# Performance of Adaptive Bit-Interleaved Polar Coded Modulation in FSOC System
## 2 Adaptive Bit-Interleaved Polar Coded Modulation (A-BIPCM)
This paper divides the transmission process into blocks, with each block spanning a duration of 2 s. Different interleaving methods are selected for each channel block based on the intensity of atmospheric turbulence. However, it should be considered that interleaving introduces additional delay and complexity to the system. Hence, a threshold judgment is necessary to determine whether interleaving should be applied. The average SNR is recorded using the coherence time \(\tau_{0}\) interval and is calculated from the SNR \(\overline{\gamma}\). The following formula represents the SNR of a signal with interference noise. It is calculated by comparing the received signal with the signal generated by the transmitter. \[SNR_{\text{sig}}=\frac{|\text{r}_{AB}(t)|^{2}}{\left|\text{r}_{AB}(t)-\left( \rho_{A}S(t)+n_{A}(t)\right)\right|^{2}} \tag{8}\]  
\(\left|\text{r}_{AB}(t)\right|^{2}\) is the total power of the received signal and represents the squared amplitude of the signal. \(\left|\text{r}_{AB}(t)-\left(\rho_{A}S_{A}(t)+n_{A}(t)\right)\right|^{2}\) represents the power of noise and interference signals, which represents the resistance of the signal source to noise and interference signals. \(\text{r}_{AB}(t)\) represents the signal that reaches the receiving end B after being transmitted through the channel. \(\rho_{A}\) is the amplification coefficient of the signal. \(\rho_{A}S_{A}(t)\) is the signal of the signal source at the transmitting end, \(n_{A}(t)\) is the received noise signal. By determining specific threshold conditions, it is possible to assess whether interleaving operations are necessary to optimize system performance. For the received signal, the SNR level of the current channel block is estimated to evaluate its quality. Within the segment, the average signal-to-noise ratio \(\overline{\gamma}\) is recorded every coherence time \(\tau_{0}\), and the signal-to-noise ratio threshold is set to \(\overline{\gamma}_{1}=8\) dB. The signal-to-noise ratio judgment principle is as follows:  
\[ \begin{cases}\overline{\gamma}<\overline{\gamma}_{1}=8\text{ dB},\text{ Marked as }1\\ \overline{\gamma}\geq\overline{\gamma}_{1}=8\text{ dB},\text{ Marked as }0\end{cases}  \tag{9}\]  
If the average SNR over a certain coherence time \(\tau_{0}\) is lower than the SNR threshold \(\gamma_{1}\), the corresponding sub-segment is marked as having a bit error. Conversely, if the average SNR exceeds the threshold, for the number of \(2/\tau_{0}\) times that needs to be detected, the sub-segment is considered error-free. In this work, the number of times that the threshold condition is met and the sub-segment is marked as having a bit error is denoted as \(Q\). Consequently, a relationship can be established between the total number of detections and the number of bit errors in the segment. \[\left\{ \begin{aligned} P_{1}=\frac{Q}{\tau_{0}}=\frac{1}{2}Q \tau_{0}\geq P_{th},\text{ Use interweaving}\\ P_{1}=\frac{Q}{\tau_{0}}=\frac{1}{2}Q\tau_{0}<P_{th},\text{ No interleaving}\\ P_{1}+P_{0}=1\end{aligned} \right. \tag{10}\]  
The decision to use interleaved polar codes to enhance error performance is based on the cumulative error detection probability. If the cumulative error detection probability surpasses a certain threshold \(P_{th}\), interleaving operations are employed. On the other hand, if the cumulative error detection probability does not exceed the threshold \(P_{th}\), there is no need for interleaving, and the received demodulation sequence can be directly passed to the decoding module for error correction. The selection of the threshold can be adjusted according to specific requirements, aiming to strike a balance between achieving optimal performance and managing system complexity. By fine-tuning the threshold, the trade-off between error correction capabilities and computational overhead can be optimized. Interleaved polar codes are constructed recursively through the structure \(x+y|y\).