is shown as  
\[E_{G}= \int_{0}^{\infty}f_{h}(h)\frac{Q-1}{2}\text{erfc}\Bigg{(}\sqrt{ \frac{SNR}{2}}\Bigg{)}dh\] \[= \int_{0}^{\infty}\left\{ \begin{array}{l}\frac{Q-1}{2}\text{erfc }\bigg{(}Gq_{e}\bar{n}_{s}\Big{(}2q_{e}^{2}G^{2}F_{a}(2\bar{n}_{b}+\bar{n}_{s} )+4q_{e}I_{D}T_{s}+\frac{8\kappa_{e}T_{e}T_{e}}{R_{e}}\Big{)}^{-1/2}\bigg{)}\\ \frac{C_{gQ}C_{p1}C_{p2}^{2}}{h_{e}T(\alpha_{t})\Gamma(\beta_{t})T(\alpha_{s} )\Gamma(\beta_{c})}\bigg{(}1-\exp\left(-\frac{\Omega_{t}^{2}}{2\sigma_{\theta} ^{2}}\right)\bigg{)}\bar{G}_{1,5}^{5,0}\bigg{[}\frac{C_{gQ}\bar{c}_{p1}}{h_{c} }h\bigg{|}\begin{array}{c}C_{p2}^{2}\\ C_{p2}^{2}-1,\alpha_{t}-1,\beta_{t}-1,\alpha_{c}-1,\beta_{c}-1\\ +\delta(h)\cdot\exp\left(-\frac{\Omega_{t}^{2}}{2\sigma_{\theta}^{2}}\right) \end{array}\right]\\ \end{array} \right\}dh\] \[= \frac{Q-1}{2}\exp\left(-\frac{\Omega_{t}^{2}}{2\sigma_{\theta}^{2} }\right)+\frac{(Q-1)C_{gQ}C_{p1}C_{p2}^{2}}{2h_{e}T(\alpha_{t})\Gamma(\beta_{t })\Gamma(\alpha_{s})\Gamma(\beta_{s})}\bigg{(}1-\exp\left(-\frac{\Omega_{t}^{2} }{2\sigma_{\theta}^{2}}\right)\bigg{)}\] \[\times\int_{0}^{\infty}\bar{G}_{1,5}^{5,0}\bigg{[}\frac{C_{gQ}C_{p1 }}{h_{c}}h\bigg{|} \begin{array}{c}C_{p2}^{2}\\ C_{p2}^{2}-1,\alpha_{t}-1,\beta_{t}-1,\alpha_{s}-1,\beta_{s}-1\end{array}