Single-step digital backpropagation for subcarrier-multiplexing transmissions

Huazhi Lun; Qunbi Zhuge; Qunbi Zhuge; Zhuopeng Xiao; Songnian Fu; Ming Tang; Deming Liu; Weisheng Hu; David V. Plant

doi:10.1364/OE.27.036680

1. Introduction

The ever-increasing capacity demands driven by new applications such as high definition video streaming, cloud computing and virtual reality push the capacity of fiber optical networks towards the nonlinear Shannon limit [1]. To overcome the obstacle of fiber nonlinearities, digital nonlinear compensation in coherent optical communication has been considered as a promising solution. Until now, several digital nonlinear compensation schemes have been proposed including digital backpropagation (DBP), perturbation-based pre-distortion, and Volterra series [2–8]. However, their complexities are generally high. Alternatively, subcarrier-multiplexing (SCM) systems were proposed and experimentally verified as an effective approach to mitigate the fiber nonlinearity [9–11]. Based on single channel transmission experiments over standard single mode fiber (SSMF), reach extensions of 23% for PDM-QPSK format and 8% for PDM-16QAM format have been achieved [11]. The combination of SCM and DBP (SCM-DBP) based on the cross-phase modulation (XPM) model [12] was proposed and experimentally demonstrated [13]. For PDM-QPSK and PDM-16QAM formats, 2-step SCM-DBP is sufficient to achieve 50% and 27% reach extension, respectively [13]. In comparison with conventional low pass filter assisted DBP (LDBP) [14], the complexity of SCM-DBP is only 16.8% of LDBP for QPSK and 57.9% of LDBP for 16QAM. Further complexity reduction has been reported by introducing infinite impulse response (IIR) filters into SCM-DBP for cross-subcarrier nonlinearity (CSN) compensation, where IIR filters are used to replace the low pass filter within the XPM model [15–16]. Nevertheless, it is reported in [13] that the complexity reduction of SCM-DBP for 16QAM format is less effective than that for QPSK format. The reason behind this is that the transmission distance of QPSK is much longer than 16QAM, and thus it requires much more steps for the SCM-DBP algorithms. Therefore, if the number of steps is reduced to similar values for QPSK and 16QAM, the complexity reduction for QPSK would be more than 16QAM. Besides, it has been identified that the SCM performance improvement decreases in the applications of shorter distances and higher order modulation formats [17–19]. Therefore, for cost-sensitive metro and regional networks with high order formats such as PDM-32QAM, a further complexity reduction of DBP is highly desired.

We proposed and numerically investigated a single-step modified SCM-DBP (M-SCM-DBP) scheme for metro or regional system in [20]. Compared to the previous schemes, an IIR filter is introduced to aid SSN compensation, which improves the performance for the single-step operation. To further reduce the complexity, the second chromatic dispersion compensation (CDC) stage is incorporated into the adaptive filter by taking the advantage of the significantly reduced pulse broadening in SCM transmissions. In this paper, we describe our work in [20] with more details and provide an experimental demonstration of the single-step M-SCM-DBP for SCM-PDM-32QAM transmissions. In particular, after 960-km SSMF transmission, the single-step M-SCM-DBP achieves 0.6-dB ${\textrm{Q}^2}$ improvement which is the same as the performance of both a conventional 2-step SCM-DBP and a 12-step LDBP. The proposed method achieves 60.4% and 86.3% complexity reduction compared to the SCM-DBP and LDBP, respectively. At a target bit-error-rate (BER) of 3×${10^{ - 2}}$ (Q²=5.49-dB), the single-step M-SCM-DBP achieves 36% SSMF reach extension to 1220-km.

2. Operational principle

The fiber nonlinear effect of SCM systems typically consists of self-subcarrier nonlinearity (SSN) and cross-subcarrier nonlinearity (CSN). As proposed in [13], the SSN can be compensated as

(1)$${v_{p/i, x/y}}(t )= {E_{p/i, x/y}}(t )exp [{ - j\varepsilon \gamma {L_{eff}}({|{{E_{p/i, x}}(t ){|^2} + } |{E_{p/i, y}}(t ){|^2}} )} ]$$

where ${v_{p/i, x/y}}(t )$ is the probe subcarrier or interfering subcarrier after SSN compensation and ${E_{p/i, x/y}}(t )$ is the probe subcarrier or interfering subcarrier before compensation. The subscript p and i refer to the index of the probe subcarrier and interfering subcarrier, respectively. The subscript x and y refer to the x-polarization and y-polarization, respectively. The ${L_{eff}}$ is the effective fiber length, $\gamma $ is the nonlinear coefficient and $\varepsilon $ is the parameter that needs to be optimized.

Afterwards, the CSN compensation is performed and the compensation matrix $M(t )$ is expressed as

(2)$$M(t )= \left[ {\begin{array}{{l}} {{e^{ - j{\phi_x}(t )}}}\\ {{w_{yx}}(t ){e^{ - j\frac{{[{{\phi_x}(t )+ {\phi_y}(t )} ]}}{2}}}} \end{array}} \right.\left. {\begin{array}{{l}} {{w_{xy}}(t ){e^{ - j\frac{{[{{\phi_x}(t )+ {\phi_y}(t )} ]}}{2}}}}\\ {{e^{ - j{\phi_y}(t )}}} \end{array}} \right]$$

where ${w_{xy/yx}}(t )$ is the total CSN-induced cross-polarization modulation and ${\phi _{x/y}}(t )$ is the phase noise [13]. They are respectively given by

(3)$$\begin{array}{l} {w_{xy/yx}}(t )= {v_{i, y/x}}(t )\cdot {v_{i, x/y}}^\ast (t )\otimes \\ IFFT\left[ {j\mathop \sum \nolimits_{k ={-} \frac{N}{{2 + 1}}}^{\frac{N}{2}} exp ({ - j\Delta \beta \omega k{L_{span}}} )\times H(\omega )} \right] \end{array}$$

(4)$$\begin{array}{l} \; {\phi _{x/y}}(t )= ({2{{|{{v_{i, x/y}}(t )} |}^2} + {{|{{v_{i, y/x}}(t )} |}^2}}) \otimes \\ IFFT\left[ {j\mathop \sum \limits_{k ={-} \frac{N}{{2 + 1}}}^{\frac{N}{2}} exp ({ - j\Delta \beta \omega k{L_{span}}} )\times H(\omega )} \right] \end{array}$$

where $\Delta \beta $ is the group velocity difference between the probe and interfering subcarriers, ${L_{span}}$ is the span length, and the symbol ⊗ denotes convolution. $H(\omega )$ is the CSN low pass filter which can be expressed as

(5)$$H(\omega )= \frac{8}{9}\gamma \times \frac{{1 - exp ({ - \alpha {L_{span}} + j\Delta \beta \omega {L_{span}}} )}}{{\alpha - j\Delta \beta \omega }}$$

where $\alpha $ is the fiber attenuation coefficient and $\omega $ is the angular frequency of the signal. For the purpose of complexity reduction, the terms $IFFT\left[ {j\mathop \sum \nolimits_{k ={-} \frac{N}{{2 + 1}}}^{\frac{N}{2}} exp ({ - j\Delta \beta \omega k{L_{span}}} )\times H(\omega )} \right]$ in Eq. (3) and Eq. (4) are replaced with a first order Butterworth IIR filter [11]. As can be seen from Eqs. (2)–(4), ${\phi _x}(t )$, ${\phi _y}(t )$, ${W_{xy}}(t )$, and ${W_{yx}}(t )$ need to be filtered by the IIR filters. Due to the fact of ${w_{xy}}(t )= {({{w_{yx}}(t )} )^\ast }$, only three IIR filters are needed for each subcarrier during the calculation of the CSN.

In this work, as illustrated in Fig. 1, we propose to introduce an IIR filter in the SSN compensation, which improves the SSN estimation accuracy in the one-step operation. Because the CD is much easier to compensate in SCM systems than in SC systems, it is possible to move the second CDC to the adaptive filter for further complexity reduction. One thing should be noticed is that the IIR filter is used in both the SSN compensation and CSN compensation. However, it is known that it is difficult for IIR filters to achieve exact linear phase response. To address this issue, the data is first filtered in the forward direction and then reversed and filtered by the same filter. The proposed method is implemented in digital domain. The minimum sampling rate should at least satisfy the Nyquist sampling theorem and the samples per symbol should be determined according to practical requirements of the system.

Fig. 1. Block diagram of NLC of the single-step M-SCM-DBP.

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In SCM systems, the required number of time domain taps to compensate CD is reduced by N² compared with single carrier systems. The required number of taps for compensating 50% accumulated CD is estimated as follows [21]:

(6)$$N = 50\%\times 2({1 + h} )|{{\beta_2}} |\pi {({2{R_S}} )^2}L$$

where L and h is the transmission distance and roll-off factor of the root raised cosine (RRC) filter, respectively, β₂ is the second order dispersion coefficient, and R_S is the symbol rate of each subcarrier. For a 34.94-GBd PDM-32QAM SCM system with 8 subcarriers, which is the configuration of our experiment setup, it requires only 5 taps for a 1000-km SSMF link. Since more taps are required to implement both CD compensation and other inter-symbol interferences (ISI) mitigation, the complexity of the adaptive filter may increase. Nevertheless, in practice a transceiver is often designed to cover various scenarios from metro to long-haul (sometimes even submarine) applications. In this case, the number of taps in the adaptive filter is usually more than 15 in order to completely compensate PMD for the worst case, i.e. longest link. In the targeted metro and regional applications, where PMD is relatively small, we will demonstrate in the following simulations and experiments that the post 50% CD can be simultaneously compensated using the existing adaptive filter with less than 15 taps.

3. Complexity analysis

Considering that the first CDC block is always needed in a coherent system, when calculating the complexity in this section, we remove the first CDC block and refer the complexity of the other blocks as the DBP complexity. The complexity of the original M-step SCM-DBP, evaluated by the number of complex multiplications per sample, is [16]

(7)$$M\frac{{{K_{CD}}({{{\log }_2}{K_{CD}} + 1} )}}{{{K_{CD}} - {P_{CD}}}} + M[{5.5 + 3{N_s}} ]$$

where K_CD is the FFT size and P_CD is the memory length for CD compensation based on the overlap-and-add method, whose expressions are the same as the Eq. (14) in [14]. The N_s is the number of subcarriers. According to [16], ${P_{cd}}$ is first determined and then ${K_{cd}}$ is set to ${2^k}$ that minimize the complexity. The first term of Eq. (7) is the complexity of the CDCs. For the M-step SCM-DBP, M + 1 CDC stages are required. The complexity of the first CDC is not included in (7) as described above. The second term is the complexity of NLC. The CSN is calculated from the interfering subcarriers. Each subcarrier is filtered by three IIR filters as described in Section 2 in forward direction and backward direction, which require 3 complex multiplications. The symmetric characteristic of the CSN, i.e. the CSN from the i-th subcarrier to (i+k)-th subcarrier is identical to that from the i-th subcarrier to (i − k)-th subcarrier, is exploited to reduce the calculation complexity of the IIR filtering [16]. In addition, the compensation of SSN and CSN for the probe subcarrier requires 5.5 complex multiplications [16]. The complexity of the proposed single-step M-SCM-DBP is expressed as

(8)$$5.5 + 3{N_s} + 1$$

The second term of (8), which is the complexity of the CSN calculation, is the same as that of SCM-DBP in (7). The third term of (8) is the additional one complex multiplication for the IIR filter in the SSN compensation. In our work, a First-order Butterworth filter is used and the transfer function of the filter in z-domain is $b({1 + {z^{ - 1}}} ){({1 + {a_1}{z^{ - 1}}} )^{ - 1}}$, where b and ${a_1}$ are constants. Because of the Hermitian symmetry [15], the tap coefficients of the IIR filter in the time domain are real values. When the multiplication between the tap and the sample (a complex value) is evaluated, the number of required complex multiplications is 0.5. Considering forward and backward filtering, 1 complex multiplication is needed. Last, the complexity of LDBP is calculated based on the analysis in [13]. Because the optimum subcarrier number is 8 in our experiment, the following complexity comparison is based on this value. As described above, ${P_{cd}}$ is calculated as 10 and ${K_{cd}}$ is 64. For a similar compensation gain, the complexity of the singe-step M-SCM-DBP, 2-step SCM-DBP and 12-step LDBP [13] is summarized in Table 1. It is shown that, the proposed M-SCM-DBP achieves 60.4% and 86.3% complexity reduction compared to the SCM-DBP and LDBP, respectively.

Table 1. Number of complex multiplications per sample of the DBP schemes

View Table

4. Simulation results

The performance of the proposed M-SCM-DBP is first evaluated through numeric simulations. The simulation setup is depicted in Figs. 2(a) and (b).

Fig. 2. (a) Simulation setup. (b) Receiver DSP.

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At the transmitter side, three channels are generated with 50-GHz spacing. Both 48-GBd PDM-32QAM and 40-GBd PDM-64QAM signals are evaluated. The number of subcarriers is 8, which gives the optimal performance in our scenarios. Each subcarrier is pulse shaped by a root-raised cosine (RRC) filter with a roll-off factor of 0.01. The spacing between subcarriers is 1.01 times of the subcarrier symbol rate. The WDM signal is launched into a SSMF fiber with 80-km span length. At the end of each span, an Erbium-doped fiber amplifiers (EDFA) is used to compensate the fiber loss completely. At the receiver side, an optical bandpass filter (OBPF) is applied to extract the center channel. After IQ imbalance compensation and frequency offset compensation (FOC), the subcarrier is demultiplexed in frequency domain. Each subcarrier is successively shifted to the baseband and filtered out through the RRC matched filter. After subcarrier demultiplexing, 50% CD is compensated by the CDC module. After that the proposed M-SCM-DBP is applied followed by a decision-directed least mean square (DD-LMS) adaptive filter and carrier phase recovery (CPR). Finally, after decision, each subcarrier’s bit error ratio (BER) is calculated and averaged to get the ${\textrm{Q}^2}$ factor.

We simulate the transmission of the PDM-32QAM signals over 1200-km SSMF fiber and the PDM-64QAM signals over 800-km SSMF. The launch power is optimized in all cases. We first evaluate the required number of taps before and after incorporating CDC in the adaptive filter in Fig. 3. For both formats, the required numbers of taps are increased as expected. Nevertheless, in both cases 13 taps are sufficient to provide a saturated performance. This result will be further validated in the following experimental results. Therefore, 13 taps are used in both the simulations and experiments to evaluate transmission performance. It should be noted that the tap number should be optimized according to practical scenario. If the transmission distance changes, the optimal number of taps will also change.

Fig. 3. Required number of filter taps for (a) PDM-32QAM and (b) PDM-64QAM.

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The performance of different DBP algorithms is compared in Figs. 4(a) and (b). For PDM-32QAM, as shown in Fig. 4(a), the performance of the SCM systems with only linear compensation (LC-SCM) outperforms the single carrier (SC) systems with only linear compensation (LC-SC) by 0.2-dB. Compared with the LC-SCM scheme, the single-step M-SCM-DBP achieves 0.7-dB Q² improvement and outperforms 2-step SCM-DBP, 15-step LDBP (1 step/span) and 1-step SCM-DBP by 0.1-dB, 0.2-dB, and 0.3-dB, respectively. For PDM-64QAM, compared with the SC systems as shown in Fig. 4(b), the SCM achieves 0.3-dB Q² improvement. The single-step M-SCM-DBP outperforms 1-step SCM-DBP by 0.2-dB. The single-step M-SCM-DBP, 2-step SCM-DBP and 10-step LDBP (1 step/span) have similar performance, achieving approximately 0.6-dB Q² improvement when compared with the LC-SCM system.

Fig. 4. Transmission results of (a) PDM-32QAM and (b) PDM-64QAM.

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5. Experimental setup

The experimental setup is shown in Fig. 5(a). The optimum subcarrier number is 8, which is found based on average ${\textrm{Q}^2}$ factor. 8-subcarrier PDM-32QAM signals with an aggregate symbol rate of 34.94-GBd are generated in the offline transmitter-side (Tx) DSP, which is the same as that in the simulations. The obtained waveforms are sent to a Ciena WaveLogic3 (WL3) transmitter which consists of an external cavity laser (ECL) with linewidth less than 100-kHz and a dual-polarization IQ modulator driven by four 39.4-GSa/s digital-to-analog converters (DACs). The ECL is operated at 1550.2-nm. The pre-amplified optical signals are fed into a recirculating loop which consists of four 80-km SSMF spans and four Erbium-doped fiber amplifiers (EDFAs). An optical bandpass filter (OBPF) within the loop is used to remove out-of-band amplified spontaneous emission (ASE) noise. At the receiver, an ECL is employed as a local oscillator (LO) for coherent detection. The four electrical outputs are captured by an 80-GSa/s real-time oscilloscope followed by the offline DSP.

Fig. 5. (a) Experimental setup and spectrum of the generated signal. VOA: variable optical attenuator. SW: switch. (b) Receiver-side DSP flow.

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The receiver-side DSP is shown in Fig. 5(b). First, the in-phase/quadrature (IQ) imbalance and frequency offset compensation (FOC) are performed. After that subcarrier de-multiplexing is performed in frequency domain as described in Section 4. Then 50% bulk CD compensation is applied, after that the proposed single-step M-SCM-DBP is implemented. A decision directed least mean square (DD-LMS) based adaptive filter is used to compensate for the rest 50% CD and to perform polarization de-multiplexing. Carrier phase recovery based on blind phase search (BPS) is implemented within the iteration of DD-LMS [22]. After the decision, BER is evaluated as an average over all the subcarriers and then converted to ${\textrm{Q}^2}$ factor. At a transmission distance of 960-km, the optimal number of subcarriers with the single-step M-SCM-DBP is found to be 8. Therefore, all the SCM systems employ 8 subcarriers in the following investigations.

6. Experiment results and discussions

We first evaluate the back-to-back (BTB) performance of the SCM-PDM-32QAM transmissions. SC transmission is also conducted for the purpose of comparison. The results are plotted in Fig. 6. The required OSNR for single carrier at the target BER of 3×10⁻² (Q²=5.49-dB) is 23.0-dB, which has a 3.4-dB implementation penalty in comparison with the theoretical limit. Relative to the SC transmission, the SCM transmission has an additional 0.6-dB OSNR penalty. Due to the limited DAC resolution, the additional OSNR penalty mainly comes from the increased peak-to-average power ratio (PAPR) of the SCM signals.

Fig. 6. BTB performance of PDM-32QAM signals.

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The results of various DBP schemes after 960-km SSMF transmission are shown in Fig. 7. The single-step M-SCM-DBP achieves 0.6-dB Q² improvement, in comparison to the SCM transmission with linear compensation (LC) only. Meanwhile, with LC only, the 8-subcarrier SCM transmission outperforms the SC scheme with 0.15-dB Q² improvement. In comparison with the 1-step SCM-DBP, the proposed single-step M-SCM-DBP can provide 0.1-dB Q² improvement. The performance gain in the experiment is less than the simulation results (0.3dB). This is because that the PDM-32QAM signal in the experiment is also affected by transceiver implementation noise, resulting in a smaller ratio of nonlinear noise. The single-step M-SCM-DBP, 2-step SCM-DBP, and 12-step LDBP (1 step/span) present similar performances. However, the complexity of the single-step M-SCM-DBP is 39.6% of the 2-step SCM-DBP and 13.7% of the 12-step LDBP as described in the previous section. One thing that should be noticed is that the linear penalty may be partially compensated by the DBP algorithms. In our experiment, the curves with and without the various DBP algorithms are based on the same measured waveform trace. Therefore, the performance gain observed in the linear regime indeed comes from the adoption of the DBP algorithms. Since our DBP algorithms were optimized in an adaptive manner, we speculate that the operations in them were converged to compensate some amount of transceiver nonlinearities from components such as RF driver and modulator. The reason behind it is left for future exploration.

Fig. 7. Transmission performance over 960-km SSMF.

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Next, in order to explore the capability of the adaptive filter for CD compensation, the Q² factor of M-SCM-DBP at optimal launch power (3-dBm) over 960-km SSMF is evaluated when the number of filter taps varies. For comparison, we implement a second stage CDC after the NLC of single-step M-SCM-DBP to completely compensate the residual CD. The results, as shown in Fig. 8, verify that the required number of taps is 13, no matter whether the adaptive filter is used to compensate the rest 50% CD or not. We infer that the adaptive filter with 13 taps is sufficient to compensate most of the residual CD, leading to a negligible Q² penalty of 0.05-dB relative to the scheme with the second CDC. However, when the tap number decreases, the performance of the system without the second CDC degrades quickly, because the filter length is less sufficient to compensate all ISI including the half CD.

Fig. 8. Required number of taps for the adaptive filter.

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Polarization tracking is another important issue of adaptive filters to ensure the stable transmission performance. Therefore, the impact of integrating CD compensation on the polarization tracking ability of the adaptive filter is also investigated, as shown in Fig. 9. The polarization rotation is implemented in digital domain before the receiver-side DSP. The performance of M-SCM-DBP with or without the second CDC over 960-km SSMF at the optimal launch power of 3-dBm is evaluated, when the polarization rotation speed ranges from 1-krad/s to 100-krad/s. It is observed that the performance is stable when the polarization rotation speed is less than 10-krad/s, and the CD compensation in the adaptive filter does not affect the polarization tracking ability.

Fig. 9. Polarization tracking ability of DD-LMS.

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Finally, Fig. 10 shows the achievable transmission distance of various fiber nonlinearity compensation schemes at a target BER 3×10⁻² (Q²=5.49-dB). The use of the SCM technique can extend the SSMF reach by 4% for PDM-32QAM. The single-step M-SCM-DBP can further extend the reach by 36% with the aid of NLC. Besides, the single-step M-SCM-DBP, 2-step SCM-DBP and 1 step/span LDBP schemes have similar maximum transmission distances. We can see that the reach extension of SCM for PDM-32QAM is less than that for PDM-16QAM and PDM-QPSK. However, although the SCM benefit decreases for higher order formats at shorter distances, the SCM enables the design of DBP with lower complexities.

Fig. 10. Transmission distance for various DBP schemes.

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7. Conclusion

The single-step M-SCM-DBP is proposed and experimentally demonstrated for single channel SCM-PDM-32QAM transmission. Additional IIR filters are introduced to the SSN compensation for more accurate NLI estimation. And the adaptive filter is used to compensate the residual 50% CD with further complexity reduction. The experiments demonstrate a 0.6-dB Q² improvement achieved by the single-step M-SCM-DBP over 960-km SSMF transmission. With 86.3% complexity reduction compared with LDBP, our scheme is able to extend the transmission distance over SSMF by 36% to 1220-km.

Funding

National Natural Science Foundation of China (61801291); Shanghai Rising-Star Program (19QA1404600); Key Technologies Research and Development Program (2018YFB1801203).

Disclosures

The authors declare no conflicts of interest.

References

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Single-step digital backpropagation for subcarrier-multiplexing transmissions

Abstract

1. Introduction

2. Operational principle

3. Complexity analysis

4. Simulation results

5. Experimental setup

6. Experiment results and discussions

7. Conclusion

Funding

Disclosures

References

Cited By

Figures (10)

Tables (1)

Equations (8)

Optics Express