## 4 PPM SER Performance Analysis  
In deep space optical uplink communication systems constrained by power budgets, PPM is strategically implemented due to its superior photon efficiency under average transmission power constraints. To quantify communication reliability, we adopt the SER as the critical performance metric, defined as the probability of erroneous identification of the PPM symbol. This section presents a comprehensive SER analysis based on both Webb and Gaussian-approximated APD output models, specifically incorporating channel impairments: atmospheric and coronal turbulence-induced fading, pointing error fluctuations, AOA fluctuations, link attenuation, and background noise. Comprehensive analysis provides critical insights for designing robust deep space optical uplink communication systems.  
The conditional SER for PPM systems under channel fading coefficient \(h\) is analytically formulated as  
\[E_{p|h}=1-\int_{-\infty}^{\infty}\!\!\!p_{\epsilon}(i_{x}|1)di_{x}\bigg{(}\int_ {-\infty}^{i_{x}}p_{\epsilon}\big{(}i_{y}|0\big{)}di_{y}\bigg{)}^{\mathrm{Q}- 1}, \tag{24}\]  
where \(Q\) denotes the PPM symbol size. Conditional probability densities \(p_{\epsilon}(i_{x}|1)\) and \(p_{\epsilon}(i_{y}|0)\) characterize the APD output under pulse transmission (bit 1) and silence states (bit 0), respectively. These distributions are formulated through both Webb (21) and Gaussian (23)-approximated stochastic APD output. The statistical characterization of the deep space optical uplink channel is analytically formulated as \(f_{h}(h)\) in (17). Under the influence of various channel impairments, the unconditioned SER of the PPM system is obtained by  
\[E_{p}=\int_{0}^{\infty}f_{h}(h)E_{p|h}dh. \tag{25}\]  
Consequently, through Webb-modeled APD statistical output, the PPM SER performance is derived as  
\[E_{W}=\int_{0}^{\infty}f_{h}(h)\left(1-\mathrm{InS}\left(i_{x},\frac{\frac{1} {c_{2}\sqrt{x}}\Big{(}1+\frac{c_{x}}{c_{f}x}\Big{)}^{2/3}}{\exp\left(\frac{c_ {x}^{2}}{c_{1}\mu_{h}+\frac{c_{x}\sigma_{m}^{2}}{c_{1}^{2}}}+\frac{\left(i_{x }-\mu_{m}\right)^{2}}{2\sigma_{h}^{2}}\right)}\right)\mathrm{InS}^{\mathrm{Q}- 1}\left(i_{y},\frac{\frac{1}{c_{2}\sqrt{h}}\Big{(}1+\frac{c_{y}}{c_{f}x_{h}} \Big{)}^{2/3}}{\exp\left(\frac{c_{y}^{2}}{c_{1}\mu_{h}+\frac{c_{y}\sigma_{m}^ {2}}{c_{1}^{2}}}+\frac{\left(i_{y}-\mu_{m}\right)^{2}}{2\sigma_{h}^{2}}\right) \right)}\right)dh. \tag{26}\]where \(\text{InS}(i,\cdot)\) represents the operation of \(\int_{-\infty}^{\infty}\sum\limits_{m=0}^{\infty}\left(\cdot\right)di\). Parameters \(x=hn_{s}+n_{b}\), \(c_{x}=m-Gx\), \(c_{y}=m-G\eta_{b}\), \(c_{1}=2G^{2}F_{a}\), \(c_{2}=\pi\sigma_{m}\sqrt{2c_{1}}\). The complete mathematical derivation of this SER formulation is provided in Appendix A.2. The Webb-modeled SER formulation (26) encounters analytical intractability in resolving this multidimensional integration complexity. We can use trapezoidal rules and simulation tools to obtain significant numerical computation results.  
For the Gaussian-approximated APD output model operating under composite channel conditions, the SER of PPM framework is analytically formulated as  
\[E_{G}= \frac{Q-1}{2}\exp\left(-\frac{\Omega_{F}^{2}}{2\sigma_{\theta}^{2 }}\right)+\frac{(Q-1)\text{C}_{\text{a}\beta}\text{C}_{p1}\text{C}_{p2}^{2}}{2 h_{c}\Gamma(\alpha_{t})\Gamma(\beta_{t})\Gamma(\alpha_{s})\Gamma(\beta_{s})} \left(1-\exp\left(-\frac{\Omega_{F}^{2}}{2\sigma_{\theta}^{2}}\right)\right)\] \[\times\int_{0}^{\infty}\text{C}_{1S}^{5,0}\left[\frac{\text{C}_{ \text{a}\beta}\text{C}_{p1}}{h_{c}}h\right|\begin{array}{c}C_{p2}^{2}\\ C_{p2}^{2}-1,\alpha_{t}-1,\beta_{t}-1,\alpha_{s}-1,\beta_{s}-1\end{array}\right] \text{erfc}\left(\frac{Gq_{e}n_{b}h}{\sqrt{c_{g1}(2n_{b}+h\bar{n}_{s})+c_{g2}}} \right)dh. \tag{27}\]  
where \(c_{g1}=2q_{e}^{2}G^{2}F_{a}\), \(c_{g2}=4q_{e}l_{D}T_{s}+\frac{8\text{c}_{e}T_{e}T_{s}}{R_{L}}\), and \(\text{erfc}(\cdot)\) is the complementary error function. Correspondingly, the mathematical derivation process underlying these results is rigorously presented in Appendix A.2.