# 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  
The average BER of a single link in a system under binary modulation can be represented as [35]:  
\[P_{e}=\int_{0}^{\infty}P_{e}(\epsilon|\gamma)f(\gamma)d\gamma=\int_{0}^{\infty }P_{e}(\epsilon|\gamma)dF(\gamma) \tag{35}\]  
in the above expression, \(P_{e}(\epsilon|\gamma)=\frac{\Gamma(p,q\cdot\gamma)}{2\Gamma(p)}\) represents the conditional error rate given in the SNR \(\gamma\), where \(\Gamma(p,q\cdot\gamma)\) denotes the upper incomplete Gamma function. Parameters \(p,q\) are modulation-related factors, with \(p=1/2,q=1/2\) for On-Off Keying (OOK) modulation. Applying integration by parts to (35) and utilizing \(\frac{\partial\Gamma(a,z)}{\partial z}=-z^{a-1}e^{-z}\)[36] (Equation 6.5.25), the average BER simplifies to:  
\[P_{e}=\frac{q^{p}}{2\Gamma(p)}\int_{0}^{\infty}F(\gamma)\gamma^{p-1}e^{-q \gamma}d\gamma \tag{36}\]  
In the case of dual-hop DF relaying, it is straightforward to derive the end-to-end system BER as:  
\[ \begin{array}{ll}P_{e}&=P_{e,1}(1-P_{e,2})+(1-P_{e,1})P_{e,2}\\ &=P_{e,1}+P_{e,2}-2P_{e,1}P_{e,2}\end{array}  \tag{37}\]  
where \(P_{e,1}\) represents the average BER of the S-R link, and \(P_{e,2}\) represents the average BER of the R-D link. And, when \(\mu_{r}=\rho_{r}=\overline{\gamma}\rightarrow\infty\), \(P_{e}\) can be asymptotically expressed as:  
\[P_{e}^{\infty}\approx P_{e,1}^{\infty}+P_{e,2}^{\infty} \tag{38}\]  
In this expression, \(P_{e,1}^{\infty}\) and \(P_{e,2}^{\infty}\) denote the asymptotic values of \(P_{e,1}\) and \(P_{e,2}\) when \(\overline{\gamma}\rightarrow\infty\). Substituting Equations (14) and (25) into Equation (36), using the identity \(\mathrm{e}^{-q\gamma}=G_{0,1}^{1,0}\left|q\gamma\right|^{-}_{0}\)(29)(Equation 8.4.3.1), and simplifying using [29](Equations 8.4.3.1, 2.24.1.1, 8.2.2.9, and 8.2.2.8), the BER for the S-R link can be written as:  
\[P_{e,1}=\frac{\delta_{1}^{2}r^{\alpha+\beta-2}}{2(2\pi)^{r-1}\Gamma(p)\Gamma( \alpha)\Gamma(\beta)}G_{3,2r+2}^{2r+12}\left[\frac{(\eta_{1}\alpha\beta)^{r}}{ \mu_{r}qr^{2r}}\left| \begin{array}{c}1,1-p;\frac{\delta_{1}^{2}+r}{r}\\ \frac{\delta_{1}^{2}}{r},\mathrm{M}_{1},\mathrm{M}_{2};0\end{array} \right.\right] \tag{39}\]  
where \(\mathrm{M}_{1}=\left\{\frac{\alpha}{r},\ldots,\frac{\alpha+r-1}{r}\right\}\), \(\mathrm{M}_{2}=\left\{\frac{\beta}{r},\ldots,\frac{\beta+r-1}{r}\right\}\) are simplified notations for vectors, with each vector containing \(r\) terms. Similarly, the average BER of the R-D link can be formulated as:  
\[P_{e,2}=\frac{\delta_{2}^{2}r^{\alpha-\frac{3}{2}}c^{p-\frac{3}{2}}c^{p-\frac {1}{2}}}{2c(2\pi)^{\frac{\alpha+r-2}{2}}\Gamma(p)\Gamma(a)}G_{c+2r+2}^{r+12} \left[\frac{c^{c}}{(bA_{0})^{rc}(p_{r}q)^{rc}r^{\prime}}\left| \begin{array}{ c}1,\mathrm{M}_{3};\frac{\delta_{2}^{2}/c+r}{r}\\ \frac{\delta_{2}^{2}/c}{r},\mathrm{M}_{4};0\end{array} \right.\right] \tag{40}\]  
where \(\mathrm{M}_{3}=\left\{\frac{1-p}{c},\ldots,\frac{c-p}{c}\right\}\), \(\mathrm{M}_{4}=\left\{\frac{a}{r},\ldots,\frac{a+r-1}{r}\right\}\), each vector contains \(r\) terms. By substituting Equations (39) and (40) into Equation (37), the specific closed-form expression for the end-to-end average \(P_{e}\) can be obtained. Similarly, when the average links with SNR, i.e., \(\mu_{r}=\rho_{r}=\overline{\gamma}\rightarrow\infty\), the system's average BER also has an asymptotic value.