Nonlinear Frequency Division Multiplexing Systems Research Articles

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.

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 Using Distributed Raman Amplification and NFT-Domain Equalization

Transmission systems based on the nonlinear Fourier transform (NFT) can potentially address the limitations in transmission reach and throughput set forth by the onset of Kerr-induced nonlinear distortion. Whereas this technique is at a preliminary research stage, a rapid progress has been shown over the past few years leading to experimental demonstrations of dual-polarization systems carrying advanced modulation formats. The lossless transmission required by the NFT to ensure the theoretical validity of the scheme is a fairly strong requirement considering practical transmission links. Here, we address it by using optimized distributed Raman amplification to minimize the power variations to approximately 3 dB over 200 km, thus approaching the lossless transmission requirement. Additionally, we experimentally evaluate the improvement provided by equalization schemes applied to the signals in the NFD. By combining distributed Raman amplification and NFD equalization, we show transmission reaches for dual-polarization nonlinear frequency division multiplexing systems transmitting both two eigenvalues (8 bit/symbol) up to 2200 km and three eigenvalues (12 bit/symbol) up to more than 600 km at hard-decision and soft-decision forward error correction threshold, respectively.

Time-Domain Neural Network Receiver for Nonlinear Frequency Division Multiplexed Systems

The nonlinear Fourier transform is a new approach for addressing the capacity limiting Kerr nonlinearities in optical communication systems. It exploits the property of integrability of the lossless nonlinear Schrodinger equation and thus incorporates nonlinearities as an element of the transmission. However, practical links employing erbium-doped fiber amplifiers include losses/gains and introduce noise which breaks the integrability of the nonlinear Schrodinger equation. Although the lossless path average approximation proposes an integrable model, its imprecision still leads to unintended distortions and thus performance degradation. We propose an alternative receiver for nonlinear frequency division multiplexing optical communication systems using techniques from machine learning. It is highly adaptive as it learns from previously transmitted pulses and thus holds no assumptions on the system and noise distribution. The detection method presented is fully applied in time-domain and omits the nonlinear Fourier transform. The numerical results provide a benchmark for nonlinear Fourier transform based detection of high order solitons for fiber links with losses and noise present.

Decision-feedback detection strategy for nonlinear frequency-division multiplexing

By exploiting a causality property of the nonlinear Fourier transform, a novel decision-feedback detection strategy for nonlinear frequency-division multiplexing (NFDM) systems is introduced. The performance of the proposed strategy is investigated both by simulations and by theoretical bounds and approximations, showing that it achieves a considerable performance improvement compared to previously adopted techniques in terms of Q-factor. The obtained improvement demonstrates that, by tailoring the detection strategy to the peculiar properties of the nonlinear Fourier transform, it is possible to boost the performance of NFDM systems and overcome current limitations imposed by the use of more conventional detection techniques suitable for the linear regime.

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.