## 4 Enabling Technologies for FSO Communication
### Error Correction Codes and Modulated Signal Optimization
#### 4.7.3 Probabilistic Shaping  
PS with non-uniform probabilities has been shown to reduce capacity gaps and facilitate fine-grained rate adaptation [296]. This approach has become a significant topic in optical communications. By adjusting the signal probability distribution, PS can modify the entropy and improve system performance [314]. Additionally, it can be used in bit-interleaved CM (BICM) systems utilizing PAS, whose advantages have been experimentally confirmed [308; 314]. PAS integrates CM with PS [296; 315]. This technique has been utilized in coherent communications to enhance SNR in AWGN channels, providing greater flexibility and precise control over bitrate. Recently, time-adaptive PAS has been presented for FSO links, leading to significant capacity improvements under time-varying power fading conditions [316; 317]. Nevertheless, the PAS cannot be directly applied to FSO communications because it relies on the symmetry of bipolar signals, whereas FSO systems use unipolar (asymmetric) signals. To address this limitation, the modified Maxwell-Boltzmann distribution based on PAS (PAS-MMB) can be utilized; however, its distribution remains pairwise, which is not ideal for unipolar symbols [309; 311]. Additionally, unipolar M-PAM are promising candidates for achieving near-capacity in FSO systems [311]. The non-uniform distributions of M-PAM can further enhance the capacity of these systems. Research in [297; 300] has introducednumerical simulations of LDPC and polar-coded PS PAM schemes to optimize the capacity of FSO channels with unipolar signals. In this context, pairwise distributions have proven effective for PAM8, resulting in only minor performance losses [308]. In addition to PAS, another approach utilizing PS is Sparse-Dense Transmission (SDT). Research indicates that SDT-based OOK modulation can achieve approximately 1 dB of shaping gain in AWGN channels [309]. Furthermore, the capacity can be enhanced through entropy-loading DMT systems based on PS. Entropy-loading is a robust and adaptable method for capacity enhancement, allowing continuous adjustments to the source entropy for each subcarrier. However, implementing entropy-loading DMT presents a trade-off between optimization complexity and achievable capacity [318; 319]. In comparison, Hybrid Constellation Shaping (HCS) offers a lower-complexity solution that yields greater shaping gains by combining PS with geometric shaping (GS), particularly for non-Gray mapped QAM [315; 320]. Nevertheless, determining appropriate probabilities and geometric shapes to align with existing FEC schemes poses challenges. To address some of the implementation issues associated with HCS designs, a rate-flexible HCS scheme has been presented for DMT-FSO systems [164; 321]. Bitrate Adaptation Using PCS  
PCS is recognized as an effective method to boost system capacity by reducing the energy of the transmitted signal. In AWGN channels, the lowest transmitted signal energy for a specified bitrate is obtained by adjusting the probabilities of QAM symbols to match the Maxwell-Boltzmann distribution [316; 322] given as [316; 317]  
\[P_{x_{n}}=\frac{\exp(-\lambda_{sp}|x_{n}|)^{2}}{\sum_{n=1}^{M}\exp(-\lambda_{ sp}|x_{n}|)^{2}}, \tag{45}\]  
where \(\lambda_{sp}\geq 0\) denotes the shaping parameter, \(M\) represents the constellation size of the chosen QAM template for PCS, and \(x_{n}\) represents the \(n\)-th symbol in the square \(M\)-QAM constellation alphabet, expressed as [317]  
\[x_{n}=2\Big{(}(n-1)\text{ mod }\sqrt{M}\Big{)}-\sqrt{M}+1+j\bigg{(}2\bigg{\lceil} \frac{n}{\sqrt{M}}\bigg{\rceil}-\sqrt{M}-1\bigg{)}. \tag{46}\]  
In addition to reducing the average signal energy, PCS can also be utilized to adapt the entropy of the transmitted signal, defined as [316]  
\[H_{\text{PCS}}=-\sum_{n=1}^{M}P_{x_{n}}\log_{2}(P_{x_{n}}). \tag{47}\]  
Thus, by utilizing Equations (45) and (47), the net bitrate can be adjusted by modifying the shaping parameter \(\lambda_{sp}\).