Digital Backpropagation Algorithm Research Articles

Nonlinear impairment compensation in multi-channel communication systems based on correlated digital backpropagation with separation of walk-off effect

Digital backpropagation (DBP) is a commonly used method to compensate for the nonlinear impairments of optical fiber channels in the electrical domain, offering a new approach to enhancing the data capacity without affecting the system performance. However, due to the repeated Fourier transform and inverse transform of the split-step Fourier method (SSFM), the complexity of DBP algorithm has increased significantly. In this paper, a correlated backpropagation algorithm with the separation of the walk-off effect is developed, which improves the computational step and significantly reduces the complexity of the algorithm. Simulation results demonstrate that in a 6-channel PDM-16QAM system, the proposed hybrid algorithm reduces the computational complexity by 73.96 % compared to the traditional DBP algorithm for an 800 km transmission.

Design of fully interpretable neural networks for digital coherent demodulation.

In this paper, we propose a digital coherent demodulation architecture using fully interpretable deep neural networks (NNs). We show that all the conventional coherent digital signal processing (DSP) is deeply unfolded into a well-structured NN so that the established training algorithms in machine learning can be applied. In contrast to adding or replacing certain algorithms of existing DSP in coherent receivers, we replace all the coherent demodulation algorithms with a fully interpretable NN (FINN), making the whole NN interpretable. The FINN is modular and flexible to add or drop modules, including chromatic dispersion compensation (CDC), the digital back-propagation (DBP) algorithm for fiber nonlinearity compensation, carrier recovery and residual impairments. The resulted FINN can be quickly initialized by straightforwardly referring to the conventional DSP, and can also enjoy further performance enhancement in the nonlinear fiber transmissions by NN. We conduct a 132-Gb/s polarization multiplexed (PM)-16QAM transmission experiment over 600-km standard single mode fiber. The experimental results show that without fiber nonlinearity compensation, FINN-CDC obtains less than 0.06-dB SNR gain than chromatic dispersion compensation (CDC). However, with fiber nonlinearity compensation, 2-steps per span FINN-DBP (FINN-2sps-DBP) and FINN-1sps-DBP bring about 0.59-dB and 0.53-dB SNR improvement compared with the conventional 2sps-DBP and 1sps-DBP, respectively.

Advanced Digital Signal Processing and Variable-Rate Coding for Unrepeatered Optical Transmission

In this paper, we evaluate the selective combination of algorithms for fiber nonlinearity compensation, transceiver impairments mitigation and variable-rate coding in unrepeatered optical transmission. A post-emphasis filter and a two-stage 4×4 multiple-input and multiple-output equalizer compensate the impact of bandwidth limitations and in-phase and quadrature skew. Nonlinear compensation is accomplished by a fast-converging adaptive digital back-propagation algorithm and maximum likelihood sequence estimation. Forward error correction is provided by a variable-rate spatially-coupled low-density parity-check code. The performance and complexity of the proposed digital subsystems are experimentally evaluated by the unrepeatered wavelength division multiplexing transmission of 17×200-Gb/s (DP-16QAM 32-GBd) channels over 350-km of large effective area and low-loss single-mode fibers. The experimental results for different combinations of algorithms elucidate the trade-off of complexity and performance in unrepeatered optical transmission.

Nonlinearity mitigation of self-phase modulation effect in coherent optical system

Abstract In this paper, the proficiency of nonlinearity compensation studied on the polarization division multiplexing 0.3 return-to-zero quadrature phase-shift keying (PDM-0.3RZ-QAM) coherent optical system. The considerable nonlinear effect is the self-phase modulation (SPM) in the presence of chromatic dispersion. The nonlinearity mitigation technique based on the digital backpropagation (DBP) algorithm is implemented on the receiver side. The performance of the system was investigated in different terms as the optical signal-noise ratio, bit error rate, and the maximum transmission distance. The results demonstrated that the penalties were decreased by 3 dB. The maximum achievable distance was extended to 2000 km.

Effect of optical pulse shaping and adaptive equalization on the performance of 100G DP-QPSK WDM system

Abstract Dual polarization quadrature phase shifting keying (DP-QPSK) modulation format along with coherent receiver helps in increasing the data carrying capability of existing optical networks without any major change in existing transmission infrastructure. The various linear and nonlinear fiber effects, frequency and phase errors are corrected in electrical domain at the receiver end with digital back propagation algorithms (DBP) instead of in-line compensation. In such a case the selection of optimum values of system parameters make the task easier for DBP algorithms. This paper highlights the importance of finding optimum operating point of continuous modulus algorithm (CMA) for better adaptive equalization (AE). The paper also discusses the optical pulse shaping using Gaussian optical band-pass filter to improve the spectral characteristics of DP-QPSK signal.

Combined Neural Network and Adaptive DSP Training for Long-Haul Optical Communications

Machine Learning (ML) algorithms have shown to complement standard digital signal processing (DSP) tools in mitigating fiber nonlinearity and improving long-haul transmission performance. However, dynamic transmission impairments such as polarization effects and carrier phase noise corrupt the training data and conventional cost functions for neural network (NN) training become unsuitable. Simple cascade of ML and standard adaptive DSP blocks will also result in suboptimal transmission performance or require impractical training methodologies in presence of such dynamic transmission impairments. We show how the adaptive DSP blocks can be treated as extra stateful NN layers and be combined with the main NN so that standard backpropagation-like training algorithms in ML can be applied. In this case, the adaptive filters are viewed as NN states which are updated in the forward pass of the backpropagation. We study the combined training of linear and nonlinear parameters in the digital backpropagation (DBP) algorithm for fiber nonlinearity compensation (named generalized DBP (GDBP) hereafter), residual impairments, polarization effects, frequency offsets and carrier phase noise compensation filters as a single NN in a 7 × 288 Gb/s polarization multiplexed (PM)-16QAM transmission experiment over 1125 km. We derived the complete set of backpropagation-like gradients and state update equations for the static and dynamic parameters of the combined NN. We further proposed and open-sourced a JAX-based coding framework for their easy and practical implementation. GDBP is more generalized than other DBP variants proposed in literature and for a given total number of steps, GDBP is the first experimental demonstration of optimal single-channel DBP based-fiber nonlinearity compensation algorithm. In addition, for complexity constrained situations with shortened filter taps, GDBP enables a 1 dB performance improvement over other DBP variants.

A Layer-Reduced Neural Network Based Digital Backpropagation Algorithm for Fiber Nonlinearity Mitigation

A layer-reduced neural network based digital backpropagation algorithm called smoothing learned digital backpropagation (smoothing-LDBP), is proposed in this paper. The smoothing-LDBP smooths the power terms in nonlinear activation functions to limit the bandwidth. The limited bandwidth of the power terms generates fewer in-band distortions, thus reduces the required layer for a given equalization performance. Simulation results show that the required layers of smoothing-LDBP are reduced by approximately 62% at 6.7% HD-FEC compared with learned digital backpropagation. Owing to the layer reduction, the latency and the complexity are reduced by 69% and 51%, respectively. The layer-reduced property of smoothing-LDBP is also validated by a proof-of-concept experiment.

Inter-Channel Fiber Nonlinearity Mitigation in High Baud-Rate Optical Communication Systems

Kerr nonlinearity in long distance fiber-optic link imposes a fundamental limitation to the capacity of wavelength division multiplexed (WDM) optical communication systems. Solving the inverse-propagating nonlinear Schrodinger equation (NLSE) is in principle a fundamental methodology to mitigate fiber nonlinear impairments. However, solving multi-channel coupled NLSE is too complexed to be implemented commercially. It is of great importance to compensate for nonlinear distortions in high baud-rate WDM systems based on the information of one single target channel. In this article, we experimentally study fiber nonlinearity mitigation in an 8 × 512 Gb/s (64Gbaud) polarization division multiplexed (PDM) 16-ary quadrature amplitude modulation (16-QAM) Nyquist-WDM system with 960 km standard single mode fiber (SSMF) loop transmission. The digital back-propagation (DBP) algorithm based on the target channel is applied, which can compensate for intra-channel nonlinearity. The inter-channel nonlinearity of cross phase modulation (XPM) induces nonlinear polarization scattering and nonlinear phase noise, which are partially correlated over adjacent pulses in high baud-rate systems. We propose an advanced nonlinear polarization crosstalk canceller (NPCC) with a novel decision feedback (DF) update strategy that is called the DF-NPCC, which incorporates the iterative process of the least mean square (LMS) algorithm and can thus mitigate the correlated nonlinear polarization crosstalk and nonlinear phase noise. Through nonlinearity mitigation, the DBP together with the DF-NPCC can achieve a 0.77 dB Q2 factor promotion. The DF-NPCC can suppress the remaining nonlinear distortions after the single channel DBP, which corresponds to a 0.36 dB Q2 factor improvement due to inter-channel nonlinear mitigation. Our work indicates XPM distortions can be mitigated with low complexity digital signal processing such as the DF-NPCC based on the single channel information, which provides an attractive solution to compensate inter-channel fiber nonlinearity in high baud-rate Nyquist-WDM systems.

Chaotic Optical Communication Over 1000 km Transmission by Coherent Detection

Chaotic optical communication was originally proposed to provide high-level physical layer encryption. For high-speed and long-distance transmission, chaotic signal is very sensitive to channel impairments such as dispersion and Kerr fiber nonlinearity. In the traditional chaotic optical communications, these impairments must be compensated in optical domain before chaos synchronization. However completely compensating the high-order dispersion and fiber nonlinearity in the optical domain has great challenges, therefore limiting the transmission distance of high-speed chaotic optical communications less than 150 km. Here we propose a method aiming to break the limit. Thanks to coherent detection, channel impairments can be compensated in the digital domain using various algorithms. Digital back propagation algorithm is used for jointly compensating linear and nonlinear impairments of the chaotic signals, constant modulus algorithm is used for channel equalization and extended Kalman filter is adopted for carrier phase recovery. After digital processing, the recovered chaos signal is converted back to the optical domain for chaos synchronization. By this means, we demonstrate a 10 Gb/s phase-modulation signal encrypted by phase chaos transmission over record-breaking 1000 km single-mode fiber with bit-rate error less than 1.0 × 10−3 by simulation, and support the results with extensive numerical analysis.

Single-step digital backpropagation for subcarrier-multiplexing transmissions.

We propose and demonstrate a single-step digital back propagation (DBP) algorithm for metro and regional transmissions with high order modulation formats. Based on subcarrier-multiplexing (SCM)-DBP, two modifications are made to improve performance and reduce complexity for the targeted link scenarios. First, an infinite impulse response (IIR) filter is adopted in self-subcarrier nonlinear compensation. Second, the second stage chromatic dispersion (CD) compensation is incorporated into an existing adaptive filter. Through experiment, we demonstrate the performance of the proposed scheme, denoted as M-SCM-DBP, for single channel 34.94-GBd SCM PDM-32QAM transmission. With 86.3% complexity reduction compared with the low-pass filter assisted DBP, the proposed M-SCM-DBP achieves 0.6-dB Q2 improvement for SCM-PDM-32QAM transmission over 960-km standard single mode fiber (SSMF). The reach extension of 36% to 1220-km is achieved with only 30.5 complex multiplications per sample, in comparison with the linear compensation scheme. Since the adaptive filter is used to simultaneously compensate 50% CD and other linear impairments, we also investigate the required number of filter taps and its polarization tracking ability.

Impact of InterChannel Interference in gridless nyquist-wdm systems with and without nonlinear impairments compensation


 
 
 
 This paper presents a characterization of interchannel interference (ICI) effects in gridless Nyquist-wdm systems due to two contributions, namely, overlapping among optical carriers and stimulation of nonlinear optical fiber impairments. ICI is assessed regarding bit error rate (BER) at 16 and 32 Gbaud, as a function of several system parameters. Nonlinear impairment compensation based on the digital back-propagation algorithm is implemented in the DSP-based coherent receiver. Results demonstrated that ICI due to channels overlapped has a higher impact concerning BER than nonlinear optical fiber impairments. Besides, the use of the back-propagation algorithm improves system performance, reducing up to 3 and 0.7 orders of magnitude of BER in QPSK and 16QAM cases, respectively. It means that this algorithm can minimize nonlinear effects under varying system parameters, but its performance is limited in sub-Nyquist cases.
 
 
 

How to Increase the Achievable Information Rate by Per-Channel Dispersion Compensation

Deploying periodic inline chromatic dispersion compensation enables reducing the complexity of the digital back propagation (DBP) algorithm. However, compared with nondispersion-managed (NDM) links, dispersion-managed (DM) ones suffer a stronger cross-phase modulation (XPM). Utilizing per-channel dispersion-managed (CDM) links (e.g., using fiber Bragg grating) allows for a complexity reduction of DBP, while abating XPM compared to DM links. In this paper, we show for the first time that CDM links enable also a more effective XPM compensation compared to NDM ones, allowing a higher achievable information rate (AIR). This is explained by resorting to the frequency-resolved logarithmic perturbation model and showing that per-channel dispersion compensation increases the frequency correlation of the distortions induced by XPM over the channel bandwidth, making them more similar to a conventional phase noise. We compare the performance (in terms of the AIR) of a DM, an NDM, and a CDM link, considering two types of mismatched receivers: one neglects the XPM phase distortion and the other compensates for it. With the former, the CDM link is inferior to the NDM one due to an increased in-band signal--noise interaction. However, with the latter, a higher AIR is obtained with the CDM link than with the NDM one owing to a higher XPM frequency correlation. The DM link has the lowest AIR for both receivers because of a stronger XPM.

Analytical method for four wave mixing in space-frequency multiplexing optical fibers

In recent years, the transmission capacity of wavelength division multiplexing (WDM) communication systems has gradually approached to the nonlinear Shannon limit. To meet the increasing demand for communication capacity, space division multiplexing (SDM) has become one of the most concerned technologies. In this paper, the four-wave mixing process (FWM) in fibers is considered from the frequency domain to the mode division multiplexing (MDM) spatial domain under pump depletion and the exact analytical solution to the FWM coupled-mode equations in the space-frequency domain is in detail deduced. The analytical method is verified by numerically calculating the amplitude and phase evolution of the idler wave in non-degenerate four-wave mixing. We discuss three new applications of the analytical solution as follows. 1) Using the phase matching condition we select the terms in the multi-wave coupling equation, and only retain the coupling term that plays a major role. According to the analytical solution in this paper, the phase matching percentage parameter is introduced to determine the FWM coupling terms necessary for multi-wave coupling equations, thus simplifying the multi-wave coupling problem in the study. 2) Combining the analytical solution with the numerical calculation results, we find the initial phase relationship between the output idler and the input guided wave for phase-insensitive FWM, and we provide the analytical expression for a theoretical basis to efficiently design the FWM-based phase arithmetic devices in parallel operating at WDM and MDM systems. 3) We propose a nonlinear compensation algorithm based on analytical solution, which can be used in the few-mode transmission system. The algorithm can fast evaluate or compensate for the fiber nonlinearity by taking into account the pump depletion of the FWM effect. Compared with the traditional digital back propagation (DBP) algorithm, our algorithm has the advantage of lower complexity.

Enabling Technologies for Fiber Nonlinearity Mitigation in High Capacity Transmission Systems

Fiber nonlinearity has become a major limiting transmission impairment factor. In this paper, we discuss various nonlinearity mitigation techniques in electrical and optical domains. In electrical domain, multiple reduced complexity digital back-propagation algorithms were developed, including perturbation back-propagation. One drawback is that they can compensate for intra-channel self-phase modulation (SPM) effects only. The inter-channel fiber nonlinearity mitigation can be done in optical domain. For example, digital subcarrier multiplexing (SCM) can effectively mitigate both SPM and cross-phase modulation (XPM) effects. It can become even more beneficial for higher symbol rate systems (e.g., 64 Gbaud and 128 Gbaud) and has a capability to deliver longer reach than single carrier 32 Gbaud DP-16QAM signals and only slightly shorter reach (by up to 7%) compared to 32 Gbaud SCM systems. Another technique, the total intensity directed phase modulation, can provide per-span SPM and XPM compensation. However, this technique is limited to compensation of a few channels only. On the other hand, the proposed enhanced pre-dispersed spectral inversion can effectively compensate for SPM/XPM effects of multiple channels and become more practical by removing limitation to link symmetry even in nonuniform transmission links. Finally, subcarrier power pre-emphasis in optical superchannels can equalize performance of all subcarriers by mitigating intersubcarrier XPM and extend transmission reach by 20%.

Improved Lower Bounds on Mutual Information Accounting for Nonlinear Signal–Noise Interaction

In fiber-optic communications, evaluation of mutual information (MI) is still an open issue due to the unavailability of an exact and mathematically tractable channel model. Traditionally, lower bounds on MI are computed by approximating the (original) channel with an auxiliary forward channel. In this paper, lower bounds are computed using an auxiliary backward channel, which has not been previously considered in the context of fiber-optic communications. Distributions obtained through two variations of the stochastic digital backpropagation (SDBP) algorithm are used as auxiliary backward channels and these bounds are compared with bounds obtained through the conventional digital backpropagation (DBP). Through simulations, higher information rates were achieved with SDBP, {which can be explained by the ability of SDBP to account for nonlinear signal--noise interactions

On the Impact of Fixed Point Hardware for Optical Fiber Nonlinearity Compensation Algorithms

Nonlinearity mitigation using digital signal processing has been shown to increase the achievable data rates of optical fiber transmission links. One especially effective technique is digital back propagation (DBP), an algorithm capable of simultaneously compensating for linear and nonlinear channel distortions. The most significant barrier to implementing this technique, however, is its high computational complexity. In recent years, there have been several proposed alternatives to DBP with reduced computational complexity, although such techniques have not demonstrated performance benefits commensurate with the complexity of implementation. In order to fully characterize the computational requirements of DBP, there is a need to model the algorithm behavior when constrained to the logic used in a digital coherent receiver. Such a model allows for the analysis of any signal recovery algorithm in terms of true hardware complexity which, crucially, includes the bit-depth of the multiplication operation. With a limited bit depth, there is quantization noise, introduced with each arithmetic operation, and it can no longer be assumed that the conventional DBP algorithm will outperform its low complexity alternatives. In this work, DBP and a single nonlinear step DBP implementation, the \textit{Enhanced Split Step Fourier} method (ESSFM), were compared with linear equalization using a generic software model of fixed point hardware. The requirements of bit depth and fast Fourier transform (FFT) size are discussed to examine the optimal operating regimes for these two schemes of digital nonlinearity compensation. For a 1000 km transmission system, it was found that (assuming an optimized FFT size), in terms of SNR, the ESSFM algorithm outperformed the conventional DBP for all hardware resolutions up to 13 bits.

NSNI Mitigation in Bi-Directional Raman Amplified Unrepeatered System Using Split-DBP

The nonlinear signal–noise interaction (NSNI) is analyzed in the bi-directional Raman amplified unrepeatered system with digital back-propagation (DBP) algorithm. When the DBP is applied at the receiver side, because in the DBP process the spontaneous Raman scattering is not dealt with, extra NSNI occurs, and because of the high power in the system, the extra NSNI is severe. By adopting split-DBP, the interaction of noise and signal in the high-power region could be avoided. Thus, the extra NSNI was mitigated and the performance was improved. The performance of split-DBP algorithm in such a system was investigated through simulations and experiments, using a single-channel 300-km bi-directional Raman amplified unrepeatered system. The simulations were carried out with 16 QAM modulation and the modulation rates were 32 GBd and 10 GBd. The results showed that the split-DBP can deliver 0.8-dB Q 2-factor improvement for the case of 32 GBd and 1.3 dB for the case of 10 GBd, compared to the traditional DBP. The impact of the nonlinear transform function of modulator and power profile mismatch on the performance of split-DBP is discussed through the simulation, which is critical to the practical implementation of split-DBP. An experimental demonstration of the split-DBP with 10 GBd 16 QAM modulation was carried out. The experimental results presented a 1.2-dB Q 2-factor improvement of the signal due to split-DBP.

Low-Complexity Time-Domain DBP Based on Random Step-Size and Partitioned Quantization

We propose and experimentally validate a low-complexity time-domain (TD) digital backpropagation (DBP) algorithm for fiber nonlinearity compensation, targeting an optimized hardware implementation. To counteract the coherent accumulation of numerical quantization errors between DBP steps, we propose a random step-size distribution along the optical link (with $\pm$ 5% interval around the optimal step-size). In addition, to further reduce the average quantization bit precision requirements, we propose a partitioned quantization technique, enabling to quantize the finite-impulse response (FIR) filter tail coefficients with significantly lower precision. The proposed low-complexity DBP algorithm is experimentally demonstrated over a 2592 km long-haul wavelength division multiplexing transmission system with 21 $\times$ 32 GBaud PM-16QAM optical channels. Employing the proposed step-size randomization together with dual-time-slot quantization, we demonstrate penalty-free operation at an average of $\sim$ 4 b per FIR coefficient, leading to a 60% complexity reduction when compared to the standard TD-DBP implementation.

Fiber Nonlinearity Mitigation in Single Carrier 400 G and 800 G Nyquist-WDM Systems

Fiber Kerr nonlinearity limits the achievable transmission distance and channel capacity of high speed optical communication systems. In this paper, we study the performance of C-band single carrier 400 G and 800 G 16-ary quadrature amplitude modulation (16 QAM) or 64-ary quadrature amplitude modulation (64 QAM) Nyquist wavelength division multiplexed (WDM) systems with emphasis on fiber nonlinearity and its mitigation. The nonlinearity mitigation methods of digital back-propagation (DBP) and coupled-equation digital back-propagation (CE-DBP) are discussed for single channel and multi-channel systems, respectively. The DBP algorithm is a general approach suitable for compensating intra-channel nonlinearity. The CE-DBP is an extension of the DBP, which can deal with both intra- and inter-channel nonlinearity by solving the coupled multi-channel Manakov equations. Based on the first order perturbation theory, we also analyze the application of Kalman filter (KF) to nonlinearity compensation, which utilizes correlation between adjacent symbols to compensate for Kerr nonlinearity. With numerical simulations, the performance of three nonlinearity equalization schemes of CE-DBP, DBP + KF, and CE-DBP + KF are compared. For nonlinearity mitigation in single carrier 400 G and 800 G WDM systems, the DBP + KF is more efficient than the CE-DBP considering five coupled channels. The CE-DBP + KF method has slightly better performance than the DBP + KF. However, the CE-DBP is quite complex due to multi-channel Manakov equation calculation, which demands broad-band joint detection with a single coherent receiver or several individual coherent receivers with time synchronization. In contrast, the DBP + KF only need the information of the target channel. We thus conclude that the DBP + KF is an attractive solution to mitigate fiber nonlinearity for single carrier 400 G and 800 G Nyquist-WDM systems. With the combination of the DBP and the KF, the transmission distance extensions are achieved as 12.5%, 14.3%, 22.6% and 28% for 400 G_64 QAM, 800 G_64 QAM, 400 G_16 QAM and 800 G_16 QAM WDM systems, respectively.

The Impact of Transceiver Noise on Digital Nonlinearity Compensation

The efficiency of digital nonlinearity compensation (NLC) is analyzed in the presence of noise arising from amplified spontaneous emission noise (ASE) as well as from a non-ideal transceiver subsystem. Its impact on signal-to-noise ratio (SNR) and reach increase is studied with particular emphasis on split NLC, where the digital back-propagation algorithm is divided between transmitter and receiver. An analytical model is presented to compute the SNR's for non-ideal transmission systems with arbitrary split NLC configurations. When signal-signal nonlinearities are compensated, the performance limitation arises from residual signal-noise interactions. These interactions consist of nonlinear beating between the signal and co-propagating ASE and transceiver noise. While transceiver noise-signal beating is usually dominant for short transmission distances, ASE noise-signal beating is dominant for larger transmission distances. It is shown that both regimes behave differently with respect to the optimal NLC split ratio and their respective reach gains. Additionally, simple formulas for the prediction of the optimal NLC split ratio and the reach increase in those two regimes are reported. It is found that split NLC offers negligible gain with respect to conventional digital back-propagation (DBP) for distances less than 1000 km using standard single-mode fibers and a transceiver (back-to-back) SNR of 26 dB, when transmitter and receiver inject the same amount of noise. However, when transmitter and receiver inject an unequal amount of noise, reach gains of 56% on top of DBP are achievable by properly tailoring the split NLC algorithm. The theoretical findings are confirmed by numerical simulations.