Nonlinear Frequency Division Multiplexing Systems Research Articles

Efficient and reliable digital sub-band multiplexed nonlinear frequency division multiplexing system.

Nonlinear frequency division multiplexing (NFDM) is a promising optical transmission system that avoids fiber nonlinear effects, but suffering from unsatisfied spectral efficiency due to the guard interval (GI) devoting to combat the cumulative chromatic dispersion (CD), whose contribution to efficiency declination is quadratic to the bandwidth. In this paper, we propose what we believe to be a novel digital sub-band multiplexed (DSBM) NFDM system that can diminish the minimal required guard interval and therefore realize considerable information rate without reliability degradation. Through numerical simulation, we show that root raised cosine (RRC) shaped subcarriers are more suitable than the commonly used orthogonal frequency division multiplexing (OFDM) shaped ones in our DSBM-NFDM system. By multiplexing 15 sub-bands, we achieve a 129.7% improvement in effective information rate (EIR) compared to the single channel (SC) NFDM system with the same effective bandwidth. Meanwhile, we achieve a net information rate of 531.0 Gbps NFDM transmission over 960 km with SE of 3.86 bit/s/Hz, which is below the hard decision forward error correction (HD-FEC) threshold. This work is the first to introduce the DSBM technique into NFDM system, and it can be considered as a novel subcarrier shaping scheme for NFDM that has potential to advance its practical implementation.

Narrowband nonlinear frequency division multiplexing system based on Hermite–Gaussian-carriers

Nonlinear frequency division multiplexing (NFDM) is a promising technique for optical fiber communication that can overcome the Kerr nonlinear effects of optical fibers. Hermite–Gaussian (HG)-carriers are a potential candidate for constructing the nonlinear spectrum in NFDM systems, as they have lower time–bandwidth product and higher tolerance to truncation noise than conventional sinc-carriers. However, HG-carriers are affected by interference and truncation error between subcarriers, which degrade the system performance. In this paper, we propose a HG-carriers reconstruction method based on Gram–Schmidt Orthogonalization, which can effectively restore the orthogonality between subcarriers and reduce the distortion. Through a numerical simulation, we explore the use of HG-carriers and sinc-carriers to construct the nonlinear spectrum, and analyze their performance differences under different system bandwidth parameters. We also implement HG-carriers in a narrowband (6.8 GHz) b-modulation dual-polarization NFDM system and optimize them to achieve a maximum spectral efficiency (SE) of 8.89 bits/s/Hz (4.44 bits/s/Hz per polarization) over 960 km with bit error rate (BER) below the hard decision forward error correction (HD-FEC) threshold. Our results demonstrate that HG-carriers outperform sinc-carriers in narrowband NFDM systems, and that our proposed HG-carriers reconstruction method can significantly improve the Q-factor and SE.

An eigenvalue-shift and matching-signal based FOE method for discrete spectrum modulated NFDM system

We propose an improved frequency offset estimation (FOE) algorithm based on eigenvalue-shift and matching-signal (ESMS) for discrete spectrum modulated nonlinear frequency division multiplexing (NFDM) system, which can realize high estimation accuracy without the constraint of the Baud-rate. Meanwhile, its computational complexity is lower in contrast with nonlinear frequency domain (NFD) method. Therefore, the proposed method is more powerful for eigenvalue NFDM transmission systems with large FO variation.

ASE noise mitigation with digital frequency offset loading for discrete spectrum nonlinear frequency division multiplexing systems.

We propose an amplified spontaneous emission (ASE) noise mitigation scheme utilizing digital frequency offset loading (DFO-loading) for discrete spectrum nonlinear frequency division multiplexing (DS-NFDM) systems. Firstly, based on the one-to-one mapping relationship between frequency offsets and eigenvalue positions, the transmitter side encodes 4-bit information onto 16 kinds of different digital frequency offsets. Then, a sliding window-assisted eigenvalue position (SWA-EP) decoding technology is further proposed to substitute the classical channel equalization and carrier phase recovery processes, with the purpose of recovering the original information. The numerical and experimental results demonstrate that, compared with b-coefficient 16 quadrature amplitude modulation (QAM) scheme, Q-factor gains are 2.1 dB under 15 dB optical signal-to-noise ratio (OSNR) and 1.8 dB after 800 km fiber transmission, respectively. Moreover, its complexity is only 0.6% of the b-coefficient scheme. The DFO-loading scheme offers an effective and low-complexity way to mitigate ASE noise of DS-NFDM system.

Autoencoder assisted subcarrier optimization for nonlinear frequency division multiplexing.

Nonlinear frequency division multiplexing (NFDM) is a novel optical communication technique that can achieve nonlinear free transmission. However, current design of NFDM is analogous to orthogonal frequency division multiplexing (OFDM), where sinc function is utilized as subcarriers, which may not be optimal for nonlinear spectrums. In this paper, we propose an auto-encoder (AE) assisted subcarrier optimization scheme for dual-polarized (DP) NFDM systems. Numerical verifications show that our scheme can improve the Q-factor by 1.54 dB and 0.62 dB compared to sinc subcarrier and linear minimum mean square error (LMMSE) equalization, respectively, in a 960 km transmission scenario. We also analyze the characteristics of the optimized subcarriers and discuss how they enhance the performance. Furthermore, we demonstrate the robustness of the optimized subcarriers to different modulation formats, transmission distances and bandwidth. Our work provides a new idea in subcarrier design for NFDM.

Design of low PAPR 2-Soliton NFDM transmit signals with optimal constellations for NFT-based high-speed optical communication

Optical Fiber Communication is known for its extremely high data-carrying capacity among all the existing transmission technologies. However, its capacity to the fullest is restrained by the effects of nonlinearity, thereby urging nonlinear compensation. In this paper, we have attempted to enhance the performance of Nonlinear Frequency Division multiplexing (NFDM) based nonlinearity compensation, a recent attraction due to its high throughput nature. Here, we design optimal transmit signals, emphasizing lowering the Peak to Average Ratio (PAPR). With appropriate investigation, optimal eigenvalue and Quadrature Amplitude Modulation (QAM) constellations for encoding have been procured that aid in constructing low PAPR NFDM signals. To upgrade the optimal QAM design, two novel Spiral QAM models, Model I and Model II are proposed that diversify signal envelopes in addition to lowering the PAPR. This improves the overall fault tolerance of the NFDM system. For a 2-eigenvalue NFDM system, using the proposed constellations, the PAPR is reduced by 0.24 dB (Model I) and 1.42 dB (Model II), compared to the conventional scheme. Thus, we have designed optimum NFDM transmit pulses and subsequently accomplished a high performing 10 Gbps NFDM system using the proposed constellation models.

Optimal Design of Eigenvalues for the Full-Spectrum Modulated Nonlinear Frequency Division Multiplexing Transmission System

Nonlinear frequency division multiplexing (NFDM) system is an optional candidate to overcome the fiber nonlinearity limit. A full-spectrum modulated NFDM system, modulating data on combined continuous spectrum (CS) and discrete spectrum (DS) together, was proposed in recent years to improve the data rates and spectral efficiency (SE) by exploiting all the degrees of freedom offered in the nonlinear spectrum. However, the selection of discrete eigenvalues greatly affects the performance of both CS and DS. Designing appropriate eigenvalues of DS is an important issue to ensure the high SE and excellent performance of the system. In this paper, we discussed the selcection principle of eigenvalues and analyzed it from multiple perspectives, 11 eigenvalues with 64-quadrature amplitude modulation (64QAM) are selected for DS. Besides optimizing the eigenvalues at the transmitter, the linear minimum mean-square estimate (LMMSE) method was used at the receiver to furtherly improve the performance of DS. Through the numerical simulation, a 113 Gb/s (SE of 2.8 bits/s/Hz) full-spectrum modulated NFDM system was set up and transmitted 1120 km distance, where the Q-factors of both CS and DS are above the hard-decision forward error correction (HD-FEC) threshold. The results provide a way to design an efficiently full-spectrum modulated NFDM system.

Full-Spectrum INFT Algorithm for Dual-Polarization NFDM Transmission

Nonlinear Fourier transform (NFT)-based nonlinear frequency division multiplexing (NFDM) system has the potential to overcome the limit of fiber nonlinearity. In the NFDM system, the symbol information is modulated onto the nonlinear spectra, which consist of continuous and discrete spectrum. In theory, the dual-polarization (DP) NFDM system with the continuous and discrete spectrum (full spectrum, FS) modulation has the potential to achieve higher spectral efficiency. However, due to the complexity of the NFT and inverse NFT (INFT) numerical algorithms, the DP-FS modulated NFDM system is rarely reported. For the FS-NFDM system, the existing INFT algorithms either only support q-modulation and present poor performance, or have large discrete spectral error as the continuous spectrum energy increases. In this paper, we propose an INFT algorithm design theory. Based the theory, an accurate DP-INFT algorithm is proposed. The algorithm supports not only q-modulation, but also b-modulation with better performance. Employing the proposed INFT algorithm, we demonstrate, for the first time, a 174.5Gbit/s FS modulated NFDM transmission system with 1440km.

Frequency offset modeling in presence of ASE noise and corresponding low-complexity solution for discrete spectrum modulated nonlinear frequency division multiplexing system

For discrete spectrum modulated nonlinear frequency division multiplexing (DS-NFDM) transmission, the combined effect of laser frequency offset (FO) and amplifier spontaneous emission (ASE) noise would cause a perturbative shift of eigenvalues and corresponding spectral distortion. Moreover, the classical 4th-power feedforward frequency offset estimation (FOE) scheme is not suitable for DS-NFDM system with low baud rate. In this paper, we firstly deduce an FO impairment model with ASE noise in nonlinear Fourier domain (NFD). Meanwhile, we have proposed a training sequence-assisted FOE (TS-FOE) scheme, which effectively reduces the impact of ASE noise. The 2 GBaud DS-NFDM 16QAM simulation results illustrate that this scheme achieves an FOE error within 1 MHz using a training sequence (TS) with length of 64. Besides, its FOE range achieves [−Rs/2, +Rs/2], where Rs denotes the baud rate of signal. Furthermore, the effectiveness of TS-FOE scheme has been verified by 2 GBaud NFDM QPSK experimental system. The results demonstrate that when optical signal-to-noise ratio (OSNR) is set 13 dB, the TS-FOE scheme presents stable FO compensation performance for different FOs within 1 GHz, and the corresponding bit error rates (BERs) are all below the 7% hard decision forward error correction (HD-FEC) threshold. Meanwhile, its complexity is on the order of OL and related with the length L of TS, which is as one thousandth as that of angle search FOE scheme.

Noise equalization scheme based on complex-valued ANN for multiple-eigenvalue modulated nonlinear frequency division multiplexing systems.

In multiple-eigenvalue modulated nonlinear frequency division multiplexing (NFDM) systems, the noise degrades the accuracy of the nonlinear Fourier transform (NFT) algorithm, resulting in perturbations in the received eigenvalues and the corresponding discrete spectrum. Moreover, with the increase in the number of eigenvalues and the order of the modulation formats, the impact of noise on the performance of the system is even more. A noise equalization scheme based on complex-valued artificial neural network (c-ANN) for multiple-eigenvalue modulated NFDM systems is proposed. This sceheme inputs the eigenvalues perturbation and the impaired discrete spectrum corresponding to the eigenvalues into the c-ANN in complex form. The scheme constructs a complex-valued logic structure with both amplitude and phase information, overlapping reuse input features and, thereby, effectively reducing the effect of noise on the multiple-eigenvalue NFDM system. The effectiveness of the scheme is verified in long-haul seven-eigenvalue modulated single-polarization NFDM simulation systems with 1GBaud 16APSK/16QAM/64APSK/64QAM modulation formats, and the results show that the scheme outperforms the NFT receiving without equalization by 1 to 2 orders of magnitude in terms of bit error rate (BER). Among them, the transmission distance of the 64APSK signal after equalization exceeds 800km while the BER meets 7% forward error correction (FEC) threshold, which is 600km longer than that of the disequilibrium case, and the spectral efficiency (SE) can reach 1.85bit/s/Hz. Compared with other schemes, the proposed scheme has better equalization performance under the same complexity, and the complexity can be reduced by half or even under the same performance.

Nonlinear Spectrum Modulation in the Anomalous Dispersion Regime Using Second- and Third-Order Solitons

We study the robustness of a nonlinear frequency-division multiplexing (NFDM) system, based on the Zakharov-Shabat spectral problem (ZSSP), that is comprised of two independent quadrature phase-shift keyed (QPSK) channels modulated in the discrete spectrum associated with two distinct eigenvalues. Among the many fiber impairments that may limit this system, we focus on determining the limits due to third-order dispersion, the Raman effect, amplified spontaneous emission (ASE) noise from erbium-doped fiber amplifiers (EDFAs), and fiber losses with lumped gain from EDFAs. We examine the impact of these impairments on a 1600-km system by analyzing the Q-factor calculated from the error vector magnitude (EVM) of the received symbols. We found that the maximum launch power due to these impairments is: 13 dBm due to third-order dispersion, 11 dBm due to the Raman effect, 3 dBm due to fiber losses with lumped gain, and 2 dBm due to these three impairments combined with ASE noise. The maximum launch power due to all these impairments combined is comparable to that of a conventional wavelength-division multiplexing (WDM) system, even though WDM systems can operate over a much larger bandwidth and, consequently, have a much higher data throughput when compared with NFDM systems. We find that fiber losses in practical fiber transmission systems with lumped gain from EDFAs is the most stringent limiting factor in the performance of this NFDM system.

Enhancing the Spectral Efficiency of Nonlinear Frequency Division Multiplexing Systems via Hermite-Gaussian Subcarriers

The nonlinear frequency division multiplexing (NFDM) is a promising concept in the field of optical fiber communications. Based on a well established mathematical technique of the nonlinear Fourier transform (NFT), a data modulation scheme has been suggested, which is intrinsically immune to Kerr nonlinearity and chromatic dispersion. However, most NFDM systems still suffer from low values of spectral efficiency due to the large burst sizes and the nonlinear interaction of the signal and the inline amplifier noise. Here, we show that this problem can be partially circumvented by using the Hermite-Gaussian spectral carriers. Using this approach, we are able to report a significantly higher spectral efficiency for NFDM schemes approaching 4.5 bits/s/Hz for a single polarization while keeping the bit error rate of uncoded input below the 7% hard decision forward error correction threshold.

10.83 Tb/s Over 800 Km Nonlinear Frequency Division Multiplexing WDM Transmission

Nonlinear frequency division multiplexing (NFDM) transmission scheme has attracted great interest in optical fiber communication systems due to its potential for surpassing the Kerr nonlinearity limit. There have been very active studies on single-channel NFDM transmission systems recently. However, up to now, few experiments have validated the feasibility of applying NFDM into wavelength division multiplexing (WDM) optical communication systems toward high-capacity transmission for wideband coverage. Moreover, the overall performance of the NFDM system is restricted due to many challenges in theoretical design and practical implementation. Novel neural network-based equalization methods contribute to eliminating multi-dimensional crosstalk, hence prompting the NFDM system performance. In this article, we experimentally demonstrate a high-capacity NFDM-WDM transmission system enabled by the newly developed digital signal processing (DSP) algorithms. With data modulation on the continuous spectrum in nonlinear frequency domain, a 25-channel WDM system with polarization division multiplexing (PDM) 16-ary quadrature amplitude modulation (16-QAM) transmitting over 800 km standard single-mode fiber (SSMF) is successfully achieved. To mitigate the undesirable correlations between the nonlinear continuous spectral components, a nonlinear spectrum equalization scheme based on artificial neural network (ANN) is utilized in the receiver side DSP. Through this ANN-based spectrum equalization, a 0.63 dB Q <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> -factor gain is achieved compared to the conventional equalization scheme. A record transmission capacity of 10.83 Tb/s is achieved in a 25-channel WDM system over 800 km SSMF, with a remarkable single-channel line rate of 433.3 Gb/s. Our work reports the highest capacity NFDM-WDM transmission up to date and helps to further explore the inherent ability of NFDM to resist fiber channel nonlinearity.

Improvement for a full-spectrum modulated nonlinear frequency division multiplexing transmission system.

Nonlinear frequency division multiplexing (NFDM), as a possible technique to overcome the limit imposed by Kerr nonlinearity in conventional coherent optical communication systems, has attracted widespread attention in the communication community in recent years. In order to fully utilize the available degrees of freedom in the nonlinear spectrum, this paper focuses on the full-spectrum (FS) modulated NFDM system. First, we maximize the data rate of discrete spectrum (DS) by optimizing the distribution of eigenvalues in DS part of FS. Then through introducing the probabilistic shaping (PS) into the FS system, and combined with linear minimum mean square (LMMSE) estimators, a 1120 km transmission with BER below the hard decision forward error correction (HD-FEC) threshold at 112 Gbps is achieved, where 128 subcarriers with PS-64QAM are used in the continuous spectrum (CS) and 13 eigenvalues with 64QAM are adopted in the discrete spectrum (DS). The achievable data rate is about 12% higher than that of pure CS modulation. Our work achieves the current FS NFDM system with the largest number of multiplexed eigenvalues, and provides a way to improve the performance of FS systems.

Probabilistic shaping and neural network-based optimization for a nonlinear frequency division multiplexing system.

A joint scheme introducing probabilistic shaping (PS) at the transmitter and utilizing a neural network (NN) equalizer at the receiver is proposed to improve the performance of the b-modulated nonlinear frequency division multiplexing (NFDM) system. Through a numerical simulation, we demonstrate that PS plays a leading role for low launch power case, which improves the performance of the system effectively, while the NN equalizer's superiority appears in a high launch power region, whose main role is to weaken the correlation among subcarriers for improving system performance. The proposed scheme would enlighten the optimum modulation and detection schemes of the NFDM system.

Geometric shaping optimization of 64-APSK constellation in discrete nonlinear frequency division multiplexing systems.

We experimentally demonstrated a geometric shaped (GS) 64-ary amplitude phase shift keying (64-APSK) eigenvalue transmission. The signal is modulated on the scatter coefficient of a single eigenvalue and linear minimum mean square error (LMMSE) estimator is used to reduce the noise. The channel response is estimated by transmitting a normally distributed 64-APSK constellation through a communication link. Based on the polar coordinates distribution of the received constellation, the diameter distributions for each circle can be obtained so that circles with larger noise can obtain larger judgment width. After optimization, the experimental results show that the Q-factor gain is 1.13 dB under 22 dB received optical signal to noise ratio (OSNR) configuration and 0.88 dB after 900 km transmission compared with normally distributed APSK configuration.

Two-stage artificial neural network-based burst-subcarrier joint equalization in nonlinear frequency division multiplexing systems.

We propose an artificial neural network (ANN)-based scheme to improve the performance of nonlinear frequency division multiplexing (NFDM) optical transmission systems in both time and frequency domain. Through two-stage ANN equalization at the receiver side, time-domain distortions between adjacent bursts and frequency-domain cross talk between neighboring subcarriers can be jointly mitigated. Burst ANN and Subcarrier ANN equalizers are characterized and validated by numerical simulations of a dual-polarization NFDM transmission system. Compared with the basic detection scheme, the proposed two-stage ANN achieves a Q-factor gain of 3.01 dB for an NFDM system with 256 Gb/s gross data rate transmitting over 960 km standard single-mode fiber (SSMF). The two-stage ANN approach offers an effective way to jointly equalize the signal in multiple dimensions.

Bounds on the Transmit Power of b-Modulated NFDM Systems in Anomalous Dispersion Fiber.

The performance of various nonlinear frequency division multiplexed (NFDM) fiber-optic transmission systems has been observed to decrease with increasing signal duration. For a class of NFDM systems known as b-modulators, we show that the nonlinear bandwidth, signal duration, and power are coupled when singularities in the nonlinear spectrum are avoided. When the nonlinear bandwidth is fixed, the coupling results in an upper bound on the transmit power that decreases with increasing signal duration. Signal-to-noise ratios are consequently expected to decrease, which can help explain drops in performance observed in practice. Furthermore, we show that there is often a finite bound on the transmit power of b-modulators even if spectral singularities are allowed.

Robust neural network receiver for multiple-eigenvalue modulated nonlinear frequency division multiplexing system

Nonlinear frequency division multiplexing (NFDM) has been shown to be promising in overcoming the fiber Kerr nonlinearity limit. In multiple-eigenvalue modulated NFDM systems, the transmission capacity increases with the number of modulated eigenvalues. However, as the number of modulated eigenvalues increases, the complexities of the signal waveform and the nonlinear Fourier transform (NFT) algorithm for demodulation increase dramatically as well, while the accuracy drops significantly. Meanwhile, impairments such as amplifier spontaneous emission noise and phase noise in practical channels would perturb the eigenvalues and the corresponding nonlinear spectra during transmission. Coupled with an increase in the modulation format order, it is difficult for NFT algorithm-based receivers to recover information. To enable the use of multiple-eigenvalue modulated NFDM systems, we propose an innovative receiver based on regression neural networks (NNs), which can demodulate information correctly for both single- and dual-polarization NFDM systems. The results show that it has strong robustness and has a certain tolerance to the impairments of communication systems. In the contrast that the poor demodulation performance of the NFT and the Euclidean minimum distance (MD) receivers for multi-eigenvalue modulated NFDM systems, our proposed NN receiver can achieve low bit error rate with 2 GBaud 16QAM modulation over 1,000 km transmission in four-eigenvalue modulated single-polarization NFDM systems. The performance of three receivers (NFT, MD and NN) in a two-eigenvalue modulated NFDM system are also compared, the NN receiver shows the best performance and appears more suitable for higher-order modulation formats.

Nonlinear-frequency-packing nonlinear frequency division multiplexing transmission.

A nonlinear frequency division multiplexing (NFDM) transmission system, designed specifically for nonlinear fiber channel, has the potential to overcome the nonlinear Shannon capacity limit. However, the spectral efficiency (SE) of the current proven NFDM transmission systems is still lower than that of the analogous orthogonal frequency division multiplexing system. It is extremely necessary to explore effective modulation scheme for the aim of increasing the SE of NFDM system. In this study, we first propose the nonlinear-frequency-packing nonlinear frequency division multiplexing (NFP-NFDM) transmission system. In NFP-NFDM, the spacing of nonlinear subcarriers is squeezed and more nonlinear subcarriers can be packed, but the inter carrier interference (ICI) is introduced. The method of NFP in nonlinear Fourier domain is carefully designed to reduce the complexity of ICI cancellation. Through numerical simulation, we illustrate the feasibility of NFP-NFDM transmission, and higher SE in NFP-NFDM than that of NFDM system is also demonstrated. The upper bound of the normalized SE for NFP-NFDM is estimated, which is higher than that of current NFDM system. Besides, we find out that the NFP scheme may have the advantage of reducing the signal-noise interaction in fiber transmission scenario, which indicates there may be a better way to load the data into the nonlinear Fourier domain.