# Performance of Adaptive Bit-Interleaved Polar Coded Modulation in FSOC System
## 2 Adaptive Bit-Interleaved Polar Coded Modulation (A-BIPCM)
\[\alpha=\left[\exp\left(\frac{0.49\sigma_{Rytov}^{2}}{\left(1+1.11\sigma_{Rytov}^{12 /5}\right)^{7/6}}\right)-1\right]^{-1},\beta=\left[\exp\left(\frac{0.51\sigma_{ Rytov}^{2}}{\left(1+0.69\sigma_{Rytov}^{12/5}\right)^{7/6}}\right)-1\right]^{-1} \tag{3}\]  
The progressive state of the atmosphere can be characterized by the Rytov variance [14]:  
\[\sigma_{Rytov}^{2}=1.23C_{n}^{2}k^{7/6}L^{11/6} \tag{4}\]  
The Rytov variance can measure the intensity of atmospheric turbulence, and the atmospheric coherence constant can be obtained based on the Rytov variance:  
\[r_{0}=1.89\left(k^{-5/6}L^{5/6}\sigma_{Rytov}^{2}\right)^{-3/5}=\left(0.423k^{ 2}C_{n}^{2}L\right)^{-3/5} \tag{5}\]  
Adaptive Bit-Interleaving Polar Coding Modulation Method Based on Approximate Error Probability Upper Bound  
In fading channels, atmospheric turbulence can cause a long string of bit errors, leading to an increased system transmission BER. To address this issue, this paper proposes a new method of adaptive bit-interleaved polar coding modulation (A-BIPCM) The encoded sequence is sent to the interleaver, and the polarization spectrum (PS) and the pairwise error probability (PEP) upper bound are used to study the error correction code scheme. An upper bound on the average bit error probability of the coding system is obtained, and the interleaver depth is varied based on a numerical evaluation of the performance bound. The core principle of A-BIPCM is to adaptively adjust the parameters of bit-interleaving and polar coding based on these error probability upper bounds. The introduction of A-BIPCM effectively mitigates the interference caused by a long string of bit errors induced by atmospheric turbulence, thereby enhancing the reliability and performance of the system; this work designs an interleaver tailored to different turbulence intensities. It upsets and changes the order of the original binary coding sequence, interleaving the erroneous bits between the information bits. Additionally, the original binary coding sequence is split into corresponding positions of the error correction code group, resulting in an interleaved sequence group. The average PEP is a measure of the distance between all-zero codewords and non-zero codewords in a polar code. It quantifies the likelihood of pairwise error events occurring between different codewords. In this work, the average PEP is denoted by the symbol \(P_{N}^{(i)}(d)\). The polar spectrum \(A_{N}^{(i)}(d)\) is a collection of weighted enumerators, each associated with a specific Hamming weight [15]. It is used to count the number of pairwise error events and evaluate the performance of polar subcodes. The polar spectrum is defined as a collection of weight distributions \(\left\{A_{N}^{(i)}(d)\right\}\), \(d\in[1,N]\), where \(d\) represents the Hamming weight of non-zero codewords. Considering a code length of N and assuming transmission over a fading channel, the minimum weight enumerator is used to approximate the upper bound mentioned above. The MLUW metric is expressed by the following formula [16]:  
\[MLUW_{N}^{(i)}=L_{N}^{(i)}\left(d_{\min}^{(i)}\right)-d_{\min}^{(i)}\ln\frac{E _{S}}{N_{0}} \tag{6}\]  
In this work, the upper bound of the minimum Hamming distance of the polar subcode \(D_{N}^{(i)}\) is denoted by the symbol \(d_{\min}^{(i)}\). For a given polar subcode \(D_{N}^{(i)}\), under a specific SNR \(\gamma=E_{S}/N_{0}\gg 1\), given an index set \(\mathcal{A}\subseteq[1,N]\). Given the fixed configuration \((N,K,\mathcal{A})\), the approximate BER upper bound of the polar code in the fading channel is expressed as \(P_{e}(N,K,\mathcal{A})\)[17]:  
\[P_{e}(N,K,\mathcal{A})\leq\frac{1}{2}\sum_{i\in\mathcal{A}}\mathcal{A}_{N}^{(i )}\left(d_{\min}^{(i)}\right)\left(\frac{E_{S}}{N_{0}}\right)^{-d_{\min}^{(i)}} \tag{7}\]By setting the upper bound of the BER to 3.8 \(\times\) 10\({}^{-3}\) in the above formula, the corresponding SNR threshold \(\gamma_{1}\) can be determined. This threshold aids in determining whether interleaving should be employed in the transmission process.