# Performance Analysis of Dual-Hop Hybrid RF-UOWC NOMA Systems
## Appendix A  
As the constraints in (29b) and (29c) are convex, consequently, we need to prove that the objective function (29a) is convex. Equation (14) is rewritten as  
\[OP_{\text{sys}}=1-(1-\theta_{1}\tau)(1-\theta_{2}\tau^{\frac{1}{r}}-\theta_{3} \tau^{\frac{\varpi}{r}})(1-\theta_{2}\tau_{1}^{\frac{1}{r}}-\theta_{3}\tau_{1} ^{\frac{\varpi}{r}}), \tag{16}\]  
where \(\theta_{1}=\frac{1}{\rho_{2}d^{-\varpi}}\), \(\theta_{2}=\frac{\varpi}{\Lambda}(\frac{1}{\rho_{2}R\mu})^{\frac{1}{r}}\), and \(\theta_{3}=\frac{1-\varpi}{\Gamma(a+1)}\left(\frac{1}{\rho_{2}R\mu}\right)^{ \frac{\varpi}{r}}\). As \(\tau=\max(\tau_{1},\tau_{2})\), we can write (16) as a piece-wise function based on the value of \(a_{1}\) as  
* For \(\tau_{1}>\tau_{2}\) or \(\frac{\gamma_{1}}{1+\gamma_{1}}<a_{1}<\frac{\gamma_{1}(1+\gamma_{2})}{\gamma_ {1}+\gamma_{2}+\gamma_{1}\gamma_{2}(1+\eta)}\), we rewrite (16) as \[OP_{\text{sys}}=1-(1-\theta_{1}\tau_{1})(1-\theta_{2}\tau_{1}^{\frac{1}{r}}- \theta_{3}\tau_{1}^{\frac{\varpi}{r}})^{2}\] (17)  
To differentiate \(OP_{\text{sys}}\) with respect to \(a_{1}\), we used the chain rule, \(\frac{\partial OP_{\text{sys}}}{\partial a_{1}}=\frac{\partial OP_{\text{sys }}}{\partial\tau_{1}}\times\frac{\partial\tau_{1}}{\partial a_{1}}\), which result in  
\[ \begin{split}&\frac{\partial OP_{\text{sys}}}{\partial a_{1}}=- \frac{(1+\gamma_{1})\gamma_{1}}{(a_{1}(1+\gamma_{1})-\gamma_{1})^{2}}[\theta_ {1}(1-\theta_{2}\tau_{1}^{\frac{1}{r}}-\theta_{3}\tau_{1}^{\frac{\varpi}{r}}) ^{2}\\ &+2(1-\theta_{1}\tau_{1})(1-\theta_{2}\tau_{1}^{\frac{1}{r}}- \theta_{3}\tau_{1}^{\frac{\varpi}{r}})(\frac{\theta_{2}}{r}\tau_{1}^{\frac{1 -r}{r}}+\frac{ac\theta_{3}}{r}\tau_{1}^{\frac{\varpi-r}{r}})],\end{split}  \tag{18}\]  
which is always negative valued, this result indicates a monotonically decreasing function in this interval. * For \(\tau_{1}<\tau_{2}\) or \(\frac{\gamma_{1}(1+\gamma_{2})}{\gamma_{1}+\gamma_{2}+\gamma_{1}\gamma_{2}(1 +\eta)}<a_{1}<\frac{1}{1+\eta\gamma_{2}}\), we rewrite (16) as \[OP_{\text{sys}}=1-\underbrace{(1-\theta_{1}\tau_{2})}_{\psi_{1}}\underbrace{ (1-\theta_{2}\tau_{2}^{\frac{1}{r}}-\theta_{3}\tau_{2}^{\frac{\varpi}{r}})} _{\psi_{2}}\underbrace{(1-\theta_{2}\tau_{1}^{\frac{1}{r}}-\theta_{3}\tau_{1 }^{\frac{\varpi}{r}})}_{\psi_{3}}.\] (19)To differentiate \(OP_{sys}\) with respect to \(a_{1}\) in this interval, we use the chain rule, \(\frac{\partial OP_{sys}}{\partial a_{1}}=\frac{\partial OP_{sys}}{\partial T_{2}} \times\frac{\partial T_{2}}{\partial a_{1}}+\frac{\partial OP_{sys}}{\partial a _{1}}\times\frac{\partial T_{1}}{\partial a_{1}}\), as \(\tau_{1}\) and \(\tau_{2}\) are independent. Using this rule, we obtain  
\[ \begin{split}\frac{\partial OP_{sys}}{\partial a_{1}}& =\theta_{1}\frac{(1+\eta\tau_{2})\tau_{2}}{(1-a_{1}(1+\eta\tau_{ 2}))^{2}}\psi_{2}\psi_{3}\\ &+(\frac{\theta_{2}}{r}\tau_{2}^{\frac{1+r}{2}}+\frac{ac\delta_{3 }}{r}\tau_{2}^{\frac{1+r}{2}})\frac{(1+\eta\tau_{2})\tau_{2}\psi_{1}\psi_{3}}{ (1-a_{1}(1+\eta\tau_{2}))^{2}}\\ &+(\frac{\theta_{2}}{r}\tau_{1}^{\frac{1+r}{2}}+\frac{ac\delta_{3 }}{r}\tau_{1}^{\frac{1+r}{2}})\frac{(1+\gamma_{1})\tau_{1}\psi_{1}\psi_{2}}{ (a_{1}(1+\gamma_{1})-\gamma_{1})^{2}},\end{split}  \tag{10}\]  
which is clear that \(\frac{\partial OP_{sys}}{\partial a_{1}}\) is always positive valued. This result indicates a monotonically increasing function in this interval. Now, we can observe that \(OP_{sys}\) is monotonically decreasing function in the interval \(\frac{\gamma_{1}}{1+\gamma_{1}}<a_{1}<\frac{\gamma_{1}(1+\gamma_{2})}{\gamma_ {1}+\gamma_{2}+\gamma_{1}\gamma_{2}(1+\eta)}\) with the minimum value at the upper limit of this interval and monotonically increasing function in the interval \(\frac{\gamma_{1}(1+\gamma_{2})}{\gamma_{1}+\gamma_{2}+\gamma_{1}\gamma_{2}(1 +\eta)}<a_{1}<\frac{1}{1+\eta\tau_{2}}\) with the minimum at the lower limit of this interval.