Performance vs. complexity in NN pre-distortion for a nonlinear channel.

Optical communications at high bandwidth and high spectral efficiency rely on the use of a digital-to-analog converter (DAC). We propose the use of a neural network (NN) for digital pre-distortion (DPD) to mitigate the quantization and bandlimited impairments from a DAC in such systems. We experimentally validate our approach with a 64 Gbaud 8-level pulse amplitude modulation (PAM-8) signal. We examine the NN-DPD training with both direct and indirect learning methods. We compare the performance with typical Volterra, look-up table (LUT) and linear DPD solutions. We sweep regimes where nonlinear quantization becomes more prominent to highlight the advantages of NN-DPD. The proposed NN-DPD trained via direct learning outperforms the Volterra, LUT and linear DPDs by almost 0.9 dB, 1.9 dB and 2.9 dB, respectively. We find that an indirect learning recurrent NN offers better performance at the same complexity as Volterra, while a direct learning recursive NN pushes performance to a higher level than a Volterra can achieve.

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.

Blind and low-complexity modulation format identification scheme using principal component analysis of Stokes parameters for elastic optical networks.

We propose a blind and low-complexity modulation format identification (MFI) scheme for elastic optical networks (EONs). Since the square operation reduces half the number of the clusters in Stokes space, the scheme directly performs principal component analysis (PCA) on Stokes parameters after square operation. This greatly reduces the dimensionality of received signals from 3 × N to 3 × 3. Subsequently, three obtained principal components (PCs) are employed synthetically to identify the modulation formats. The effectiveness is first verified through 28 GBaud polarization division multiplexing (PDM)-BPSK/-QPSK/-8QAM/-16QAM/-32QAM/-64QAM simulation systems. Only using 2048 symbols, the required minimum optical signal-to-noise ratio (OSNR) values to achieve 100% MFI success rate are all equal to or lower than their corresponding 7% forward error correction (FEC) thresholds. Besides that, the scheme also obtains significant tolerances to residual chromatic dispersion (CD) and differential group delay (DGD). Finally, the proposed scheme is further verified by 20 GBaud PDM-QPSK/-16QAM/-32QAM long-haul transmission experiments. The results demonstrate that the scheme exhibits good resilience towards fiber nonlinear impairments. More importantly, compared with other four kinds of MFI schemes, the used symbol number to achieve 100% MFI success rate notably equals to at most 2/5 as that of other schemes, and its time complexity can be reduced to O(N).

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.

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.

Joint blind equalization of CD and RSOP using a time-frequency domain Kalman filter structure in Stokes vector direct detection system.

We propose a joint blind equalization method for chromatic dispersion (CD) and ultra-fast rotation of state-of-polarization (RSOP) in a Stokes vector direct detection (SV-DD) system based on a new time-frequency domain Kalman filter structure. In an SV-DD system, the impairments induced by CD and RSOP possess a nonlinear form. Therefore, CD and RSOP cannot be treated sequentially, which causes difficulty in jointly equalizing these two impairments using an ordinary algorithm. The Kalman filter was proven to be effective in equalizing polarization effects in a coherent receiver. However, this approach has inherent limitations given that the Kalman filter was originally presented as a method implemented in the time domain whereas CD is eventually induced in the frequency domain. In this report, the proposed time-frequency domain Kalman filter can facilitate CD compensation in the frequency domain and RSOP equalization in the time domain by exploiting a sliding window structure. Both the CD compensation and the RSOP equalization are conducted in Stokes space when the proposed method is utilized, which is specially designed for an SV-DD system. The presented approach was checked using a 28 Gbaud 16-QAM SV-DD system simulation platform. The simulation results confirm that the method is very effective and has strong tolerance to CD (more than 2550 ps/nm, equivalent to a 150 km G. 652 fiber) combined with ultra-fast RSOP (up to 2 Mrad/s) for application in extreme polarization environments, like the transient lightning in a rainy day.

Joint equalization scheme of ultra-fast RSOP and large PMD compensation in presence of residual chromatic dispersion.

Polarization demultiplexing is generally carried out by a multiple-input multiple-output (MIMO) based algorithm in polarization division multiplexing (PDM) coherent systems. However, in some extreme environments, the MIMO algorithm becomes inapplicable due to the ultra-fast rotation of the state of polarization (RSOP) and large polarization mode dispersion (PMD). In addition, the residual chromatic dispersion (RCD) is always present because of the mismatch of the compensated chromatic dispersion and real value induced in the optical fiber channel. According to the literature, the Kalman filter-based polarization demultiplexing algorithms possess very weak RCD tolerance. Faced with this dilemma, in this paper, a new Kalman filter structure is proposed, which can jointly compensate ultra-fast RSOP, large PMD and RCD. This Kalman filter structure enables the equalization of the RSOP in the time domain and compensation for RCD and PMD in the frequency domain. We verified the performance of the proposed Kalman scheme in the 28 Gbaud PDM-QPSK/16 QAM coherent system, with a comparison to constant modulus algorithm/multiple modulus algorithm (CMA/MMA). The simulation results confirm that, compared with CMA/MMA, the proposed Kalman scheme can provide a significant performance enhancement to cope with ultra-fast RSOP (up to 3 Mrad/s) and large PMD (more than 200 ps) with a large tolerance to RCD (over the range of ± 820 ps/nm in PDM-QPSK and ± 500 ps/nm in PDM-16 QAM).

Window-split structured frequency domain Kalman equalization scheme for large PMD and ultra-fast RSOP in an optical coherent PDM-QPSK system.

A window-split frequency domain Kalman scheme is proposed in this paper for the equalization of large polarization mode dispersion (PMD) and ultra-fast rotation of state-of-polarization (RSOP) which is an extreme environment due to the Kerr effect and the Faraday effect under the lightning strike near the fiber cables. In order to carry out the proposed Kalman scheme, we give a simplified and equivalent fiber channel model as a replacement for the general model of the polarization effect of the co-existence of PMD and RSOP. With this fiber channel model, we can conduct compensation for PMD in the frequency domain and tracking RSOP in time domain. A half analytical and half empirical theory for the initialization of the process and measurement noise covariance is also presented in theory and verified by the numerical simulation. The performance of the proposed Kalman scheme is checked in the 28Gbaud PDM-QPSK coherent system built on both simulation and experiment platforms. The simulation and experiment results confirm that compared with the generally used constant modulus algorithm (CMA), the proposed scheme provides excellent performance and stability to cope with large range DGD from 20ps to 200ps and RSOP from 200krad/s to 2Mrad/s, with less computational complexity.

Two-parameter-SOP and three-parameter-RSOP fiber channels: problem and solution for polarization demultiplexing using Stokes space.

In this paper, we highlight that it is inadequate to describe the rotation of the state of polarization (RSOP) in a fiber channel with the 2-parameter description model, which was mostly used in the literature. This inadequate model may result in problems in polarization demultiplexing (PolDemux) because the RSOP in a fiber channel is actually a 3-parameter issue that will influence the state of polarization (SOP) of the optical signal propagating in the fiber and is different from the 2-parameter SOP itself. Considering three examples of the 2-parameter RSOP models typically used in the literature, we provide an in-depth analysis of the reasons why the 2-parameter RSOP model cannot represent the RSOP in the fiber channel and the problems that arise for PolDemux in the coherent optical receiver. We present a 3-parameter solution for the RSOP in the fiber channel. Based on this solution, we propose a DSP tracking and equalization scheme for the fast time-varying RSOP using the extended Kalman filter (EKF). The proposed scheme is proved to be universal and can solve all the PolDemux problems based on the 2- or 3-parameter RSOP model and exhibits good performance in the time-varying RSOP scenarios.