Nonlinear Equalization Research Articles

Machine Learning in Short-Reach Optical Systems: A Comprehensive Survey

Recently, extensive research has been conducted to explore the utilization of machine learning (ML) algorithms in various direct-detected and (self)-coherent short-reach communication applications. These applications encompass a wide range of tasks, including bandwidth request prediction, signal quality monitoring, fault detection, traffic prediction, and digital signal processing (DSP)-based equalization. As a versatile approach, ML demonstrates the ability to address stochastic phenomena in optical systems networks where deterministic methods may fall short. However, when it comes to DSP equalization algorithms such as feed-forward/decision-feedback equalizers (FFEs/DFEs) and Volterra-based nonlinear equalizers, their performance improvements are often marginal, and their complexity is prohibitively high, especially in cost-sensitive short-reach communications scenarios such as passive optical networks (PONs). Time-series ML models offer distinct advantages over frequency-domain models in specific contexts. They excel in capturing temporal dependencies, handling irregular or nonlinear patterns effectively, and accommodating variable time intervals. Within this survey, we outline the application of ML techniques in short-reach communications, specifically emphasizing their utilization in high-bandwidth demanding PONs. We introduce a novel taxonomy for time-series methods employed in ML signal processing, providing a structured classification framework. Our taxonomy categorizes current time-series methods into four distinct groups: traditional methods, Fourier convolution-based methods, transformer-based models, and time-series convolutional networks. Finally, we highlight prospective research directions within this rapidly evolving field and outline specific solutions to mitigate the complexity associated with hardware implementations. We aim to pave the way for more practical and efficient deployment of ML approaches in short-reach optical communication systems by addressing complexity concerns.

A Stochastic Inexact Sequential Quadratic Optimization Algorithm for Nonlinear Equality-Constrained Optimization

A stochastic algorithm is proposed, analyzed, and tested experimentally for solving continuous optimization problems with nonlinear equality constraints. It is assumed that constraint function and derivative values can be computed but that only stochastic approximations are available for the objective function and its derivatives. The algorithm is of the sequential quadratic optimization variety. Distinguishing features of the algorithm are that it only employs stochastic objective gradient estimates that satisfy a relatively weak set of assumptions (while using neither objective function values nor estimates of them) and that it allows inexact subproblem solutions to be employed, the latter of which is particularly useful in large-scale settings when the matrices defining the subproblems are too large to form and/or factorize. Conditions are imposed on the inexact subproblem solutions that account for the fact that only stochastic objective gradient estimates are employed. Convergence results are established for the method. Numerical experiments show that the proposed method vastly outperforms a stochastic subgradient method and can outperform an alternative sequential quadratic programming algorithm that employs highly accurate subproblem solutions in every iteration. Funding: This material is based upon work supported by the National Science Foundation [Awards CCF-1740796 and CCF-2139735] and the Office of Naval Research [Award N00014-21-1-2532].

Analysis of numerical modeling of steady-state modes of methane–hydrogen mixture transportation through a compressor station to reduce CO2 emissions

This study presents a mathematical model to evaluate the performance of gas pipelines during hydrogen injection in a gas pipeline-compressor station. The developed model presents the calculation of methane–hydrogen mixture (CH4/H2) transportation through the compressor station, where the compensation of pressure drops in the mass and energy balance takes place. Simultaneously, in the operation of the centrifugal blower system of gas compressor stations, the emissions of CO2 are considered, considering the mixing of gas media and the compression of CH4/H2. This mathematical model is realized for the pipeline transportation of hydrogen, at which the principle of mixture expansion occurs. The aim is to solve the problem of CO2 emissions at compressor stations. The optimization procedure has been formulated using a system of nonlinear algebraic equalities. The research focuses on the adaptation of existing gas transportation systems to CH4/H2 transportation and the impact of environmental risks on the operation of compressor station equipment. In this case, it is possible to determine the quantitative amount of hydrogen that can be added to natural gas. By solving the problem of finding the inner point of sets using the system of nonlinear algebraic equalities, it is possible to obtain the control parameters for safety control of technological modes of CH4/H2 mixture transportation. The study findings reveal that the consumption of gas charger and hydrogen was 50.67 and 0.184 kg/s, respectively, and the estimated efficiency resulting from the modified turbine design was 75.1 percent. These results indicate that the equipment operates more efficiently when hydrogen is being transported. The numerical analytical results indicated in this study hold practical significance for design applications. It will assist in identifying and evaluating the restrictions that may develop during the technological, operational, and design stages of decision-making.

Global Receptive Field Designed Complex-Valued Convolutional Neural Network Equalizer for Optical Fiber Communications

In this paper, an improved complex-valued convolutional neural network (CvCNN) structure to be placed at the received side is proposed for nonlinearity compensation in a coherent optical system. This complex-valued global convolutional kernel-assisted convolutional neural network equalizer (CvGNN) has been verified in terms of Q-factor performance and complexity compared to seven other related nonlinear equalizers based on both the 64 QAM experimental platform and the QPSK numerical platform. The global convolution operation of the proposed CvGNN is more suitable for the calculation process of perturbation coefficients, and the global receptive field can also be more effective at extracting effective information from perturbation feature maps. The introduction of CvCNN can directly focus on the complex-valued perturbation feature maps themselves without separately processing the real and imaginary parts, which is more in line with the waveform-dependent physical characteristics of optical signals. Based on the experimental platform, compared with the real-valued neural network with small convolutional kernel (RvCNNC), the proposed CvGNNC improves the Q-factor by ∼2.95 dB at the optimal transmission power, while reducing the time complexity by ∼44.7%.

Nonlinear States-Truncated BCJR Equalization With States and Complexity Reduction for 100-Gaud PAM-8 IM/DD Transmission

Nonlinear States-Truncated BCJR Equalization With States and Complexity Reduction for 100-Gaud PAM-8 IM/DD Transmission

Distributed reservoir computing based nonlinear equalizer for VCSEL based optical interconnects

We propose a distributed reservoir computing based nonlinear equalizer for 106 Gbps VCSEL based optical interconnects. By comparing with neural network and Volterra series based equalizers, the computational complexity of this proposed nonlinear equalizer is significantly reduced in the training, prediction and inference process. 106 Gbps PAM4 short-range optical transmission is demonstrated by experiment, using a VCSEL-based co-packaged optical transmitter to verify the performance of the equalizer and key parameters that affect the nonlinear compensation performance are carefully investigated. We believe that by incorporating such simplified nonlinear equalizers, it is possible to compensate for nonlinear impairments while preserving the inherent advantages of VCSEL-based optical interconnects. This optical interconnect will become a promising paradigm to overcome I/O performance penalty when communication leaving packages.

Constructing order-2 information granules of linguistic expressions with the aid of the principle of justifiable granularity

To capture collective opinions/evaluations in a collection of individual linguistic expressions, this study proposes an approach to construct order-2 information granules by extending the numerical data-based principles of justifiable granularity to a linguistic data-based one. First, the two key criteria of the principle of justifiable granularity, namely coverage and specificity, are formally defined in the context of order-2 information granules. Second, three order-2 information granules construction models by maximizing the product of coverage and specificity are developed for coping with one-dimensional direct linguistic expressions, linguistic preference relations, and multi-dimensional direct linguistic expressions, respectively. Third, considering that the developed order-2 information granules construction models exhibit a non-linear uncertain objective function and equality constraints, the constrained multi-swarm PSO without velocity is improved with the use of the ε constrained handling technique for effectively solving the models. Case studies on the considered three types of linguistic expressions show the applicability of the proposed models and ensuing algorithm. The superiority of the improved algorithm in handling the proposed models is demonstrated by comparison with the original one. The effectiveness of the proposed models in terms of abnormal corrective ability and balance of coverage and specificity is verified by comparing with a family of Top-n methods. The originality of this study lies in the construction of an operational entity, namely order-2 information granule, that reflects the group opinion without specifying the formalism of individual linguistic expressions, which provides an effective and efficient way to aggregate ubiquitous linguistic information for subsequent computation, reasoning and decision-making.

Overfitting Problem of ANN- and VSTF-Based Nonlinear Equalizers Trained on Repeated Random Bit Sequences

Overfitting Problem of ANN- and VSTF-Based Nonlinear Equalizers Trained on Repeated Random Bit Sequences

ExtraTrees-hidden Markov model based equalizer for mode division multiplexing ring-core fiber communication with non-orthogonal MultiCAP modulation

This paper presents an orbital angular momentum mode division multiplexing (OAM-MDM) ring-core fiber transmission method based on non-orthogonal multiband carrierless amplitude and phase (NMCAP) modulation with an extremely randomized trees-hidden Markov model (ExtraTrees-HMM)-based equalizer. The ExtraTrees-HMM equalizer uses the statistical characteristics of the received distorted signals to model the nonlinear channel of the system to classify these distorted signals into corresponding constellation classes. Experiments were conducted using a 216 Gbit/s OAM-MDM NMCAP modulation optical fiber communication system with 2 km ring-core fiber transmission and the results show that compared with a conventional Volterra nonlinear equalizer (VNE), the proposed ExtraTrees-HMM equalizer could improve the receiver sensitivity by 1 dB for OAM mode l = + 2, and 0.6 dB for OAM mode l = + 3. In addition, the computational complexity of the proposed equalizer was reduced by 43.94% compared with the VNE. In brief, the ExtraTrees-HMM is a promising equalization candidate for ultra-high-capacity inter-data-center interconnections.

Chance-Constrained Optimization for a Green Multimodal Routing Problem with Soft Time Window under Twofold Uncertainty

This study investigates a green multimodal routing problem with soft time window. The objective of routing is to minimize the total costs of accomplishing the multimodal transportation of a batch of goods. To improve the feasibility of optimization, this study formulates the routing problem in an uncertain environment where the capacities and carbon emission factors of the travel process and the transfer process in the multimodal network are considered fuzzy. Taking triangular fuzzy numbers to describe the uncertainty, this study proposes a fuzzy nonlinear programming model to deal with the specific routing problem. To make the problem solvable, this study adopts the fuzzy chance-constrained programming approach based on the possibility measure to remove the fuzziness of the proposed model. Furthermore, we use linear inequality constraints to reformulate the nonlinear equality constraints represented by the continuous piecewise linear functions and realize the linearization of the nonlinear programming model to improve the computational efficiency of problem solving. After model processing, we can utilize mathematical programming software to run exact solution algorithms to solve the specific routing problem. A numerical experiment is given to show the feasibility of the proposed model. The sensitivity analysis of the numerical experiment further clarifies how improving the confidence level of the chance constraints to enhance the possibility that the multimodal route planned in advance satisfies the real-time capacity constraint in the actual transportation, i.e., the reliability of the routing, increases both the total costs and carbon emissions of the route. The numerical experiment also finds that charging carbon emissions is not absolutely effective in emission reduction. In this condition, bi-objective analysis indicates the conflicting relationship between lowering transportation activity costs and reducing carbon emissions in routing optimization. The sensitivity of the Pareto solutions concerning the confidence level reveals that reliability, economy, and environmental sustainability are in conflict with each other. Based on the findings of this study, the customer and the multimodal transport operator can organize efficient multimodal transportation, balancing the above objectives using the proposed model.

Optimized pilot structure for PS-PDM ultrahigh-order QAM coherent optical transmission

We proposed and experimentally demonstrated a general pilot structure for probabilistic shaped (PS)-polarization division multiplexing (PDM) M-ary quadrature amplitude modulation (MQAM) coherent optical transmission, where a portion of PS-MQAM symbols is exploited as the pilot symbols with the same information entropy as the transmitted signal. The pilot symbols are simultaneously used in the entire digital signal processing (DSP) modules for polarization de-multiplexing, frequency offset estimation, carrier phase recovery, nonlinear equalization, and linear equalization. Compared to the conventional quadrature phase shift keying (QPSK) pilot structure, the proposed MQAM pilot structure can yield the nonlinear properties of the overall signal so that nonlinear equalization can effectively improve the performance of the normalized generalized mutual information. The remarkable performance has been achieved at the shaping parameters of 3.25 and 4.35 for 1024QAM and 4096QAM signals based on the proposed pilot scheme, which corresponds to the raw spectral efficiency of 16.190 and 20.381 bit/s/Hz, respectively. The pilot ratio is optimized to 5% for higher achievable information rates (AIRs).

The Regularization Continuation Method for Optimization Problems with Nonlinear Equality Constraints

The Regularization Continuation Method for Optimization Problems with Nonlinear Equality Constraints

Transformer-Based Nonlinear Equalization for DP-16QAM Coherent Optical Communication Systems

Transformer-Based Nonlinear Equalization for DP-16QAM Coherent Optical Communication Systems

DACNN-aided nonlinear equalizer for a probabilistic shaping coherent optical communication system.

The probabilistic shaping (PS) technique is a key technology for fiber optic communication systems to further approach the Shannon limit. To solve the problem that nonlinear equalizers are ineffective for probabilistic shaping optical communication systems with non-uniform distribution, a distribution alignment convolutional neural network (DACNN)-aided nonlinear equalizer is proposed. The approach calibrates the equalizer using the probabilistic shaping prior distribution, which reduces the training complexity and improves the performance of the equalizer simultaneously. Experimental results show nonlinear equalization of 120Gb/s PS 64QAM signals in a 375km transmission scenario. The proposed DACNN equalizer improves the receiver sensitivity by 2.6dB and 1.1dB over the Volterra equalizer and convolutional neural network (CNN) equalizer, respectively. Meanwhile, DACNN converges with fewer training epochs than CNN, which provides great potential for mitigating the nonlinear distortion of PS signals in fiber optic communication systems.

Learning bayesian network parameters from limited data by integrating entropy and monotonicity

High-accuracy parameter learning in Bayesian Networks (BNs) is a key challenge in real-time decision support applications, particularly when the available data are limited. Prior/Expert knowledge was introduced to eliminate the drawbacks of insufficient information; however, this method is subjective. In this study, we explored the use of monotonicity constraints to control the causal relationships between the nodes and their parents-nodes in BNs and proposed a new learning algorithm called global domain monotonicity based on maximum information entropy (GDM-MIE), which was designed for parameter learning in uncertain discrete BNs with nonlinear equality constraints when only finite data are available. In the proposed algorithm, a class of monotonicity is encoded as a constraint on information entropy and the parameter learning problem is transformed into constraints among the node parameters based on a known network. Furthermore, we considered the parameters as uncertain entropy information and discussed the monotonicity among parameters in the global spatial domain, proving the accuracy of the logical relationships of the model, system reliability, and time complexity. Finally, the proposed method was validated using standard BNs, and its performance was analyzed by comparing the proposed method with the existing learning algorithms. The results showed that the proposed method is more accurate and has better Kullback–Leibler divergence. To revalidate the rationality of the proposed method, the Alarm and Asia networks were employed as special cases. The GDM-MIE was found to achieve the intended goal of observing the estimated parameters by closely approximating the original real parameters with a small sample size, indicating that the proposed algorithm can served as an efficient and feasible method for learning Bayesian parameter.

Polyhedral Relaxations for Optimal Pump Scheduling of Potable Water Distribution Networks

The classic pump scheduling or optimal water flow (OWF) problem for water distribution networks (WDNs) minimizes the cost of power consumption for a given WDN over a fixed time horizon. In its exact form, the OWF is a computationally challenging mixed-integer nonlinear program (MINLP). It is complicated by nonlinear equality constraints that model network physics, discrete variables that model operational controls, and intertemporal constraints that model changes to storage devices. To address the computational challenges of the OWF, this paper develops tight polyhedral relaxations of the original MINLP, derives novel valid inequalities (or cuts) using duality theory, and implements novel optimization-based bound tightening and cut generation procedures. The efficacy of each new method is rigorously evaluated by measuring empirical improvements in OWF primal and dual bounds over 45 literature instances. The evaluation suggests that our relaxation improvements, model strengthening techniques, and a thoughtfully selected polyhedral relaxation partitioning scheme can substantially improve OWF primal and dual bounds, especially when compared with similar relaxation-based techniques that do not leverage these new methods. History: Accepted by David Alderson, Area Editor for Network Optimization: Algorithms & Applications. Funding: This work was supported by the U.S. Department of Energy (DOE) Advanced Grid Modeling project, Coordinated Planning and Operation of Water and Power Infrastructures for Increased Resilience and Reliability. Incorporation of the PolyhedralRelaxations Julia package was supported by Los Alamos National Laboratory’s Directed Research and Development program under the project Fast, Linear Programming-Based Algorithms with Solution Quality Guarantees for Nonlinear Optimal Control Problems [Grant 20220006ER]. All work at Los Alamos National Laboratory was conducted under the auspices of the National Nuclear Security Administration of the U.S. DOE, Contract No. 89233218CNA000001. This work was also authored in part by the National Renewable Energy Laboratory, operated by the Alliance for Sustainable Energy, LLC, for the U.S. DOE, Contract No. DE-AC36-08GO28308. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.0233 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0233 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Coherent all-optical reservoir computing for nonlinear equalization in long-haul optical fiber communication systems

Photonic reservoir computing (RC), as a promising paradigm of optical signal processing, has attracted a great deal of attention in fiber nonlinearity equalization recently. For real long-haul transmission scenarios, we propose an all-optical RC operated in a coherent scheme to mitigate the nonlinear distortion of high-speed signals. Compared with the direct decision architecture, the proposed RC can dramatically improve the Q2 factor by 2.6 dB for a single-wavelength system and 1.9 dB for a 7-channel wavelength-division multiplexing (WDM) system over 3960-km fiber transmission. Furthermore, the robustness of the coherent all-optical RC is also verified by conducting studies on different training sequences. The coherent all-optical RC shows great potential in real-time nonlinearity mitigation for long-haul fiber transmissions.

Wide and Deep Learning-Aided Nonlinear Equalizer for Coherent Optical Communication Systems

In this study, we developed a wide and deep network-based nonlinear equalizer to compensate for nonlinear impairment in coherent optical communication systems. In our proposed equalizer, the power feature factor and inter-symbol feature sequence in the received signal are analyzed by two combined networks, wide and deep, respectively, so that the information contained in the signal can be fully utilized. We designed an experiment using a 120 Gbit/s 64-quadrature amplitude modulation (64-QAM) coherent optical communication system over a 375 km standard single-mode fiber (SSMF) to verify the performance of the proposed wide and deep network-based nonlinear equalizer. The experimental results showed that the proposed wide and deep network-based nonlinear equalizer achieved better performance at lower complexity compared with the traditional neural network-based nonlinear equalizer. The proposed equalizer significantly improved the equalization effect at a cost of a 0.3% increase in parameters, which indicates the potential of the proposed method for application in coherent optical communication systems.

OAM mode-division multiplexing IM/DD transmission at 4.32 Tbit/s with a low-complexity adaptive-network-based fuzzy inference system nonlinear equalizer.

Stochastic nonlinear impairment is the primary factor that limits the transmission performance of high-speed orbital angular momentum (OAM) mode-division multiplexing (MDM) optical fiber communication systems. This Letter presents a low-complexity adaptive-network-based fuzzy inference system (LANFIS) nonlinear equalizer for OAM-MDM intensity-modulation direct-detection (IM/DD) transmission with three OAM modes and 15 wavelength division multiplex (WDM) channels. The LANFIS equalizer could adjust the probability distribution functions (PDFs) of the distorted pulse amplitude modulation (PAM) symbols to fit the statistical characteristics of the WDM-OAM-MDM transmission channel. Therefore, although the transmission symbols in the WDM-OAM-MDM system are subjected to a stochastic nonlinear impairment, the proposed LANFIS equalizer can effectively compensate the distorted signals. The proposed equalizer outperforms the Volterra equalizer with improvements in receiver sensitivity of 2, 1.5, and 1.3 dB for three OAM modes at a wavelength of 1550.12 nm, respectively. It also outperforms a CNN equalizer, with improvements in receiver sensitivity of 1, 0.5, and 0.3 dB, respectively. Moreover, complexity reductions of 67%, 74%, and 99.9% are achieved for the LANFIS equalizer compared with the Volterra, CNN, and ANFIS equalizers, respectively. The proposed equalizer has high performance and low complexity, making it a promising candidate for a high-speed WDM-OAM-MDM system.

Multichannel Nonlinear Equalization in Coherent WDM Systems Based on Bi-Directional Recurrent Neural Networks

Multichannel Nonlinear Equalization in Coherent WDM Systems Based on Bi-Directional Recurrent Neural Networks