1. Introduction

According to Shannon’s theorems, in order to satisfy the increasing requirement of communication capacity, an available solution is to raise optical signal to noise ratio (OSNR) [1,2]. However, increasing the launch power to improve the OSNR will introduce nonlinear distortions due to the Kerr effect. Many techniques have been developed to overcome the nonlinear capacity limitation, such as digital backpropagation (DBP) [3,4], perturbation theory [5], optical phase conjugation [6], etc. However, these techniques treat nonlinearity as an undesired factor and mostly rely on approximate models derived from nonlinear Schrödinger equation (NLSE) [7,8], which means that they cannot completely eliminate nonlinear impairments. In recent years, an innovative technique called nonlinear frequency division multiplexing (NFDM) has attracted widespread attention [9]. The core of NFDM is the powerful mathematical tool named nonlinear Fourier transformation (NFT), which can linearize the nonlinear distance evolution of signal in the nonlinear spectrum defined by NFT [10]. Thus, it has the potential to realize nonlinear-free optical transmission.

As stated by NFT theory, the nonlinear spectrum of a given time domain signal consists of continuous and discrete part. According to the degree of freedom (DOF) used to encode the signal, NFDM can be divided into discrete spectral (DS) [11–15], continuous spectral (CS) [16–20] and full spectral (FS) [21–25] modulation, and the researches evolve from single polarization to dual polarization [26–32]. In this paper, we focus on dual-polarized CS-NFDM (DP-CS-NFDM), which has wider DOF than DS and less algorithmic complexity than FS [23,30,32]. CS modulation is implemented by nonlinear inverse synthesis (NIS) scheme, which encodes signals on the nonlinear spectrum and generates the corresponding time domain waveforms by inverse NFT (INFT). This scheme is analogous to orthogonal frequency division multiplexing (OFDM), and sinc function is used as subcarriers [16]. Except for sinc, other subcarriers such as root-raised-cosine (RRC) [33], Hermite-Gaussian (HG) [34,35] and faster than Nyquist (FTN) [36] subcarriers are reported, and show their advantages in different system conditions. Apparently, they are still an extension of linear systems, but may not be optimal options for nonlinear spectrum.

In recent years, applications of machine learning (ML) in optical communications have been widely discussed [37], as well as NFDM [38]. Various types of neural networks (NN), including artificial NN (ANN) [39–41], convolutional NN (CNN) [42–44] and recurrent NN (RNN) [45], have been used to equalize inter-carrier interference (ICI) and shorten guard interval. Nowadays, a learning model, called auto-encoder (AE), is innovatively applied in communication systems. An AE consists of an encoder and a decoder, and aims to recover input data, which resembles a communication system [46]. In [47], AE is firstly utilized in DS-NFDM to optimize the imaginary part of eigenvalues, radii of the constellations and phase difference between two solitons. This inspires the idea of optimizing subcarriers in CS-NFDM by AE.

In this paper, we present an AE assisted subcarrier optimization scheme for DP-CS-NFDM. We introduce an inter-polarized matrix at the transmitter side and jointly optimize the encoding and decoding matrices to make the received signal converge to the transmitted signal. This leads to a set of optimal sinc-like subcarriers naturally. We numerically verify the superiority of our scheme over standard sinc with linear minimum mean square error (LMMSE) equalization. We also analyze the features of the optimal subcarrier that account for the additional performance gain. Furthermore, we validate the generalizations of the optimized subcarriers to different modulation formats, transmission distances and bandwidths. Our proposed optimization scheme exhibits high robustness to variations in system parameters and offers a new perspective for designing subcarriers in NFDM.

The rest of this paper is organized as follows. Section 2 gives a brief introduction to the AE model. Section 3 first reviews the basis of NFT theory, and then explains the fundamental principles of NFDM and the AE assisted optimization scheme. Section 4 presents numerical verifications and further discussions. Section 5 concludes the work.

2. Brief review of AE

As is well known, a general communication system consists of five parts: information source, transmitter, channel, receiver and destination. In a coherent end-to-end optical communication system, the transmitter (Tx), the receiver (Rx) and their digital signal processing (DSP) work together to recover the signal at the receiver side with low error rate. Specially for NFDM, TxDSP encodes the signal on the nonlinear spectrum and generates a fiber-friendly time domain waveform by INFT. The fiber channel acts as an impairments source that degrades the transmission performance. In RxDSP, NFT decodes the nonlinear spectrum from the received waveform and then compensates for the channel response to recover the transmitted signal. The above process matches well with an AE.

AE is a special type of unsupervised learning model that uses the input data itself as supervision, forcing the output equal to the input [48]. In this way, the model is guided to establish a mapping relationship, and a reconstructed output of input is obtained. In [46], AE is first applied to the design of radio communication systems, also known as end-to-end optimization. Figure 1 illustrates a typical AE model. The encoder ${f(} \cdot {)}$ and decoder ${g(} \cdot {)}$ are trainable functions, while the channel response ${h(} \cdot {)}$ is the key component in optical communications, which is an untrainable function in the AE model. Therefore, an accurate channel model is the prerequisite of end-to-end optimization. For the DP-NFDM we studied in this paper, the numerical simulation channel model is the split step Fourier method (SSFM) base on the Manakov equation. The goal of AE is to jointly optimize ${f(} \cdot {)}$ and ${g(} \cdot {)}$ so that the output ${\mathbf{X}_{\mathbf{r}}}$ is close to the input $\mathbf{X}_{\mathbf{t}}$, and thus obtain the optimal channel-adapted encoder and decoder.

 

Fig. 1. A typical AE model for a communication system.

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Now, back to our concerned CS-NFDM, it encodes signals on the continuous spectrum by NIS. This process refers to OFDM, and the subcarriers used in linear systems are directly applied, which may not be optimal. To find an optimal set of subcarriers, we treat the NIS scheme as the encoder and decoder in an AE model. By making the received signals equal to the transmitted ones, a pair of optimal encoding and decoding subcarriers can be naturally obtained. In the next section, we will give a detailed introduction to the AE assisted subcarrier optimization scheme.

3. Basis of NFT theory and AE assisted optimized NFDM

3.1 Review of NFT theory

A DP signal transmitting along standard single mode fiber (SSMF) links follows the Manakov equation [49]

Here, ${\mathbf{Q}(\tau ,l)\ =\ [}{{Q}_{1}}{(\tau ,l)\;\ }{{Q}_{2}}{(\tau ,l)}{{]}^{T}}$ is the complex envelope of optical field, ${\mathbf{N}(\tau ,l)} =$ $[{{N}_{1}}{(\tau ,l)}\ {{N}_{2}}{(\tau ,l)}{{]}^{T}}$ is the noise vector, the subscripts denotes the two polarization components, ${\tau }$ and ${l}$ are retarded time and distance along fiber. ${\alpha }$, ${{\beta }_{2}}$ and ${\gamma }$ are attenuation, group velocity dispersion (GVD), and nonlinear coefficient respectively. The attenuation in Eq. (1) will destroy the integrability of Manakov equation, so the lossless path-averaged (LPA) model is used for approximation [50]. Meanwhile, normalized parameters are introduced as below

Omitting the noise component, the normalized form of Manakov equation is obtained

Equation (3) is an integrable partial differential equation, its NFT is defined by solving the Zakharov–Shabat problem [51,52]

On the condition that $\mathbf{q}$ has vanishing boundary, Eq. (4) exists a pair of canonical eigenvector solutions as below

Here, $\boldsymbol{\mathrm{\phi}} {(\lambda ,t)\ =\ [}\phi {(\lambda ,t)\;\ }\bar{\phi }{(\lambda ,t)}{]}^T$ and $\boldsymbol{\mathrm{\psi}} (\lambda ,t)\ =\ [\psi (\lambda ,t)\;\ \bar{\psi }(\lambda ,t){]}^{T}$ are 2 solution basis of the solution space. According to the inverse scattering theory, the scattering coefficients ${a(\lambda )}$ and ${\mathbf{b}(\lambda )\ =\ [b_1(\lambda )\;b_2(\lambda )}{]}^{T}$ are defined by

In our studied CS modulation, ${\lambda } \in \mathbb{R}$, and they satisfy

In numerical computations, ${\lambda =\ }{{T}_{n}}{\pi f}$, where ${f}$ is the linear frequency. The process of solving the scattering coefficients is called NFT. Accordingly, given the scattering coefficients, the corresponding time domain waveform is determined, which known as INFT. In our numerical verifications, Ablowitz-ladik (AL) algorithm is used for (I)NFT operations [26]. As stated in NFT theory, the ${z}$-evolution of the scattering coefficients are written by

As can be inferred from Eq. (9), the ${z}$-evolution of ${\mathbf{b}(\lambda )}$ manifests as a phase rotation, which is the theoretical basis of nonlinear free transmission in NFDM.

3.2 AE assisted optimized NFDM system

Figure 2(a) illustrates the setup of DP-NFDM, which consists of three stages. First, in TxDSP stage, signals are encoded on the ${\mathbf{b}(\lambda )}$, and the INFT generates the corresponding electrical time domain waveform. Then, the waveform is modulated on the optical carriers by a DP in-phase and quadrature modulator (DP-IQM), and transmitted through ${{N}_{s}}$ spans of lumped amplification (LA). The received optical signals are converted to electrical waveforms by the coherent receiver. Finally, in the RxDSP stage, the NFT recovers the ${\mathbf{b}(\lambda )}$ and then compensates for the channel response. It is worth mentioning that the channel of AE model should contain the DP-IQM, fiber link and coherent receiver in an actual optical communication system as shown in Fig. 2(a). We will explain the TxDSP, RxDSP and AE assisted optimization in detail below.

 

Fig. 2. (a) Setup of DP-NFDM; (b) Illustration of AE assisted end-to-end optimization scheme.

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The flowchart of TxDSP and RxDSP can be extracted in Fig. 2(b). To begin with, after serial-to-parallel conversion and symbol mapping, the DP signals can be expressed in vector form as follows

Here, ${k}$ denotes the index of subcarriers, ${{T}_{0}}$ is the signal duration, and ${A}$ is a power control factor. In our numerical simulation, we set ${A = 1}{{0}^{\mathrm{\Omega /10}}}$ and by adjusting $\mathrm{\Omega }$ to adjust power, in which for every 0.5 increasing in $\mathrm{\Omega }$, the actual power goes up by approximately 1 dB. Meanwhile, to improve the accuracy of (I)NFT algorithm and avoid inter- burst interference, the oversampling of ${{R}_{0}}$ times and guard interval (GI) insertion of ${H - 1}$ times ${{T}_{0}}$ should be performed, which can be implemented by sampling the sinc function, that is, ${\lambda }$ is sampled from ${\ -\ \pi B}{{R}_{0}}{{T}_{n}}{/2}$ to ${\pi B}{{R}_{0}}{{T}_{n}}{/2}$ with ${{N}_{c}}{{R}_{0}}{H}$ equally spaced points. Then Eq. (11) can be expressed in matrix form

Here, ${B}$ is the signal bandwidth and ${L}$ is the transmission distance. For conveniently evaluating the GI duration, it often defines ${\eta =\ T/}{{T}_{0}}$. The bandpass filters (BPF) in TxDSP and RxDSP are used to limit the bandwidth in the training stage of AE, whose application will be discussed in the next section. The RxDSP is generally the inverse process of TxDSP. It is worth mentioning that the de-truncation is performed by adding zeros on both sides, making the result length equal to the original. Moreover, the spectrum decoding is executed by a matched filter, which can be written as

Accordingly, the spectrum decoding Eq. (15) in RxDSP is extended to

Conveniently, we redefine the identifiers as

It should be noted that the decoding matrix $\mathbf{G}$ is not equal to the encoding matrix $\mathbf{F}$. Otherwise, ICI will occur inevitably. Since linear correlation can be cancelled by a decorrelation matrix $\mathbf{W}$, the ICI can be equalized by pre-multiplying $\mathbf{W}$ after spectrum decoding. Therefore, we can get ${\mathbf{G}^{\dagger }} \approx \mathbf{W}{\mathbf{F}^{\dagger}}$. In fact, LMMSE is an effective way that determine the decorrelation matrix $\mathbf{W}$. We give a detailed discussion about this in Appendix. Now, we take Eq. (16) as an encoder, Eq. (17) as an decoder, and $\mathbf{F}$ and $\mathbf{G}$ are trainable complex valued matrices. Consequently, an AE model is formed naturally as shown in Fig. 2(b), and the training objective is to minimize the mean square error (MSE)

Algorithm 1. AE training procedure.

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Then, when the loss function reaches convergence, a couple of optimal $\mathbf{F}$ and $\mathbf{G}$ is obtained. The above mentioned are our proposed AE assisted DP-NFDM subcarrier optimization scheme. We summarize the training procedure in Algorithm  1. As the training batches are created randomly in each iteration, AE can hardly overfit. Thus, AE has inherent strong generalization.

4. Numerical verification

4.1 System parameters and training process

The system parameters in numerical verifications are listed in Table. 1. Benefit by the automatic derivation mechanism of Pytorch 2.0, the optimization process is able to be performed. To ensure the loss function converges steadily, ${N}$ should not be too small. However, to store the gradient requires a large amount of memory resources. Due to our device limitation, the number of subcarriers ${{N}_{c}}$ is set as 64. Moreover, AE may exploit the remaining resources to improve the performance, so we must constrain the bandwidth during the training stage. Therefore, we set the 3 dB bandwidth of the BPF as 1.25 times the signal bandwidth, which guarantees a flat gain within the signal band.

Table 1. System parameters in numerical verification

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During the training stage, we initialize subcarriers in $\mathbf{F}$ and $\mathbf{G}$ of intra- polarization as standard sinc, and zero the inter ones. ${N}$ is set as 300 in each iteration. We use the RMSprop optimizer with an initial learning rate ${\mu }$ of 5e-4 to update the encoder and decoder for 750 iterations in total. Besides, we pre-generated 2000 additional test bursts and evaluate the performance at the start of each iteration. We plot the training performances in Fig. 3. As can be seen in Fig. 3(a), the loss curves of training and test bursts closely match, indicating no overfitting. The tendency of Q-factor curve in Fig. 3(b) fits the loss curve, and exceeds the hard decision forward error correction (HD-FEC) threshold after about 300 iterations. Besides, in the training stage, the actual power is determined by ${A}$, amplitude of $\mathbf{S}_{{in}}^{{X/Y}}$ and the optimizing $\mathbf{F}$. Here, ${A}$ is determined by setting ${\Omega }$ as -7, which is constant during training stage, and the maximum amplitude of $\mathbf{S}_{{in}}^{{X/Y}}$ is $\sqrt {{34}} $ for 32QAM is implemented. Therefore, AE will adjust the energy of subcarriers in $\mathbf{F}$, and the actual launch power will slide to the optimal. Figure 3(b) illustrates this variation, which converges after about 400 iterations, decreasing from the original -4.78 dBm to -5.57 dBm.

 

Fig. 3. (a) Loss, (b) Q-factor and power as a function of iteration during training stage.

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4.2 Performance of optimized subcarriers

The encoding and decoding subcarriers obtained by AE are shown in Fig. 4. Due to the symmetry between two polarizations, we only display that of ${x}$-polarization. Figure 4(a)(b) show the encoding and decoding subcarriers respectively. The peaks of optimized ${\mathbf{F}_{{xx}}}$ are below 1, while the corresponding ${\mathbf{G}_{{xx}}}$ is larger, because of the high initial power in training stage, as pointed out in Fig. 5(a). This phenomenon coincides the power curve shown in Fig. 3(b). The 3 pairs of dash lines indicate the bandwidth of signal, 3 dB BPF and 3 dB OBPF from the center to the ends respectively. Apparently, the falloff at the 3 dB BPF bandwidth reflects the limited available bandwidth for spectrum encoding and decoding. As the training bursts and ASE noise are randomly created, AE will generate noise-like values outside the available bandwidth as shown in Fig. 4. To further analyze the characteristics of the optimized subcarriers, we filter $\mathbf{F}$ and $\mathbf{G}$ by a same BPF, which means that we keep the band within the green dash lines. We name the resulting subcarriers as AE-sinc subcarriers in the rest of this paper.

 

Fig. 4. AE optimized subcarriers: (a)(b) are the intra- polarizaed components of $\mathbf{F}$ and $\mathbf{G}$, and (c)(d) are the inter- ones. The transparent blue dashed lines are standard sinc subcarriers, while the colorful ones are optimized subcarriers. The red, green and blue dash lines presents the bandwidth of signal, BPF and OBPF respectively.

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Fig. 5. (a) Q-factor performance for different encoding and decoding strategies; (b) BER of different subcarriers at the optimal launch powers.

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We exhibit the Q-factor versus launch power for different encoding and decoding strategies in Fig. 5(a). For AE-sinc, we obtain the optimal point directly from the training stage, and vary ${A}$ while keeping $\mathbf{F}$ and $\mathbf{G}$ constant to obtain the other points, as done for the other strategies. Compared to the standard sinc and LMMSE equalization, the optimal launch power increases by 1.12 dB. Meanwhile, AE-sinc achieves a Q-factor gain of 1.54 dB and 0.62 dB over these two methods, respectively, exceeding the HD-FEC threshold. Figure 5(b) depicts the bit error rate (BER) of each subcarrier at the optimal launch powers. Due to truncation, the outer sinc subcarriers have poor BER performance, especially the right ones, which are also affected by the cumulative error of numerical (I)NFT algorithm. LMMSE equalization alleviates these effects, but the outer subcarriers still perform slightly worse than the inner ones. On the contrary, the AE-sinc shows flat BER performance across different subcarriers, and the BER is lower than LMMSE overall. In the next section, we will try to explain these phenomena.

4.3 Characteristics of subcarriers optimized by AE

To figure out how the AE-sinc leads to its performance improvements, we discuss its characteristics here. Since the out-of-band of AE-sinc have been analyzed and filtered previously, we mainly focus on the in-band and the inter- polarized components of AE-sinc. In advance, we make an assumption that the subcarriers optimized by AE are an optimal value close to the initial subcarriers under the current system parameters. This implies that the obtained AE-sinc is sinc-like. Hence, we can use the standard sinc as a reference for analysis.

In order to investigate the unflatten gain within the signal band observed in Fig. 4(a)(b), we retrain the model under three other scenarios: (1) back-to-back (BTB) without truncation; (2) BTB with truncation; (3) 960 km transmission without truncation. Figure 6(a-c) plot the obtained AE-sinc in each scenario. Conveniently, we mark the scenario in previous sections as Scenario 4, which is 960 km transmission with truncation. Upon the assumption above, the four AE-sinc subcarriers are optimal for their respective scenarios. Focusing on the Scenario 4, which is the most practical one, we notice three features: (1) lower peaks on the left and higher peaks on the right; (2) lower side lobes within signal band, but higher within 3 dB band of BPF; (3) higher central peaks than the outer ones. We can infer their benefits by comparing the unique variable involved in the other three scenarios.

 

Fig. 6. Obtained AE-sinc in 3 scenarios: (a) BTB w/o truncation; (b) BTB w/ truncation; (c) 960 km transmission w/o truncation. The dash lines have the same meaning as that in Fig. 4.

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As shown in Fig. 6(a), in Scenario 1, the left peaks of AE-sinc are slightly lower and the outer ones are a bit higher. Considering that Scenario 1 only involves the (I)NFT inaccuracy, which is transitive and proportional to high energy [56], it can be easily deduced that this energy distribution achieves a flat inaccuracy among different ${\lambda }$. Take ${\mathbf{F}_{{xx}}}$ for instance, where INFT is performed, the inverse AL algorithm operates from right to left, which is the same direction as the error propagation. Therefore, by reducing the energy of the left subcarriers, the total error of them will be reduced. Then, it leads to flat performance among subcarriers.

In Scenario 2 shown in Fig. 6(b), truncation is introduced with respect to Scenario 1. Compared with Fig. 6(a), it is obvious that the side lobes within the signal band are well suppressed. Figure 7(a) shows the time domain waveform of sinc subcarriers, where some information may be lost due to the truncation of the tails. Figure 7(b) compares the synthesized waveforms in Scenario 1 and 2, indicating that the tails of Scenario 1 are clearly higher than those of Scenario 2. It is well known that the ${\mathbf{b}(\lambda )}$ of a rectangle waveform is approximate to sinc [10], but the nonlinear transformation ${{\varGamma }_{b}}$ will magnify the side lobes of sinc, which have lower energy. That is why undesired tails are observed after ${{\varGamma }_{b}}$. Therefore, suppressing the side lobes can make the ${\mathbf{b}(\lambda )}$ more approximate to sinc after ${{\varGamma }_{b}}$ and improve the localizability of the corresponding time domain waveform.

 

Fig. 7. Time domain waveform of (a) sinc and (b) AE-sinc in Scenario 1 and 2; synthesized ${{b}_{1}}{(\lambda )}$ of (c) sinc and (d) AE-sinc in Scenario 3 and 4, (e)(f) are the corresponding time domain waveform respectively. The green and pink dash lines are useful and truncated time window respectively. (g) Q-factor performance of AE-sinc (Scenario 4) and sinc under different ${\eta }$.

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Figure 6(c) shows the AE-sinc obtained in Scenario 3, which introduces ASE noise based on Scenario 1. However, there is no significant difference from Fig. 6(a), except for the increased side lobes between the signal band and 3 dB band of BPF. It can be explained by the analytical model of noise density of ${\mathbf{b}(\lambda )}$, where the noise power on ${\mathbf{b}(\lambda )}$ has a negative correlation with the signal power [55]. As AE considers this band available, it chooses to increase the corresponding power spectral density (PSD) to counteract the noise PSD of ${\mathbf{b}(\lambda )}$. Figure 7(c)(d) show that compared with sinc, the synthesized ${\mathbf{b}(\lambda )}$ of sinc and AE-sinc in Scenario 3 and 4 have broader bands.

Now, we return to Scenario 4. The first two features mentioned above have appeared commonly in previous scenarios, but the lower peaks of outer subcarriers are unique. Considering that the truncation may cut off the broadened tails after PDC, we attribute it to different delay among subcarriers. As we know, the essence of dispersion is different propagation constant for different spectral components, which results in a time delay to the non-zero frequency subcarriers. Since the outer subcarriers are far from zero frequency, their delays are even longer. Therefore, together with the tails generated by ${{\varGamma }_{b}}$, the truncation operation will affect the outer subcarriers more than the inner ones. It leads to the BER distribution of sinc subcarriers shown in Fig. 5(b). Thus, the AE-sinc in Scenario 4 is a deliberate design choice to minimize the useful energy that is truncated. Figure 7(e)(f) show the time domain waveform of sinc and AE-sinc after PDC in Scenario 3 and 4. As can be seen, the waveform of Scenario 4 is more concentrated within the truncation window. The fluctuations outside the window can be mainly attributed to the randomness of training stage. To further demonstrates the inhibition of the tails of AE-sinc, we exhibit the Q-factor performance under different ${\eta }$ in Fig. 7(g). As can be seen, the Q-factor improvements of AE-sinc is about 0.31 dB when ${\eta }$ increases from the theoretical value (${\eta } \approx {1}{.26}$) to 2, while 0.87 dB improvements for standard sinc. On the other hand, when ${\eta }$ is smaller than the theoretical value, AE-sinc degrades significantly, as more energy of tails moves into the truncation window. It further proves that AE-sinc has shorter tails than sinc.

At last, we exhibit the correlation matrices of ${\mathbf{F}_{{xx}}}$, ${\mathbf{F}_{{xy}}}$, ${\mathbf{G}_{{xx}}}$ and ${\mathbf{G}_{{xy}}}$ in Fig. 8, to inspect the characteristics of the inter- polarized components of AE-sinc shown in Fig. 4(c)(d). As shown in Fig. 8(a)(b), AE-sinc induces weak correlation among adjacent subcarriers in both the encoder and decoder. Compared with the diagonal elements, there exists about -10 dB energy transfer. Figure 8(c)(d) exhibit the correlation of ${\mathbf{F}_{{xy}}}$ and ${\mathbf{G}_{{xy}}}$, where almost no correlation exists among adjacent subcarriers, but strong diagonal elements are observed. Focusing on the value of diagonal ones, it suggests that AE-sinc transfers about -9 dB energy between two polarizations in TxDSP.

 

Fig. 8. Correlation among AE-sinc subcarriers of (a) ${\mathbf{F}_{{xx}}}$, (b) ${\mathbf{G}_{{xx}}}$, (c) ${\mathbf{F}_{{xy}}}$ and (d) ${\mathbf{G}_{{xy}}}$. We take the logarithm of their magnitudes.

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4.4 Generalization validation

One of the major challenges for ML applications in optical communications is the generalization ability. To evaluate the generalization performance of AE-sinc trained in the system listed in Table. 1, we vary one parameter at a time to ensure the unique variable, while keeping the dimension of $\mathbf{F}$ and $\mathbf{G}$ unchanged. The parameters that validated involve modulation format, distance and bandwidth. That means although the AE-sinc is trained under the condition of 32QAM, 12${\ \times }$80 km and 16 GHz, it is utilized to perform 16/64QAM or 10${\ \times }$80 km and 14${\ \times }$80 km or 24/32 GHz transmission. In each case, GI is set as theoretical threshold given by Eq. (14). Figure 9(a) illustrates the Q-factor versus launch power for different modulation formats. AE-sinc achieves a Q-factor improvement of 0.85 dB and 0.62 dB for 16/32QAM compared with sinc equalized by LMMSE. For 64QAM, the Q-factor improvement is 0.40 dB and exceeds the soft decision FEC (SD-FEC) threshold. Then, Fig. 9(b) presents the validation results of distance, where AE-sinc obtains about 0.6 dB Q-factor gain in the validated distances and surpasses the HD-FEC threshold in the extended transmission of 1120 km. Finally, Fig. 9(c)(d) illustrate the validation of bandwidth of 24 GHz and 32 GHz respectively. It is evident that the generalization performance of AE-sinc worsens as the bandwidth deviates from the trained condition. This can be attributed to two reasons. First, the ICI is more sensitive to bandwidth and $\mathbf{F}$ and $\mathbf{G}$ no longer match well, so LMMSE brings considerable improvements for AE-sinc. Secondly, the AE-sinc is trained on the condition of 16 GHz, while the validated 24 GHz and 32 GHz have longer pulse broadening due to dispersion. Therefore, the suppression of outer subcarriers obtained in 16 GHz does not suit the validated scenario. In summary, our proposed AE assisted subcarrier optimization scheme exhibits excellent generalization for modulation format and distance, while that for bandwidth near the training condition.

 

Fig. 9. Generalization validation of (a) modulation format, (b) transmission distance, and (c)(d) bandwidth.

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5. Conclusion

In this paper, we propose and numerically verify an AE assisted subcarrier optimization scheme for DP-NFDM. Taking into account the inter-polarization correlation in DP-NFDM, we introduce an inter-polarized component to the spectrum encoding and decoding process. The resulting AE-sinc subcarriers surpass HD-FEC threshold in a 960 km transmission system and outperform sinc by 1.54 dB in Q-factor. Furthermore, by analyzing the characteristics of AE-sinc, we observe that AE tends to suppress side lobes and outer subcarriers to mitigate the effects of truncation and balance the transitive error of numerical (I)NFT by adjust the energy higher at right. In addition, the generalization validation shows that AE-sinc obtained in the training stage has excellent robustness to modulation format and transmission distance, while to bandwidth near the training condition. Our work provides a new perspective for subcarrier design in NFDM in the future.