# Two-Dimensional Constellation Shaping in Fiber-Optic Communications
## 2 Typical Constellation Shaping Schemes
### Geometric Shaping  
GS-QAM is generated by optimizing certain criterion under a given signal-to-noise ratio (SNR). Such criteria can be maximizing MI [40], maximizing GMI [41], maximizing constellation  
Figure 1: The conceptual diagram of probabilistic amplitude shaping (PAS)-M-ary quadrature amplitude modulation (MQAM) generation. figure of merit [53], or minimizing mean-square error of Gaussian distribution [54], etc. Typically, the lower-amplitude constellation points will be more concentrated around the origin. Generalized cross constellations and Voronoi constellations were proposed decades ago [38; 39]. GMI-optimized constellations have also been proposed recently to enhance the capacity in a binary FEC coding featured fiber-optic communication systems. GS could potentially simplify the process to generate Gaussian-like constellations, but it also brings several issues. Firstly, due to the unavailability of Gray-mapping in most cases, GMI performance hardly approach MI performance. Although nonbinary FEC coding is a solution to close such gap, the high complexity hinders its commercial development [55]. Secondly, the common asymmetry of the constellations may result in more complex PAM constellations in both in-phase and quadrature branches. Therefore, the ADC/DAC is required to be implemented with higher resolutions. Thirdly, the DSP circuits to recover GS-MQAM formats are not compatible with the ones to recover regular-MQAM. As a result, new DSP algorithms are suggested to be developed to efficiently recover the GS-MQAM signals. Fourthly, it is hard to reach standard agreements between optical transceiver manufacturers due to the variety of GS-QAM formats and the matched DSP algorithms. Figure 2 shows the conceptual diagram of a GS-QAM-based communication system. If binary FEC coding is used, the suboptimal B2S mapping table has to be found to minimize the gap between GMI and MI, in order to minimize the Non-Gray mapping penalty. Brutal force algorithm may be used to find such mapping rule at a cost of high computational complexity. Alternatively, binary data can be mapped to the GS-QAM symbols through any B2S look-up table, followed by a reasonable complexity nonbinary FEC encoder to sustain reliable communications [56]. Figure 3 shows 16/32/64 QAM formats generated by maximizing constellation figure of merit [53], minimizing mean-square error of Gaussian distribution [57], and maximizing GMI [41], respectively. The GS-16 QAM constellation shown in Figure 3a is obtained by maximizing the minimum Euclidean distance under the same average power of the 16-ary constellation. Figure 3: GS-QAM formats based on (**a**) maximizing constellation figure of merit, (**b**) minimizing mean-square error of Gaussian distribution, and (**c**) maximizing generalized mutual information (GMI). Figure 2: The conceptual diagram of a geometric shaping (GS)-QAM-based communication system. The GS-32 QAM constellation shown in Figure 3b can be obtained by the following steps. 1. Choosing 2D Gaussian distribution as the optimal source distribution, and select the uniformly distributed regular-32 QAM as the initial constellation. 2. Generating a symbol sequence following the Gaussian distribution. 3. Distributing the symbols into 32 clusters, where the decision is made as per the minimum Euclidean distance from the 32 QAM constellation points obtained in previous iteration. 4. Finding the average central positions from the symbols labeled by each cluster. Such 32 points located on the central positions are used as the new MQAM constellation points. 5. Iterating over Steps 2 and 4 until convergence. The GS-64 QAM constellation shown in Figure 3b is designed with the constraint of Gray mapping, which can maximize the GMI. Due to the Gray mapping constraint, such scheme cannot fully explore the 2D space, but it can also reduce capacity gap towards the Shannon limit. It is an open question about the optimal GS solution, since it depends on lots of implementation scenarios.