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This paper studies a learning approach to geometric shaping (GS) for concatenated two-level multi-level codes (MLC). Unlike standard bit-interleaved coded modulation (BICM), the proposed MLC protects only the least reliable bits (LRBs) by an inner code, and thereby efficiently reduces complexity associated with inner soft-decision (SD) decoding compared to the BICM without performance degradation. We propose a new loss function for training a geometric constellation shape in the proposed MLC, leading to maximization of the achievable rate. More specifically, considering the different coding structures for the most reliable bits (MRBs) and the LRBs, the proposed loss function combines two different functions: the union bound on a bit error rate (BER) for the MRBs and the generalized mutual information (GMI) for the LRBs. We demonstrate by simulations of wavelength division multiplexing (WDM) optical fiber systems that the proposed loss function offers a performance gain over the conventional cross entropy-based loss function. Furthermore, it is demonstrated that the proposed MLC with GS outperforms the conventional BICM with GS in terms of a transmission distance, even with significantly lower SD decoding complexity. [ABSTRACT FROM AUTHOR]

Approaching Shannon's capacity via geometric shaping has usually been regarded as challenging due to modulation and demodulation complexity, requiring look-up tables to store the constellation points and constellation bit labeling. To overcome these challenges, in this article, we study lattice-based geometrically shaped modulation formats in multidimensional Euclidean space. We describe and evaluate fast and low complexity modulation and demodulation algorithms that make these modulation formats practical, even with extremely high constellation sizes with more than $10^{28}$ points. The uncoded bit error rate performance of these constellations is compared with the conventional quadrature amplitude modulate (QAM) formats in the additive white Gaussian noise and nonlinear fiber channels. At a spectral efficiency of 2 bits/sym/polarization, compared with 4-QAM format, transmission reach improvement of more than 38% is shown at the hard-decision forward error correction threshold of $2.26\times 10^{-4}$. [ABSTRACT FROM AUTHOR]

In this article, we propose a geometric shaping (GS) strategy to design 8, 16, 32, and 64-ary modulation formats for the optical fibre channel impaired by both additive white Gaussian (AWGN) and phase noise. The constellations were optimised to maximise generalised mutual information (GMI) using a mismatched channel model. The presented formats demonstrate an enhanced signal-to-noise ratio (SNR) tolerance in high phase noise regimes when compared with their quadrature amplitude modulation (QAM) or AWGN-optimised counterparts. By putting the optimisation results in the context of the 400ZR implementation agreement, we show that GS alone can either relax the laser linewidth (LW) or carrier phase estimation (CPE) requirements of 400 Gbit/s transmission links and beyond. Following the GMI validation, the performance of the presented formats was examined in terms of post forward error correction (FEC) bit-error-rate (BER) for a soft decision (SD) extended Hamming code (128, 120), implemented as per the 400ZR implementation agreement. We demonstrate gains of up to 1.2 dB when compared to the 64-ary AWGN shaped formats. [ABSTRACT FROM AUTHOR]

We use the generalized pair-wise optimization (PO) algorithm to optimize the geometrically shaped 32-ary quadrature amplitude modulation (GS-32-QAM) constellation and adopt the probabilistic fold shaping (PFS) scheme to generate the probabilistically shaped 32-QAM (PS-32-QAM) format. We present a detailed comparison among the application of GS-32-QAM, PS-32-QAM and uniform 32-QAM in an intensity modulation/direct detection (IM/DD) system with a bandwidth-limited digital-to-analog converter (DAC). We experimentally demonstrate the 32-QAM discrete Fourier transform-spread (DFT-S) discrete multi-tone (DMT) signal transmission over 1-km standard single mode fiber (SSMF) in an IM/DD system with a net data rate of 108.29-Gb/s/λ, and achieve 0.6 dB and 0.9 dB receiver sensitivity gain using PS and GS constellations, respectively. In the high signal to noise ratio (SNR) case with a net bit rate of 69.61-Gb/s/λ, the PS-32-QAM DMT signal can provide the receiver sensitivity gain of 0.5 dB while the receiver sensitivity gain of the GS-32-QAM DMT signal is 0.4 dB, compared to the uniform 32-QAM DMT signal. Although PS-32-QAM achieves better receiver sensitivity gain than GS-32-QAM in the high SNR case, the proposed GS-32-QAM format outperforms PS-32-QAM in the low SNR case. [ABSTRACT FROM AUTHOR]

This article proposes a novel 3-D probabilistically shaped carrier-less amplitude phase (3D-PS-CAP) modulation based on constellation design using regular tetrahedron cells, which is capable of being applied in short reach communication. Geometric shaping based on regular tetrahedron cells for 3D constellation, combined with probabilistic shaping, can increase the constellation figure of merit (CFM) of the 3D constellation and improve the bit error rate (BER) performance of the 3D-PS-CAP modulation scheme. An experiment demonstrating 4 Gb/s 3D-PS-CAP-16 data transmission over 25 km standard single-mode fiber (SSMF) is successfully carried out. Results show that at the BER of $\text{1} \times \text{10}^{ -\text{3}}$ , the proposed uniform 3D-CAP-16 outperforms the traditional counterparts by 1.1 dB receiver sensitivity gain. Moreover, in the entropy of 3.6 bits/symbol, the proposed 3D-PS-CAP-16 obtains 0.6 and 1.0 dB receiver sensitivity gains compared to the two traditional 3D-PS-CAP-16, respectively. The experimental results indicate the proposed scheme a bright prospect for future short reach communication. [ABSTRACT FROM AUTHOR]

Generalized mutual information (GMI) has been comprehensively studied in multidimensional constellation and probabilistic-shaped (PS) constellation together with different forward error correction (FEC) coding schemes. The simulation results confirm that GMI is an efficient and accurate tool to compare their post-FEC performance. In particular for uniformly geometric-shaped constellation, the pre-FEC Q-factor is highly correlated with GMI though the correlation is reduced at lower FEC coding rate. Furthermore, GMI can be used to design optimized constellation together with generalized pairwise optimization algorithm to mitigate the GMI loss in non-Graymapped constellation. The GMI-optimized 32QAM (opt32) shows ~0.5 dB signal-to-noise ratio improvement between 3 and 4 b/s GMI in both simulated and experimental results. Optimized two-dimensional 8 QAM is also designed to show the consistent GMI improvement over multi-dimensional 8 QAM-equivalent formats. In simulations, PS-64 QAM outperforms opt32 when a long sequence block is used in the distribution matcher. [ABSTRACT FROM AUTHOR]