On the Capacity of Correlated MIMO Phase-Noise Channels: An Electro-Optic Frequency Comb Example

The capacity of a discrete-time multiple-input-multiple-output channel with correlated phase noises is investigated. In particular, the electro-optic frequency comb system is considered, where the phase noise of each channel is a combination of two independent Wiener phase-noise sources. Capacity upper and lower bounds are derived for this channel and are compared with lower bounds obtained by numerically evaluating the achievable information rates using quadrature amplitude modulation constellations. Capacity upper and lower bounds are provided for the high signal-to-noise ratio (SNR) regime. The multiplexing gain (pre-log) is shown to be $M-1$, where $M$ represents the number of channels. A constant gap between the asymptotic upper and lower bounds is observed, which depends on the number of channels $M$. For the specific case of $M=2$, capacity is characterized up to a term that vanishes as the SNR grows large.

On the Performance of Multidimensional Constellation Shaping for Linear and Nonlinear Optical Fiber Channel

Multidimensional constellation shaping of up to 32 dimensions with different spectral efficiencies are compared through AWGN and fiber-optic simulations. The results show that no constellation is universal and the balance of required and effective SNRs should be jointly considered for the specific optical transmission scenario.

Low-Complexity Soft Decision Detection for Combating DFE Burst Errors in IM/DD Links

The deployment of non-binary pulse amplitude modulation (PAM) and soft decision (SD)-forward error correction (FEC) in future intensity-modulation (IM)/direct-detection (DD) links is inevitable. However, high-speed IM/DD links suffer from inter-symbol interference (ISI) due to bandwidth-limited hardware. Traditional approaches to mitigate the effects of ISI are filters and trellis-based algorithms targeting symbol-wise maximum a posteriori (MAP) detection. The former approach includes decision-feedback equalizer (DFE), and the latter includes Max-Log-MAP (MLM) and soft-output Viterbi algorithm (SOVA). Although DFE is easy to implement, it introduces error propagation. Such burst errors distort the log-likelihood ratios (LLRs) required by SD-FEC, causing performance degradation. On the other hand, MLM and SOVA provide near-optimum performance, but their complexity is very high for high-order PAM. In this paper, we consider a one-tap partial response channel model, which is relevant for high-speed IM/DD links. We propose to combine DFE with either MLM or SOVA in a low-complexity architecture. The key idea is to allow MLM or SOVA to detect only 3 typical DFE symbol errors, and use the detected error information to generate LLRs in a modified demapper. The proposed structure enables a tradeoff between complexity and performance: (i) the complexity of MLM or SOVA is reduced and (ii) the decoding penalty due to error propagation is mitigated. Compared to SOVA detection, the proposed scheme can achieve a significant complexity reduction of up to 94% for PAM-$8$ transmission. Simulation and experimental results show that the resulting SNR loss is roughly 0.3 to 0.4 dB for PAM-4, and becomes marginal 0.18 dB for PAM-8.

Analytical Model of Nonlinear Fiber Propagation for General Dual-Polarization Four-Dimensional Modulation Format

Coherent dual-polarization (DP) optical transmission systems encode information on the four available degrees of freedom of an optical field: the two polarization states, each with two quadrature components. Such systems naturally operate based on a four-dimensional (4D) signal space. Having a general analytical model to accurately estimate nonlinear interference (NLI) is key to analyze such transmission systems as well as to study how different DP-4D formats are affected by NLI. However, the available models in the literature are not completely general. They either do not apply to the entire DP-4D formats or do not consider all the NLI contributions. In this paper we develop a model that applies to the entire class of DP-4D modulation formats. Our model takes self-channel interference, cross-channel interference and multiple-channel interference effects into account. As an application of our model, we further study the effects of signal-noise interactions in long-haul transmission via the proposed model. When compared to previous results in the literature, our model is more accurate at predicting the contribution of NLI for both low and high dispersion fibers in single- and multi-channel transmission systems. For the NLI, we report an average gap from split step Fourier simulation results below 0.15dB. The simulation results further show that by considering signal-noise interactions, the proposed model in long-haul transmission improves the NLI power accuracy prediction by up to 8.5%.

Data-driven Enhancement of the Time-domain First-order Regular Perturbation Model

A normalized batch gradient descent optimizer is proposed to improve the first-order regular perturbation coefficients of the Manakov equation, often referred to as kernels. The optimization is based on the linear parameterization offered by the first-order regular perturbation and targets enhanced low-complexity models for the fiber channel. We demonstrate that the optimized model outperforms the analytical counterpart where the kernels are numerically evaluated via their integral form. The enhanced model provides the same accuracy with a reduced number of kernels while operating over an extended power range covering both the nonlinear and highly nonlinear regimes. A $6-7$~dB gain, depending on the metric used, is obtained with respect to the conventional first-order regular perturbation.

Geometrically-Shaped Multi-Dimensional Modulation Formats in Coherent Optical Transmission Systems

Shaping modulation formats in multi-dimensional (MD) space is an effective approach to harvest spectral efficiency gains in both the additive white Gaussian noise (AWGN) channel and the optical fiber channel. In the first part of this paper, existing MD geometrically-shaped modulations for fiber optical communications are reviewed. It is shown that large gains can be obtained by exploiting correlation in the dimensions or/and by increasing the cardinality of the modulation format. Practical limitations and challenges are also discussed together with efficient solutions. In the second part, we extend the recently proposed four-dimensional (4D) modulation format family based on the constraint of orthant-symmetry to high spectrum efficiencies up to 10 bit/4D-sym by maximizing generalized mutual information for AWGN channel. Reach increases of up to 25% for a multi-span optical fiber transmission system are reported. Lastly,with the help of a recently introduced nonlinear interference (NLI) model, an optimization for designing nonlinear-tolerant 4D modulation formats is introduced for a single-span optical fiber system. Simulation results show that the proposed NLI model-based 4D modulation format could increase the effective SNRs by 0.25 dB with respect to the AWGN channel-optimal 4D modulation format.

Extension and Validation of 4D Model for Improving the Accuracy of Modulation-Dependent Nonlinear Interference

Existing nonlinear interference (NLI) model underestimates the NLI of dual polarization four-dimensional (4D) modulation in long-haul transmission, due to the ignorance of nonlinearity caused by signal-ASE interaction. We propose an enhanced 4D model by lifting an underlying assumption, which could improve the performance prediction accuracy.

High-Cardinality Geometrical Constellation Shaping for the Nonlinear Fibre Channel

This paper presents design methods for highly efficient optimisation of geometrically shaped constellations to maximise data throughput in optical communications. It describes methods to analytically calculate the information-theoretical loss and the gradient of this loss as a function of the input constellation shape. The gradients of the \ac{MI} and \ac{GMI} are critical to the optimisation of geometrically-shaped constellations. It presents the analytical derivative of the achievable information rate metrics with respect to the input constellation. The proposed method allows for improved design of higher cardinality and higher-dimensional constellations for optimising both linear and nonlinear fibre transmission throughput. Near-capacity achieving constellations with up to 8192 points for both 2 and 4 dimensions, with generalised mutual information (GMI) within 0.06 bit/2Dsymbol of additive white Gaussian noise channel (AWGN) capacity, are presented. Additionally, a design algorithm reducing the design computation time from days to minutes is introduced, allowing the presentation of optimised constellations for both linear AWGN and nonlinear fibre channels for a wide range of signal-to-noise ratios.

Shaped Four-Dimensional Modulation Formats for Optical Fiber Communication Systems

We review the design of multidimensional modulations by maximizing generalized mutual information and compare the maximum transmission reach of recently introduced 4D formats. A model-based optimization for nonlinear-tolerant 4D modulations is also discussed.

High-Cardinality Hybrid Shaping for 4D Modulation Formats in Optical Communications Optimized via End-to-End Learning

In this paper we carry out a joint optimization of probabilistic (PS) and geometric shaping (GS) for four-dimensional (4D) modulation formats in long-haul coherent wavelength division multiplexed (WDM) optical fiber communications using an auto-encoder framework. We propose a 4D 10 bits/symbol constellation which we obtained via end-to-end deep learning over the split-step Fourier model of the fiber channel. The constellation achieved 13.6% reach increase at a data rate of approximately 400 Gbits/second in comparison to the ubiquitously employed polarization multiplexed 32-QAM format at a forward error correction overhead of 20%.