Nonlinear Frequency Division Multiplexing Transmission 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.

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.

PDM-SHD NFDM transmission with envelope-detection feedback polarization tracking and optical injection locking.

Nonlinear frequency division multiplexing (NFDM) systems, especially the eigenvalue communications have the potential to overcome the nonlinear Shannon capacity limit. However, the baud rate of eigenvalue communications is typically restricted to a few GBaud, making it challenging to mitigate laser frequency impairments such as the phase noise and frequency offset (FO) using digital signal processing (DSP) algorithms in intradyne detections (IDs). Therefore, we introduce the polarization division multiplexing-self-homodyne detection (PDM-SHD) into the NFDM link, which could overcome the impact of phase noise and FO by transmitting a pilot carrier originating from the transmitter laser to the receiver through the orthogonal polarization state of signal. To separate the signal from the carrier at the receiver, a carrier to signal power ratio (CSPR) unrestricted adaptive polarization controlling strategy is proposed and implemented by exploiting the optical intensity fluctuation of the light in a particular polarization rather than its direct optical power as the feedback. Optical injection locking (OIL) is used subsequently to amplify optical power of pilot carrier and mitigate the impact of signal-signal beat interference (SSBI). Additionally, the effects of cross-polarization modulation (XPolM) and modulation instability (MI) in long haul transmission are explored and inhibited. The results show that the tolerable FO range is about 3.5 GHz, which is 17 times larger than the ID one. When 16-amplitude phase shift keying (APSK) or 64-APSK constellations are used, identical Q-factor performance can be obtained by using distributed feedback (DFB, ∼10 MHz) laser, external cavity laser (ECL, ∼100kHz), or fiber laser (FL, ∼100 Hz), respectively, which demonstrates that our proposed PDM-SHD eigenvalue communication structure is insensitive to the laser linewidth. Under the impact of cycle slip, the Q-factor difference of 16-APSK signal between the ECL-ID system and ECL-SHD system can be up to 8.73 dB after 1500 km transmission.

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.

100 Gbaud PDM 16QAM NFDM transmission with neural network-based equalization

We demonstrate ultrahigh-speed data transmission at a symbol rate of 100 Gbaud in a nonlinear frequency division multiplexing (NFDM) system with continuous spectrum (CS) modulation. Neural network (NN)-based equalization schemes enable the transmission of 100 Gbaud polarization division multiplexed (PDM) 16-ary quadrature amplitude modulation (16-QAM) over 560 km of standard single-mode fiber (SSMF), with the bit-error-rate (BER) below the 25% overhead soft-decision forward error-correction (SD-FEC) threshold of 4 × 10−2.

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.

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.

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.

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.

Frequency offset estimation for nonlinear frequency division multiplexing with discrete spectrum modulation.

Although fruitful studies have been conducted on carrier frequency offset (CFO) estimations in linear coherent optical fiber communication systems, there are few studies on CFO estimations and recoveries in the systems based on the nonlinear Fourier transform (NFT). Although the CFO is originated from the linear frequency domain, it definitely has effects on nonlinear spectra, including the shift of the nonlinear frequency and the phase rotations of the scattering data, which are similar to its effects on linear spectra. This work indicates that it is feasible to estimate frequency offset (FO) by capturing symbol variations in the nonlinear frequency domain (NFD) rather than in the linear frequency domain; the latter was usually exploited in the literature. Based on a thorough investigation of the FO induced behavior that appears in a nonlinear frequency division multiplexing (NFDM) system, we proposed a nonlinear frequency domain estimation method aided by training symbols (TS) using an angle search algorithm after NFT operations at the receiver. The discussions in this paper prove that the proposed method is generally applicable to the NFDM systems regardless of whether using single or multiple eigenvalues. A performance comparison between the NFD method and the conventional method in the linear frequency domain is performed with different modulation formats for both single and multiple eigenvalue NFDM transmission systems. The analysis results show that the proposed method holds the better stability and estimation accuracy in contrast with the linear domain estimation method. The TS overhead can also be deduced dramatically, which implies better transmission efficiency. Therefore, the NFD method is more powerful for eigenvalue NFDM transmission systems, especially for the scenarios where high order modulation formats and multiple eigenvalues are utilized.

Dual-Polarization NFDM Transmission With Continuous and Discrete Spectral Modulation

Nonlinear distortion experienced by signals during their propagation through optical fibers strongly limits the throughput of optical communication systems. Recently, a strong research focus has been dedicated to nonlinearity mitigation and compensation techniques. At the same time, a more disruptive approach, the nonlinear Fourier transform (NFT), aims at designing signaling schemes more suited to the nonlinear fiber channel. In a short period, impressive results have been reported by modulating either the continuous spectrum or the discrete spectrum. Additionally, very recent works further introduced the opportunity to modulate both spectra for single polarization transmission. Here, we extend the joint modulation scheme to dual-polarization transmission by introducing the framework to construct a dual-polarization optical signal with the desired continuous and discrete spectra. After a brief analysis of the numerical algorithms used to implement the proposed scheme, the first experimental demonstration of dual-polarization joint nonlinear frequency division multiplexing (NFDM) modulation is reported for up to 3200 km of low-loss transmission fiber. The proposed dual-polarization joint modulation schemes enables to exploit all the degrees of freedom for modulation (both polarizations and both spectra) provided by a single-mode fiber (SMF).

Dual Polarization Nonlinear Frequency Division Multiplexing Transmission

We experimentally demonstrate dual-polarization nonlinear frequency division multiplexing (NFDM) using the continuous spectrum in 1680 km of normal dispersion fiber, at the net data rate of 25.6 Gb/s. NFDM exhibits a gain of 0.4 dB in $Q$ -factor and 1 dB in total launch power when compared with burst mode OFDM. DGD penalties are shown to be negligible in NFDM transmission.

64 × 0.5 Gbaud Nonlinear Frequency Division Multiplexed Transmissions With High Order Modulation Formats

Nonlinear frequency division multiplexing (NFDM) with the modulation of the nonlinear Fourier spectrum (both discrete and continuous parts) has been recently considered as a promising transmission method to combat fiber nonlinearity impairments. In this paper, we experimentally demonstrate the generation, detection, and transmission performance of NFDM systems where the transmitted information is encoded directly onto the continuous part of the nonlinear spectrum using 64 overlapped 0.5 Gbaud channels with high order QAM formats (e.g., 16 QAM, 32 QAM, and 64 QAM). In comparison with the conventional orthogonal frequency division multiplexing transmission, which is the linear counterpart of NFDM transmission, clear performance advantage of NFDM systems, up to 1.3 dB, has been observed. This result indicates for the first time the performance benefit of NFDM transmission over its linear counterpart, showing the high potential of this emerging technology.