## Abstract

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.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Kerr nonlinearity is a significant factor limiting the transmission capacity of fiber optic communication systems [1,2]. To suppress the influence of nonlinear effects, various nonlinear compensation (NLC) and mitigation schemes have been proposed [3]. In recent years, a revolutionary scheme, the nonlinear Fourier transform (NFT)-based nonlinear frequency division multiplexing (NFDM) system, in which the intrinsic nonlinearity of fiber was treated as a constructive factor [1,2,4–10], was presented to overcome the limitations of Kerr nonlinearity. In NFDM systems, the information is encoded on the nonlinear spectra, which are obtained by NFT and propagate independently (i.e. without crosstalk between components) under the ideal nonlinear Schrödinger equation (NLSE), the model for describing optical field evolution in fiber links. The nonlinear spectrum $\lambda$ is comprised of two parts: the discrete spectrum (soliton components) where $\lambda$ lies on the upper-half complex plane, and the continuous spectrum where $\lambda$ is real [4,5]. The information can be modulated on the discrete spectrum, the continuous spectrum, or both. In this work, we focus on the discrete spectrum modulation NFDM systems.

Although a number of proof-of-concept experimental demonstrations of discrete spectrum modulated NFDM systems have been reported [7–9], there are still many challenges to overcome. One of the challenges is to find the optimal receivers for multiple-eigenvalue NFDM systems [11–16]. The spectral efficiency (SE) of the NFDM system increases with the number of eigenvalues [17]. However, the demodulation complexity also increases dramatically with the increasing of the eigenvalue numbers. Additionally, the accuracy of the NFT algorithm for multiple-eigenvalue demodulation will decrease significantly. These two issues may result in the failure of the receiver NFT algorithm to demodulate multiple eigenvalues because of insufficient or spurious roots when recovering the modulated discrete eigenvalues. Further, the amplifier spontaneous emission (ASE) noise, the phase noise (PN) introduced by the lasers, and the application of the polarization division multiplexing (PDM) technique [10,18–21] make the demodulation even more difficult. Due to the difficulty in demodulation, the maximum number of modulated eigenvalues is only two in dual-polarization (DP) NFDM systems [8,21,22]. An alternative method of using machine learning (ML) instead of NFT for demodulation has been proposed. In [11], an artificial neural network (ANN) based demodulation method was experimentally demonstrated in on-off encoded 4-eigenvalue modulated NFDM system. On the basis of [11], literature [12] proposed an eigenvalue-domain ANN demodulation scheme, which realized 3,000 km transmission distance with 2.5 Gb/s on-off encoded eigenvalues modulation. Due to on-off encoding belongs to amplitude modulation, the NFDM system in [11,12] are insensitive to phase noise, while the spectrum efficiency is low. In [13], the two-eigenvalue QPSK-modulated signals was transmitted and a ML-based receiver was applied in NFDM systems, where only the case of ideal lasers was considered. However, in the actual transmission system, the phase noise introduced by the transmitter laser and the local oscillator laser is a non-negligible factor. And the higher order the modulation format, the greater the impact of phase noise on the system. To improve spectrum efficiency and the tolerance to phase noise, based on the simulation platform with 16 quadrature amplitude modulation (16-QAM) modulated the discrete spectrum b-coefficients, we proposed a neural network (NN)-based receiver in [15], which can implement demodulation of four and five eigenvalues in 2 GBaud single-polarization (SP) NFDM systems. Another receiver using Euclidean minimum distance (MD) [16] was also proposed, which is used for detection in a 7-eigenvalue QPSK-modulated over 1440 km NFDM system.

In this work, we further developed the structure of NN receiver in [15], and compared the performance of NN receiver with other receivers in NFDM systems. By studying the efficacy of ML methods, we adopt a regression neural network receiver which learns from the received waveforms directly without the need for root finding. By selecting and optimizing the input and output data of NN receiver reasonably, the NN receiver can realize the prediction of continuous variables and obtain a higher robustness against impairments, which is more suitable for practical NFDM transmission systems. Compared with the NFT and MD receivers, the NN receiver realizes improved bit error-rate (BER) and error vector magnitude (EVM), which can achieve higher-order multiple-eigenvalue demodulation in single- and dual-polarization NFDM transmission systems.

The rest of the paper is organized as follows. Section 2 gives a brief introduction to NFDM communication systems using NFT receivers and their challenges. The basic principles of NN and MD receivers, and the complexity analysis of neural networks are introduced in Section 3. In Section 4, the simulation setup for the NFDM transmission system using different detection schemes is described. In Section 5, the demodulation performance of the NN receiver in the multiple-eigenvalue modulated NFDM system is shown. The performances of three receivers in single-polarization and dual-polarization NFDM systems are also compared. The BER and EVM performance under different transmission distances are evaluated and the numerical simulation results are shown. The hyperparameter selection of the NN model and the analysis of processing speed are also presented. Conclusions are drawn in Section 6.

## 2. NFDM communication system with discrete spectrum modulation

Figure 1 shows the schematic of discrete spectrum modulated NFDM system, where the information is modulated onto the nonlinear spectrum (either eigenvalues ${\lambda _i}$or discrete spectrum amplitude $\tilde{q}({{\lambda_i}} )$) at the transmitter by inverse NFT (INFT). After propagating through the fiber, the nonlinear spectrum is obtained by NFT at the receiver. The discrete nonlinear spectrum is estimated by locating the discrete eigenvalues${\lambda _i}$and calculating the corresponding spectral amplitude$\tilde{q}({{\lambda_i}} )$. In this work, Darboux transform [4,8] (i.e., INFT) was used to generate time-domain waveform signals$q({t,0} )$. The data was modulated on the b-coefficients of the discrete spectrum to mitigate the adverse effects of noise. At the receiver, NFT was implemented by the fast computation of nonlinear Fourier transform software package [23], which used fast and backward stable computation of roots of polynomials [24].

For ideal cases, the NLSE is integrable, and the eigenvalues${\lambda _i}$of multiple soliton signals remain constant during transmission, $i.e.,{\lambda _i}(z) = {\lambda _i}(0)$ for any $0 \le z \le {\cal \textrm{L}}$. After propagating through a distance *z*, the discrete spectral amplitudes will evolve according to

In the presence of ASE noise, the eigenvalue${\lambda _i}$is distorted and becomes ${\lambda _d} = {\lambda _i} + \Delta \lambda$. Eigenvalue noise causes spectral amplitude noise. Such spectral amplitude perturbations accumulate with the propagation distance $z$ [25] and can be written as

Equation (3) shows that the eigenvalue drift during transmission disturbs the spectrum amplitude. The disturbance becomes significant impairments for transmission through long-haul links and lead to inaccurate discrete spectral estimation that makes the NFT-based receiver suboptimal. To illustrate this, we used 2,000 data points to simulate the distribution of two eigenvalues $\lambda \in \{{0.3j,0.6j} \}$ and the corresponding constellation at the NFT receiver after 1,000 km transmission with 16QAM modulation. The simulation parameters are shown in Table 1. The normalized time window duration is 20, and physical time window duration of each soliton pulse is 0.5 ns. The optical signal-noise ratio (OSNR) is around 20 dB. The fiber link consists of 20 spans, each of which contains 50 km non-zero dispersion shift fiber (NZ-DSF). The launch power is set to 1.1 dBm for two eigenvalues $\lambda \in \{{0.3j,0.6j} \}$. The ASE noise from the Raman amplifiers is added to the signal in the fiber channel, and the ASE noise density can be expressed as [26,27]:

Where $\alpha$ is the fiber attenuation coefficient,*L*is the transmission length, $h{f_s}$ is the photon energy, and ${K_T}$ is the photon occupancy factor, which is about 1.13 for the Raman amplification of the optical fiber communication systems at room temperature [26,27].

As shown in Fig. 2(a) and Fig. 2(b), the NFT demodulation eigenvalues deviate from the standard positions and that the constellation distribution of $\tilde{q}({{\lambda_i}} )$is distorted. In multiple-eigenvalue NFDM systems, the complexities of the NFT algorithm mathematical model and the demodulation difficulty increase with the number of eigenvalues. This is reflected in the root search problem and the difficulty of calculating the spectrum amplitude. It should be noted that when the number of modulated eigenvalues exceeds four, insufficient root-searching will occur, which cannot be shown in the figure. The greater the number of modulated eigenvalues, the more severe the deviation of the demodulated eigenvalues and the constellation distribution, and the higher the bit error rate of the system.

In addition to the ASE noise, the adverse effects of impairments will cause the constellation points to rotate and deviate from the ideal position, which also greatly increases the difficulty of NFT demodulation. These impairments include frequency offset (FO), phase noise, and so on. Compared with amplitude modulation, phase noise has strong influences on the QAM-modulated system. In addition, the phase noise follows the Wiener distribution and is more difficult to equalize, so the main impact of phase noise on the NFDM system is analyzed.

We simulate the time-domain waveforms and constellation perturbation caused by phase noise, as shown in Figs. 3(a)-(c). Due to two eigenvalues 16QAM modulated signal corresponds to 256 time-domain waveform types, they will overlap seriously when they are drawn together. To show the effects of PN on the system clearly, here, we only considered the scenario with single-eigenvalue $\{{0.3j} \}$ 16QAM-modulated and drew 16 different waveform types. The system parameters are same as listed in Table 1. According to the actual parameters of laboratory lasers and experimental reports [7,28], the linewidth (LW) used here is 100kHz. In order to express the variation in the waveforms more intuitively, we divide them into real and imaginary parts. In each of the plots that follow, the abscissa denotes the sampling points of the waveforms, and the ordinate denotes the interval between the 16 types of waveforms. Figure 3(a) shows the ideal back-to-back time domain waveforms and Fig. 3(b) the corresponding time domain waveforms distorted by 100 kHz PN. Figure 3(c) is the constellation diagram obtained from the signal of Fig. 3(b) by NFT demodulation. We observe that the PN causes the time domain waveforms of the signal to be severely distorted, resulting in the signal being improperly demodulated.

## 3. Machine learning receivers

Machine learning is a powerful tool that has been widely used in recent optical communication systems [29–31]. Compared to NFT algorithms, ML methods in NFDM systems can achieve improved performance with reduced complexity and execution time.

#### 3.1 Neural network receiver

Neural networks can learn nonlinear relationships and provide estimates based on the input features accurately. The aim of the NN is to form the nonlinear mapping between the input features and the expected outputs. Unlike NFT algorithms, the neural network algorithm in this work does not need to search for eigenvalues or calculate the corresponding discrete spectra. It learns from the huge amount of received waveforms and generates a black-box transmission model. Benefitting from their powerful learning ability, NNs can approximate any nonlinear function in principle. This is the key to realizing demodulation in NFDM systems with NNs in place of NFT.

The rotation and offset of constellation points are important factors that affect the performance of the neural network. To improve the tolerance of the regression neural network model to impairments, we need to choose the NN input and output data reasonably and adapt the network model to the adverse effects of impairments in practical applications. Figure 4 shows the structure of the NN using three hidden layers. In the training process, the sampling points of distorted time domain waveforms are taken as the NN inputs, and the NN outputs are the complex values corresponding the offset constellation points. The NN inputs and outputs are both decomposed into real and imaginary parts. Therefore, the number of input nodes is double that of sampling points, and the number of output nodes is double that of the eigenvalues. For reasonable input and output data, the performance of a NN in finding the nonlinear and complex relationships between input and output data depends mainly on its hyperparameters, including the training datasets, number of hidden layers, number of hidden layer nodes, and activation function. The effects of different hidden layers and nodes on the system are discussed in Section 5. Since the input and output of the neural network have negative values, after testing a variety of activation functions, we finally chose tangent sigmoid activation function for the hidden layer neurons, and a linear function for the output layer neurons. The tangent sigmoid activation function is expressed as $f(\tau ) = ({{e^\tau } - {e^{ - \tau }}} )/({{e^\tau } + {e^{ - \tau }}} )$, where $\; \tau$ is the input to a hidden layer neuron. In order to prevent the vanishing gradient problem, we adopted the clipping gradient method. The idea is to set an appropriate threshold. If the gradient exceeds the threshold when updating the gradient, it is forced to be limited to the range, and the updated gradient is the threshold.

In this work, to effectively detect test data and output recognition results, it is usually necessary to train the model to realize good generalization ability. Therefore, conjugate gradient backpropagation with Powell-Beale restarts [32] is used for training the ANN. This involves gradually adjusting the NN algorithm parameters until the error is minimized. Specifically, the ANN outputs are compared with the desired outputs and the errors and their derivatives are calculated. The weights of the neurons are then updated by the gradient descent method, and training is continued until the optimal set of weights with the smallest error is found. The NN saves the trained models during training and loads the saved models during prediction to demodulate signals.

For the waveform recognition in this work, the complexity of the neural network demodulation is mainly reflected in the following two aspects. Firstly, with the increase in the number of eigenvalues *n* and the order of the modulation format *q*, the types of time-domain waveforms *m* increase exponentially. The number of waveform types in a dual-polarization NFDM system is the square of that in a single-polarization NFDM system:

Secondly, as the number of eigenvalues increases, the soliton pulses become more complex, which further increases the demodulation difficulty in the NN receiver. Figure 5 shows the time-domain waveforms corresponding to different numbers of eigenvalues in 16QAM. Therefore, to improve the generalization ability of the NN training model, a large amount of training data and appropriate network hyperparameters are required. It is worth mentioning that due to the accumulation of noise during the transmission process, the time domain waveforms need to be filtered.

Neural networks are classified into classification and regression according to their functions. Classification is the prediction of discrete variables. Given a set of data, the corresponding category is obtained through the training set. Regression is the prediction of continuous variables. Given a continuous set of data, the output value is obtained. Differing from [13], this work considers the NN demodulation as a regression problem and not a classification problem. There are two reasons for this. First, the rotation offset caused by phase noise and frequency offset will significantly blur the boundaries between the demodulated constellation points, resulting in the class labels becoming continuous values. The classification neural network cannot handle this situation. The second is the amount of data mentioned above. When the number of eigenvalues is more than four, the time-domain waveform types are large; therefore, a larger training set and a more complex network are required. For classification neural networks, the number of output nodes and the number of training samples are a major test. Therefore, the benefits of choosing regression neural networks are twofold. The first point is that regression neural networks are more suitable for practical communication systems. The other benefit is that when demodulation of the NN is considered as a regression problem, the demodulation results are continuous. Even if the training set data is less than the number of time-domain waveform types, the performance will not be greatly affected and the NN is more applicable to multiple-eigenvalue NFDM systems.

In this work, we also compared the performance of the MD receiver. For the MD receiver, the reference waveforms $w_{ref}^{(i)}(t,L),i = 1..N$ are obtained by INFT. After transmission through the distance *L*, the Euclidean distances of every received pulse ${w_{rx}}(t,L)$ to the reference pulses are calculated [16], and the reference pulse with the minimum distance is selected for decoding:

## 4. Simulation platform and setup

As shown in Figs. 6(a)–(c), three detection schemes that use NFT, NN, and MD receivers in NFDM communication systems were setup. In this work, single and dual polarization NFDM systems were simulated.

At the transmitter, 2 GBaud symbols with 16QAM were modulated on the *b*-coefficients of scattering data, and multiple-eigenvalue modulated signals were generated by an inverse NFT. The corresponding time domain waveforms were used to drive the Mach-Zehnder (IQ) modulator. The multi-soliton pulses were then launched into a recirculating loop which consisted of a 50 km span of NZ-DSF and a distributed Raman amplifier (DRA). The launch power was set to 4 dBm for four eigenvalues $\lambda \in \{{0.3j - 0.3,0.6j - 0.3,0.3j + 0.3,0.6j + 0.3} \}$ and 3.2 dBm for$\lambda \in \{{0.3j - 0.\textrm{4},0.45j - 0.\textrm{4},0.45j + 0.\textrm{4},0.3j + 0.\textrm{4}} \}$, respectively, which were matched to the NLSE channel. Other simulation parameters are shown in Table 1. The split-step Fourier method (SSFM) was employed to simulate the signal evolution. A 1-nm bandwidth optical band-pass filter (OBPF) was used to filter out-band noise. After coherent detection, the received waveforms were sent to three different receivers for detection. The linewidth of both the transmitter and local lasers was 100 kHz, and the FO was set as 100 MHz. In the 16QAM modulated NFDM systems, the blind phase search (BPS) algorithm was used to realize the carrier phase estimation (CPE). The bit information was finally recovered and the error rates were calculated.

In the NN receiver, data sets were usually divided into three categories: training, validation, and testing (with a ratio of 6: 2: 2). The training data set was used to fit model. In general, the larger the training data set, the better the performance will be. However, the learning time also increases with the size of the training data. Therefore, the minimum required training data is usually determined based on the specific situation. The validation set was used to optimize the model to avoid overfitting and the test set was used to assess model performance. In this work, the NN and MD receivers were trained with 40,000 symbols. The validation and testing data sets were both about one-third of the training data sets for each distance. Here, training, testing, and validation were carried out using the MATLAB R2018b Neural Network toolbox. In the training phase, the input data after down sampling and the output data are used for training and obtaining the trained model. To improve the robustness of the model, multiple sets of time-domain waveform data were usually combined for training. During prediction, new data is then input and the model is loaded for prediction; frequency offset estimation and carrier phase estimation is performed to obtain a modified constellation.

## 5. Results and analysis

In this section, we present the results of NN demodulation with different settings of eigenvalues and hidden layers in multiple-eigenvalue NFDM communication systems. To verify the performance of the NN receiver, the demodulation results are compared with those from the NFT and MD receivers in single- and dual-polarization NFDM systems. In addition, the EVM and complexity analysis of different receivers are given.

#### 5.1 Multiple-eigenvalue demodulation results of the NN receiver

In view of the poor demodulation performance of the other receivers in multiple-eigenvalue modulated NFDM systems, only the simulation results of NN receivers for four-eigenvalue modulated NFDM systems are shown. There are no fixed rules for choosing the optimal number of hidden nodes and layers in the ANN. We simulated NN detection schemes in a single-polarization NFDM system with different hidden layers to determine the optimal structure.

Figure 7 shows the BER versus transmitted distance for the NN receiver on two different combinations of four eigenvalues:$\lambda \in \{{0.3j - 0.3,0.6j - 0.3,0.3j + 0.3,0.6j + 0.3} \}$and$\lambda \in \{{0.3j - 0.\textrm{4},0.45j - 0.\textrm{4},0.45j + 0.\textrm{4},0.3j + 0.\textrm{4}} \}$. We selected three different hidden layer structures with 32, 64, and 128 hidden nodes and compared the two scenarios with or without phase noise. The results show that the NN receiver has a certain tolerance to PN. The transmission distance over which the BER falls below the soft-decision forward error correction (SD-FEC) limit can reach 1,000 km, and the three-hidden layer structure with 64 nodes has the best performance. The degraded performance of the 128-node complex network structure is due to overfitting of the neural network. It is necessary to select a suitable network structure to optimize performance and minimize training time. It should be noted that, for comparison, the demodulation performance of the classification neural network was also studied, however, its performance is poor and the bit error rate is very high, so the comparative simulation results are not given here.

#### 5.2 Comparison and analysis of the three receivers

In our simulation platform, due to the interference of noise, the NFT receiver [23] behaves extremely poor performance with more than four modulated eigenvalues. To compare the performance of the three receivers, we chose a two-eigenvalue 16QAM modulated NFDM system for simulation.

As shown in Figs. 8(a) and 8(b), the relationship between the BER and transmission distance of the three receivers on $\lambda \in \{{0.3j,0.6j} \}$ was simulated in single- and dual-polarization systems. In dual-polarization NFDM systems, we adopted a neural network with three hidden layers, each of which had 64 nodes, as the network had turned out to be sufficient for transforming the input features into accurate predictions. In the 750–1500 km range, the NN receiver has significantly improved performance over the NFT and MD receivers. The performance can be further improved to some extent by optimizing the hidden layer structure.

The EVM as a function of transmission distance is also studied. The smaller the EVM, the better the signal quality. The EVM requirements increase with the modulation order. To ensure correct demodulation in the receivers, it is reasonable to set the EVM limit for 16QAM to 12.5% and the EVM limit for 64QAM to 8%. As shown in Fig. 9(a), we simulated the EVM with two-eigenvalue 16QAM modulation in single- and dual-polarization NFDM systems. The EVM of the NN receiver is lower than that of the NFT receiver. In the dual-polarization NFDM system, the NFT receiver was unable to recover information correctly for transmissions over 1,000 km. The EVM with 64QAM modulation in single-polarization NFDM system was also simulated in Fig. 9(b). The results show that the outer constellation points for NFT demodulation are severely diffused and the EVM is high, while the inner constellation points are more convergent. Compared with NFT, the extension of the NN demodulation external constellation points is not obvious and all the constellation points are basically equal. We can hence take advantage of the unique features of NN to achieve higher order modulation.

Besides accuracy, data processing speed is also an important indicator for comparison. The time required for training depends on the amount of training data, the training algorithm, and the structure of the neural network. Since it is not necessary to train the model again after the training is completed, the test time, rather than the training time, is the key factor affecting the processing speed. The NN test time is far shorter than the NFT demodulation time because demodulation via the NFT algorithm requires iterative calculations while the NN model obtained through training involves parallel calculations. Comparing the three detection schemes, NFT is the most time-consuming algorithm.

## 6. Conclusion

Impairments such as the ASE noise and laser phase noise have detrimental impacts on the demodulation in multiple-eigenvalue modulated NFDM systems. The NFT receiver cannot work normally when the eigenvalues and spectra amplitudes are distorted, which makes it unsuitable for practical communication systems. In view of these considerations, we proposed a robust regression NN-based receiver to achieve multiple-eigenvalue demodulation in this work. The demodulation recognizes the signal time-domain waveforms without the need for root searching and can achieve a higher impairment tolerance. The structural optimization of the NN receiver was also simulated and studied. Through reasonable selection of input and output data and parameters, the NN receiver achieved low bit error rate demodulation and the transmission distance reached up to 1,000 km in four eigenvalue-modulated NFDM systems. In the two eigenvalue-modulated single- and dual-polarization NFDM systems, the NN receiver has the best performance compared to other receivers and can achieve higher-order modulation. The results show that the proposed NN receiver has broad application prospects in future practical multi-eigenvalue modulated NFDM communication systems.

## Funding

State Key Laboratory of Information Photonics and Optical Communications (IPOC2019ZZ02); Beijing Excellent Ph.D. Thesis Guidance Foundation (CX2018113).

## Disclosures

The authors declare no conflicts of interest.

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