1. Introduction

In the digital age with the rapid development of the Internet, the explosive growth of network traffic puts forward higher requirements for optical fiber communication networks in terms of high speed and large capacity [1]. However, the capacity and transmission rate of optical fiber communication are constrained by linear and nonlinear impairments, making it difficult to meet the requirements. The linear impairment, including chromatic dispersion (CD) and polarization-mode dispersion (PMD), can be effectively compensated by digital signal processing (DSP) algorithms, while the impact of the nonlinear impairment brought by Kerr nonlinear effect escalates when the signal power and baud-rate increase. Especially, nonlinear effects in WDM systems include not only self-phase modulation (SPM) from the same channel, but also cross-phase modulation (XPM) and four-wave mixing (FWM) from the other channels, where the nonlinear phase shift caused by SPM and XPM will cause critical signal distortions. In addition, the optical fiber communication system is not a pure linear system or nonlinear system, and linear impairments will interfere with nonlinear effects to a certain extent during the signal transmission. Therefore, overcoming nonlinear effects of fiber is not only the key to optimize the system performance, but also a major difficulty [2].

Researchers have proposed several effective nonlinear compensation algorithms to alleviate the distortion caused by nonlinear effects. The digital back propagation (DBP) algorithm and its improved scheme [3–6] achieve alternating compensation of CD and nonlinearity by solving the reverse signal transmission equation based on the split-step Fourier method (SSFM). However, the iteration of DBP requires multiple Fourier transform pairs, and the performance improves as the number of steps increases, which means more high-level performance requires higher computational complexity. Also, this algorithm is a theoretical method to compensate nonlinear distortions, which requires transparent parameters of fiber links, and faces great challenges when directly applied to practice. In addition, optical phase conjugation (OPC) [7] for nonlinear compensation in the optical domain, the nonlinear equalization method based on Volterra series [8] and the nonlinear compensation algorithm based on perturbation theory [9,10] have also been proved to be effective. However, OPC has a high cost and low conversion efficiency in practical applications, resulting in performance limitations. Given that the method based on Volterra series requires Fourier transform modules, the complexity increases with the accumulation of CD. Nonlinear compensation based on perturbation theory requires higher computational complexity to achieve the expected quantization accuracy.

In recent years, with the rapid development of machine learning, the powerful learning ability of neural network has drawn worldwide attention. It can complete operations without consuming too much prior link information of systems. Therefore, artificial neural network (ANN), convolutional neural network (CNN), etc. have been introduced into the field of fiber nonlinearity compensation to further improve system performance [11]. The proposal of the correlation between neighboring symbols of triples has made memory neural networks a research hotspot. Taking long short-term memory (LSTM) networks and their variants [12–16] as examples, they can effectively realize nonlinear impairment compensation of coherent optical communication systems by memorizing the correlation between neighboring symbols. However, most of the nonlinear compensation methods based on neural networks mentioned above are “black box” approaches that only focus on performance improvement, the output results and learning process are difficult to be explained. Therefore, researchers combined theoretical models with neural networks and proposed interpretable learned DBP (LDBP), which addressed the limitations of “black box” neural network on nonlinear compensation. C. Häger et al. used deep neural networks to simulate the linear and nonlinear steps of DBP and conducted simulations in a single-channel single-polarization system [17]. Q. Fan et al. deeply studied DBP based on deep neural networks, which alleviated nonlinear impairments in single-channel and WDM systems by optimizing input parameters and neural network structure [18]. D. Tang et al. combined DBP and nonlinear polarization crosstalk compensation (NPCC) with neural networks based on their physical meanings to address the nonlinear impairments in WDM systems [19]. T. Inoue et al. proposed a LDBP scheme that considers SPM and XPM to compensate for the nonlinear distortion with a reasonable calculation cost [20]. O. Sidelnikov et al. used deep convolutional neural network to alleviate nonlinear distortions in long-haul fiber communication systems [21]. P. He et al. proposed a LDBP algorithm based on layer-reduced neural network, which smoothed the power term in the nonlinear compensation model [22]. However, most of the above methods did not fully consider the complex correlation between linear and nonlinear impairments and ignored the disturbance of signal nonlinearity caused by pulse-broadening effect within the channel and inter-channel walk-off effect in WDM systems.

This article proposes a novel joint compensation scheme for both intra and inter-channel nonlinearity based on improved LDBP. In this scheme, CD is compensated by linear filters in the time-domain convolutional layer. According to the time series and dispersion properties of the signal, the input characteristics are adjusted to achieve the appropriate combination of the overlap-and-save method and pulse-broadening effect. Considering that the influence of CD on nonlinearity varies with factors such as fiber length and cannot be quantified, we improve the nonlinear compensation model for this uncertainty interference. In the case of considering the nonlinear interaction between neighboring symbols, we use the enhanced split-step Fourier method (ESSFM) [23] to improve the SPM compensation model, which effectively improves the accuracy of nonlinear compensation. At the same time, the XPM compensation model is improved by factorizing walk-off effects between neighboring channels, which effectively solves the problem of asynchronous transmission of signal pulses.

The rest of the paper is organized as follows. In Section 2, we describe and analyze the channel model, the principle of time-domain CDC and nonlinear compensation of the proposed scheme. Section 3 is the construction of the simulation system, the analysis and discussion of the corresponding results. Moreover, the computational complexity is analyzed in this section. Section 4 further verifies the effectiveness of this scheme through experiments. Section 5 is a summary of the entire paper.

2. Principle of the improved LDBP

2.1. Channel model of PDM-WDM systems

For WDM systems, when we take the channel k as target channel and focus on the two polarizations of it, taking into account the influence of neighboring channels, the signal propagation can be represented by the following coupled nonlinear Schrödinger equation (NLSE) [24,25]:

CD and nonlinear phase shift caused by SPM and XPM are solved by alternating linear compensation and nonlinear compensation. Taking any three neighboring channels of the WDM system as an example, the corresponding architecture of the improved LDBP is shown in Fig. 1. The entire neural network structure is presented Fig. 1(a). It should be pointed out that a dispersion compensation layer and a nonlinear compensation layer form a hidden layer in the neural network. The blue squares represent the dual-polarized signals of different channels sent to the neural network. Green neurons denote the data that needs to be conveyed to the nonlinear layer after time-domain CDC. The orange neurons represent the data that needs to be conveyed to the next layer after nonlinear compensation. In nonlinear operations, the purple lines denote the process of compensating XPM effects between the target channel and other channel, the black lines denote compensation for the SPM effect on each channel. After the alternating compensation of linear and nonlinear is completed, the data is conveyed to other operations by the yellow and gray neurons, and the dark red neurons following them represent filters that compensate for nonlinear interactions related to polarizations. The yellow and gray neurons labeled with signals in the output layer are signal sequences of different channels obtained through learning and training.

 

Fig. 1. The architecture of improved LDBP. (a) The entire neural network structure; (b) Nonlinear compensation section.

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2.2. Principle of time-domain chromatic dispersion compensation

Ignoring the nonlinear effects in NLSE, we can obtain the propagation equation of channel k with only linear impairments as follows:

CDC of long pulses can be realized effectively by the compensation operator $C{D^{ - 1}} = \textrm{exp} \left( { - j\frac{{{\beta_2}}}{2}\omega_k^2h + j\frac{{{\beta_3}}}{6}\omega_k^3h} \right)$ in the frequency domain. However, the proposed scheme is similar to DBP, the real-time compensation of accumulated CD corresponding to the steps taken for a span can be more efficiently implemented in time-domain [26]. Moreover, the time-domain CD compensation does not require the operation of Fourier transform, which significantly reduces the computational complexity. The practical implementation of one-dimensional convolution operation is based on the linear layer shown in Fig. 1. Due to the fact that CDC of the proposed scheme is based on the time-domain filter [26], the convolution kernel in the linear layer is equivalent to the CDC filter. [27] And initial values of the convolution kernel are dependent on the inverse Fourier transform of the CD compensation operator $C{D^{ - 1}}$ derived from Eq. (4). According to the physical properties of the time-domain CDC filter, the taps of the filter should be symmetric [21], and we set the CDC filter width as 2R + 1. In addition, given that the long-haul accumulated CD will lead to crosstalk between neighboring pulses, which results in optical pulse broadening, we appropriately add overlapping waveforms into overlap-and-save method to effectively compensate for the accumulated CD in the transmission link [26].

2.3. Theoretical model of nonlinear compensation

In PDM-WDM systems, we reverse transfer the coupling NLSE (1) and (2) and ignore the linear part, the nonlinear compensation theoretical model can be obtained as follows:

The interaction between CD and nonlinearity in WDM systems cannot be ignored, and CD plays an important role in nonlinear compensation. On the one hand, CD will cause the intra-channel pulse broadening which is inherent. On the other hand, due to group velocity mismatch between channels, pulses are not synchronized in time sequence, resulting in walk-off effects owing to CD [25]. The pulse broadening and the walk-off effect will respectively generate additional uncertain phase noise for $\varphi _{k,x/y}^{SPM}$ and $\varphi _{k,x/y}^{XPM}$ brought by nonlinear effects, resulting in inaccurate nonlinear compensation.

In the same channel, both CD-induced pulse-broadening effects and nonlinear effects are present in the fiber, and the existence of CD can cause nonlinear phase mismatch at various points of the fiber, thereby reducing the nonlinear intensity. However, this issue is often overlooked in every step of classical SSFM, resulting in overestimation of nonlinear effects [4], which reduces the accuracy of the algorithm. So based on ESSFM, we consider the interaction between CD and nonlinearity and reconstruct more accurate nonlinear compensation models along each step of the fiber. Taking the nonlinear interaction of neighboring symbols into account, which compensates for nonlinear impairments and alleviates interference related to pulse broadening, the SPM nonlinear phase shift can be described by the formula:

Among different channels, the walk-off effects can make the signal of each channel asynchronous during transmission. But the existing nonlinear compensation methods assume that they are synchronous, which makes it impossible to accurately compensate the XPM effect to a large extent. If the channel k is taken as the target channel and focuses on XPM effects, its nonlinear phase shift in the time domain can be written by [25]:

Take the Fourier transform of Eq. (7), the nonlinear phase shift in the frequency domain is expressed as follows:

Since both the transmitted signal and the nonlinear phase shift related to SPM are in the time domain, we need to transform the XPM phase shift given by Eq. (9) into the time domain as follows:

To demonstrate the principle of nonlinear compensation in the proposed scheme in a straight-forward way, we further refine the nonlinear compensation section of the architecture, as shown in Fig. 1(b). The light blue part in the figure represents the solving of SPM-induced nonlinear phase shift, while the light green part is the solving of XPM phase shift. In the light orange part, nonlinear compensation is performed and the corrected signals are outputted.${g_{SPM,k}}$, ${\delta _i}$ and ${g_{XPM,nk}}$ are joint optimized parameters of the neural network when the knowledge of transmission link is unknown. In addition, random polarization rotation and PMD in PDM-WDM systems can lead to polarization-dependent nonlinear interactions [18]. The magnitude and direction of PMD vary along the fiber. And polarizations of different channels rotate randomly at different rates and directions, which can distort the transmission signal and reduce the ability of DBP to compensate for nonlinear impairments [28]. Therefore, we add time-domain filters to the back end of the neural network to reduce the random influence of polarization-dependent impairments on LDBP, which is like the process of the blind constant mode algorithm.

3. Simulation system and result analysis

3.1. Description of the simulation setup

We construct an 11-channel WDM system with 36 GBaud PDM uniform 16 quadrature amplitude modulation (PDM-16QAM) 1600 km and 64 GBaud PDM-64QAM 400 km based on VPI Design Suite 11.1 and Matlab, as shown in Fig. 2. To avoid that the neural network learns the rules of generating bits, we use random function of Matlab to randomly generate bit sequences [29], with a symbol length of 65536 for each polarization. The frequency offset and laser linewidth are set to 100 MHz and 100 KHz, respectively. The waveforms are shaped by a root raised cosine (RRC) filter with a roll-off factor of 0.1. The wavelength of the central channel is 1550 nm, and the channel spacings for PDM-16QAM and PDM-64QAM are 50 GHz and 75 GHz, respectively. Each transmission span of the system consists of 80 km/span standard single mode fiber (SSMF) and an erbium-doped fiber amplifier (EDFA). The attenuation, CD, dispersion slope, PMD and nonlinearity coefficient of the fiber are set to 0.2 dB/km, 16 ps/nm/km, 0.08 ps/nm²/km, 0.1 ${{\textrm{ps}} / {\sqrt {\textrm{km}} }}$ and 1.3 W^-1/km, respectively. Here, we consider the dispersion slope in order to make nonlinear compensation in WDM systems more accurate, while at the same time being more consistent with the real-world optical transmission system. EDFA works in gain control mode with noise figures of 5 dB (PDM-16QAM) and 4 dB (PDM-64QAM). Finally, On the receiver side, we use the coherent receiver to receive the signal and then perform offline processing.

 

Fig. 2. The simulation setup of 11-channel WDM system and the flowchart of DSP. (i) The constellation diagrams after resampling; (ii) The constellation diagrams after the improved LDBP; (iii) The constellation diagrams after carrier phase recovery. MUX: multiplexer. SSMF: standard single-mode fiber. EDFA: erbium-doped fiber amplifiers. DEMUX: demultiplexer.

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The improved LDBP in this paper is located before polarization demultiplexing, and the complete offline DSP is shown in Fig. 2. ${U_n} = {[{{U_{nx}},{U_{ny}}} ]^T}$ in the figure represents the complex envelope signal with x and y polarization of any channel n, n = 1, 2, …, 11. Firstly, we resample the received signals to 2 samples per symbol, Fig. 2(i) is the constellation diagram corresponding to the signal that needs to be equalized after resampling. Then, they are fed into the improved LDBP for training and testing to achieve alternating compensation of CD and nonlinearity. The compensated constellation is shown in Fig. 2(ii). Then the compensated signals are sent into other digital processing algorithms, in the successive order of polarization demultiplexing, down sampling, frequency offset estimation (FOE) and carrier phase recovery (CPR). Then the constellation diagram as shown in Fig. 2(iii) is obtained. Finally, the bit error rate (BER) is calculated after CPR.

For the received discrete data, whether it is PDM-16QAM or PDM-64QAM, the data length of each channel is 131072 samples, with the first 50% of 65536 samples used for training and the remaining 65536 samples used for testing. We divide 65536 training samples into training data blocks composed of input vectors. The batch size of each data block is 32 and the layer width is 128. In order to compensate for CD more effectively in time-domain convolutional layers, we combine overlap-and-save method with pulse-broadening effects. Neighboring data affected by CD are added to both ends of original data blocks, and then they are formed into input features which will be sent to the neural network for training and learning. In order to train the neural network, we choose the Adam optimizer with relatively fast convergence speed, which is suitable for large-scale data and parameter scenarios. The learning rate of the optimizer is set as 0.001 to ensure the stability of the neural network training performance. Moreover, the mean square error (MSE) is selected as the loss function to measure the similarity between the labels and the data trained through the neural network.

3.2. Parameter optimization

In order to obtain favorable initial conditions before training to have the more efficient improved LDBP scheme, we take the parameter to be optimized as the only variable to analyze the CDC filter width W = 2R + 1, the filter width Z = 2T + 1 for polarization-dependent interference, and the SPM filter width M = 2S + 1 that corresponds to the number of samples associated with nonlinear interactions owing to CD in the same channel. Since nonlinear impairments will increase with the launch power, we choose to optimize the parameters under the higher power of 0 dBm for PDM-16QAM and 2 dBm for PDM-64QAM according to the linear compensation results. It is worth noting that the Q-factor of each parameter during the optimization process does not represent the final result of the proposed scheme, as the other two initial parameters may not necessarily reach the optimal value of the univariate analysis.

Figure 3 shows the optimization curves of CDC filters and PMD filters under different modulation formats of 36 GBaud PDM-16QAM and 64 GBaud PDM-64QAM. It can be seen from Fig. 3(a) and (b) that the Q-factors of both PDM-16QAM and PDM-64QAM tend to stabilize after W = 81. Both PDM-16QAM and PDM-64QAM have no significant change in Q factor after Z = 3 for the filters that compensate for polarization-dependent interference. Considering the trade-off between the performance and computational complexity for different modulation formats of the proposed scheme, W = 81 and Z = 3 are selected as optimal parameters for both PDM-16QAM and PDM-64QAM. In order to further analyze the relationship between the nonlinear interaction and neighboring samples in the same channel, we present a comparison diagram between the compensation performance and the computational complexity of different number of neighboring samples in Fig. 4. When the number of neighboring samples in the channel is set to 0, 1, 2, …, 6, …, the corresponding value of M is 1, 3, 5, …, 13, …. The Q-factor of M tends to converge after 11 for PDM-16QAM, while for PDM-64QAM, the Q-factor increases slowly after 11. In summary, the Q-factor gradually increases and eventually stabilizes as M increases, while the corresponding complexity continues to rise. After comprehensive consideration, we eventually choose M = 11 with stable performance as the optimal parameter applied to the neural network for both PDM-16QAM and PDM-64QAM.

 

Fig. 3. Different filter width and Q-factor for different modulation formats in the WDM simulation system. (a) CDC filter width; (b) Polarization-dependent filter width.

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Fig. 4. Performance and complexity of different SPM filter width for different modulation formats in the WDM simulation system. (a) 36 GBaud PDM-16QAM; (b) 64 GBaud PDM-64QAM.

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Next, we optimize the number of hidden layers based on the trade-off between performance and complexity. The corresponding simulation results are shown in Fig. 5, which reveal the variation trend of performance and complexity corresponding to the different number of hidden layers. As can be seen from the figure, in the range of the number of hidden layers of 6-12, the complexity is increasing while the overall performance of the neural network gradually increases and eventually stabilizes. After comprehensive consideration, we take the number of hidden layers equal to 10 as the optimal parameter for both PDM-16QAM and PDM-64QAM.

 

Fig. 5. Performance and complexity of different number of hidden layers for different modulation formats in simulation scenario: (a) PDM-16QAM; (b) PDM-64QAM.

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In order to ensure the fairness of the comparison of different schemes in the next section, we continue to optimize the parameters applied in the deep convolutional neural network (DCNN) and the step size which is the dominant parameter that determines the performance of DBP. The optimization method and process of parameters in DCNN are the same as those in the improved LDBP, and in order to make a better comprehensive comparison between DCNN and the improved LDBP, the parameters of the two schemes are consistent, and this also means that the parameters of the DCNN have been optimized at the same time. For DBP step size, its optimization is affected by a variety of system parameters. As the symbol rate, launch power, the number of channels, and fiber nonlinear effects increase, the DBP step size decreases, while the number of steps per span increases, resulting in an increase in system complexity proportional to the number of steps. Therefore, the optimization of DBP step size needs to consider a variety of parameters, and finally determine the optimal value according to the trade-off between performance and complexity. Figure 6 shows the optimization curve of DBP with different steps per span (StPS) considering the various parameters described above. For the PDM-16QAM signals, we try to analyze and verify the performance and complexity of the DBP with 1, 2, 3, 5, 8, and 10 steps per span using a linear compensation scheme (represented by 0 steps per span in Fig. 6) as a benchmark. From Fig. 6(a), it can be seen that the complexity increases significantly after the DBP selection of 5 steps per span, but the performance does not improve significantly, so we ultimately choose DBP-5StPS. For the PDM-64QAM signals, after trying DBP with steps of 1, 3, 5, 8, 10, 15, and 20 per span, according to Fig. 6(b), we ultimately choose DBP-10StPS.

 

Fig. 6. The optimization curves of DBP with different steps per span (StPS) considering the various parameters in the simulation scenario: (a) PDM-16QAM; (b) PDM-64QAM. (Notes: The complexity of the linear compensation scheme is relatively low, and the unit value of the vertical axis is large due to the complexity of the DBP that needs to be fully displayed, so the complexity of the linear compensation scheme seems to be almost zero.).

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3.3. Analysis and discussion of simulation results

In order to further analyze the effect of the improved LDBP in compensating nonlinear impairments, we compare the performance and computational complexity of the proposed scheme with other typical schemes from the perspectives of only considering SPM and simultaneously considering SPM and XPM. The detailed complexity analysis is shown in Section 3.4. Figure 7 shows the performance curves for PDM-16QAM and PDM-64QAM under different launch powers for the channel of interest. It can be seen from Fig. 7(a) that for PDM-16QAM, compared with the linear compensation scheme, the optimal launch power of the LDBP scheme which only compensates SPM with 0.5 steps per span is increased from -1 dBm to 0 dBm, and the performance is better than that of DBP-1StPS. In Fig. 7(b), for PDM-64QAM, compared with the linear compensation scheme, the optimal launch power of the LDBP scheme which only compensates SPM with 2 steps per span is increased from 1 dBm to 2 dBm, and the performance is comparable to that of DBP-10StPS. The above situation signals the LDBP scheme that compensates SPM alone can reduce the number of steps and computational complexity while achieving the same performance as classical DBP.

 

Fig. 7. Performance curves of different schemes for different modulation formats in the WDM simulation system. (a) 36 GBaud PDM-16QAM with the transmission distance of 1600 km; (b) 64 GBaud PDM-64QAM with the transmission distance of 400 km. (Notes: Due to the limited space in the label, we use LDBP to represent the improved LDBP, and other figures in this paper deal with this problem in the same way, and will not explain them one by one.).

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Next, we consider both SPM and XPM to analyze the joint compensation performance for the nonlinear impairments of the improved LDBP scheme in WDM systems. In order to be fair, the classical DBP scheme is still used as the reference. In Fig. 7(a), for PDM-16QAM, compared with the linear compensation scheme, the optimal launch power of the improved LDBP scheme is increased from -1 dBm to 0 dBm. Compared with the linear compensation scheme and DCNN scheme [21], the signal to noise ratio (SNR) can be improved by about 3.1 dB and 1 dB at the optimal launch power, respectively, and the Q-factor gain is about 0.75 dB and 0.15 dB. Moreover, compared to DBP-5StPS, the proposed joint compensation scheme has better performance, with the SNR improvement of about 1 dB and the Q-factor gain of about 0.15 dB at the optimal launch power. In the PDM-16QAM transmission scenario, the Q-factor of the proposed scheme satisfies the 7% forward error correction (FEC) threshold with the range of -2.5 dBm to 2.5 dBm for launch power. As shown in Fig. 7(b), for PDM-64QAM, compared to the linear compensation scheme, the optimal launch power of this scheme increases from 1 dBm to 2 dBm. Compared with linear compensation scheme and DCNN scheme, the SNR under optimal launch power can be improved by about 3 dB and 1 dB, respectively, and the Q-factor gain is about 0.54 dB and 0.1 dB. The performance of the proposed joint compensation scheme exceeds that of DBP-10StPS. Compared with DBP-10StPS, the improved LDBP scheme can improve the SNR by about 1.5 dB at the optimal launch power, and the Q-factor gain is about 0.2 dB. In the PDM-64QAM transmission scenario, the range of launch power under the proposed scheme with the Q-factor meeting the 20% FEC threshold is -3 dBm to 6 dBm. Based on the above analysis and discussion, the joint compensation performance of intra and inter-channel nonlinear impairments of the proposed scheme is not only better than that of LDBP considering only SPM compensation, but also better than that of DBP with more steps, and the computational complexity is significantly reduced.

Figure 8 shows the relationship between the Q-factor and the different transmission distances of two modulation formats at the optimal launch power. It can be seen from the figure that the performance of the linear compensation scheme is the worst. The transmission distance of PDM-16QAM is 1720km when meeting the 7% FEC threshold condition, and the transmission distance of PDM-64QAM is about 700 km at the Q-factor of 20% FEC. However, under the same condition, our proposed scheme can reach approximately 2080km for PDM-16QAM. The proposed scheme increases the effective transmission distance by about 360 km compared with the linear compensation scheme, and increases 160 km more than the LDBP scheme that only compensates for SPM. PDM-64QAM can effectively transmit about 880 km under the improved LDBP, which extends the distance by about 180 km compared with the linear compensation scheme, and increases the distance by about 120 km compared with the LDBP scheme that only compensates for SPM.

 

Fig. 8. The curves of Q-factor and transmission distance of different schemes for different modulation formats in the WDM simulation system. (a) 36 GBaud PDM-16QAM; (b) 64 GBaud PDM-64QAM.

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Next, we make a detailed and comprehensive comparison between the proposed improved LDBP and DCNN from three different perspectives in the same scenario.

Firstly, with the same transmission parameters and identical neural network parameters, we compare the performance and complexity of the two schemes. As shown in Fig. 7, for PDM-16QAM and PDM-64QAM signals, improved LDBP has a Q-factor gain of 0.15 dB and 0.1 dB compared with DCNN, respectively. And the complexity of DCNN is 3.2873 × 10⁸ RMS, while the improved LDBP is 2.9596 × 10⁸ RMS, which is a 9.97% reduction in complexity. (The specific complexity calculations are described in Section 3.4). Therefore, we can draw the conclusion that compared with DCNN, the proposed improved LDBP scheme not only reduces complexity but also improves performance.

Secondly, we carry out the comprehensive analysis of neural network parameters and performance metrics of two schemes with the same level of complexity. For PDM-16QAM and PDM-64QAM, the complexity of the two neural networks can be consistent by changing the number of hidden layers. The proposed scheme can be implemented with 10 layers and DCNN with 9 layers when keeping other parameters consistent. At this point, the complexity of both schemes is 2.9596 × 10⁸ RMS. Figure 9 shows the performance curve obtained by comparing the improved LDBP and DCNN at an equivalent level of complexity through simulation. As shown in Fig. 9(a), for PDM-16QAM, compared with DCNN scheme, the improved LDBP improves the SNR by about 1.7 dB and the Q-factor gain is about 0.43 dB at the optimal launch power. As shown in Fig. 9(b), for PDM-64QAM, also compared with DCNN scheme, the SNR of the improved LDBP is improved by about 1.8 dB and the Q-factor gain is about 0.41 dB at the optimal launch power. Therefore, the proposed scheme outperforms the DCNN scheme in performance at the same level of complexity.

 

Fig. 9. Performance curves of the improved LDBP and DCNN at an equivalent level of complexity: (a) PDM-16QAM; (b) PDM-64QAM.

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Finally, there is the comprehensive analysis of neural network parameters and complexity for both schemes at the same performance. We use the performance of DCNN as a criterion and take the input parameter CDC filter width W of the improved LDBP as the only variable. The parameters with the same performance of the two schemes are obtained by traversing the variables W in the improved LDBP under the fixed transmission power, and there is a verification for the parameters through different launch power. As shown in Fig. 10(a), in the PDM-16QAM transmission scenario, the black dashed line represents the performance of the DCNN scheme when the CDC filter width W is equal to 81, which is a fixed value. The red curve represents the performance of the improved LDBP scheme under condition of different parameters, and when the parameter R is set as 35, the performance of the improved LDBP is similar to that of DCNN. Figure 10(b) shows that the improved LDBP (R = 35) and DCNN (R = 40) have similar performance under different launch powers, confirming the rationality of the selected parameters. Similarly, we choose R = 36 as the parameter of the improved LDBP for PDM-64QAM. Based on the selected parameters in Fig. 10 and the unified expression in Table 2 of Section 3.4, it can be calculated that the complexity of the improved LDBP is reduced by 17.94% for PDM-16QAM compared with DCNN, and reduced by 16.35% for PDM-64QAM. Therefore, at the same performance, the computational complexity of the proposed scheme is lower than that of the DCNN scheme.

 

Fig. 10. Selection and validation of input parameter R in the improved LDBP and DCNN with the same performance for: (a) (b) PDM-16QAM signals; (c) (d) PDM-64QAM signals. The “?” in the label indicates that W of the improved LDBP is the variable.

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Table 1. Complexity analysis for the improved LDBP scheme

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Table 2. Complexity analysis of three different schemes for PDM-16/64QAM

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In order to explain the learning process and optimization results of the neural network, we take SPM and XPM effects at the optimal launch power of PDM-16QAM as an example to analyze the evolution of corresponding nonlinear parameters at different segments. The different segments mentioned here correspond to different layers of nonlinear compensation. Figure 11(a) shows the learning process of nonlinear parameters for SPM effects at different segments. Through the observation from the figure, we can see that the values of nonlinear parameters $g_{SPM}^{(i )}$ are smaller at the beginning and end segments, and larger at middle segments. The evolution curve shows an inverted “U” shape, which is consistent with Ref. [18]. However, the transmission model simulated by the VPI in this paper is opposite to the model in the reference, so their opening directions are opposite. In addition, we can see that the value of $g_{SPM}^{(i )}$ on the right is larger than that on the left, indicating that the dominant segment for nonlinear phase shift is close to the transmitter, which is the same conclusion with Ref. [20]. In Fig. 11(b), we visually demonstrate the correlation of neighboring samples within the same channel. The weight ${\delta _p}$ corresponding to each sample is not equal to 0. And they are symmetrically distributed on both sides with ${\delta _0}$ of the sample to be equalized as the center. This indicates that the correlation of neighboring samples exists and closely depends on the distance between samples. If the distance is the same, the degree of correlation is the same.

 

Fig. 11. Optimized parameters at different segments of the improved LDBP for SPM effects at the optimal launch power. (a) Nonlinear parameters $g_{SPM}^{(i )}$; (b) Correlation ${\delta _p}$ of neighboring samples in the same channel.

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Figure 12 shows the evolution of nonlinear parameters for XPM effects between any two neighboring channels at different segments. Assuming the channel of interest is labeled with k, it can be seen from Fig. 12(a) that the nonlinear parameter between the channel k-1 and channel k stabilizes in the third segment and the value is less than 0. Figure 12(b) shows the evolution of nonlinear parameter between channel k + 1 and channel k, which is opposite to the parameter in Fig. 12(a) and has a value greater than 0 after stabilizing in the third segment. Next, we will take Fig. 12(a) as an example to analyze the reason why the nonlinear parameter of XPM is less than 0. Since the wavelength of channel k-1 is greater than that of channel k, when transmitting the same distance, the time required for channel k-1 is longer, and the walk-off parameter between the two channels is positive. Therefore, the nonlinear parameters including the walk-off parameter should be less than 0 to correct the influence caused by the walk-off effect when the signal is transmitted in the reverse direction. Moreover, if we apply the known information that the walk-off parameter is positive to the theoretical model, the conclusion that the nonlinear parameter of XPM is less than 0 can also be obtained. This principle also explains the trend of the curve in Fig. 12(b).

 

Fig. 12. Optimized parameters at different segments of the improved LDBP for XPM effects at the optimal launch power. (a) Nonlinear parameters ${g_{XPM}}_{,({k - 1} )k}$; (b) Nonlinear parameters ${g_{XPM}}_{,k({k + 1} )}$.

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3.4. Complexity analysis

Complexity is a key index to evaluate the effectiveness of a scheme. We take the improved LDBP to equalize all samples of the target channel with two polarizations, as measured by the total number of real multiplications (RMs) required. Let's consider a WDM system with n channels. In general, the parameters and eigenvalues learned by neural networks are real values. However, complex signals are transmitted in WDM systems, which means that complex logic should be used in the neural network of the proposed scheme. The product of two complex numbers $a \times b = ({\textrm{Re} [a ]\textrm{Re} [b ]- {\mathop{\rm Im}\nolimits} [a ]{\mathop{\rm Im}\nolimits} [b ]} )+ j({\textrm{Re} [a ]{\mathop{\rm Im}\nolimits} [b ]+ {\mathop{\rm Im}\nolimits} [a ]\textrm{Re} [b ]} )$ is equivalent to four real multiplications and two real additions. The weights of CDC filters in the convolution layer are symmetric, which reduces the number of RMs required. For a complex sample, it needs 4(R + 1) RMs to complete the operation in convolution layer with the size of 2R + 1 [21]. From Eq. (6), we can see that the compensation for SPM in the nonlinear compensation layer is multiplied by the square of the signal amplitude and the real-valued parameters of filters. And the filter width required to compensate SPM for a sample is 2S + 1, so the computational complexity is (S + 1) RMs. The compensation of XPM requires both direct and inverse Fourier transforms, and there are $\frac{N}{2}({{{\log }_2}N - 2} )$ complex multiplications in a fast Fourier transform for the data with length N. Therefore, both direct and inverse Fourier transform operations require $2N({{{\log }_2}N - 2} )$RMs. The data after the Fourier transform of each channel is provided to the other channels to calculate the XPM nonlinear phase shift. In the proposed scheme, the required Fourier transforms for the target channel can be provided by calculation units for other channels without considering their complexity [20]. Taking into account the Fourier transform of the target channel and the inverse Fourier transform to transform XPM phase shift from frequency domain to time domain, there are $2N({{{\log }_2}N - 2} )\times 2$RMs required. In addition, the complexity of rough compensation for polarization-dependent impairment is 8N (T + 1). Table 1 lists the RMs required for each layer and the total complexity of the proposed scheme. According to optimized parameters in Section 3.2, the R, S, and T for PDM-16/64QAM are 40, 5, and 1, respectively. N_layer represents the number of neural network layers in the improved LDBP scheme. The number of layers of the neural network used to train PDM-16QAM in this scheme is 10, which is equivalent to 0.5 steps per span of the classical DBP, and for PDM-64QAM the number is 10, which is equivalent to 2 steps per span of the classical DBP.

Next, we analyze and compare the complexity of the proposed scheme with other typical schemes. As known from the previous discussion, whether it is the improved LDBP scheme or the DCNN scheme, the total computational complexity consists of three parts, which are linear layer, nonlinear layer and polarization correlation filter. Thereinto, the complexity of the linear layer and rough compensation for polarization-dependent impairment in DCNN are same with the improved LDBP, are 4(R + 1)N × 2 RMs and 8N(T + 1) RMs, respectively. For the nonlinear layer, the complexity of DCNN is 2Nn(S + 2) + 18N RMs and improved LDBP is (4n + S + 17)N + 2N(log₂N-2) × 2 RMs. And for DBP scheme, we refer to the calculation method of complexity for DBP in Refs. [16,19,30] and obtain its total complexity as 8NN_spanN_step(log₂N + 2) RMs, where N_span represents the number of fiber spans, and N_step represents the number of steps per span. In Table 2, the complexity of the three schemes is summarized. And the unified expression formula in the second column corresponds to each scheme in the first column. The number of real multiplications required for PDM-16/64QAM corresponding to the different schemes is given in the third column, respectively. As can be seen from Fig. 7 and Table 2, the performance of the proposed scheme with 0.5 steps per span for PDM-16QAM is higher than that of DBP-5StPS, but the complexity is only 31.36% of that of DBP-5StPS. And for PDM-64QAM, the performance of the proposed scheme with 2 steps per span is higher than DBP-10StPS and the complexity is 62.72% of DBP-10StPS. The complexity of the two modulation formats in this scheme is 90.03% of that of DCNN, and the performance is improved.

4. Experimental system and result analysis

4.1. Description of the experimental system

To further verify the proposed scheme, we build a 28 GBaud PDM-16QAM 806.4 km 5-channel WDM experimental transmission system, as shown in Fig. 13. At the transmitter, continuous waves emitted by five external cavity lasers (ECLs) with center frequencies of 193.3 THz, 193.35 THz, 193.4 THz, 193.45 THz, and 193.5 THz are multiplexed by a multiplexer. The frequency offset and linewidth of the laser are approximately 100 MHz and 100 KHz. As in the simulation, the symbol length is 65536. We use Matlab built-in function to construct random bit sequence, and generate signals from 65 GSa/s arbitrary waveform generator (AWG, Keysight M8195A). The pulse waveform is shaped by a RRC filter with the roll-off factor of 0.1, and then modulated with an IQ modulator. Each transmission span of the fiber link is composed of 100.8 km/span SSMF, EDFA with the noise figure of 6.5 dB, optical band-pass filter (OBPF) and optical loop controller. Moreover, the signal needs to pass through an EDFA and variable optical attenuator (VOA) to adjust to the appropriate optical power before entering the fiber link. The attenuation coefficient, CD coefficient, nonlinearity coefficient, and PMD coefficient of the fiber are 0.19 dB/km, 16.7 ps/nm/km, 1.27 W^-1/km and 0.2 ${{\textrm{ps}} / {\sqrt {\textrm{km}} }}$, respectively. Prior to entering the receiver, the 5-channel multiplexed waveforms enter the demultiplexer for demultiplexing based on different center frequencies, so that the coherent receiver can achieve the reception of the single for each channel individually. On the receiver side, the received optical signal is sampled by a real-time oscilloscope with the sampling rate of 80 GSa/s to obtain discrete signals, and then processed by offline DSP.

 

Fig. 13. 5-channel WDM experimental system. ECL: external cavity laser. PBS: polarization beam splitter. PBC: polarization beam combiner. AWG: arbitrary waveform generator. MUX: multiplexer. SSMF: standard single-mode fiber. EDFA: erbium-doped fiber amplifiers. VOA: variable optical attenuator. OBPF: optical band-pass filter. DEMUX: demultiplexer.

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4.2. Analysis of experimental results

We adopt the same method as Section 3.2 to optimize the step size of DBP, the number of hidden layers and the three initial parameters of CDC filter width W = 2R + 1, polarization-dependent filter width Z = 2T + 1 and SPM filter width M = 2S + 1 in the WDM experimental system. According to the results of linear compensation, parameters are optimized for 28 GBaud PDM-16QAM with the launch power of 0 dBm. Figure 14 shows the optimization curves of CDC filters and polarization-dependent filter for PDM-16QAM. In Fig. 14(a), the Q-factor tends to be stable after the filter width W = 61 for CDC, and for the filter that compensates polarization-dependent interference, in Fig. 14(b) the Q-factor fluctuates very little after Z = 3. Therefore, we choose W = 61 and Z = 3 as the optimal parameter for neural network through comprehensive consideration. The relationship between the performance of the SPM compensation improved by ESSFM and the number of neighboring samples is shown in Fig. 15. After the number of neighboring samples M = 9, the Q-factor tends to be stable and the complexity continues to increase. Therefore, in the case of comprehensive consideration of performance and complexity, we choose M = 9 as the optimal filter width for neural network. As for the optimization of the number of hidden layers and DBP step size, we can still learn from the methods used by the simulation system, and according to the trade-off between performance and complexity in Fig. 16, we choose the number of hidden layers equal to 4 as the optimal parameter for PDM-16QAM in the WDM experimental system. In addition, we try DBP with 1, 2, 3, 5, 8, and 10 steps per span and compare them with linear compensation schemes in the experiment system, and obtain optimization curves of DBP with different StPS for PDM-16QAM, as shown in Fig. 17. Based on the trade-off between performance and complexity corresponding to different StPS of DBP, we choose DBP-5StPS to compare performance with the improved LDBP later.

 

Fig. 14. Different filter widths and Q-factors for PDM-16QAM in the WDM experimental system. (a) CDC filter width; (b) Polarization-dependent filter width.

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Fig. 15. Performance and complexity of different SPM filter widths for PDM-16QAM in the WDM experimental system.

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Fig. 16. Performance and complexity of different number of hidden layers for PDM-16QAM in the experiment system.

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Fig. 17. The optimization curves of DBP with different StPS for PDM-16QAM in the experiment system. (Notes: The complexity of the linear compensation scheme is relatively low, and the unit value of the vertical axis is large due to the complexity of the DBP that needs to be fully displayed, so the complexity of the linear compensation scheme seems to be almost zero.).

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Based on the above optimized parameters, we proceed to verify the effectiveness of the improved LDBP scheme in the experimental system, as shown in Fig. 18. Compared with the linear compensation scheme, the optimal launch power of the proposed scheme is increased from -1 dBm to 0 dBm. The SNR is improved by about 3.5 dB and the Q-factor gain is about 0.86 dB under the optimal launch power. Under the scenario of 28 GBaud PDM-16QAM 806.4 km, the range of launch power for the target channel with Q-factor meeting the 7% FEC threshold is -1.5 dBm to 1.8 dBm. Under the same conditions, the improved LDBP scheme only needs 0.5 steps per span to reach and even exceed the performance of DBP-5StPS, and the complexity is only 23.68% of DBP-5StPS. It is worth noting that although they are all compared with DBP-5StPS, the complexity of the simulation is higher than that of the experiment due to the different parameter settings of the simulation and experimental scenarios.

 

Fig. 18. Performance curves of different schemes for PDM-16QAM signals in the WDM experimental system.

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5. Conclusion

We propose a joint nonlinear compensation scheme based on improved LDBP for WDM systems. By optimizing the physical model of signal transmission, the scheme achieves alternating compensation of CD and nonlinear effects. The intra and inter-channel nonlinear compensation in WDM systems can be realized without the specific parameters. In this scheme, linear filters are used to compensate CD in the time domain, and the corresponding network structure is designed as one-dimensional convolution. And in the convolution layer, the pulse-broadening effect is properly introduced into the overlap-and-save method. The physical model of nonlinear compensation is improved by considering the interference of pulse broadening in the channel and walk-off effects between channels, so as to achieve the purpose of better compensation for intra-channel SPM and inter-channel XPM. The effectiveness of the proposed scheme is verified by an 11-channel WDM simulation system of 36 GBaud PDM-16QAM 1600 km and 64 GBaud PDM-64QAM 400 km, as well as a 5-channel WDM experimental system of 28 GBaud PDM-16QAM 806.4 km. The simulation and experimental results show that the optimal launch power of this scheme can be increased by 1 dB, compared to the linear compensation scheme. The performance of the improved LDBP is higher than that of DBP-5StPS for PDM-16QAM while reducing the number of steps. At the same time, the performance of PDM-64QAM is higher than DBP-10StPS, and both the two modulation formats have lower computational complexity. Taking the FEC threshold as a measurement standard, this scheme significantly extends the effective transmission distance. The improved LDBP scheme exhaustively leverages the inherent random interaction between CD and nonlinearity in optical fiber communication systems to compensate the phase shift caused by nonlinear effects. It is an effective method to equalize the nonlinear distortion of signals and control computational complexity, and it is expected to have good application prospects in large-capacity WDM systems for long-haul transmission.