Nonlinear Frequency Division Multiplexing Research Articles

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.

Expectation maximization channel estimation for nonlinear OFDM systems

This paper introduces the expectation maximization (EM)-based channel estimation for the high power amplifier (HPA)-incurred nonlinear orthogonal frequency division multiplexing (OFDM) systems based on the linearization using extended Bussgang decomposition. Analyses and numerical simulations show that the proposed algorithm only requires reasonable computation complexity with relatively small number of iterations while vastly improves the estimation performance compared to the other conventional estimation methods such as least square error (LSE) or minimum mean square error (MMSE) applied to such system. Moreover, the EM-LSE estimator could give almost the same performance as the EM-MMSE counterpart while does not require channel statistics, forming a robust estimator for both fading and nonlinear channels with reduced computation complexity. This makes the estimator more applicable.

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.

Serial and parallel convolutional neural network schemes for NFDM signals

Two conceptual convolutional neural network (CNN) schemes are proposed, developed and analysed for directly decoding nonlinear frequency division multiplexing (NFDM) signals with hardware implementation taken into consideration. A serial network scheme with a small network size is designed for small user applications, and a parallel network scheme with high speed is designed for places such as data centres. The work aimed at showing the potential of using CNN for practical NFDM-based fibre optic communication. In the numerical demonstrations, the serial network only occupies 0.5 MB of memory space while the parallel network occupies 128 MB of memory but allows parallel computing. Both network schemes were trained with simulated data and reached more than 99.9% accuracy.

Nonlinear frequency domain PMD modeling and equalization for nonlinear frequency division multiplexing transmission.

Polarization mode dispersion (PMD) is one of the fundamental properties of a standard single-mode fiber. It affects the propagating signals and degrades the performance of high-speed optical fiber communication systems. PMD also gives an effect on the nonlinear spectra or scattering data in nonlinear frequency division multiplexing (NFDM) systems. However, PMD is usually described in the linear frequency domain, and there are few investigations about the influence of PMD in the nonlinear frequency domain (NFD). An NFD-PMD model is needed to understand the impact of PMD in the NFD. In this work, using a linear approximation method, we first propose an NFD-PMD model and verify its effectiveness. With the guide of the NFD-PMD model, a blind NFD-PMD equalization scheme is designed. The simulation results indicate that the proposed NFD-PMD equalization scheme has better performance than the training sequence method based on linear frequency domain equalization.

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.

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.

Mitigating the Impact of Noise on Nonlinear Frequency Division Multiplexing

In the past years, nonlinear frequency division multiplexing (NFDM) has been investigated as a potentially revolutionary technique for nonlinear optical fiber communication. However, while NFDM is able to exploit the Kerr nonlinearity, its performance lags behind that of conventional systems. In this work, we first highlight that current implementations of NFDM are strongly suboptimal, and, consequently, oversensitive to noise: the modulation does not ensure a large minimum distance between waveforms, while the detection is not tailored to the statistics of noise. Next, we discuss improved detections strategies and modulation techniques, proposing some effective approaches able to improve NFDM. Different flavors of NFDM are compared through simulations, showing that (i) the NFDM performance can be significantly improved by employing more effective detection strategies, with a 5.6 dB gain in Q-factor obtained with the best strategy compared to the standard strategy; (ii) an additional gain of 2.7 dB is obtained by means of a simple power-tilt modulation strategy, bringing the total gain with respect to standard NFDM to 8.3 dB; and (iii) under some parameters range (rate efficiency η≤30%), the combination of improved modulation and detection allows NFDM to outperform conventional systems using electronic dispersion compensation.

Enhanced-power NFDM transmission system with midpoint optical phase conjugation.

We investigate the combination of a nonlinear frequency division multiplexed (NFDM) transmission scheme with midpoint optical phase conjugation (OPC), and show that midpoint OPC introduces power enhancement by compensating for nonlinear impairments. It offers a degree of freedom to have a flexible power normalization factor, Pn. Optimizing Pn helps minimize the signal-noise mixing in nonlinear Fourier transform processing for a specific launch power, resulting in improving the system performance significantly. The mid-OPC NFDM system can provide 4.5 dB and 5.6 dB advantages in Q-factor as compared to the conventional NFDM system when the transmission fiber is standard single-mode fiber and a fiber with optimum dispersion, respectively. Mid-OPC NFDM can also offer higher spectral efficiency at a longer transmission reach due to the shorter guard interval.

Experimental Demonstration of Nonlinear Frequency Division Multiplexing Transmission With Neural Network Receiver

Nonlinear frequency division multiplexing (NFDM) communication systems that are based on the nonlinear Fourier transform (NFT), have seen a rapid improvement in performance and transmission reach over just a few years. However, such an improvement is now being slowed down by fundamental challenges such as fiber loss and noise. As the NFT theory is defined over a lossless transmission fiber, a strong research focus has been dedicated to either improve the lossless assumption for practical fibers, by adapting the theory to approximately account for the fiber loss, or by devising encoding schemes that increase the robustness of the NFT to the fiber attenuation. However, the proposed solutions provide only minimal benefits to the system performance, especially for long fiber spans as in deployed links. Alternatively, a detection strategy based on replacing a conventional NFT receiver with a time-domain Neural network (NN)-based symbol decisor has been numerically proposed. Here, we extend such an idea by validating it experimentally. In order to apply the method in an experimental environment, the impact of phase noise, and receiver frequency offset needs to be addressed. We, therefore, propose a novel time-domain receiver architecture that combines a two-stage iterative carrier recovery with a NN-based symbol decisor. The carrier recovery, itself based on aNN for phase estimation, is numerically, and experimentally characterized. The proposed receiver has been evaluated for single-polarization two-eigenvalue transmission at 1 GBd. A two-fold increase in the transmission reach is enabled by the NN receiver (≈1600 km) compared to a conventional NFT receiver (≈560 km) for a practical link using 80-km spaced erbium-doped fiber amplifier (EDFAs).

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.

Exact NFDM Transmission in the Presence of Fiber-Loss

Nonlinear frequency division multiplexing (NFDM) techniques encode information in the so called nonlinear spectrum which is obtained from the nonlinear Fourier transform (NFT) of a signal. NFDM techniques so far have been applied to the nonlinear Schrodinger equation (NLSE) that models signal propagation in a lossless fiber. Conventionally, the true lossy NLSE is approximated by a lossless NLSE using the path-average approach which makes the propagation model suitable for NFDM. The error of the path-average approximation depends strongly on signal power, bandwidth, and the span length. It can degrade the performance of NFDM systems and imposes challenges on designing high data rate NFDM systems. Previously, we proposed the idea of using dispersion decreasing fiber (DDF) for NFDM systems. These DDFs can be modeled by a NLSE with varying-parameters that can be solved with a specialized NFT without approximation errors. We have shown in simulations that complete nonlinearity mitigation can be achieved in lossy fibers by designing an NFDM system with DDF if a properly adapted NFT is used. We reported performance gains by avoiding the aforementioned path-average error in an NFDM system by modulating the discrete part of the nonlinear spectrum. In this article, we extend the proposed idea to the modulation of continuous spectrum. We compare the performance of NFDM systems designed with dispersion decreasing fiber to that of systems designed with a standard fiber with path-average model. Next to the conventional path-average model, we furthermore compare the proposed system with an optimized path-average model in which amplifier locations can be adapted. We quantify the improvement in the performance of NFDM systems that use DDF through numerical simulations.

Laser linewidth tolerance for nonlinear frequency division multiplexing transmission with discrete spectrum modulation.

Although fruitful investigations of carrier phase estimation (CPE) have been conducted for a traditional coherent fiber optical transmission, there are few studies on the CPE for a nonlinear Fourier transform (NFT) based transmission. A laser linewidth induced phase noise leads to a phase rotation of the nonlinear spectra and the scattering data, which is similar to its effect on the linear spectra. Here, we first identify that both feed forward the M-th power, and the blind phase search (BPS)-based CPE can function well in the nonlinear frequency division multiplexing (NFDM) transmission with discrete spectrum modulation. Then, a performance comparison between two CPE schemes is presented for various modulation formats under the scenario of a single eigenvalue NFDM transmission. Our simulation results indicate that the laser linewidth tolerances of 2 GBaud quadrature phase shift keying (QPSK), 8-phase shift keying (8-PSK), and 16-amplitude phase shift keying (16-APSK) are 2.3 MHz, 1.05 MHz, and 250 KHz, respectively, given a 1-dB optical signal to noise ratio (OSNR) penalty at BER = 10-3. Finally, the BPS algorithm is experimentally verified under the same scenario of a 2 GBaud back-to-back transmission, due to the use of a semiconductor laser with a 100 KHz linewidth.

Impact of Transmitter Phase Noise on NFDM Transmission With Discrete Spectral Modulation

Nonlinear frequency division multiplexing (NFDM) systems have been showing remarkable progress in the past few years. However, the majority of the demonstrations have neglected the impact of laser phase noise by employing narrow-linewidth lasers and self-homodyne receivers. The impact of transmitter laser linewidth on NFDM transmission is here numerically and experimentally investigated for dual-polarization discrete NFDM systems. The scaling of linewidth tolerance is analyzed for different signal symbol rates and the results are validated experimentally at 250 MBd. Numerical and experimental analysis show a limited degradation due to laser phase noise with relevant penalty appearing only for linewidth-symbol duration products ( $\Delta \nu \times T_{s}$ ) above $10^{-3}$ . The 250-MBd experimental results indeed show limited penalty for laser linewidths up to 750-kHz and 100-kHz, for back-to-back and 2000-km transmission, respectively.

Linear and Nonlinear Frequency-Division Multiplexing

Two signal multiplexing schemes for optical fiber communication are considered: Wavelength-division multiplexing (WDM) and nonlinear frequency-division multiplexing (NFDM), based on the nonlinear Fourier transform. Achievable information rates (AIRs) of NFDM and WDM are compared in a network scenario with an ideal lossless model of the optical fiber in the defocusing regime. It is shown that the NFDM AIR is greater than the WDM AIR subject to a bandwidth and average power constraint, in a representative system with one symbol per user. The improvement results from nonlinear signal multiplexing.