Open Access
Add to BibSonomy
Abstract
A novel joint intra and inter-channel nonlinearity compensation method is proposed, which is based on interpretable neural network (NN). For the first time, conventional cascaded digital back-propagation (DBP) and nonlinear polarization crosstalk canceller (NPCC) are deep unfolded into an NN architecture together based on their physical meanings. Verified by extensive simulations of 7-channel 20-GBaud DP-16QAM 3200-km coherent optical transmission, deep-unfolded DBP-NPCC (DU-DBP-NPCC) achieves 1 dB and 0.36 dB Q factor improvement at the launch power of −1 dBm/channel compared with chromatic dispersion compensation (CDC) and cascaded DBP-NPCC, respectively. Under the bit error rate threshold of 2 × 10−2, DU-DBP-NPCC extends the maximum transmission reach by 28% (700 km) compared with CDC. Besides, 3 different training schemes of DU-DBP-NPCC are investigated, implying the effective signal-to-noise ratio is not the proper evaluation metric of nonlinearity compensation performance for DU-DBP-NPCC. Moreover, DU-DBP-NPCC costs 26% lower computational complexity compared with DBP-NPCC, providing a better choice for joint intra and inter-channel nonlinearity compensation in long-haul coherent systems.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
With the enormous growth of the internet traffic data flow, optical fiber communication networks are facing an unprecedented challenge for extending the upper bound of capacity and transmission distance even further [1]. It is acknowledged that linear distortions can be equalized by means of advanced digital signal processing (DSP) algorithms with satisfactory performance improvement. However, since the optical fiber communication system is not a pure linear system, the nonlinearity of optical fiber also causes serious signal distortions, becoming the most significant factor limiting the capacity and maximum transmission distance of optical communication systems [2].
In order to mitigate nonlinear distortions, several nonlinear compensation algorithms have been proposed and proved to be very effective. Digital back-propagation (DBP) via split-step Fourier method (SSFM) plays a pivotal role in nonlinear distortion compensation. The signal with linear and nonlinear distortions is re-propagated through a virtual fiber by simply inverting the channel parameters to compensate for chromatic dispersion (CD) and self-phase modulation (SPM) [3–5]. The performance of this method improves with the increase of steps per span (StPS), which means superior performance requires more complexity, becoming the main obstacle for actual realization. Although high StPS DBP requires unacceptable complexity, it has an effective equalization performance, so it always serves as a benchmark for nonlinear compensation. Besides DBP, nonlinear compensation methods based on Volterra series transfer functions (VSTF) [6,7] and perturbation-based pre-distortion [8,9] have also been proved to be effective. It has been verified in [6] that VSTF based backpropagation extends the fiber quasi-linear regime by approximately 2 dB compared to the SSFM-based DBP. However, this method acquires $O(N^{2})$ complexity and still needs to face the major challenge of high complexity when the FFT block length is larger due to the increased accumulated chromatic dispersion. Though a simplified Volterra series nonlinear equalizer has been proposed in [7] and acquires lower complexity even compared with SSFM-based DBP, there still exists a trade-off between the frequency-flat approximation error and the incomplete kernel representation error, which inevitably limits its performance. Besides, the perturbation-based pre-distortion nonlinear equalization method [8,9] has also been shown effective which can be implemented under the condition of one sample per symbol. However, this kind of method adds computational and implementation complexity to the transmitter and requires higher accuracy for quantization.
Besides, in wavelength-division-multiplexed (WDM) systems, with inter-channel nonlinearity, just performing single-channel DBP will have a limited improvement with the lack of other channels’ information. Although using multi-channel DBP will be beneficial, in most cases, the knowledge of the neighbouring channels at receiver is unknown and there exist unpredictable add-and-drop operations in the dynamic elastic optical network, making the implementation of multi-channel DBP impossible [10]. Hence, the complete elimination of inter-channel nonlinearity is considered as an impractical mission and the inter-channel nonlinerity is always treated as a noise [10]. In [11], inter-channel nonlinearity is simply divided into two kinds of distortions: cross-phase modulation-induced (XPM-induced) nonlinear phase noise (NPN) and XPM-induced nonlinear polarization cross-talk (NPC). During the traditional cascaded DSP chains, most NPN can be cancelled by properly tuning the parameters of carrier phase recovery (CPR) algorithms or by pilot tone-based mitigation methods [12]. For NPC mitigation, derived from the modeling theory in [11], a method called nonlinear polarization crosstalk canceller (NPCC) [13] has been proposed, which is hardware implementation-friendly and provides obvious performance improvement for a 40-channel 112 Gb/s dual-polarization QPSK system. The methods proposed in [14,15] aimed to further improve the compensation performance of NPC by introducing generalized maximum likelihood (GML) into the NPC estimation [14] and decision-feedback into NPCC [15]. Although these methods contributed highly interpretable solutions for XPM and obtained obvious performance improvement compared to linear compensation only, they mostly focus on the XPM itself while the intra-channel nonlinearity compensation may have an obvious effect on their performance [13]. Besides, methods based on extended Kalman smoother [16] have also been proposed to mitigate NPN and NPC by introducing the time-varying intersymbol interference (ISI) model of inter-channel nonlinear interference (NLIN) and utilizing the Turbo equalization and extended Kalman smoothing (EKS) algorithm. Although the gain it provides is higher than any intra-channel nonlinearity mitigation scheme, the computational complexity is expensive and increases quadratically with the equalization order. Therefore, there is still room for the study of these inter-channel nonlinearity compensation methods to further improve their nonlinearity compensation capability or reduce the computational complexity.
Recently, with universal approximation properties [17], neural network (NN) has ignited massive applications both in industry and academia [18]. For optical fiber communication systems, NN has also been investigated to be applied in different tasks [19], especially the nonlinearity compensation. However, most NN-based nonlinear equalization methods focus on the performance themselves, neglecting the interpretability of the models. To solve this problem, guided by the deep unfolding theory in [20,21] which establishes the relationship between the iterative algorithm and NN architecture, a learned DBP (LDBP) has been proposed in [22] which converts each step of DBP into a layer of NN and jointly optimized all the taps of CD compensation filters and nonlinear coefficients of each step. This method is highly clear in physical meanings compared with "black-box" NN-based nonlinear equalizers, with higher performance and lower complexity compared with DBP, bringing the nonlinearity mitigation into a new level. Authors in [23–25] also further investigated LDBP from hardware implementation to mathematical insights. Additionally, a deep-unfolded version of multi-channel DBP has also been proposed in [26]. If this method can be modified to jointly mitigate intra and inter-channel nonlinearity with knowing the channel of interest only, it will be a great step in nonlinearity compensation.
In this paper, we propose an interpretable NN-based joint intra and inter-channel nonlinearity compensation method by deep unfolding the overall conventional cascaded DPB-NPCC structure into an NN, which is called deep-unfolded DBP-NPCC (DU-DBP-NPCC). The DBP section of DU-DBP-NPCC is similar to the conventional DBP, with alternating CDC and nonlinear compensation but both with trainable parameters, which is the same as the structure of LDBP. The NPCC section of DU-DBP-NPCC is similar to the conventional NPCC but with a different data structure and a trainable option. Numerical results of 7-channel 20-GBaud dual-polarization 16QAM (DP-16QAM) 3200-km coherent optical transmission simulations show that, at the launch power of −1 dBm/channel, DU-DBP-NPCC provides 1 dB and 0.36 dB Q factor gain compared with chromatic dispersion compensation (CDC) and DBP-NPCC, respectively. Under the bit error rate (BER) threshold of $2\times 10^{-2}$, DU-DBP-NPCC extends the maximum reach by 28% (700 km) and 7% (210 km) compared with CDC and DBP-NPCC, respectively. What’s more, we investigated 3 different training schemes of DU-DBP-NPCC. The results imply that it’s not proper to train the smoothing filter in NPCC layer directly in DU-DBP-NPCC, which will destroy the averaging operation in NPCC and make the effective signal-to-noise ratio (SNR) not a proper evaluation metric of nonlinearity compensation performance for DU-DBP-NPCC. The complexity of the proposed method is also analyzed, verifying that the proposed method has 26% lower complexity compared with DBP-NPCC, providing a promising choice for joint intra and inter-channel nonlinearity mitigation for future long-haul coherent optical systems.
2. Proposed joint intra and inter channel nonlinearity equalizer
2.1 Brief principle of the theoretical model
Based on the Manakov equation, the WDM optical signal characteristics can be analysed using a simple fiber model for determination of XPM effects [11], which is in the following form when taking 2 channels in one span into consideration:
The first, second and third term in the right side of (3) and (4) represents SPM, XPM-induced nonlinear phase noise (NPN) and XPM-induced nonlinear polarization crosstalk (NPC), respectively. As is mentioned above, SPM can be effectively compensated by DBP while XPM is difficult to mitigate. According to Volterra expansion theory, the XPM impact, including both the NPN and NPC, can be modeled as a time-varying Jones matrix [11]
Assuming that all the other linear distortions are neglected and an ideal CPR is performed, the XPM Jones matrix then degenerates and the signals interfered by NPC can be modeled as
2.2 Principle of conventional NPCC and DU-DBP-NPCC
The block diagram of conventional NPCC is shown in Fig. 1. Since the nonlinear NPC weights $|w_{yx/xy}(t)|^2<<1$, Eq. (10) can be further simplified to
Then, the NPC weights $w_{yx/xy}(t)$ can be easily calculated with $w_{yx}(t)=(R_{x}-T_{x})/T_{y}$ and $w_{xy}(t)=(R_{y}-T_{y})/T_{x}$. However, it is impossible to acquire the exact transmitted signal $T_{x/y}$ at the receiver side. Hence, the hard decision result of $R_{x/y}$ is adopted to be the approximate $T_{x/y}$. After NPC weights calculation, an averaging operation is performed, which is to mitigate the influence of hard decision error induced by amplified spontaneous emission (ASE) and residual nonlinear noise. The averaging window length depends on the strength of ASE noise and the nonlinearity, which need to be optimized. Finally, after averaging operation, we get the NPC weights $w_{yx/xy}(t)$ and the nonlinear polarization crosstalk cancelling is performed by $R_{x}'=R_{x}-R_{y}w_{yx}(t)$ and $R_{y}'=R_{y}-R_{x}w_{xy}(t)$ where $R_{x/y}'$ denotes the signal after NPCC. The block diagram in Fig. 1 depicts that NPCC is a simple feed-forward algorithm without any feedback circle and any iteration process, which means NPCC is easy to be implemented and cascaded after other conventional equalization algorithms.
According to the above model, SPM can be equalized using DBP and NPC can be partly cancelled by the conventional NPCC. A simple cascaded structure of the above two algorithms can be adopted to jointly mitigate intra and inter-channel nonlinearity. When DBP is cascaded with NPCC, the whole part of this structure, the cascaded DBP-NPCC, can be deep unfolded since NPCC can be generally considered as one fixed iteration of the whole cascaded algorithm structure just with a different function from DBP. The essential structure of DU-DBP-NPCC is actually the deep-unfolded version of cascaded DBP-NPCC which is shown in Fig. 2. The DBP section of DU-DBP-NPCC is similar to conventional DBP, with alternating CDC and nonlinear compensation but both with trainable parameters, which is the same as the structure of LDBP [22]. The NPCC section of DU-DBP-NPCC is similar to conventional NPCC but with a different data structure and a trainable option. Hard decision is performed to obtain the $T_{x/y}$ in Eq. (11); NPC is calculated with $W_{yx}=(R_{x}-T_{x})/T_{y}$ and $W_{xy}=(R_{y}-T_{y})/T_{x}$. After averaging operation, the data after NPCC is obtained by $R_{x}'=R_{x}-R_{y}W_{yx}$ and $R_{y}'=R_{y}-R_{x}W_{xy}$. Since the data structure of DU-DBP-NPCC is redesigned to take full advantage of the tensors, reshaped to a 3-dimensional (3D) tensor with the structure (batch size, data length, 4) where 4 represents the real and imaginary part of x and y polarization, $w_{yx/xy}(t)$ in conventional NPCC is also reshaped to a 3D matrix $W_{yx/xy}$ ($(t)$ is omitted) and this is the first difference between conventional NPCC. Moreover, inspired by smoothing filters in image processing, the averaging operation in the conventional NPCC is converted to a convolution operation with a 3D smoothing filter. The taps of the 3D smoothing filter are optional to be trainable or not and this is the second difference between conventional NPCC. Since the real and imaginary parts of data are separated, the calculation during the NPCC section is a little different from conventional NPCC which is only an engineering realization problem and has been explained briefly in Fig. 3 and its caption. During the training stage, the DBP section and NPCC section are optimized together, which means they should be considered as a whole part rather than two separated parts. Hence, considering the proposed method is modified based on cascaded DBP-NPCC, we name this proposed algorithm as "DU-DBP-NPCC" and the deep unfolding procedure is applied on the overall cascaded DBP-NPCC algorithm, not on NPCC itself.
Fig. 2. The entire NN structure of DU-DBP-NPCC. $\alpha$: fiber attenuation. $h$: taps of CDC filter. $\gamma$: fiber nonlinear coefficient. Circles with different colors represent the neurons with different functions. Dark gray: adjust the power of data. Orange: convey the data to the next layer. Light gray: 1-dimensional convolution, perform CDC (3 circles) or matched filter and down sampling (2 circles). Purple: activate data with a nonlinear activation function. Cyan-blue with "W": calculate the NPC weights, corresponding to $w_{yx/xy}(t)$. Orange with "R": convey the received signal to the next layer, corresponding to $R_{x/y}$. Red: perform NPCC.
Fig. 3. The NPCC section of DU-DBP-NPCC. $R_{x/y}$: the received signal of x/y-polarization after the mitigation of intra-channel nonlinearity, $T_{x/y}$: the signal of x/y-polarization after hard decision, $R_xI$: the in-phase component of $R_x$, $R_xQ$: the quadrature component of $R_x$, the same for $R_yI$, $R_yQ$, $T_xI$, $T_xQ$, $T_yI$ and $T_yQ$. $W_{yx/xy}$: the 3D matrix of $w_{yx/xy}$, $w_{yx}R$: the real part of $w_{yx}$, $w_{yx}I$: the imaginary part of $w_{yx}$. The calculation of real and imaginary part of $w_{yx}$ is shown in the upper right corner and the calculation of $w_{xy}$ is omitted. $N_t$: the averaging length of NPCC.
Since NPCC can be generally considered as one fixed iteration of the whole cascaded algorithm, it’s natural to consider that whether the performance of the cascaded algorithm can be further improved by training the parameters in NPCC. We have investigated 3 different training schemes for NPCC layer: scheme 1 (S1): a non-trainable smoothing filter in NPCC section; scheme 2 (S2): a trainable smoothing filter in NPCC section; scheme 3 (S3): a non-trainable smoothing filter and two trainable linear filters for 3D matrix $W_{yx/xy}$ in NPCC section. A simple diagram is shown in Fig. 4 in order to distinguish these three methods more intuitively. The results of the above 3 training schemes will be discussed in section 4.
3. System setup
3.1 Transmission simulation
The simulation setup is depicted in Fig. 5. We set up a 7-channel 20-GBaud DP-16QAM WDM system with 50 GHz channel spacing using root-raised cosine pulses (0.1 roll-off factor). The probe channel is centered at 1550 nm with no linewidth added to the lasers in order to neglect the effect of carrier phase noise. Each transmission span includes one 100-km single mode fiber with $\alpha$ = 0.2 dB/km, $D$ = 17.01 ps/nm/km and $\gamma$ = 1.4 /W/km as well as an erbium-doped fiber amplifier (EDFA) working at gain controlled mode with a 5-dB noise figure. The bandwidth of the OBPF is set to 50 GHz and an ideal EDFA without noise is used to control the optical power before coherent receiver. After coherent receiving, signals with 2 samples/symbol are processed offline. The offline DSP functions include two technical routes: the first one is the traditional cascaded DSP algorithms, which includes frequency DBP with 1 step per span (StPS), down sample, ideal phase rotation, NPCC and data recovery; the second one is the proposed DU-DBP-NPCC which is equivalent to the corresponding function of the first route but with an optimization process. In order to make the training process of DU-DBP-NPCC more clear, the details of the training setup will be elaborated in the following subsection.
Fig. 5. The diagram of a 7-channel 20-GBaud DP-16QAM WDM system. PBS: polarization beam splitter. PBC: polarization beam combiner. OBPF: optical band-pass filter.
3.2 Training setup of DU-DBP-NPCC
Training process is performed using 65536 random symbols (131072 samples) generated in MATLAB. According to [27], for a 3200-km transmission, the overlap of CD compensation should be at least 512 samples. Hence, we put 512 neighboring samples to the head and tail of a 64 samples block of interest, forming the final (1519, 1088, 4) 3D training data set (75% of data set) and (495, 1088, 4) 3D validation data set (25% of data set). The validation data set is utilized to guarantee the network having the best performance on new data, avoiding overfitting problems. The initial values of CD compensation filter taps are set to the values of the inverse discrete Fourier transform of the frequency CD compensation operator using a rectangle window to cut off the redundant taps. According to the physical properties, taps of the filter designed for CD compensation should be symmetric. Hence, we set symmetric limitation to the trainable filters, making the filters after training also meets the physical properties. Since the filter taps are complex values, in order to perform convolution with 3D data, we separated the real and imaginary part of filter taps, forming the final ($K$,4,4) 3D CDC filters where $K$ denotes the number of filter taps. Nonlinear coefficient $\gamma$ of each layer is set to the ideal value 1.4 /W/km. As is mentioned above, the taps of 3D smoothing filter in NPCC section is optional to be trainable and the initial value of it is set to $1/N_{t}$ as is depicted in Fig. 3 where $N_{t}$ denotes the NPCC averaging length. The linear filter in S3 is set to an impulse function initially. Besides, the learning rate is set to 0.001 in order to guarantee a stable training performance of the NN. During the training stage, the batches are randomly shuffled also in order to avoid overfitting problems. Another data set of 65536 symbols generated by a different random seed is set as the testing set, evaluating the performance of the trained model. Necessary parameters needed for training are summarized in Table 1.
Table 1. The main parameters of the training stage.
Moreover, we also explored the relationship between the taps of NPCC and the performance of DU-DBP-NPCC S1 which is shown in Fig. 6(a). It is unfavorable to the performance of both DBP-NPCC and DU-DBP-NPCC when NPCC taps are too short. For DBP-NPCC, the gain of Q factor tends to be converged above 25 taps while for DU-DBP-NPCC 55 taps and above are needed. Hence, we set $N_{t}$ to 55 in order to obtain a stable performance both in DBP-NPCC and DU-DBP-NPCC. Besides, we made a further study of the training process by observing the change of the Q factor and loss within 10 epochs under a 3000-km transmission. As is shown in Fig. 6(b), after only about 2 epochs, the performance tends to be converged. This fast converged characteristic can be attributed to the mini-batch gradient descent, saving plenty of time when verifying the effectiveness of the adjustment of parameters. However, it should be pointed out that there still exists a little gain after the next several thousands of epochs. We found that with more training epochs, the gain of Q factor will also increases but with a lower and lower speed. Hence, we made a trade off between the training time and the performance of DU-DBP-NPCC and finally fixed the epoch number to 10000.
Fig. 6. (a) Gain of Q factor as a function of the number of NPCC taps. (b) The loss and corresponding Q factor as a function of the epoch.
4. Results and discussions
4.1 Performance of DU-DBP-NPCC with different training schemes
First, we scanned the launch power per channel from −3 dBm to 1 dBm for 3200 km long-haul transmissions. The Q factor performance of CDC, DBP-NPCC (1StPS), DBP-NPCC (50 StPS), DU-DBP-NPCC S1 and the corresponding constellation diagrams are shown in Fig. 7(a) and (b), respectively. Q factor is defined as $Q=20log_{10}[\sqrt {2}erfc^{-1}(2BER)]$ where $erfc^{-1}$ represents the inverse error function complement. It should be noted that if there are no special instructions, the DBP-NPCC and DU-DBP-NPCC in this article are implemented under 1-StPS DBP precision with $K=13$ and $N_t=55$. The blue dash line shows the performance of cascaded DBP-NPCC with a higher precision of DBP (50-StPS), which serves as a benchmark. Compared with CDC under the launch power of −1 dBm per channel, DU-DBP-NPCC S1 obtained 1-dB Q factor gain and performs similar to the high computational 50 StPS cascaded DBP-NPCC. What’s more, DU-DBP-NPCC S1 also brings another 0.36 dB Q factor improvement compared with cascaded DBP-NPCC. When comparing the performances of different methods under the optimal launch power, DU-DBP-NPCC S1 still outperforms over CDC with 0.81 dB Q factor improvement. The constellation diagrams in Fig. 3(b) also depict that DU-DBP-NPCC S1 performs better over CDC and 1 StPS cascaded DBP-NPCC with the clearest display of constellation. Besides, in order to make the performance improvement of the proposed algorithm more intuitive, we scanned the transmission distance at the launch power of −1 dBm/channel to obtain the maximum transmission distance at the BER threshold of $2\times 10^{-2}$. The proposed DU-DBP-NPCC S1 increases the maximum transmission distance from about 2500 km to about 3200 km, which is 28% extension compared with CDC. Moreover, DU-DBP-NPCC S1 also brings another 210 km extension compared with the cascaded DBP-NPCC, which is about 7% improvement. Above all, the proposed DU-DBP-NPCC S1 contributes a lot to further improving the transmission distance for long-haul coherent optical systems.
Fig. 7. (a) Performance of different equalization methods. (b) The constellation diagram of different equalization methods.
Next, we briefly analyze the evolution of the parameters trained in the DBP section. The combined amplitude and phase response of the CDC filter is shown in Fig. 8(a). There exist ripples in the initial amplitude response since the initial weights are generated using a rectangle window as mentioned above. After 10000 epochs training, a low-pass ("M"-shaped) characteristic is exhibited, which is already reported in previous works [24,25]. The phase response is almost the same during the training process. This phenomenon shows that in addition to CD, the CDC filter learns to mitigate the effect of out-of-band noise during the nonlinearity compensation process. Moreover, Fig. 8(b) depicts that the nonlinear coefficient $\gamma$ shows a similar "U" shape during the training process which is also pointed out in [24]. Since all these phenomenons have already been well studied, we will focus on the training schemes in the NPCC section.
Fig. 8. The evolution of (a) the combined amplitude and phase response of the CDC filter and (b) the nonlinear coefficient $\gamma$.
Then, we found that the effective SNR, which is defined as $\|\mathbf {T}\| / \| \mathbf {T}-\hat {\mathbf {T}}\|$ where $\mathbf {T}$ denotes the transmitted symbol and $\hat {\mathbf {T}}$ denotes the recovered symbol after DSP, is not a proper metric to evaluate the performance of DU-DBP-NPCC with a certain training scheme (S2), since the decision process in trainable NPCC will lead to a supreme effective SNR performance while the Q factor actually drops. In Fig. 9(a) and (b), it is obvious that S2 shows the best effective SNR performance and the best constellation display, while in Fig. 9(c), S2 shows a worse Q factor performance compared to S1, S3, and even LDBP alone. Figure 10(a) depicts the smoothing filter trained by S2. After training, the smoothing filter taps have changed from a flat shape to an impulse-like shape, which implies the averaging operation has been destroyed after the training process in S2 and the smoothing filter after trained performs like a impulse function which has only little impact on the 3D matrix $W_{yx/xy}$. In other words, NPCC in S2 nearly degenerated to hard decision. This phenomenon implies that it’s not a good idea to train the smoothing filter in the NPCC layer directly in DU-DBP-NPCC, which will destroy the averaging operation in NPCC, affect the optimization of the parameters in DBP and even lead to a worse performance than LDBP alone.
Fig. 9. (a) The effective SNR performance of different equalization methods. (b) The constellation diagrams after DU-DBP-NPCC with S1, S2, and S3 training schemes under the launch power of −1 dBm/channel. (c) The Q factor performance of different equalization methods.
Fig. 10. (a) The magnitude of smoothing filter before and after training. (b) The magnitude of linear filters before and after training.
Moreover, we found that DU-DBP-NPCC with non-trainable NPCC layer performs the best. Although we also tried training two linear filters (S3) for 3D matrix $W_{yx/xy}$ inspired by [15], the nonlinearity equalization performance under this training scheme is almost the same as the one of DU-DBP-NPCC with a non-trainable NPCC layer (S1). We also found the shapes of linear filters after trained, as shown in Fig. 10(b), have changed in order to consider the neighboring interference. The reason why S3 has almost the same Q factor performance compared to S1 is that, since DU-DBP-NPCC is an optimized version of cascaded DBP-NPCC and can be considered as a whole part, whether to train the NPCC parameters or not is almost the same for the nonlinearity equalization as long as the whole part of this cascaded structure reaches its optimal performance. From the filter structure aspect, the CDC filters and the linear filters in S3 are the same, which implies adding linear filters or not has no obvious impact on the whole optimization process since the response of the added linear filters can be integrated into the previous CDC filters in S1. It should be noted that, in our proposed method, DBP and NPCC are trained together rather than training NPCC alone like [15]. Hence, the added linear filters are optional due to the overall optimization process. Moreover, since the performance of S1 is almost the same as S3, adding redundant trainable parameters is not necessary. Therefore, our conclusion is that DU-DBP-NPCC with a non-trainable NPCC layer is a better choice since it has a little higher performance and lower complexity compared with DU-DBP-NPCC with a trainable NPCC layer. What’s more, DU-DBP-NPCC S1 still has a little more advantage over LDBP alone and even over simply cascaded LDBP-NPCC, with another 0.1 dB and 0.05 dB Q factor improvement at the optimal input power per channel, respectively.
4.2 Complexity analysis
It’s a common knowledge that during the training stage of NN, it requires massive computing resources. Since the training can be done offline before deployment, the complexity of training stage is always omitted in many studies. However, we convince that it is necessary to analyse the complexity of the training process, which significantly helps us to choose proper parameters before training and save unnecessary training trials. There already exists several complexity analysis using real multiplications as a metric [3,25]. However, in this paper, we calculated the complexity of the proposed DU-DBP-NPCC S1 at both the training stage and testing stage using complex multiplications (CMs) per symbol as a surrogate in order to analyze the proposed algorithm more comprehensively. It should be noted that DU-DBP-NPCC mentioned in this subsection refers to DU-DBP-NPCC S1.
Assuming that $2N$ samples of a complex signal of two polarizations with a sampling rate of twice the symbol rate are cached to be equalized, then for 1StPS-DBP, the frequency-domain CD compensation requires $4(log_2(N)+1)$ CMs/symbol/span when symmetric SSFM is adopted. Assuming that the nonlinear exponential function is implemented with a look-up table, then the nonlinear step of DBP requires 4 CMs/symbol/span. The calculation of NPC for 1 symbol requires 1 complex division and the canceller requires another 1 complex multiplication, which is equivalent to 3 CMs/symbol. Hence, the traditional cascaded DBP-NPCC requires
CMs/symbol where $N_{sp}$ denotes the number of spans.
To simplify the derivation, here we assume that the batch size is set to 1. Let $L$ denote the length of overlap, $M$ denote the length of samples of interest in each batch and $K$ denote the filter taps, then the overall batch number is $N_{bt}=(N-2L)/M$. For one batch, the CD compensation process requires $4K(2L+M)/N$ CMs/symbol/span and the nonlinear step requires $4(2L+M)/N$ CMs/symbol/span. Redesigned NPCC requires $3(N-2L)(2L+M)/M/N$ CMs/symbol. Hence, without the consideration of the optimization process and only focus on one training epoch, DU-DBP-NPCC at the training stage requires
CMs/symbol. For training stage, $N$ is always fixed and the ratio between $C_{train}$ and $C_{DBP-NPCC}$ as a function of $M$ and $K$ is shown in Fig. 11(a). It is obvious that the training process has inevitably high complexity even in one training epoch, which means the complexity will be extremely high with tens of thousand optimization iterations. That’s the common knowledge of NN which requires massive computing resource at the training stage. However, Fig. 11(a) is still meaningful because it helps us to choose proper parameters before training. As is shown in Fig. 11(a), $K$ makes more contribution to the complexity than $M$, giving us more intuitive choices when tuning the NN in order to reduce the training time.
Fig. 11. (a) The ratio between $C_{train}$ and $C_{DBP-NPCC}$. (b) The ratio between $C_{test}$ and $C_{DBP-NPCC}$.
However, as is mentioned above, the training process is always done offline prior to deployment. Once the training finished, we can pull out all the trained parameters and perform the time-domain DBP and NPCC which is equivalent to the NN form. Hence, the complexity of testing process is equivalent to the one of time-domain DBP cascaded with NPCC, which is
CMs/symbol. In order to make the complexity comparison more clearly, Fig. 11(b) also depicts the ratio between the complexity of the DBP-NPCC and DU-DBP-NPCC at the testing stage when $L$ is fixed to 512 and $N_{sp}$ is fixed to 32. When $K$ is within 17, the testing stage of DU-DBP-NPCC will be simpler than cascaded DBP-NPCC. For the parameters adopted in our simulations, with $N_{sp}=32$, $N=131072$, $M=64$ and $K=13$, the ratio mentioned above is only about 0.74, which means DU-DBP-NPCC is about 26% less complex than DBP-NPCC.
Above all, it can be concluded that the proposed DU-DBP-NPCC is easy and fast to train and low-cost to deploy, which shows great potential to jointly mitigate intra and inter-channel nonlinarity in the future long-haul coherent optical systems.
5. Conclusion
DU-DBP-NPCC has been proposed to jointly equalize intra and inter-channel nonlinearity which is for the first time to deep unfold the conventional cascaded DBP-NPCC into an NN structure. It has been verified via extensive simulation that, DU-DBP-NPCC provides 1 dB and 0.36 dB Q factor gain compared with CDC and DBP-NPCC at the launch power of −1 dBm/channel, respectively. Moreover, the proposed method can extend the maximum transmission distance by 28% compared with CDC and by 7% compared with cascaded DBP-NPCC. 3 different training schemes of DU-DBP-NPCC has also been investigated, which implies that it’s not proper to train the smoothing filter in NPCC layer directly in DU-DBP-NPCC and the effective SNR is not the proper evaluation metric of nonlinearity compensation performance for DU-DBP-NPCC. Finally, the analysis of the computational complexity verifies that DU-DBP-NPCC achieves about 26% lower computation complexity compared with conventional DBP-NPCC. The proposed method shows great potential for joint intra and inter-channel nonlinearity equalization, providing a better choice for future long-haul coherent optical systems.
Funding
National Natural Science Foundation of China (61771062, 61871044).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity Limits of Optical Fiber Networks,” J. Lightwave Technol. 28(4), 662–701 (2010). [CrossRef]
2. R. Dar and P. J. Winzer, “On the Limits of Digital Back-Propagation in Fully Loaded WDM Systems,” IEEE Photonics Technol. Lett. 28(11), 1253–1256 (2016). [CrossRef]
3. E. Ip and J. M. Kahn, “Compensation of Dispersion and Nonlinear Impairments Using Digital Backpropagation,” J. Lightwave Technol. 26(20), 3416–3425 (2008). [CrossRef]
4. E. Ip, “Nonlinear Compensation Using Backpropagation for Polarization-Multiplexed Transmission,” J. Lightwave Technol. 28(6), 939–951 (2010). [CrossRef]
5. L. B. Du and A. J. Lowery, “Improved single channel backpropagation for intra-channel fiber nonlinearity compensation in long-haul optical communication systems,” Opt. Express 18(16), 17075–17088 (2010). [CrossRef]
6. F. P. Guiomar, J. D. Reis, A. L. Teixeira, and A. N. Pinto, “Digital Postcompensation Using Volterra Series Transfer Function,” IEEE Photonics Technol. Lett. 23(19), 1412–1414 (2011). [CrossRef]
7. F. P. Guiomar and A. N. Pinto, “Simplified Volterra Series Nonlinear Equalizer for Polarization-Multiplexed Coherent Optical Systems,” J. Lightwave Technol. 31(23), 3879–3891 (2013). [CrossRef]
8. L. Dou, Z. Tao, L. Li, W. Yan, T. Tanimura, T. Hoshida, and J. C. Rasmussen, “A Low Complexity Pre-Distortion Method for Intra-channel Nonlinearity,” in Optical Fiber Communication Conference/National Fiber Optic Engineers Conference 2011, (Optical Society of America, 2011), p. OThF5.
9. Z. Tao, L. Dou, W. Yan, L. Li, T. Hoshida, and J. C. Rasmussen, “Multiplier-Free Intrachannel Nonlinearity Compensating Algorithm Operating at Symbol Rate,” J. Lightwave Technol. 29(17), 2570–2576 (2011). [CrossRef]
10. R. Dar, M. Feder, A. Mecozzi, and M. Shtaif, “Inter-Channel Nonlinear Interference Noise in WDM Systems: Modeling and Mitigation,” J. Lightwave Technol. 33(5), 1044–1053 (2015). [CrossRef]
11. Z. Tao, W. Yan, L. Liu, L. Li, S. Oda, T. Hoshida, and J. C. Rasmussen, “Simple Fiber Model for Determination of XPM Effects,” J. Lightwave Technol. 29(7), 974–986 (2011). [CrossRef]
12. L. Li, Z. Tao, L. Liu, W. Yan, S. Oda, T. Hoshida, and J. C. Rasmussen, “XPM tolerant adaptive carrier phase recovery for coherent receiver based on phase noise statistics monitoring,” in European Conference on Optical Communication, (IEEE, 2009), pp. 1–2.
13. L. Li, Z. Tao, L. Liu, W. Yan, S. Oda, T. Hoshida, and J. C. Rasmussen, “Nonlinear Polarization Crosstalk Canceller for Dual-Polarization Digital Coherent Receivers,” in Optical Fiber Communication Conference, (Optical Society of America, 2010), p. OWE3.
14. P. Layec, A. Ghazisaeidi, G. Charlet, J.-C. Antona, and S. Bigo, “A novel compensation method at the receiver for cross-polarization modulation effects,” in 2014 European Conference on Optical Communication, (IEEE, 2014), pp. 1–3.
15. C. Li, F. Zhang, L. Zhang, X. Chen, F. Yang, H. Ming, X. Ruan, Z. Chen, and C. Yang, “Inter-Channel Fiber Nonlinearity Mitigation in High Baud-Rate Optical Communication Systems,” J. Lightwave Technol. 39(6), 1653–1661 (2021). [CrossRef]
16. O. Golani, M. Feder, and M. Shtaif, “NLIN Mitigation Using Turbo Equalization and an Extended Kalman Smoother,” J. Lightwave Technol. 37(9), 1885–1892 (2019). [CrossRef]
17. K. Hornik, M. Stinchcombe, and H. White, “Universal approximation of an unknown mapping and its derivatives using multilayer feedforward networks,” Neural Networks 3(5), 551–560 (1990). [CrossRef]
18. Y. LeCun, Y. Bengio, and G. Hinton, “Deep learning,” Nature 521(7553), 436–444 (2015). [CrossRef]
19. F. N. Khan, Q. Fan, C. Lu, and A. P. T. Lau, “An Optical Communication’s Perspective on Machine Learning and Its Applications,” J. Lightwave Technol. 37(2), 493–516 (2019). [CrossRef]
20. J. R. Hershey, J. L. Roux, and F. Weninger, “Deep Unfolding: Model-Based Inspiration of Novel Deep Architectures,” arXiv preprint arXiv:1409.2574 (2014).
21. A. Balatsoukas-Stimming and C. Studer, “Deep Unfolding for Communications Systems: A Survey and Some New Directions,” in IEEE International Workshop on Signal Processing Systems, (IEEE, 2019), pp. 266–271.
22. C. Häger and H. D. Pfister, “Nonlinear Interference Mitigation via Deep Neural Networks,” in Optical Fiber Communication Conference, (Optical Society of America, 2018), p. W3A.4.
23. C. Fougstedt, C. Häger, L. Svensson, H. D. Pfister, and P. Larsson-Edefors, “ASIC Implementation of Time-Domain Digital Backpropagation with Deep-Learned Chromatic Dispersion Filters,” in European Conference on Optical Communication, (IEEE, 2018), pp. 1–3.
24. Q. Fan, G. Zhou, T. Gui, C. Lu, and A. P. T. Lau, “Advancing theoretical understanding and practical performance of signal processing for nonlinear optical communications through machine learning,” Nat. Commun. 11(1), 3694 (2020). [CrossRef]
25. B. I. Bitachon, A. Ghazisaeidi, M. Eppenberger, B. Baeuerle, M. Ayata, and J. Leuthold, “Deep learning based digital backpropagation demonstrating SNR gain at low complexity in a 1200 km transmission link,” Opt. Express 28(20), 29318–29334 (2020). [CrossRef]
26. O. Sidelnikov, A. Redyuk, S. Sygletos, M. Fedoruk, and S. Turitsyn, “Advanced Convolutional Neural Networks for Nonlinearity Mitigation in Long-Haul WDM Transmission Systems,” J. Lightwave Technol. 39(8), 2397–2406 (2021). [CrossRef]
27. R. Kudo, T. Kobayashi, K. Ishihara, Y. Takatori, A. Sano, and Y. Miyamoto, “Coherent Optical Single Carrier Transmission Using Overlap Frequency Domain Equalization for Long-Haul Optical Systems,” J. Lightwave Technol. 27(16), 3721–3728 (2009). [CrossRef]
