Nonlinear Frequency Division Multiplexing 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.

Experimental demonstration of optical waveform transmission with multiple-eigenvalue 16-APSK modulation.

The multi-eigenvalue multiplexing-based discrete spectrum-modulated nonlinear frequency-division multiplexing (DS-NFDM) system with higher-order modulation format has been demonstrated experimentally. After designing the coefficients of the eigenvalue set and the constellation point distribution of 16-amplitude phase shift keying (16-APSK), the realizations of 14-, 30-, and 46-eigenvalue multiplexed DS-NFDM signals have been implemented. The results show that 46-eigenvalue and 30-eigenvalue multiplexed DS-NFDM signals can transmit 50 km and 400 km over a nonzero dispersion-shifted fiber (NZDSF) under soft-decision forward error correction (SD-FEC) threshold of 2.4E-2, respectively. This demonstration shows for the first time, to the best of our knowledge, the record for multiplexed eigenvalue number and data rate of the multiple-eigenvalue-based DS-NFDM system.

Blind frequency offset estimation method based on a minimum phase correction error for a full spectrum modulated NFDM system.

To improve the spectral efficiency of a full spectrum modulated nonlinear frequency division multiplexing (FS-NFDM) system, a blind frequency offset estimation (FOE) method has been proposed. The approach based on the minimum phase correction error can achieve high estimation accuracy of sub-MHz without need of any training symbols. Furthermore, in order to reduce the computational complexity, an eigenvalue-shift method is used to get a coarse search interval of FO, and then the one-dimensional optimization algorithm based on golden section search and parabolic interpolation is used to get the optimal FOE for the coarse search interval. The feasibility and reliability of the proposed blind FOE approach have been demonstrated in both BTB and fiber transmission scenarios. Compared with the grid search method, the proposed solving scheme can save hundreds of times of the searches. The experimental results reveal that the proposed method is robust to the amplified spontaneous emission noise and phase noise and has the capabilities of a wide FOE range and a high FOE accuracy.

Noise equalization scheme based on probabilistic shaping and complex-valued ANN for dual-polarization continuous spectrum NFDM system with high-order modulation formats

In dual-polarization continuous spectrum nonlinear frequency division multiplexing (DP-CS NFDM) systems, the presence of noise destroys the independence between subcarriers in the nonlinear spectral domain (NSD). Additionally, as the modulation order and the number of subcarriers increase, the signal power increases and the effect of noise on the performance of the system becomes more severe. In this paper, we propose a noise equalization scheme based on probabilistic shaping (PS) and complex-valued neural network (c-ANN) for dual-polarization transmission CS NFDM system with high-power signals. The scheme utilizes PS to reduce the distribution probability of high-power constellation points of the signal with high-order modulation format, to decrease the peak-to-average power ratio of the continuous spectrum subcarriers and the average power of the signal, which can achieve the suppression and equalization of the amplifier spontaneous emission (ASE) noise and the processing noise. In addition, the scheme uses the c-ANN to effectively reduce the correlation between subcarriers caused by the noise and improve the system performance. The effectiveness of the proposed scheme is verified in the DP-CS NFDM system, and the simulation results show that the Q-factor gain is enhanced by roughly 1.1 dB and 0.8 dB for the number of subcarriers of 128 and 256, respectively, compared to the classic nonlinear Fourier transform (NFT) scheme. Among them, with different noise figure (NF) and satisfying the 7% forward error correction (FEC) threshold, the 512QAM signal with an entropy of 8 after equalization increase the transmission distance by 80 km longer than that of uniform 256QAM signals for the number of subcarriers of 128. Compared with the joint scheme based on PS and real-valued neural network (r-ANN), this scheme has better equalization performance at the same complexity, and the complexity can be reduced by 50% under the same performance.

Joint equalization of frequency offset and phase noise using two-stage cascaded extended Kalman filter for discrete spectrum 16/64APSK NFDM systems.

For the discrete spectrum nonlinear frequency division multiplexing (DS-NFDM) 16/64 amplitude phase shift keying (APSK) system, the inevitable laser impairments including frequency offset (FO) and carrier phase noise (CPN) would cause different rotations of the received signal constellations. In addition, the combined effect of FO and amplifier spontaneous emission (ASE) noise induces the eigenvalue shift, accordingly the residual channel impairment (RCI) is inevitably yielded. To address the above problems, we deduce the joint impairment model of FO, CPN and RCI, and then propose a joint equalization scheme using two-stage cascaded extended Kalman filter (TSC-EKF) for these impairments. It performs frequency offset compensation in the first stage, subsequently carries out joint equalization of CPN and RCI in the second stage. Meanwhile, the minimum Euclidean distance and phase difference between the received symbols and the ideal 16/64APSK constellations are ingeniously fused to calculate the innovations of TSC-EKF. The effectiveness has been verified by 2 GBaud DS-NFDM 16/64 APSK simulations and DS-NFDM 16APSK transmission experiments. The results demonstrate that when performing the joint equalization of FO, CPN and RCI, the maximum FOE range of TSC-EKF scheme achieves 1.2 and 9.6 times as that of nonlinear frequency domain (NFD) scheme and fast Fourier transform -Like (FFT-Like) scheme, respectively. Furthermore, its maximum LW tolerance reaches 3.3 times as that of the M-th power scheme. Importantly, the complexity of TSC-EKF is 63.4% as that of NFD scheme and on an order of O(N).

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.

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

Carrier frequency offset (CFO) estimation is very important for the optical fiber communications and has been studied widely in linear coherent systems, while only a few works have been reported for nonlinear Fourier transform (NFT) based systems. In continuous spectrum (CS) modulation nonlinear frequency division multiplexing (CS-NFDM) systems, frequency offset (FO) has a great influence on its performance, requiring an improved frequency offset estimation (FOE) method. We found that the oversampling rate R0 adopted in NFDM to ensure the accuracy of the NFT and inverse NFT (INFT) calculations, would cause the estimation accuracy of the traditional FFT-FOE method to decrease by R0 times. Moreover, CS-NFDM signals with higher baud rate require more subcarriers and then result in an oversampling factor greater than 16. This makes the traditional FFT-FOE method be ineffective to use the common training sequence (TS) overhead to meet the FOE error requirement of CS-NFDM system. Therefore, a modified FOE method based on FFT assisted by TS and autocorrelation has been proposed. The theoretical analysis and simulation results show that the proposed method is applicable to CS-NFDM system, no matter what modulation format is used. For 512 subcarriers, with a high rate of 70GBaud and the TS length of 8192, the proposed method can obtain a minimum FO estimation error about 0.1 MHz, which is better than the other two typical FFT-FOE and Schmidl & Cox methods. In addition, the proposed method can save at least 87.5% and 50% overhead. Thus, the proposed method has obvious improvement for CS-NFDM system with requiring high oversampling rate.

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.

On Optimally Shaped Signals for Nonlinear Frequency Division Multiplexed Fiber Systems

An approximated channel model is proposed for direct signaling on the continuous spectrum of a nonlinear frequency division multiplexed (NFDM) communication system, describing the effect of noise and nonlinearity at the receiver. The optimal input distribution that maximizes the mutual information of the proposed approximated channel under the peak amplitude constraint is then studied. We present that, considering the input-dependency of the noise, the conventional amplitude-constrained constellation designs can be geometrically shaped to provide significant mutual information gains. However, it is observed that further probabilistic shaping and constellation size optimization can provide only limited additional gains beyond the best geometrically shaped benchmark scheme, i.e., 64 Amplitude Phase Shift Keying. Then, an approximated channel model that neglects the correlation between subcarriers is proposed for the matched filtered signaling system, based on which the input constellation is geometrically shaped. We demonstrate that although the inter-subcarrier interference in the filtered case is neglected in the channel model, the shaping of the matched filtered case can provide promising gains in mismatch capacity over the unfiltered scenario.

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.

Accurate Computation of b-Coefficients in Nonlinear Fourier Transform for the Manakov Equation

In recent years, nonlinear Fourier transform (NFT) has been intensely researched as a scheme for communication over the nonlinear optical fiber and characterizing dynamics in nonlinear optical systems. For the accurate computation of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</i> -coefficients on discrete eigenvalues, the bidirectional algorithm has been proposed where <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</i> -coefficients are calculated at a cutting point within the temporal window rather than two boundaries. The performance of the bidirectional algorithm depends severely upon the selection criterion of the cutting-point, which has been investigated in nonlinear Schrodinger equation and Korteweg–De Vries equation but not yet in Manakov equation accounting for the dual-polarization signals over optical fiber. In this paper, we propose a cutting-point criterion of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</i> -coefficients for the bidirectional algorithm of Manakov equation and compare it with existing criteria. Numerical results show that, the resulting algorithm has much better accuracy compared with existing criteria, especially for signals containing a large number of eigenvalues with large imaginary parts. The bidirectional algorithm with the proposed criterion is useful for decoding the dual-polarization nonlinear frequency division multiplexing (DP-NFDM) signals for communication and characterization of vector soliton dynamics in Manakov equation described systems.

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.

Channel model for the dual-polarization b-modulated nonlinear frequency-division multiplexing optical transmission systems.

We consider optical transmission systems based on the nonlinear frequency division multiplexing (NFDM) concept, i.e., the systems employing the nonlinear Fourier transform (NFT) for signal processing and data modulation. Our work specifically addresses the double-polarization (DP) NFDM setup that utilizes the so-called b-modulation, the most efficient NFDM method proposed up-to-date. We extend the previously-developed analytical approach based on the adiabatic perturbation theory for the continuous nonlinear Fourier spectrum (b-coefficient) onto the DP case to obtain the leading order of continuous input-output signal relation, i.e., the asymptotic channel model, for an arbitrary b-modulated DP-NFDM optical communication system. Our main result is in deriving the relatively simple analytical expressions for the power spectral density of the components of effective conditionally Gaussian input-dependent noise emerging inside the nonlinear Fourier domain. We also demonstrate that our analytical expressions are in remarkable agreement with direct numerical results if one extracts the "processing noise" arising due to the imprecision of numerical NFT operations.

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.

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.