## 3 Performance Analysis
### Average Bit Error Rate  
The average BER refers to the ratio of the number of bits that are incorrectly received to the total number of bits that are transmitted. The average BER of the binary system depicts various detection techniques by adjusting two parameters, \(p\) and \(q\), as [29]  
\[\overline{P_{b}}=\frac{q^{p}}{2\Gamma(p)}\int_{0}^{\infty}\exp(-q\gamma) \gamma^{p-1}F(\gamma)d\gamma. \tag{11}\]  
Therefore, the average BER of the system under AoA fluctuations with the heterodyne detection and the IM/DD techniques, respectively, are stated as follows by substituting (10) into (11) and using Equation (2.24.3.1) in [30]  
\[\overline{P_{b}}=\frac{1}{4\Gamma(p)}\left(1-\exp\left(-\frac{\theta_{FoV}^{2} }{2\sigma_{a}^{2}}\right)\right)A\xi^{2}\sum_{i=1}^{b}b_{i}G_{3,4}^{3,2}\left[ \frac{C\sigma_{n}^{2}}{\eta P_{t}q} \begin{array}{c}1-p,1,1+\xi^{2}\\ \xi^{2},a,i,0\end{array} \right]+\frac{1}{2}\exp\left(-\frac{\theta_{FoV}^{2} }{2\sigma_{a}^{2}}\right), \tag{12}\]\[ \begin{split}\overline{P_{b}}&=\frac{2^{a-5}A\xi^{2}}{ \pi\Gamma(p)}\left(1-\exp\biggl{(}-\frac{\theta_{F,W}^{2}}{2\sigma_{a}^{2}} \biggr{)}\biggr{)}\sum\limits_{i=1}^{b}b_{i}2^{i}\\ &\cdot G_{5,8}^{6,3}\biggl{[}\biggl{(}\frac{C}{\eta\Gamma}\biggr{)} ^{2}\frac{\sigma_{a}^{2}}{16\sigma_{a}}\biggr{]}\frac{1-p,\frac{1}{2},1}{ \frac{1+\xi^{2}}{2},\frac{2+\xi^{2}}{2}}\\ &\frac{\xi^{2}}{2},\frac{\xi^{2}+1}{2}\frac{\sigma_{a}+1}{2}\frac{ i+1}{2},\frac{i+1}{2},0,\frac{1}{2}\end{split} \bigg{]}+\frac{1}{2}\exp \biggl{(}-\frac{\theta_{F,W}^{2}}{2\sigma_{a}^{2}}\biggr{)}\end{split}. \tag{13}\]  
**Remark 1**. : _For the particular situation, the average BER under the influence of atmospheric attenuation, atmospheric turbulence, and pointing errors may also be obtained according to (12) and (13), provided \(\sigma_{a}\to 0\). The result is consistent with those in Equation (22) of [31]._  
**Remark 2**. : _The average BER under the heterodyne detection technique can be computed asymptotically at high optical power using Equation (6.2.2) in [32] to transform the argument function in (12), followed by Equation (17) in [31], as shown below:_  
\[ \begin{split}\overline{P_{b}}&\underset{B_{\gamma> 1}}{\approx}\frac{1}{4\Gamma(p)}\biggl{(}1-\exp\biggl{(}-\frac{\theta_{F,W}^{2 }}{2\sigma_{a}^{2}}\biggr{)}\biggr{)}A\xi^{2}\sum\limits_{i=1}^{b}b_{i}\biggl{[} \sum\limits_{k=1}^{3}\biggl{(}\frac{\eta Bq}{C\sigma_{a}^{2}}\biggr{)}^{-v_{k }}\\ &\cdot\frac{\prod_{i=1,j\
eq k}^{3}\Gamma(v_{i}-v_{k})\Gamma(p+v_{ k})}{v_{k}\Gamma(1+\xi^{2}-v_{k})}\biggr{]}+\frac{1}{2}\exp\biggl{(}-\frac{ \theta_{F,W}^{2}}{2\sigma_{a}^{2}}\biggr{)}\end{split} \, \tag{14}\]  
_where \(v=\bigl{(}\xi^{2},\alpha,i\bigr{)}\), \(v_{k}\) represents the \(k^{th}\)-term of \(v\), and \(\Gamma(\cdot)\) is the Gamma function._  
Then, the diversity order is achieved by expanding (14) in the following manner:  
\[ \begin{split}\overline{P_{b}}&\underset{B_{\gamma> 1}}{\approx}\frac{1}{4\Gamma(p)}\biggl{(}1-\exp\biggl{(}-\frac{\theta_{F,W}^{2 }}{2\sigma_{a}^{2}}\biggr{)}\biggr{)}A\xi^{2}\sum\limits_{i=1}^{b}b_{i}\biggl{[} \biggl{(}\frac{\eta Bq}{C\sigma_{a}^{2}}\biggr{)}^{-\xi^{2}}\frac{\Gamma(a- \xi^{2})\Gamma(i-\xi^{2})\Gamma(p+\xi^{2})}{\xi^{2}}\\ &+\Bigl{(}\frac{\eta Bq}{C\sigma_{a}^{2}}\Bigr{)}^{-\alpha}\frac{ \Gamma(i-a)\Gamma(p+a)}{\alpha(\xi^{2}-a)}+\Bigl{(}\frac{\eta Bq}{C\sigma_{a} ^{2}}\Bigr{)}^{-\frac{\Gamma}{2}(a-i)\Gamma(p+i)}\biggr{]}+\frac{1}{2}\exp \biggl{(}-\frac{\theta_{F,W}^{2}}{2\sigma_{a}^{2}}\biggr{)}\end{split} . \tag{15}\]  
Therefore, the asymptotic formula for the average BER in (15) is mainly determined by the diversity order of \(\min\bigl{(}\xi^{2},a,1\bigr{)}\). In other words, the atmospheric turbulence and pointing errors determine the average BER at high power. This is because the effects of atmospheric turbulence and pointing errors become more pronounced at high optical powers.