# Two-Dimensional Constellation Shaping in Fiber-Optic Communications
## 2 Typical Constellation Shaping Schemes  
### Probabilistic Shaping  
In a Gaussian-noise-limited channel, PS-QAM yielding a Maxwell-Boltzmann (M-B) distribution is recognized as the optimal format to maximize the channel capacity [46]. In general, the low-amplitude constellation points are sent with a higher probability than the high-amplitude ones. Besides, the constellation points under the same amplitude layer are sent equally likely. Therefore, the average symbol power will be decreased, but at a cost of lower source entropy. The first PS scheme was proposed by [PERSON], which is based on many-to-one mapping [47]. Complex forward error correction (FEC) coding is required to be implemented to recover the original bits from the systematic errors after many-to-one demapping. The recently proposed arithmetic coding-based constant composition distribution matcher (CCDM) is an invertible fix-to-fix length distribution matcher, enabling maximum information rate asymptotically in the block-length [48]. Later, there are some methods proposed to further reduce the complexity of CCDM, like multiset-partition distribution matching [49] and streaming distribution matching [50]. The first PS-based coded modulation was probabilistic amplitude shaping (PAS) [51], which could seamlessly combine the binary FEC coding and CCDM in a square M-QAM format. The proposed PAS enables capacity approaching fiber-optic communications, but also brings some issues. Firstly, CCDM and the modified CCDM architectures are hard to be implemented in commercial optical transceivers. Secondly, more bit-to-symbol (B2S) mapping and symbol-to-bit (S2B)mapping modules are required to be implemented at the transceivers, leading to extra complexity. Thirdly, there will be an entropy loss by shaping on a given M-ary signal constellation format. Fourthly, intrablock error propagation will loom over the dematching procedure once the FEC coding cannot totally correct all bit errors. Fifthly, the applicable FEC code rate is limited, which will be lower-bounded by \(\left[\log_{2}(M)-2\right]/\log_{2}(M)\). Sixthly, DSP circuit will be under a higher pressure to recover Gaussian-like constellation diagrams. Thereafter, more pilot-tones are essential to be used for signal recovery. Last but not least, although could be used for reach extension, PAS-MQAM suffers more from the modulation-dependent noise in long-haul fiber-optic communications [52]. In a square MQAM format, the coordinate of each constellation can be represented by the Cartesian product of two pulse-amplitude modulation (PAM) coordinates, namely,  
\[X=\left\{\pm 1,\ \pm 3,\ldots,\ \pm\left(\sqrt{M}-1\right)\right\} \tag{1}\]  
The well-known M-B distribution is defined by  
\[P_{X_{o}}(x)=e^{-\varsigma y\lambda t^{2}}/\sum_{x^{\prime}\in X}e^{-\varsigma y \lambda^{\prime}t^{2}} \tag{2}\]  
where \(v\) is a non-negative scaling factor. If \(v\) is 0, the PAM format follows a uniform distribution. The capacity of PAS-MQAM format can be defined as [51]  
\[C=H(p)-m(1-R) \tag{3}\]  
where \(H(p)\) is the entropy of the PAS-MQAM format, \(R\) is the FEC code rate, and \(m=\log_{2}(M)\). Figure 1 shows the conceptual diagram of PAS-MQAM generation, where 16 QAM is used as an illustrative example. In such scheme, the 1D amplitude symbols labeled by 1-3-1-1-1-3..., will be probabilistically shaped according to the M-B distribution (the probabilities of Symbol-1 and Symbol-3 are indicated by the blue and black colors, respectively), and the sign bits labeled by 1-0-0-1-0..., will be used to carry the uniformly distributed parity-check bits. The S2B mapping is used to map the Symbol-1 and Symbol-3 to amplitude bits, i.e., Bit-1 and Bit-0, respectively. In the FEC encoder, the party-check bits will be appended to the sign bits, and combine with the amplitude bits. Such encoded bits are remapped to PAM-4 symbols (00\(\rightarrow\)-3, 01\(\rightarrow\)-1, 11\(\rightarrow\)+1, 10\(\rightarrow\)+3). Therefore, after FEC coding, the M-B distribution will not be changed, and Gray-mapping rule is still applicable.