# O2O: An Underwater VLC Approach in Baltic and North Sea
## 3 The Channel Condition Model
For the two different water mediums, the PDF of turbulence conditions are defined as follows with \(i\in(N,B)\)[24],  
\[f_{t}(I_{t})=\frac{\prod_{i=1}^{2}\Phi_{\alpha_{i}\beta_{i}}}{\prod_{i=1}^{2} \Phi_{\Gamma(\alpha_{i})\Gamma(\beta_{i})}}G_{0,4}^{4}\left[\prod_{i=1}^{2} \Phi_{\alpha_{i}\beta_{i}}I_{t}\left| \begin{array}{c}...\\ \mathbf{a}_{NB}\end{array} \right.\right]. \tag{8}\]  
In (8), we have \(\prod_{i=1}^{2}\Phi_{\alpha_{i}\beta_{i}}=\alpha_{N}\alpha_{B}\beta_{N}\beta_ {B}\), and \(\prod_{i=1}^{2}\Phi_{\Gamma(\alpha_{i})\Gamma(\beta_{i})}=\Gamma(\alpha_{N}) \Gamma(\alpha_{B})\Gamma(\beta_{N})\Gamma(\beta_{B})\) along with the argument \(\mathbf{a}_{NB}=[\alpha_{N}-1,\alpha_{B}-1,\beta_{N}-1,\beta_{B}-1]\). Further, replacing the average SNR and implying the changes of variable method, the PDF of (8) can be written as,  
\[f_{\gamma}(\gamma)=\frac{\prod_{i=1}^{2}\Phi_{\alpha_{i}\beta_{i}}}{2\sqrt{ \gamma\mu}\prod_{i=1}^{2}\Phi_{\Gamma(\alpha_{i})\Gamma(\beta_{i})}}G_{0,4}^{4,0}\left[\prod_{i=1}^{2}\Phi_{\alpha_{i}\beta_{i}}\sqrt{\frac{\gamma}{\mu}} \left| \begin{array}{c}...\\ \mathbf{a}_{NB}\end{array} \right.\right]. \tag{9}\]  
In the more convenient comparative study for isolated water medium, there is some mathematical modification in (9). We can formulate (10) for a single layer (SL), which can be written as,  
\[f_{\gamma}^{\mathrm{SL}}(\gamma)=\frac{\alpha\beta}{2\sqrt{\gamma\mu\Gamma}( \alpha)\Gamma(\beta)}G_{0,2}^{2,0}\left[\alpha\beta\sqrt{\frac{\gamma}{\mu}} \left| \begin{array}{c}...\\ \alpha-1,\beta-1\end{array} \right.\right]. \tag{10}\]To obtain the CDF of (9) by integrating the PDF expression as \(F_{\gamma}(\gamma)=\int\limits_{0}^{\gamma}f_{\gamma}(\gamma)d\gamma\). We utilize the properties of Wolfram Mathmatica (Equation 07.34.21.0084.01 in [25]), and we obtained (11) as follows,  
\[F_{\gamma}(\gamma)=z\sqrt{\frac{\gamma}{\mu}}G_{1,9}^{8,1}\left(\kappa\left( \frac{\gamma}{\mu}\right)\Bigg{|} \begin{array}{c}\frac{1}{2}\\ \phi\end{array} \right), \tag{11}\]  
where \(z\), \(\kappa\), and \(\phi\) can be written as,  
\[z=\frac{\left(\sum\limits_{i=1}^{2}(\alpha_{i}+\beta_{i})\right)-7\left(\prod \limits_{i=1}^{2}\Phi_{\alpha_{i}\beta_{i}}\right)}{2\pi^{2}\prod\limits_{i=1} ^{2}\Phi_{\Gamma(\alpha_{i})\Gamma(\beta_{i})}}\]  
\[\kappa=\left(\frac{\prod\limits_{i=1}^{2}\Phi_{\alpha_{i}\beta_{i}}}{16} \right)^{2},\]  
\[\phi=\left[\frac{\alpha_{N}-1}{2},\frac{\alpha_{N}}{2},\frac{\alpha_{B}-1}{2},\frac{\alpha_{B}}{2},\frac{\beta_{N}-1}{2},\frac{\beta_{N}}{2},\frac{\beta_{ B}-1}{2},\frac{\beta_{B}}{2},-\frac{1}{2}\right].\]  
Similarly, the CDF of (10) could be written as for single layer,  
\[F_{\gamma}^{SL}(\gamma)=\frac{2^{(\alpha+\beta-3)}\alpha\beta}{2\pi\Gamma( \alpha)\Gamma(\beta)}\sqrt{\frac{\gamma}{\mu}}G_{1,5}^{4,1}\Bigg{[}\frac{ \alpha^{2}\beta^{2}\gamma}{16\mu}\Bigg{|} \begin{array}{c}\frac{1}{2}\\ \frac{\alpha-1}{2},\frac{\alpha}{2},\frac{\beta-1}{2},\frac{\beta}{2},\frac{1 }{2}\end{array} \Bigg{]}. \tag{12}\]  
_Analysis of Average BER Calculation_  
The error probability under OOK modulation scheme is defined as follows: [26],  
\[P_{b}=\frac{1}{2}erfc\sqrt{\frac{SNR(\gamma)}{2}}. \tag{13}\]  
In (13), replacing the value of SNR, and further utilizing the properties of Wolfram Mathematica (Equation 06.27.26.0006.01 in [25]), the modified equation can be written as,  
\[P_{b}=\frac{1}{2\sqrt{\pi}}G_{1,2}^{2,0}\Bigg{(}\sqrt{\frac{\mu|I|^{2}}{2}} \Bigg{|} \begin{array}{c}1\\ 0,\frac{1}{2}\end{array} \Bigg{)}. \tag{14}\]  
For ABER performance calculation of the whole system under consideration IM/DD technique, the ABER for expression is written as,  
\[\overline{BER}_{\text{ook}}=\int\limits_{0}^{\infty}P_{b}f_{t_{b}}(I_{t})dI_{t}. \tag{15}\]Furthermore, replacing the values from (8) and (14) into (15), we obtained (16) as  
\[