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In this paper, we propose a new geometric-shaping design for golden angle modulation (GAM) based on the complex geometric properties of open symmetrized bidisc, termed Bd-GAM, for future generation wireless communication systems. Inspired from the circular symmetric structure of the GAM, we construct the modulation schemes, Bd-GAM1 and Bd-GAM2. Specifically, we consider MI-optimized probabilistic modulation scheme with the geometrics properties of symmetric bidisc. With minimum SNR and entropy constraint, Bd-GAM1 and Bd-GAM2 can overcome the shaping-loss. Compared with the existed golden angle modulation introduced, the new design improves the mutual information, and the distance between adjacent constellation points. [ABSTRACT FROM AUTHOR]

Four-dimensional (4D) constellations with up to 131 072 points (17 bit/4D-sym) are designed for the first time using geometric shaping. The constellations are optimized in terms of mutual information (MI) and generalized MI (GMI) for the additive white Gaussian noise (AWGN) channel, targeting a forward error correction (FEC) rate of 0.8 at finite signal-to-noise ratios. The presented 15–17 bit constellations are currently the highest-performing constellations in the literature, having a gap to the AWGN capacity as low as 0.17 dB (MI) and 0.45 dB (GMI) at 17 bit/4D-sym. For lower cardinalities, our constellations match or closely approach the performance of previously published optimized constellations. We also show that (GMI-)optimized constellations with a symmetry constraint, optimized for a FEC rate of 0.8, perform nearly identical to their unconstrained counterparts for cardinalities above 8 bit/4D-sym. A symmetry constraint for MI-optimized constellations is shown to have a negative impact in general. The proposed procedure relies on a Monte-Carlo-based approach for evaluating performance and is extendable to other (nonlinear) channels. Stochastic gradient descent is used for the optimization algorithm for which the gradients are computed using automatic differentiation. This article is part of the theme issue 'Celebrating the 15th anniversary of the Royal Society Newton International Fellowship'. [ABSTRACT FROM AUTHOR]

For high-speed optical communication systems, laser phase noise (LPN) stands as a pivotal factor influencing the quality of the received signal. Therefore, the employment of a highly accurate carrier phase recovery (CPR) algorithm at the receiving end is indispensable to ensure the reliability of transmission. While a CPR algorithm called principal component-based phase estimation (PCPE) has been proven to be capable of achieving low-complexity and high-performance phase recovery for even-bit quadrature amplitude modulation (QAM) (i.e., square QAM) signals, it is not compatible with traditional cross-shaped odd-bit QAM signals. To circumvent this problem, a signal constellation design scheme based on geometric shaping (GS) is proposed. The pair-wise optimization (PO) algorithm is used to optimize the constellation structure of 32QAM and 128QAM signals in order to obtain results that are compatible with the PCPE algorithm. Monte Carlo simulation results reveal that for odd-bit QAM signals utilizing PCPE for phase recovery, the proposed GS constellations enhance the mutual information (MI) performance across the entire measured signal-to-noise (SNR) range. Moreover, compared with regular 32QAM and 128QAM constellations using the well-known blind phase search (BPS) algorithm, the proposed GS and PCPE scheme can achieve SNR gains of 1.10 dB and 2.59 dB, respectively, when considering the 20% soft-decision forward error correction (SD-FEC) overhead. Verification through commercial simulation software corroborates these findings, demonstrating that the proposed GS constellations are particularly suitable for the PCPE algorithm, especially under conditions of high optical signal-to-noise ratio (OSNR). To the best of our knowledge, this is the first time that the incompatibility between the PCPE algorithm and odd-bit QAM signals has been investigated, and the proposed GS scheme has broadened the application scope of the low-complexity CPR algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we propose an innovative channel-sensitive autoencoder (CSAE)-aided end-to-end deep learning (E2EDL) technique for joint geometric probabilistic shaping. The pretrained conditional generative adversarial network (CGAN) is introduced in the CSAE which performs differentiable substitution of the optical fiber channel model under variable input optical power (IOP) levels. This enables the CSAE-aided E2EDL to design optimal joint geometric probabilistic shaping schemes for optical fiber communication systems at varying IOPs. The results of the proposed CSAE-aided E2EDL technique show that for a dual-polarization 64-Gbaud signal with a transmission distance of 5 × 80 km, when the modulation format is a 64-quadrature amplitude modulation (QAM) or a 128-QAM, the maximum generalized mutual information (GMI) level learned via CSAE-aided E2EDL is 5.9826 or 6.8384 bits/symbol under varying IOPs, respectively. In addition, the pretrained CGAN, as a substitution for optical fiber transmission model, accurately characterizes the distortion of signals with different IOPs, with an average bit error ratio (BER) difference of only 1.83%, an average mean square error (MSE) of 0.0041 and an average K-L divergence of 0.0046. In summary, this paper delivers new insights into the application of E2EDL and demonstrates the feasibility of joint geometric probabilistic shaping-based E2EDL for fiber optic communication systems with varying IOPs. [ABSTRACT FROM AUTHOR]

Intra-data center links are subject to transmission impairments that pose challenges for efficient scaling of per-wavelength data rates beyond 100 Gb/s. Limited electrical and optical component bandwidths, limited data converter resolution, and direct detection induce significant signal-dependent distortion, degrading receiver sensitivity (RS) and chromatic dispersion tolerance. In this paper, we present a geometric shaping (GS) scheme that optimizes transmitted intensity levels based on symbol-error statistics observed at the receiver. The proposed GS scheme adjusts these levels to achieve substantially equal symbol-error probabilities at all decision thresholds. The scheme enables the levels most affected by signal-dependent distortion to be detected with the same reliability as other levels, thereby increasing the effectiveness of linear or nonlinear equalization techniques. This can be exploited to improve RS and extend transmission distance for a fixed equalization scheme or, alternatively, to reduce the complexity of signal processing needed to achieve a target RS or transmission distance. For example, in 200 Gb/s PAM links, GS and 21-tap linear equalization achieves RS and reach similar to uniform level spacing and Volterra nonlinear equalization with 21 linear and 3 second-order taps.

Since visible-light communication (VLC) has become an increasingly promising candidate for 6G, the field of underwater visible-light communication (UVLC) has also garnered significant attention. However, the impairments introduced by practical systems and the time-varying underwater channels always limit the performance of underwater visible-light communication. In this paper, we propose and experimentally demonstrate an autoencoder-based geometric shaping model (AEGSM) framework to jointly optimize quadrature amplitude modulation (QAM) signals at the symbol-wise and bit-wise levels for underwater visible-light communication. Unlike traditional geometric shaping (GS) methods, which only give theoretically optimal shaping solutions, our framework can always obtain the globally optimal shaping scheme for a specific channel condition or different application scenarios. In our AEGSM framework, an autoencoder is used to find the optimal shaping scheme at the symbol-wise level and a revised pairwise optimization (RPO) algorithm is applied to achieve bit-wise optimization. In a real UVLC system, 2.05 Gbps transmission is achieved under the hard decision–forward error correction (HD-FEC) threshold of 3.8 × 10−3 by employing the autoencoder-based 8QAM (AE-8QAM) optimized by the AEGSM, which is 103 Mbps faster than the Norm-8QAM. The AE-8QAM also shows its resistance to nonlinearity and enables the UVLC system to operate within a larger dynamic range of driving voltages. The results substantiate the potential and practicality of the proposed AEGSM framework in the realm of underwater visible-light communication. [ABSTRACT FROM AUTHOR]

We used a hybrid probabilistic geometric shaping scheme that uses a probabilistic shaping method with labels and a pairwise optimization (PO) algorithm to transmit signals in a 2 × 2 MIMO Photon-assisted millimeter wave (MMW) radio-over-fiber (RoF) system in the D-band. This scheme has low complexity and a simple recovery method. This scheme optimizes 16-ary quadrature amplitude modulation (16QAM) to probabilistically shaped 9-ary quadrature amplitude modulation (PS-9QAM) by adding labels, and optimizes it to hybrid probabilistically and geometrically shaped 9-ary quadrature amplitude modulation (PGS-9QAM) by geometric shaping using the PO algorithm. The experimental results prove the optimization effect of this scheme. Under the bit error rate threshold of 0.0038, when the net bit rate is the same, compared with 16QAM, PS-9QAM and PGS-9QAM, the optical power at the transmitter can be reduced by 0.8 dB and 1.45 dB respectively. When the transmission power is the same, compared with 16QAM, the net rates of PS-9QAM and PGS-9QAM can be increased by 7.5Gbit/s and 9Gbit/s respectively. • The paper explores a hybrid probabilistic and geometric shaping scheme in a D-band 2 × 2 MIMO radio-over-fiber (RoF) system. • Optimizes 16QAM to 9QAM (PS-9QAM) with probabilistic shaping using labels, and further refines it to PGS-9QAM through geometric shaping. • Proposes a method to eliminate DC bias to reduce nonlinear distortion. • PS-9QAM and PGS-9QAM can reduce optical power by 0.8 dB and 1.45 dB respectively, compared to 16QAM, at the same net bit rate. [ABSTRACT FROM AUTHOR]

For high-speed optical communication systems, laser phase noise (LPN) stands as a pivotal factor influencing the quality of the received signal. Therefore, the employment of a highly accurate carrier phase recovery (CPR) algorithm at the receiving end is indispensable to ensure the reliability of transmission. While a CPR algorithm called principal component-based phase estimation (PCPE) has been proven to be capable of achieving low-complexity and high-performance phase recovery for even-bit quadrature amplitude modulation (QAM) (i.e., square QAM) signals, it is not compatible with traditional cross-shaped odd-bit QAM signals. To circumvent this problem, a signal constellation design scheme based on geometric shaping (GS) is proposed. The pair-wise optimization (PO) algorithm is used to optimize the constellation structure of 32QAM and 128QAM signals in order to obtain results that are compatible with the PCPE algorithm. Monte Carlo simulation results reveal that for odd-bit QAM signals utilizing PCPE for phase recovery, the proposed GS constellations enhance the mutual information (MI) performance across the entire measured signal-to-noise (SNR) range. Moreover, compared with regular 32QAM and 128QAM constellations using the well-known blind phase search (BPS) algorithm, the proposed GS and PCPE scheme can achieve SNR gains of 1.10 dB and 2.59 dB, respectively, when considering the 20% soft-decision forward error correction (SD-FEC) overhead. Verification through commercial simulation software corroborates these findings, demonstrating that the proposed GS constellations are particularly suitable for the PCPE algorithm, especially under conditions of high optical signal-to-noise ratio (OSNR). To the best of our knowledge, this is the first time that the incompatibility between the PCPE algorithm and odd-bit QAM signals has been investigated, and the proposed GS scheme has broadened the application scope of the low-complexity CPR algorithm.

In this paper, transmission of orthogonal chirp division multiplexing (OCDM) based on asymmetric pair mapping over a seven core fiber is proposed. High-order 32 quadrature amplitude modulation (QAM) is reduced to quadrature phase-shift keying (QPSK) and 8 QAM signal employing asymmetric pair mapping, which will not affect the rate of the system. Combined with geometric shaping (GS), the average transmission power is successfully lowered, realizing signal damage suppression. In order to verify the performance of the proposed asymmetric pair mapping scheme, a 140 Gb/s asymmetric pair mapping OCDM signal transmission over 2 km seven core fiber is experimentally demonstrated. The experimental results show that the proposed transmission scheme has the best performance when the power distribution ratio (PDR) is 4 dB. Meanwhile, compared with GS 32 QAM with QPSK+rectangle 8 QAM, GS 32 QAM with QPSK+star 8 QAM and GS 32 QAM, the proposed asymmetric pair mapping OCDM signal has improved by 1.55 dB, 1.45 dB, and 0.7 dB, respectively. It has a wide range of applications in the future optical transmission system due to its outstanding performance in the suppression of high-order modulation signal degradation.

Since visible-light communication (VLC) has become an increasingly promising candidate for 6G, the field of underwater visible-light communication (UVLC) has also garnered significant attention. However, the impairments introduced by practical systems and the time-varying underwater channels always limit the performance of underwater visible-light communication. In this paper, we propose and experimentally demonstrate an autoencoder-based geometric shaping model (AEGSM) framework to jointly optimize quadrature amplitude modulation (QAM) signals at the symbol-wise and bit-wise levels for underwater visible-light communication. Unlike traditional geometric shaping (GS) methods, which only give theoretically optimal shaping solutions, our framework can always obtain the globally optimal shaping scheme for a specific channel condition or different application scenarios. In our AEGSM framework, an autoencoder is used to find the optimal shaping scheme at the symbol-wise level and a revised pairwise optimization (RPO) algorithm is applied to achieve bit-wise optimization. In a real UVLC system, 2.05 Gbps transmission is achieved under the hard decision–forward error correction (HD-FEC) threshold of 3.8 × 10−3 by employing the autoencoder-based 8QAM (AE-8QAM) optimized by the AEGSM, which is 103 Mbps faster than the Norm-8QAM. The AE-8QAM also shows its resistance to nonlinearity and enables the UVLC system to operate within a larger dynamic range of driving voltages. The results substantiate the potential and practicality of the proposed AEGSM framework in the realm of underwater visible-light communication.

• We propose the HD-MMIM-OFDM based on geometric shaping. • Several design methods for HD constellations are proposed. • Multi-mode index modulation is used to improve SE of the system. • Pseudo-Gray coding scheme further improves the receiver sensitivity. • The designed scheme exhibits enhanced tolerance to nonlinearity. A high-dimensional multi-mode index modulation orthogonal frequency division multiplexing (HD-MMIM-OFDM) based on geometric shaping is proposed and demonstrated in cost-effective and power efficient intensity modulation direct detection (IM/DD) systems. Four geometrically shaped three-dimensional (3D-GS) constellation schemes are compared and evaluated in terms of minimum Euclidean distance (MED) and constellation figure of merit (CFM). We select the 3D constellation C3 with the optimal performance as the basis and construct three HD constellation schemes by stacking low-dimensional constellations, namely 4D, 5D, and 6D. To further improve the spectral efficiency (SE) of the system, a multi-mode index modulation (MMIM) based on OFDM is proposed and validated at different fiber lengths. The result shows that compared with traditional 2D-OFDM, the receiver sensitivity of 4D-MMIM-OFDM under the same SE increases by 0.46 dB, and the pseudo-Gray coding scheme further improves the receiver sensitivity by 0.16 dB. When the fiber length increases from 50 km to 80 km, the receiver sensitivity of 6D-MMIM-OFDM increases by 0.97 dB compared to 4D-MMIM-OFDM. It has been demonstrated that as the dimension of the constellation increases, the designed scheme exhibits enhanced tolerance to nonlinearity. [ABSTRACT FROM AUTHOR]

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]

We propose a learning-based method for the joint design of a transmit and receive filter, the constellation geometry and associated bit labeling, as well as a neural network (NN)-based detector. The method maximizes an achievable information rate, while simultaneously satisfying constraints on the adjacent channel leakage ratio (ACLR) and peak-to-average power ratio (PAPR). This allows control of the tradeoff between spectral containment, peak power, and communication rate. Evaluation on an additive white Gaussian noise (AWGN) channel shows significant reduction of ACLR and PAPR compared to a conventional baseline relying on quadrature amplitude modulation (QAM) and root-raised-cosine (RRC), without significant loss of information rate. When considering a 3rd Generation Partnership Project (3GPP) multipath channel, the learned waveform and neural receiver enable competitive or higher rates than an orthogonal frequency division multiplexing (OFDM) baseline, while reducing the ACLR by $\mathrm {10~ \text {dB}}$ and the PAPR by $\mathrm {2~ \text {dB}}$. The proposed method incurs no additional complexity on the transmitter side and might be an attractive tool for waveform design of beyond-5G systems. [ABSTRACT FROM AUTHOR]

Voronoi constellations (VCs) are finite sets of vectors of a coding lattice enclosed by the translated Voronoi region of a shaping lattice, which is a sublattice of the coding lattice. In conventional VCs, the shaping lattice is a scaled-up version of the coding lattice. In this paper, we design low-complexity VCs with a cubic coding lattice of up to 32 dimensions, in which pseudo-Gray labeling is applied to minimize the bit error rate. The designed VCs have considerable shaping gains of up to 1.03 dB and finer choices of spectral efficiencies in practice compared with conventional VCs. A mutual information estimation method and a log-likelihood approximation method based on importance sampling for very large constellations are proposed and applied to the designed VCs. With error-control coding, the proposed VCs can have higher information rates than the conventional scaled VCs because of their inherently good pseudo-Gray labeling feature, with a lower decoding complexity. [ABSTRACT FROM AUTHOR]

In coherent optical systems, optical fiber non-linearity is a consistent limiting factor towards the effective signal-to-noise ratio though being mitigated by various digital signal processing based approaches. In this paper, an intrinsic method of signal processing based on the shape of the input constellation is employed to yield a novel non-linearity tolerant geometric constellation. 16-QAM back-to-back coherent system is optimized for minimum value of non-linear interference, and a novel 2-Tier star constellation using sequential quadratic programming algorithm is proposed. The values of second and fourth order moments of input obtained for the optimized 2-Tier star constellation are 1.19 and 1.70 respectively, resulting in an overall reduction of non-linear interference. The complexity of the system proposed is found to be minimal in comparison to the other existing systems employing digital compensation. [ABSTRACT FROM AUTHOR]

The benefits of end-to-end learning has been demonstrated over AWGN channels but has not yet been quantified over realistic wireless channel models. This work aims to fill this gap by exploring the gains of end-to-end learning over a frequency- and time-selective fading channel using OFDM. With imperfect channel knowledge at the receiver, the shaping gains observed on AWGN channels vanish. Nonetheless, we identify two other sources of performance improvements. The first comes from a neural network-based receiver operating over a large number of subcarriers and OFDM symbols which allows to reduce the number of orthogonal pilots without loss of BER. The second comes from entirely eliminating orthogonal pilots by jointly learning a neural receiver together with either superimposed pilots (SIPs), combined with conventional QAM, or an optimized constellation. The learned constellation works for a wide range of signal-to-noise ratios, Doppler and delay spreads, has zero mean and does hence not contain any form of SIP. Both schemes achieve the same BER as the pilot-based baseline with 7% higher throughput. Thus, we believe that a jointly learned transmitter and receiver are a very interesting component for beyond-5G communication systems which could remove the need and associated overhead for demodulation reference signals. [ABSTRACT FROM AUTHOR]

A novel adaptive weighted geometric‐shaped (GS) 61 quadrature amplitude modulation (QAM) constellation is proposed in order to mitigate the nonlinear damage and increase the transmission distance. By introducing an adaptive weighting factor related to Kerr effect and fiber transmission distance in concentric regular hexagon constellation, the overlapping problem of the outer constellation points induced by nonlinear damage was reduced. The simulation results show that the proposed GS‐61QAM outperformed uniform 64QAM and has more obvious advantages in long‐haul optical transmission system. For a 120‐Gb/s coherent optical communication system, compared with uniform 64QAM, the proposed algorithm can obtain about 4.1 dB optical signal‐to‐noise (OSNR) gain at the BER of 10−2. Moreover, in order to further improve the system tolerance of nonlinearity, a scheme of adaptive weighted GS‐61QAM in coherent optical communication system utilizing K‐Means algorithm is proposed, the proposed scheme outperforms the traditional uniform 64QAM communication system by 5 dB in OSNR at the BER of 10−2. [ABSTRACT FROM AUTHOR]

A novel fast‐adaptive weighted hybrid‐shaped M‐ary quadrature amplitude modulation (HS‐MQAM) format is proposed in this paper. By introducing a fast‐adaptive weighting factor array related to geometric shaping and probabilistic shaping, the non‐linear impairment is mitigated and the bit error rate (BER) and mutual information (MI) performances are further improved. The simulation results show that the proposed HS‐61QAM outperforms both geometric‐shaped 61 quadrature amplitude modulation (GS‐61QAM) and regular 64QAM, and has more obvious advantages in the long‐haul optical transmission system. For a 120‐Gb/s optical communication system over 10 × 75 km fiber transmission link, compared with GS‐61QAM and regular 64QAM, the proposed approach can achieve about 4.2 and 8 dB OSNR gain at the BER of 10−2, and 0.21 and 0.55 bits/symbol MI gain at the OSNR of 22 dB. [ABSTRACT FROM AUTHOR]

A probabilistic shaping method based on reduced-exponentiation subset indexing and honeycomb-structured constellation optimization is proposed to compress the number of signal points in the constellation, so that the total number doesn't fit the traditional pattern of multiples of power exponents of 2. The proposed scheme can significantly reduce the average signal power, enhance the space utilization in view of (identifying signals in) the judgement area of the constellation, as well as improve the mutual information, owing to the combination of probabilistic shaping and constellation optimization. Moreover, an experiment of a 25 km intensity-modulation and direct-detection system (IM/DD) transmission system is successfully demonstrated to present the superiority of our proposed scheme. It is shown that the proposed probabilistic shaping 64-to-31 carrier-less amplitude and phase modulation (CAP) based on honeycomb constellation can achieve the gains of 1.5 dB and 3 dB over receiver sensitivity when compared with uniform 32-CAP and uniform 64-CAP at the bit error rate (BER) of 1*10-3, respectively. The experiment suggests the proposed scheme a promising technique for future 5G fronthaul network.

We introduce and compare typical shaping schemes suitable for optical communications. The geometrically shaped-quadrature amplitude modulation (GS-QAM) formats are characterized by the non-equidistant spacing of constellation points, transmitted uniformly, and applied to improve system capacity. On the other hand, the well-known constant composition distribution matcher (CCDM) is applied for the generation of probabilistically shaped QAM (PS-QAM) formats. Mutual information (MI) is used as a metric to analyze the performances of regular/GS/PS-MQAM formats. In a linear amplified spontaneous emission noise limited region, it can be proved by the numerical simulation that MI performances of the GS-8/16QAM are always better than regular 8/16QAM and PS-8/16QAM; the largest shaping gains can be separately reached by PS-32QAM and GS-32QAM. We continue with the experimental demonstration on the 16QAM-based transmission system, and find that GS-16QAM generally has the best MI performance. We also find that the modulation-dependent nonlinear noises of the GS-8/16/32QAM are comparable to that of the regular 8/16/32QAM and generally lower than PS-8/16/32QAM. By using the enhanced Gaussian noise model, we observe that the GS-8/16QAM formats have better performances than regular 8/16QAM and PS-8/16QAM over multi-span transmission. Meanwhile, PS-32QAM formats provide superior performance over a relatively long transmission distance.

Aiming to improve the performance of free space visible light communication (VLC) system, we combined the use of rotation coding together with geometric shaping (GS) technology in carrierless-amplitude-and-phase (CAP) modulation visible light communication system. In this paper, we designed a new GS-16QAM constellation named Triangular. Triangular GS-16QAM has a better noise resistance due to its larger value of the minimum Euclidean distance. Based on the excellent new 16QAM constellation, we further improve the performance of VLC system utilizing a proposed new coding technology named rotation coding after the quadrature-amplitude-modulation (QAM) mapping. The rotation coding can reduce the overlapping problem of the outer constellation points induced by non-linear impairment. Finally, the upper gain about 1 dB is achieved with rotation coding in 1.2 m free space VLC communication when the transmission speed is relatively low about 80 Mb/s. While the transmission speed is relatively high around 2 Gb/s in CAP-VLC system, Triangular constellation with rotation coding still has positive effect for transmission quality with about 1 dB enhancement than the square 16QAM. All of these results are verified by simulations and experiments.

Due to the rapid growth of high-resolution services, low earth orbit satellite-to-ground (LEO-StG) optical communications are expected to provide higher speed and better receiver sensitivity transmission. Here, we propose a phase uniformly distributed circular m-ary quadrature amplitude modulation (MQAM) combined with probabilistic shaping (PS) scheme for polarization multiplexing coherent optical orthogonal frequency-division multiplexing (PM-CO-OFDM) systems in LEO-StG communications to achieve high capacity transmission and improve receiver sensitivity as well as linewidth tolerance. In such scheme, the radius ratio of the circular 16/32QAM is optimized to maximize the minimum Euclidean distance (MED), and the constellation points of each ring are uniformly distributed to maximize the phase space of adjacent constellation points. Besides, the circular 16/32QAM is combined with probabilistic shaping (PS) to further increase the MED under the constraint of average transmit power. Simulation results of such circular 16/32QAM combined with PS scheme under different turbulence channels not only demonstrate its feasibility, but also show the improvements in receiver sensitivity and linewidth tolerance compared with regular 16/32QAM under same entropy. The improvement of receiver sensitivity is also experimentally validated. The superiorities of such scheme make it highly desirable for potential application in long-reach, non-relay, and high capacity LEO-StG optical communications .

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]

Analytic expressions are derived for the Wiener filter (WF), also known as the linear minimum mean square error (LMMSE) estimator, for an intensity-modulation/direct-detection (IM/DD) short-haul fiber-optic communication system. The link is purely dispersive and the nonlinear square-law detector (SLD) operates at the thermal noise limit. The achievable rates of geometrically shaped PAM constellations are substantially increased by taking the SLD into account as compared to a WF that ignores the SLD. [ABSTRACT FROM AUTHOR]

We consider a trainable point-to-point communication system, where both transmitter and receiver are implemented as neural networks (NNs), and demonstrate that training on the bit-wise mutual information (BMI) allows seamless integration with practical bit-metric decoding (BMD) receivers, as well as joint optimization of constellation shaping and labeling. Moreover, we present a fully differentiable neural iterative demapping and decoding (IDD) structure which achieves significant gains on additive white Gaussian noise (AWGN) channels using a standard 802.11n low-density parity-check (LDPC) code. The strength of this approach is that it can be applied to arbitrary channels without any modifications. Going one step further, we show that careful code design can lead to further performance improvements. Lastly, we show the viability of the proposed system through implementation on software-defined radios (SDRs) and training of the end-to-end system on the actual wireless channel. Experimental results reveal that the proposed method enables significant gains compared to conventional techniques. [ABSTRACT FROM AUTHOR]

The maximum data throughput in a single mode optical fibre is a function of both the signal bandwidth and the wavelength-dependent signal-to-noise ratio (SNR). In this paper, we investigate the use of hybrid discrete Raman & rare-earth doped fibre amplifiers to enable wide-band signal gain, without spectral gaps between amplification bands. We describe the widest continuous coherent transmission bandwidth experimentally demonstrated to date of 16.83 THz, achieved by simultaneously using the S-, C- and L-bands. The variation of fibre parameters over this bandwidth, together with the hybrid amplification method result in a significant SNR wavelength-dependence. To cope with this, the signal was optimised for each SNR, wavelength and transmission band. By using a system-tailored set of geometrically shaped constellations, we demonstrate the transmission of 660 $\times25$ GBd channels over 40 km, resulting in a record single mode fibre net throughput of 178.08 Tbit/s. [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]

A novel constellation shaped 11-QAM signaling scheme based on geometric shaping (GS) and probabilistic shaping (PS) is proposed in this paper. The 11-QAM signals are obtained by Huffman coding with an optimized 11-QAM constellation structure. Then the non-uniform 11-QAM signals are generated by bit-interleaved coded modulation (BICM), which can get a better performance of bit error ratio (BER) and approach the Shannon limit. An experiment of the proposed 11-QAM constellation is carried out to transmit data over 25-km standard single-mode fiber (SSMF) in the polarization division multiplexed coherent optical communication system. Results show that when the data rate reaches 33Gbps, the proposed GS/PS 11-QAM scheme respectively outperforms the traditional uniformly distributed star 8-QAM and the traditional PS 9-QAM by 2.9dB and 1.2dB in OSNR. [ABSTRACT FROM AUTHOR]

Aiming to improve the performance of free space visible light communication (VLC) system, we combined the use of rotation coding together with geometric shaping (GS) technology in carrierless-amplitude-and-phase (CAP) modulation visible light communication system. In this paper, we designed a new GS-16QAM constellation named Triangular. Triangular GS-16QAM has a better noise resistance due to its larger value of the minimum Euclidean distance. Based on the excellent new 16QAM constellation, we further improve the performance of VLC system utilizing a proposed new coding technology named rotation coding after the quadrature-amplitude-modulation (QAM) mapping. The rotation coding can reduce the overlapping problem of the outer constellation points induced by non-linear impairment. Finally, the upper gain about 1 dB is achieved with rotation coding in 1.2 m free space VLC communication when the transmission speed is relatively low about 80 Mb/s. While the transmission speed is relatively high around 2 Gb/s in CAP-VLC system, Triangular constellation with rotation coding still has positive effect for transmission quality with about 1 dB enhancement than the square 16QAM. All of these results are verified by simulations and experiments. [ABSTRACT FROM AUTHOR]

Constellation shaping has been widely used in optical communication systems. We review recent advances in two-dimensional constellation shaping technologies for fiber-optic communications. The system architectures that are discussed include probabilistic shaping, geometric shaping, and hybrid probabilistic-geometric shaping solutions. The performances of the three shaping schemes are also evaluated for Gaussian-noise-limited channels. [ABSTRACT FROM AUTHOR]

• We propose a geometrically shaped 19-ary 3D constellation in IM/DD systems. • The regular octahedron cell is used to increase MED and decrease average power. • The Huffman coding is used to generate non-uniform signals. • The 3D-PS O-8 is superior to 3D C1, 2D-PS C-6, 3D C3 and 3D O-8 in terms of BER and CFM. • The proposed scheme can improve system capacity and nonlinear tolerance. A three-dimensional (3D) constellation based on geometric shaping (GS) and probabilistic shaping (PS) is proposed for an intensity modulation/direct detection (IM/DD) system. The GS of 3D constellation is based on regular octahedron cells, and combined with PS using Huffman coding, the constellation figure of merit (CFM) and bit error rate (BER) performance of 3D constellation can be significantly improved. Results demonstrate that proposed 19-ary 3D constellation achieves a 2.2 dB receiver sensitivity gain compared with traditional 16-ary 3D constellation at the BER of 3.8 × 10-3. Moreover, the probabilistically shaped 19-ary 3D constellation exhibits a BER that is reduced by around one order of magnitude compared with the non-shaped 19-ary 3D constellation for the same transmission distance. In addition, the proposed 19-ary 3D constellation exhibits superior performance in terms of transmission distance and data rate compared with traditional 16-ary 3D constellation and GS 16-ary 3D constellation. [ABSTRACT FROM AUTHOR]

A constellation-designed 32QAM method is proposed using geometric shaping (GS) and probabilistic shaping (PS). In addition, a 50-Gbaud polarization division multiplexed multispan optical fiber link for the proposed scheme is set up. The simulation results show that when the net data rate reaches 211 Gbit/s, the proposed 32QAM with GS and PS is capable of achieving about 0.1- and 0.3-dB OSNR gain over PS-32QAM and regular 32QAM (R-32QAM), respectively. When 213 Gbit/s net data rate transmission is achieved, the use of the proposed 32QAM can increase the transmission length by ~60 km over PS-32QAM and ~210 km over R-32QAM. Moreover, when OSNR is set to 19 dB and signal bandwidth is 40 Gbaud, the net data rate of the proposed 32QAM is 2 and 7.5 Gbit/s larger than PS-32QAM and R-32QAM, respectively. [ABSTRACT FROM AUTHOR]

Coded modulation (CM), a combination of forward error correction (FEC) and high order modulation formats, has become a key part of modern optical communication systems. Designing CM schemes with strict complexity requirements for optical communications (e.g., data center interconnects) is still challenging mainly because of the expected low latency, low overhead, and the stringent high data rate requirements. In this paper, we propose a CM scheme with bit-wise hard-decision FEC and geometric shaping. In particular, we propose to combine the recently introduced soft-aided bit-marking decoding algorithm for staircase codes (SCCs) with geometrically-shaped constellations. The main goal of this CM scheme is to jointly boost the coding gain and provide shaping gain, while keeping the complexity low. When compared to existing CM systems based on M-ary quadrature-amplitude modulation (MQAM, M = 64 , 128 , 256 ) and conventional decoding of SCCs, the proposed scheme shows improvements of up to 0.83 dB at a bit-error rate of 10 − 6 in the additive white Gaussian noise channel. For a nonlinear optical fiber system, simulation results show up to 24 % reach increase. In addition, the proposed CM scheme enables rate adaptivity in single-wavelength systems, offering six different data rates between 450 Gbit/s and 666 Gbit/s.

Constellation shaping has been widely used in optical communication systems. We review recent advances in two-dimensional constellation shaping technologies for fiber-optic communications. The system architectures that are discussed include probabilistic shaping, geometric shaping, and hybrid probabilistic-geometric shaping solutions. The performances of the three shaping schemes are also evaluated for Gaussian-noise-limited channels.

A systematic study, including theory, simulation and experiments, is carried out to review the generalized pairwise optimization algorithm for designing optimized constellation. In order to verify its effectiveness, the algorithm is applied in three testing cases: 2-dimensional 8 quadrature amplitude modulation (QAM), 4-dimensional set-partitioning QAM, and probabilistic-shaped (PS) 32QAM. The results suggest that geometric shaping can work together with PS to further bridge the gap toward the Shannon limit. [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]

Coded modulation (CM), a combination of forward error correction (FEC) and high order modulation formats, has become a key part of modern optical communication systems. Designing CM schemes with strict complexity requirements for optical communications (e.g., data center interconnects) is still challenging mainly because of the expected low latency, low overhead, and the stringent high data rate requirements. In this paper, we propose a CM scheme with bit-wise hard-decision FEC and geometric shaping. In particular, we propose to combine the recently introduced soft-aided bit-marking decoding algorithm for staircase codes (SCCs) with geometrically-shaped constellations. The main goal of this CM scheme is to jointly boost the coding gain and provide shaping gain, while keeping the complexity low. When compared to existing CM systems based on M-ary quadrature-amplitude modulation (MQAM, M = 64 , 128 , 256 ) and conventional decoding of SCCs, the proposed scheme shows improvements of up to 0.83 dB at a bit-error rate of 10 − 6 in the additive white Gaussian noise channel. For a nonlinear optical fiber system, simulation results show up to 24 % reach increase. In addition, the proposed CM scheme enables rate adaptivity in single-wavelength systems, offering six different data rates between 450 Gbit/s and 666 Gbit/s. [ABSTRACT FROM AUTHOR]