# Modeling of Satellite-to-Underwater Integrated FSO-PON System Using NOMA-VLC
## 3 Numerical Analysis
### BER Performance Analysis
However, in the presence of turbulence regimes, the mean SNR\(\langle SNR\rangle\) can be presented as [39]:  
\[\langle SNR\rangle=\frac{SNR_{0}}{\sqrt{\frac{P_{0}}{\langle R\rangle}+k_{0}^{ 2}(D_{r})SNR_{0}^{2}}} \tag{9}\] \begin{table}
\begin{tabular}{l l l l l} \hline \hline
**Type of Water** & \(\lambda\) **(nm)** & \(\alpha(\lambda)\) **(m\({}^{-1}\))** & \(\beta(\lambda)\) **(m\({}^{-1}\))** & \(y(\lambda)\) **(m\({}^{-1}\))** \\ \hline Clear ocean & 450–500 & 0.12 & 0.05 & 0.16 \\ Coastal ocean & 520–570 & 0.19 & 0.23 & 0.35 \\ Turbid harbor & 550–600 & 0.38 & 1.84 & 2.21 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Conventional values of scattering as well as absorption in distinct kinds of water [38]. where \(P_{t0}\) is the signal optical power without turbulence and \(\langle P_{t}\rangle\) implies the mean of \(P_{t}\). Moreover, the error probability is described as [39]:  
\[P_{e}=P(1|0)P(0)+P(0|1)P(1) \tag{10}\]  
Where \(P(0)\) and \(P(1)\) mean the binary '0' and '1' probability, respectively. Again, \(P(0|1)\) and \(P(1|0)\) imply the conditional probabilities. Assuming the transmitter is sending '0's and '1's with the alike probability of \(\frac{1}{2}\), \(P_{e}\) is written as [39]:  
\[P_{e}=\frac{1}{2}P(1|0)+P(0|1) \tag{11}\]  
Also, the Gaussian distribution for both binary '0' and '1' signals along with noise, the \(P_{e}\) or BER without turbulence is [39]:  
\[BER_{0}=\frac{1}{2}\bigg{(}\frac{SNR_{0}}{2\sqrt{2}}\bigg{)} \tag{12}\]  
Though, without taking turbulence regimes, FSO link BER is [39]:  
\[BER_{FSO}=\frac{1}{2}\int_{0}^{\infty}p_{I}(s)\text{erfc}\bigg{(}\frac{\langle SNR \rangle s}{2\sqrt{2}\langle i_{s}\rangle}\bigg{)}ds \tag{13}\]  
where \(p_{I}(s)\) implies irradiance probability distribution with a signal strength of \(s\). Again, \(\langle i_{s}\rangle\) is the signal current mean value. By using Meijer G-function using \(e^{z}=G_{1,0}^{1,0}(-z|0)\) expression, erfc in Equation (13) is given as [40]:  
\[\text{erfc}(y)=\frac{1}{\sqrt{\pi}}G_{1,2}^{2,0}\big{[}y|0,\frac{1}{2}\big{]} \tag{14}\]  
and for the FSO, the link average BER is expressed as:  
\[BER_{FSO}=\frac{1}{2}\int_{0}^{\infty}f_{h_{a}}(h_{a})\;erfc\bigg{(}\frac{ \langle SNR\rangle s}{2\sqrt{2}\langle i_{s}\rangle}\bigg{)}ds \tag{15}\]  
Furthermore, the BER performance of TWDM-PON is represented as [41]:  
\[BER_{TWDM-PON}=\frac{1}{2}erfc\bigg{(}\frac{SNR_{0}}{\sqrt{2}}\bigg{)} \tag{16}\]  
Finally, the BER performance of integrated PON-FSO utilizing VLC link can be evaluated as [41]:  
\[BER_{t}=BER_{TWDM-PON}+(1-C_{r})BER_{FSO} \tag{17}\]  
where \(C_{r}\) represents the channel correlation coefficient from 0 and 1, which is utilized to determine the performance of the system for distinct channels reception at the PD receiver end for distinct turbulence conditions. As \(C_{r}\) increases, the system output diminishes because of channels overlapping at the reception side. From Equations (7), (15) and (16) for the proposed system, the total BER with GG fading for FSO channels with \(C_{r}=0.1\) is evaluated as [42]:  
\[BER_{Total}=\frac{1}{2}erfc\bigg{(}\frac{SNR_{0}}{\sqrt{2}}\bigg{)}+\frac{1}{ 2}\int_{0}^{\infty}f_{h_{a}}(h_{a})\;erfc\bigg{(}\frac{\langle SNR\rangle s}{2 \sqrt{2}\langle i_{s}\rangle}\bigg{)}ds \tag{18}\]