## 3 Bit Error Rate Performance
 \tag{22}\]  
and  
\[\begin{split}\bar{P}_{\text{e,LN}}=&\frac{\gamma^{2 }\mu_{c}\exp(\mu_{a}-\mu_{b}^{2}\gamma^{2})}{4}\{\frac{1}{a^{2}\mu_{c}}[\exp( \gamma^{2}\mu_{c}B)\operatorname{erfc}(B)-\exp(\frac{\gamma^{2}\mu_{c}}{4}) erf(\frac{\gamma^{2}\mu_{c}}{2}-B)]\\ &-\frac{4}{\sqrt{\pi}}\sum_{k=0}^{\infty}\frac{(-1)^{k}(\gamma A _{0}h_{l}\mu_{c})^{\frac{2\pi+1}{2}}}{k!(2k+1)(2a^{2}+2k+1)\mu_{c}}\exp\biggl{(} -\frac{\mu_{b}(2k+1)}{2}\  
Here, \(t\) is a parameter related to the behavior of the PDF near the origin, and \(a\) is a constant. The diversity order \(G_{d}\) indicates how the BER improves with increasing SNR, while the coding gain \(G_{c}\) provides a measure of how efficiently the coding scheme utilizes the available SNR. Consequently, for composite LN fading with pointing errors, performing a limit operation on Equation (11) yields the PDF of the channel gain \(h\) near the origin as derived in Section 2:  
\[\underset{h\to 0}{\lim}f_{\text{LN}}(h)=\frac{\gamma^{2}}{(A_{0}h_{l})} \exp(\mu_{a})h^{\gamma^{2}-1}+g_{\gamma^{2}-1}(h). \tag{26}\]  
The asymptotic BER can then be obtained from \(P_{e}^{\infty}=(G_{c}\cdot\bar{\gamma})^{-Gd}\). Using Equations (21)-(23), the diversity order and coding gain can be determined, then the asymptotic BER can be expressed as  
\[P_{\text{e,LN}}^{\infty}=\frac{2^{\frac{\gamma^{2}}{2}-1}\Gamma(\frac{\gamma^ {2}}{2}+\frac{1}{2})\exp(\mu_{a})}{\sqrt{\pi}(A_{0}h_{l})^{\alpha^{2}}}(\bar{ \gamma})^{-\frac{\gamma^{2}}{2}}. \tag{27}\]  
For the GG channel model, using Equations (12)-(14), the average BER can be written as  
\[P_{\text{e,GG}}=\int_{0}^{\infty}\{\frac{1}{2}\operatorname{erfc}(\sqrt{\bar {\gamma}h})\frac{2\gamma^{2}(\alpha\beta)^{\frac{\alpha+\beta}{2}}}{(A_{0}h_{ l})^{\gamma^{2}}\Gamma(\alpha)\Gamma(\beta)}h^{\gamma^{2}-1}\int_{\frac{ \beta}{A_{0}\bar{\gamma}}}^{\infty}h_{\alpha^{2}-1-\alpha^{2}}K_{\alpha+ \beta}(2\sqrt{\alpha\beta h_{\alpha}})dh_{\alpha}\}dh. \tag{28}\]  
The composite PDF of the GG model can be approximated [29] by finite series as follows:  
\[f_{\text{GG}}(h)\approx\sum_{j=0}^{J}\{\frac{1}{j!}(\frac{\alpha\beta}{A_{0}h _{l}})^{j}(v_{j}(\alpha,\beta)h^{\beta-1+j}-v_{j}(\beta,\alpha)h^{\alpha-1+j })\} \tag{29}\]  
where \(J=\left\lfloor\gamma^{2}-\alpha\right\rfloor\) and  
\[v_{j}(\alpha,\beta)=\frac{\gamma^{2}\pi\left(\frac{\alpha\beta}{A_{0}h_{l}} \right)^{\beta}\sin^{-1}((\alpha-\beta)\pi)}{\Gamma(\alpha)\Gamma(\beta)\Gamma (j-(\alpha-\beta)+1)\left\lfloor-(\beta-\alpha^{2}+j)\right\rfloor}. \tag{30}\]  
It should be noted that although Equation (29) is an approximation of the composite PDF for GG fading with pointing error, it is valid under the condition \(\gamma^{2}>\alpha\) and converges rapidly as the value of \(J\) increases. Utilizing an integral identity ([27], Equation (6.281.1)),  
\[\int_{0}^{\infty}(1-\Phi(px))x^{2q-1}dx=\frac{\Gamma(q+\frac{1}{2})}{2\sqrt{ \pi}qp^{2q}} \tag{31}\]  
one can obtain an approximate BER with finite series as follows:  
\[P_{\text{e,GG}}=\sum_{j=0}^{J}\left\{\frac{1}{j!}\left(\frac{\alpha\beta}{A_{0 }h_{l}}\right)^{j}\left(v_{j}(\alpha,\beta)\frac{\Gamma(\beta+j+\frac{1}{2})} {2\sqrt{\pi}(\beta+j)\bar{\gamma}^{\beta+j}}-v_{j}(\beta,\alpha)\frac{\Gamma( \alpha+j+\frac{1}{2})}{2\sqrt{\pi}(\alpha+j)\bar{\gamma}^{\alpha+j}}\right) \right\}. \tag{32}\]  
We notice that Equation (32) is obtained without additional approximation in Equation (27). The PDF power series expansion near the origin for GG composite fading channels is derived as follows:  
\[\underset{I\to 0}{\lim}=v_{0}(\alpha,\beta)I^{\beta-1}+g_{\beta-1}(h) \tag{33}\]where \(v_{0}(\alpha,\beta)\) is defined as in Equation (29), and it is assumed that \(\gamma^{2}>\alpha\).