# The BER Performance of the LDPC-Coded MPPM over Turbulence UWOC Channels
## 5 Results and Discussion  
This section demonstrates the effect of LDPC code on alleviating turbulence-induced fading by comparing the BER performance of the coded and uncoded MPPM UWOC system in different fading scenarios modeled by various statistical distributions. An illustration of the system flowchart is given in Figure 2. \begin{table}
\begin{tabular}{c c} \hline \(x_{i}\) & \(w_{i}\) \\ \hline \(0.13779347054049237\) & \(0.308441\) \\ \hline \(0.7294545495031706\) & \(0.40112\) \\ \hline \(1.8083429017403165\) & \(0.218068\) \\ \hline \(3.4014336978549595\) & \(0.0620875\) \\ \hline \(5.552496140063418\) & \(0.00950152\) \\ \hline \(8.330152746764144\) & \(0.000753008\) \\ \hline \(11.843785837899944\) & \(0.0000282592\) \\ \hline \(16.279257831377613\) & \(4.24931\times 10^{-7}\) \\ \hline \(21.99658581198083\) & \(1.83956\times 10^{-9}\) \\ \hline \(29.92069701227372\) & \(9.91183\times 10^{-13}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Zeros of \(L_{10}(x)\) and corresponding weight coefficients. Figure 3 shows the BER and SER of uncoded (5, 2) MPPM for the AWGN-only channel. The simulation results are utilized to obtain the constants \(b_{1}\), \(b_{2}\) and \(b_{3}\) of the conditional PDF \(p(\varepsilon|\gamma)\), which is verified by contrastive analysis. By comparison, the SER, based on the simplified LLRs of Equation (17) is close to, but lower than, that based on finding the largest slots as the symbol slot, i.e., \(SER=0.836\exp(-0.5231\gamma)\), which is obtained using exponential fitting to the analytical SER in [31]. Meanwhile, the corresponding BER, based on simplified LLRs, is not greater than the BER upper bound defined in [27]. Hence, the correctness of the simulation for the system is demonstrated, resulting in \(b_{1}=0.4251\), \(b_{2}=0.5861\) and \(b_{3}=0.9655\). Figure 3: Bit error rate and symbol error rate of the (5, 2) MPPM UWOC system in the absence of fading. Figure 2: The flowchart for studying the effect of turbulence on UWOC using MPPM scheme. Figure 4 shows the BER performance of the (5, 2) MPPM UWOC system in the presence of weak, moderate, and strong turbulences modeled by GG distribution. The channel parameters of GG distribution are obtained from [7] and given in Table 2, where aperture averaging was taken into account, as the experiments were carried out using AAL, and \(\sigma_{I}^{2}=\Gamma(d/p)\Gamma\left(\frac{d+2}{p}\right)/\Gamma^{2}\left(\frac {d+1}{p}\right)-1\) represents the scintillation index. The theoretical BERs without LDPC code are obtained using (20). As expected, the simulation and theoretical results demonstrate excellent matching, validating the mathematical analysis. The BERs increase with increase of the scintillation index \(\sigma_{I}^{2}\) (i.e., turbulence strength). The LDPC code substantially improves the BER performance for all cases. For example, the LDPC coded system can achieve a BER of \(10^{-5}\) at the SNRs of 14.4 dB and 19.9 dB, respectively, at moderate (\(\sigma_{I}^{2}=0.5782\)) and strong (\(\sigma_{I}^{2}=2.0399\)) turbulence. In the absence of the LDPC code, the BERs are higher than \(10^{-2}\) at the SNR of 22 dB, demonstrating significant coding gain. In weak turbulence with \(\sigma_{I}^{2}=0.2073\), a code gain of \(\sim\)12.1 dB is obtained at the BER of \(10^{-4}\). Figure 5 presents the BER performance of (6, 3) MPPM UWOC system over the same turbulence channel, but modeled by different popular statistical distributions, including log-normal, gamma-gamma, EW and GG. Log-normal and gamma-gamma generally model fading induced by weak and moderate/strong turbulence, respectively [44]. Here, the four distributions were used to fit the same experimental data suffering weak fading described in Table 2, Line 2 (the detailed parameters are given in [7], Table IV, Line 5). Uncoded UWOC system shows a similar BER performance at low SNRs irrespective of channel model. A difference in BER performance is observed at higher SNR.