# BICM-ID with Physical Layer Network Coding in TWR Free Space Optical Communication Links +
## 4 Simulation Results
### Convergence Behaviour  
At the commencement of the BICM iterative decoding process, the TWR-FSO PNC systems were presented as the ExIT function. The main idea when utilizing the ExIT chart is to track the pdf of information messages that are exchanged in the iterative decoding process, where the decoder uses the Bahl--Cocke--Jelinek--Ravivi (BCJR) algorithm. Furthermore, links have been made between ExIT charts and the corresponding BER curves. The ExIT chart provides an accurate prediction of the convergence behaviour, which is verified and visualized by the simulated decoding trajectory. Consequently, ExIT charts visualize the density evolution of the extrinsic LLRs over the iterations using a single parameter of the average mutual information between the coded bits at the transmitter and the LLRs at the receiver. To predict the evolution of the investigated parameter, each SISO receiver component has close to ideal a priori information from the other receiver components. To evaluate and optimize the convergence behaviour of iterative receivers, a number of blocks is simulated for different values of \(\sigma^{2}\) in order to estimate the output mutual information for the decoder. When natural (anti-Gray) mapping is utilized in the transfer function of the demapper, the demapper is affected by log-normal and gamma-gamma channel models as it depends on the SNR in terms of \(E_{b}/N_{0}\). Therefore, the demapper transfer characteristics almost approach non-convex shapes depending on the SNR values. Figure 11 shows the ExIT chart function when the channel is log-normal for 0, 2, 4 and 6 dB of SNR values, whereas Figure 12 shows the ExIT chart function when the channel is gamma-gamma for 0, 2, 4, 6 and 8 dB of SNR values. Observing that the transfer characteristic has a steep slope, it can be seen that it yields a strong potential for performance improvement in the PNC with BICM-ID system. Moreover, a decrease/increase in the value of SNR will shift the ExIT curve down/up, respectfully. The ExIT function of the outer code is depicted in Figure 13, which shows the transfer characteristics of code specification. For example, the stronger code of a constraint length of nine has a steeper ExIT function than the weaker code of a constraint length of three and therefore needs lower mutual information at the input to produce high quality LLRs at the output. Furthermore, the decoder curve shows the relationship between the a priori information and predicts the BER for the system by using the ExIT function, as shown in Figure 14. As part of the research to assess the effectiveness of the BICM-ID TWR-FSO PNC system, we can observe that the simulation results along with the Monte Carlo simulation match with the exact analytical results of the ExIT charts. However, the system with the strong turbulence condition affects the convergence behaviour more than the weak turbulence condition. Therefore, we evaluated the \(I_{E}\) with respect to the range of SNR by Monte Carlo simulation based on the demapper TWR-FSO PNC system under the weak and strong turbulence model. More specifically, in Figures 15 and 16, there is a one-to-one correspondence between SNR and \(I_{E}\) in the demapper TWR-FSO PNC system. The relationship between the mutual information  
Figure 10: BER performance of TWR-FSO with PNC over the gamma-gamma channel when employing uncoded, E2E convolutional code and E2E BICM-ID codes. and the variance of LLRs was formally described in [27]. With an increase in SNR, we increase the knowledge of information in the mutual information of \(I_{E}\). Therefore, the trajectory of the iteration system is evaluated by the progress of the data components in terms of mutual information, and both characteristic functions can be plotted in a single figure. Consequently, in the graph, the convergence of curves for the system gives the approaches between the demapper and the inverted ExIT function of the decoder. Hence, the trajectory of the system can be predicted as the stair case between the two curves of the demapper and decoder.