# Blind Detection for Serial Relays in Free Space Optical Communication Systems
## 4 Numerical Results and Discussion  
In the obtained simulation results, \(10^{8}\) bits were transmitted for each depicted SNR value and the Hermite polynomial order was \(n\leq 50\).  
Now, we evaluate the ABER performance of the dual hop blind detection FSO system for various locations of the relay. The ABER performance was calculated over weak and moderate turbulence in order to investigate the degradation effects. The direct link simulation results, investigated in [13], were used as a benchmark to show the performance difference between the direct link and the proposed model. For the direct link, the power value was multiplied by two to achieve a fair comparison [6]. Table 2 shows the system parameters under investigation. Table 3 shows the standard deviation values with the multiple scenarios under weak and moderate atmospheric turbulence.  
The derived analytical results were verified with Monte Carlo simulations. Figures 2-4 show the weak turbulence. Figure 5 shows the combined analytical values for moderate turbulence. Figures 2-5  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Symbol** & **Value** \\ \hline \(A_{r}\) & \(2\pi(0.1)^{2}\) \\ \hline \(\beta_{0}\) & \(0.1\,\text{km}^{-1}\) \\ \hline \(C_{n}^{2}\) & \(\begin{array}{c}0.5\times 10^{-14}\text{m}^{-2/3}\\ 2\times 10^{-14}\text{m}^{-2/3}\end{array}\) \\ \hline \(d_{0}\) & \(1\) km \\ \hline \(\theta\) & \(1\) mrad \\ \hline \hline \end{tabular}
\end{table}
Table 2: System configuration.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Atmospheric** & **Link Type** & \
\begin{tabular}{c} **Standard Deviation of** \\ **the First Hop** \\ \end{tabular} &
\begin{tabular}{c} **Standard Deviation of** \\ **the Second Hop** \\ \end{tabular} \\ \hline \multirow{3}{*}{**Weak Turbulence**} & Direct & 0.1572 \\ \cline{2-4}  & Dual-hop 500-500 & 0.0833 & 0.0833 \\ \cline{2-4}  & Dual-hop 600-400 & 0.0984 & 0.0679 \\ \cline{2-4}  & Dual-hop 400-600 & 0.0679 & 0.0984 \\ \hline \multirow{3}{*}{**Moderate Turbulence**} & Direct & 0.3145 & \\ \cline{2-4}  & Dual-hop 500-500 & 0.1666 & 0.1666 \\ \cline{2-4}  & Dual-hop 600-400 & 0.196 & 0.1358 \\ \cline{2-4}  & Dual-hop 400-600 & 0.1358 & 0.196 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Standard deviation values.  
show that the ABER decreased when a relay is added due to decreasing the distance. When comparing the figures with weak turbulence, Figure 2 with Figures 3 and 4, it can be seen that the best location for the relay is in the middle of the channel. We have tried all the ranges for the relay, which showed that the equidistant relay is the best location. Figure 5 shows that the best location for the relay for moderate turbulence is also in the middle of the channel.  
Figure 3: ABER performance for the direct link and a single relay after 400 m from the source, over weak turbulence.  
Figure 2: ABER performance for the direct link and a single equidistant relay, over weak turbulence.  
The generated analytical results are comparable to the CSI results as shown in Figures 2-4. The generated simulation results show the accuracy of the analytical results in Figures 2-4. Figure 2 shows an enhancement of around 10 dB between the direct link and using the relay.  
Figures 3 and 4 also show an improvement of the proposed relay system by 6 dB. Figure 2 shows the equidistant relay outperforms its counterparts in Figures 3 and 4 by 3.5 dB.  
Figure 4: ABER performance for the direct link and a single relay after 600 m from the source, over weak turbulence.  
Figure 5: ABER performance for the direct link and a single relay with different locations, over moderate turbulence.  
As for the moderate turbulence, shown in Figure 5, an enhancement of 17 dB between the direct link and the equidistant relay can be seen. An enhancement of 12.5 dB was also obtained between the direct link and the other relays (600 m-400 m and 400 m-600 m).