# Performance Analysis of FSO-UWOC Mixed Dual-Hop Relay System with Decode-and-Forward Protocol
## 4 System Performance Analysis
### Average Bit Error Rate and Its Asymptotic Value
Utilizing [34](Equation 07.34.06.0006.01), the asymptotic representation of \(P_{e,1}\) can be formulated as:  
\[P_{e,1}^{\infty}=\frac{\Theta_{1}}{\overline{\gamma}^{\frac{c^{2}}{r}}}+ \frac{\Theta_{2}}{\overline{\gamma}^{\frac{c^{2}}{r}}}+\frac{\Theta_{3}}{ \overline{\gamma}^{\frac{c^{2}}{r}}} \tag{41}\]  
where  
\[\Theta_{1}=\frac{r^{\alpha+\beta-2\alpha_{1}^{2}-1}(\eta_{1}\alpha\beta)^{ \frac{c^{2}}{r}}\Gamma\left(\frac{\alpha-\delta_{1}^{2}}{r}\right)\cdots \Gamma\left(\frac{\alpha-\delta_{1}^{2}+r-1}{r}\right)\Gamma\left(\frac{ \beta-\delta_{1}^{2}}{r}\right)\cdots\Gamma\left(\frac{\beta-\delta_{1}^{2} +r-1}{r}\right)\Gamma\left(p+\frac{\delta_{1}^{2}}{r}\right)}{2(2\pi)^{r-1} \Gamma(p)\Gamma(\alpha)\Gamma(\beta)q^{\frac{c^{2}}{r}}}.\]  
\[\Theta_{2}=\frac{\zeta_{1}^{2}r^{\alpha+\beta-2\alpha}(\eta_{1}\alpha\beta)^{ \alpha}\Gamma\left(\frac{1}{r}\right)\cdots\Gamma\left(\frac{r}{r}\right) \Gamma\left(\frac{\beta-\alpha}{r}\right)\cdots\Gamma\left(\frac{\beta- \alpha+r-1}{r}\right)\Gamma\left(p+\frac{\delta}{r}\right)}{2(2\pi)^{r-1} \Gamma(p)\Gamma(\alpha)\Gamma(\beta)(\delta_{1}^{2}-\alpha)\alpha q^{\frac{c ^{2}}{r}}}\]  
\[\Theta_{3}=\frac{\zeta_{1}^{2}r^{\alpha+\beta-2\beta}(\eta_{1}\alpha\beta)^{ \beta}\Gamma\left(\frac{\alpha-\beta}{r}\right)\cdots\Gamma\left(\frac{\beta -\delta+r-1}{r}\right)\Gamma\left(\frac{1}{r}\right)\cdots\Gamma\left(\frac{ r}{r}\right)\Gamma\left(p+\frac{\beta}{r}\right)}{2(2\pi)^{r-1}\Gamma(p)\Gamma( \alpha)\Gamma(\beta)(\delta_{1}^{2}-\beta)\beta q^{\frac{c^{2}}{r}}}\]  
Similarly, when \(\mu_{r}=\rho_{r}=\overline{\gamma}\rightarrow\infty\), the asymptotic representation of \(P_{e,2}\) can be written as:  
\[P_{e,2}^{\infty}=\frac{\Theta_{4}}{\overline{\gamma}^{\frac{c^{2}}{r}}}+\frac {\Theta_{5}}{\overline{\gamma}^{\frac{c^{2}}{r}}} \tag{42}\]  
where  
\[\Theta_{4}=\frac{r^{\alpha-\frac{1}{2}-\frac{c^{2}}{c}}c^{p-\frac{1}{2}+\frac{ c^{2}}{r}}\Gamma\left(\frac{\alpha-\delta_{2}^{2}/c}{r}\right)\cdots\Gamma\left( \frac{\alpha-\delta_{2}^{2}/c+r-1}{r}\right)\Gamma\left(1-\frac{1-p}{c}+\frac {\delta_{2}^{2}}{\alpha}\right)\cdots\Gamma\left(1-\frac{c-p}{c}+\frac{\delta _{2}^{2}}{\alpha}\right)}{2(2\pi)^{\frac{\alpha+r-2}{2}}\Gamma(p)\Gamma(a)(bA_{0 })^{\delta_{2}^{2}}q^{\frac{c^{2}}{r}}}\]  
\[\Theta_{5}=\frac{\zeta_{2}^{2}r^{\frac{1}{2}}c^{p-\frac{1}{2}+\frac{c^{2}}{r}} \Gamma\left(\frac{1}{r}\right)\cdots\Gamma\left(\frac{r}{r}\right)\Gamma \left(1-\frac{1-p}{c}+\frac{a}{r}\right)\cdots\Gamma\left(1-\frac{c-p}{c}+\frac {a}{r}\right)}{2(2\pi)^{\frac{\alpha+r-2}{2}}\Gamma(p)\Gamma(a)(\delta_{2}^{2}- ac)\left(bA_{0}\right)^{\alpha}c^{\alpha}q^{\frac{c^{2}}{r}}}\]Therefore, combining Equations (41) and (42), when \(\mu_{r}=\rho_{r}=\overline{\gamma}\rightarrow\infty\), \(P_{\varepsilon}\) can be asymptotically expressed as:  
\[P_{e}^{\infty}\rightarrow\frac{\Theta_{1}}{\overline{\gamma}\overline{\gamma} }+\frac{\Theta_{2}}{\overline{\gamma}\overline{\gamma}}+\frac{\Theta_{3}}{ \overline{\gamma}\overline{\gamma}}+\frac{\Theta_{4}}{\overline{\gamma} \overline{\gamma}}+\frac{\Theta_{5}}{\overline{\gamma}\overline{\gamma}}+ \frac{\Theta_{5}}{\overline{\gamma}\overline{\gamma}} \tag{43}\]  
Similar to the OP, when the detection technique is fixed, the BER is also determined by the magnitude of pointing errors of the two hops and the parameters characterizing atmospheric and oceanic turbulence.