Low complexity digital backpropagation for high baud subcarrier-multiplexing systems

In this paper, we propose two modifications to reduce the complexity of the subcarrier-multiplexing (SCM) based digital backpropagation (DBP) for high symbol rate SCM systems. The first one is to reduce the number of interfering subcarriers (RS-SCMDBP) when evaluating the cross-subcarrier nonlinearity (CSN). The second one is to replace the original frequency domain CSN filters with the infinite impulse response (IIR) filters (IIRRS-SCM-DBP) in the CSN compensation. The performance of the proposed schemes are numerically evaluated in three-channel dual-polarization (DP) 16QAM wavelength-division multiplexing (WDM) transmissions. The aggregate symbol rate for each channel is 120 GBaud and the transmission distance is 1600 km. For the SCM system with 16 subcarriers, the IIR-RS-SCM-DBP with only 4 interfering subcarriers and 2 steps can achieve a 0.3 dB Qfactor improvement in the WDM transmission. Compared to the original SCM-DBP, the proposed IIR-RS-SCM-DBP reduces the complexity by 48% at a performance loss of only 0.07 dB. ©2016 Optical Society of America OCIS codes: (060.1660) Coherent communications; (060.2330) Fiber optics communications References and links 1. X. Liu, S. Chandrasekhar, and P. J. Winzer, “Digital signal processing techniques enabling multi-Tb/s superchannel transmission [An overview of recent advances in DSP-enabled superchannels],” IEEE Signal Process. Mag. 31(2), 16–24 (2014). 2. J. Zhang, J. Yu, Z. Jia, and H.-C. Chien, “400 G transmission of super-Nyquist-filtered signal based on singlecarrier 110-GBaud PDM QPSK with 100-GHz grid,” J. Lightwave Technol. 32(19), 3239–3246 (2014). 3. S. Randel, D. Pilori, S. Corteselli, G. Raybon, A. Adamiecki, and A. Gnauck, “All-electronic flexibly programmable 864-Gb/s single-carrier PDM-64-QAM,” in Proc. Opt. Fiber Commun. Conf. (2014), paper Th5C.8. 4. G. Raybon, A. Adamiecki, P. J. Winzer, S. Randel, L. 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Vol. 24, No. 15 | 25 Jul 2016 | OPTICS EXPRESS 17027 #263546 http://dx.doi.org/10.1364/OE.24.017027 Journal © 2016 Received 21 Apr 2016; revised 18 Jun 2016; accepted 18 Jun 2016; published 19 Jul 2016 13. W. Yan, Z. Tao, L. Dou, L. Li, S. Oda, T. Tanimura, T. Hoshida, and J. C. Rasmussen, “Low complexity digital perturbation back-propagation,” in Proc. European Conference and Exposition on Optical Communications (2011), paper Tu.3.A.2. 14. F. Zhang, Q. Zhuge, M. Qiu, W. Wang, M. Chagnon, and D. Plant, “XPM model based digital backpropagation for subcarrier-multiplexing systems,” J. Lightwave Technol. 33(24), 5140–5150 (2015). 15. D. Marcuse, A. R. Chraplyvy, and R. W. Tkach, “Dependence of cross-phase modulation on channel number in fiber WDM systems,” J. Lightwave Technol. 12(5), 885–890 (1994). 16. E. Mateo, L. Zhu, and G. Li, “Impact of XPM and FWM on the digital implementation of impairment compensation for WDM transmission using backward propagation,” Opt. 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Introduction
Coherent optical transmission systems have been commercialized and deployed in regional, long-haul and submarine backbone networks [1].Coherent transceivers typically deliver symbol rates of 30~35 GBaud, supporting 50, 100, 150 and 200 Gb/s with BPSK, QPSK, 8QAM and 16QAM formats, respectively.Next generation coherent transceivers are expected to carry higher symbol rates and spectrally-efficient formats for 400 Gb/s and 1 Tb/s applications.For example, 110 GBaud single-carrier transmission with dual-polarization (DP)-QPSK was recently demonstrated in [2].Moreover, 80 GBaud signal-carrier transmissions with DP-16QAM format and 72 GBaud single-carrier transmissions with DP-64QAM were demonstrated [3,4].For the multi-carrier or superchannel signal transmission, the aggregate symbol rate can be higher than 100 GBaud and the bit rate can be higher than 1 Tb/s [5].On one hand, high spectral-efficiency formats are more vulnerable to fiber nonlinearities and are limited to shorter distances.On the other hand, with higher symbol rates, the intra-channel nonlinearities will become more dominant, which suggests that the digital nonlinear compensation will be beneficial in a wavelength-division multiplexing (WDM) transmission systems.In this context we anticipate a heightened role for digital nonlinear compensation in next generation 400 Gb/s and 1 Tb/s systems.Recently, it was numerically [6,7], experimentally [8], and theoretically [9] demonstrated that subcarriermultiplexing systems can achieve a higher nonlinear tolerance by optimizing the symbol rate of each subcarrier compared to single carrier (SC) systems.This improvement is more significant in high symbol rate systems, since the total symbol rate is further from the optimal symbol rate [6][7][8][9].
Digital backpropagation (DBP) is one of the widely investigated methods for fiber nonlinearity compensation.However, due to the large required number of steps, each of which involves a fast Fourier transform (FFT) and an inverse FFT (IFFT), the computational complexity of the conventional DBP is prohibitively high [10,11].Many modified DBPs such as low-pass-filter (LPF) assisted DBP (LDBP) [12] and perturbation based DBP [13] were proposed to reduce the complexity.However, as the symbol rate increases, more steps will be needed because of the enhanced chromatic dispersion (CD) walk-off between frequency components.
Recently, we proposed a cross-phase modulation (XPM) model based DBP algorithm for subcarrier-multiplexing (SCM) systems, denoted as SCM-DBP [14].In this scheme, the selfsubcarrier nonlinearity (SSN) of each subcarrier and the cross-subcarrier nonlinearity (CSN) between subcarriers are compensated separately.It was demonstrated in a 34.94 GBaud single channel SCM experiment with transmission distances of 6400 km for QPSK and 2560 km for 16QAM that SCM-DBP only required two steps to achieve sizable improvements.
In this paper, we propose further complexity reductions for SCM-DBP in high symbol rate WDM transmission systems.Firstly, we reduce the number of interfering subcarriers involved in the processing of SCM-DBP, denoted as RS-SCM-DBP, without significantly degrading the system performance.Then a first order Butterworth infinite impulse response (IIR) filter is utilized to replace the CSN LPF in the CSN compensation in order to further reduce the complexity, which is denoted as IIR-RS-SCM-DBP.In a 3 × 120 GBaud WDM simulation of DP-16QAM 16-subcarrier SCM signals at a transmission distance of 20 × 80 km, we show that IIR-RS-SCM-DBP with only 4 interfering subcarriers in CSN compensation only reduces the Q factor by 0.07 dB compared to SCM-DBP at 2 steps, while the complexity of the former is reduced by 48% compared to the latter.The performance of LDBP with and without cross polarization modulation (XPolM) is also included for comparison, which diminishes rapidly as the number of steps decreases.Finally, the IIR-RS-SCM-DBP with only 2 steps achieves ~0.3 dB Q-factor improvement in the SCM WDM transmission.The overall performance improvement compared to SC systems with linear compensation only is ~0.8 dB.

Reduction of the interfering subcarrier number in SCM-DBP
SCM-DBP was proposed in [14] to significantly reduce the required number of stages in DBP based nonlinearity compensation for long-haul SCM transmission systems.In the SCM-DBP scheme, SSN and CSN are similar to the self-phase modulation (SPM) and cross-phase modulation (XPM) in WDM transmission systems, respectively.They are compensated separately for each subcarrier in parallel as illustrated in Fig. 1, which shows the spectrum of an eight-subcarrier SCM signal with the leftmost subcarrier as the probe subcarrier.For the CSN compensation, the amounts of CSN-induced phase noise and polarization crosstalk need to be evaluated first [14].Then the contributions from all the interfering subcarriers are summed up and applied to the probe subcarrier.This requires a complexity order of O(N SCM 2 ) for all the subcarriers in one step, where N SCM is the number of subcarriers.Apparently the required complexity increases for higher symbol rate systems, where the optimal number of subcarriers for nonlinearity mitigation becomes larger [6,8,14].If we only consider the contributions of the closest N RS interfering subcarriers in the CSN compensation, the complexity can be reduced to the order of O(N SCM N RS ).In fact, it has been well understood that the influence of nonlinear interference decreases as channel spacing increases in nonlinear propagation [15] and WDM nonlinear compensation [16].This property can be leveraged to achieve a complexity reduction of the CSN compensation but the tradeoff between performance and complexity should be carefully quantified.In the following discussion, we show that by reducing the number of interfering subcarriers and utilizing the symmetric property of CSN LPF the complexity can be reduced without severely degrading the system performance.
In a transmission link with N spans, the CSN compensation transfer function, which can be modeled as a LPF, is expressed as [14] where H N (ω) and H 1 (ω) are CSN LPFs for a transmission link with N spans and one span [14,17], respectively.∆β = β 2 (ω I -ω P ) is the group velocity difference between the probe subcarrier and interfering subcarrier, with β 2 , ω P , and ω I being the second order dispersion coefficient, the central angular frequency of the probe subcarrier, and the central angular frequency of the interfering subcarrier, respectively.L span , γ, and α are the span length, nonlinear coefficient, and attenuation coefficient, respectively.Figure 2 shows the CSN LPF obtained from Eqs. ( 1) and ( 2) for different interfering subcarriers relative to the probe subcarrier illustrated in Fig. 1.We see that the bandwidth of the CSN LPF decreases as the frequency spacing between the interfering subcarrier and probe subcarrier increases, resulting in a reduced CSN impact.Consequently, if we only consider the CSN of the neighboring interfering subcarriers in the CSN compensation, the computational complexity can be reduced without significantly sacrificing the system performance.Therefore, we propose a SCM-DBP scheme with a reduced number of interfering subcarriers (RS-SCM-DBP), which only considers the N RS nearest interfering subcarriers in the CSN compensation.In Eq. ( 2), the span length usually satisfies exp(-αL span )<<1.Consequently, the CSN LPF for one span per step can be approximated as As a result, combining Eq. ( 1) and Eq. ( 3), H N (ω) can be expressed as For the interfering subcarrier with a small subcarrier spacing to the probe subcarrier, the second term in the phase in Eq. ( 4), namely tan −1 (∆βω/α), can be assumed to be linear since (∆βω/α) is a relative small value.For a large subcarrier spacing case, the second term in the phase can be ignored when compared to the first term.Consequently, the CSN LPF has a linear phase response.For the two interfering subcarriers that are symmetrical about the probe subcarrier in the frequency domain as plotted in Fig. 3(a), the corresponding ∆β has an opposite sign, resulting in the same amplitude response and an opposite linear phase response of the CSN LPF.Consequently, the CSN LPFs of the symmetrical interfering subcarriers can be assumed to be the same.When calculating the accumulated CSN effects of each pair of the symmetrical subcarriers in the frequency domain, the intensity (phase noise) and polarization crosstalk can be summed together first, and then multiplied by their common CSN LPF.By doing so, the complex multiplications of this part can be approximately halved.If there are no interfering subcarriers symmetrically located about the probe subcarrier in the spectrum, as shown as Fig. 3(b), the complexity cannot be further reduced.Overall, in RS-SCM-DBP the number of interfering subcarriers is reduced and the symmetry of the interfering subcarriers is leveraged in the CSN compensation algorithm in order to reduce the computational complexity.

Application of the IIR filter
It was proposed in the perturbation DBP [13,18] to use an IIR filter in the nonlinear compensation (NLC) block to replace the FIR filter for complexity reduction.In the CSN compensation of SCM-DBP, IIR LPFs can also be used to replace the frequency domain CSN LPF in Eqs. ( 1) and (2).One obstacle of applying IIR filters is that IIR filters cannot deliver exact linear phase response.Fortunately, the problem can be solved by processing the data in the forward and backward direction to achieve zero-phase distortion [19].Figure 4(b) shows the implementation of zero-phase IIR filtering.In particular, after filtering the data in the forward direction by the IIR filter, the filtered sequence is reversed and passed through the filter again.Assuming that the Z domain transfer function for the forward filter is H(Z), the transfer function for the backward filter is H(Z −1 )Z -d , where d is the delay in the implementation.As a consequence, the transfer function for the IIR filter is H(Z)H(Z −1 )Z -d , implying that poles and zeros exist in mirror-image pairs, which achieves the linear phase response [19,20].
In this work, we propose to use the IIR filter to replace the CSN LPF in the CSN compensation step, which is denoted as IIR-RS-SCM-DBP, in order to further reduce the computational complexity.A Butterworth IIR filter will be used for the following investigations.It has been found through simulations that other IIR filter types such as Chebyshev Type 1 deliver similar performances.The analytical expression of the n th order Butterworth IIR filter is [20] 2 0 where ω 0 is the 3 dB cut-off frequency.The cut-off frequency is roughly proportional to the inverse of the frequency spacing between the interfering subcarrier and the probe subcarrier.This is because the larger the frequency spacing, the smaller CSN effect will have on the probe subcarrier, leading to a smaller cut-off frequency.Hence, the CSN IIR filter can be written as where ξ is the bandwidth parameter which is adjusted to optimize the system performance when IIR-RS-SCM-DBP is applied.Through our investigations we have observed that the order of the Butterworth IIR filter has very little impact on system performance.Therefore, in the following discussions we only consider the first order Butterworth IIR filter, which has a comparable performance to the frequency domain LPF with a relatively small complexity demand.

Analysis of computational complexity
Reference [14] gives the detailed analysis of the complexity requirement for SCM-DBP.In order to verify the effectiveness of both RS-SCM-DBP and IIR-RS-SCM-DBP schemes in reducing the computational complexity, we investigate the required number of the complex multiplications for the two schemes.In addition, the complexity requirement of LDBP with cross-polarization modulation (XPolM) compensation, denoted as LDBP XPolM, is also evaluated as a reference.
The required number of complex multiplications per sample for RS-SCM-DBP with a compensation of M steps is obtained as ) where K CD,R is the block size per subcarrier for the chromatic dispersion compensation (CDC) filter in the frequency domain using RS-SCM-DBP.P CD,R is the overhead of the corresponding CDC filter per subcarrier, which has a length that is larger than the minimum tap length required for compensating the corresponding CD.We have , where T S is the sampling rate per subcarrier.K NL,R is the block size per subcarrier per polarization for the CSN LPF.P NL,R , which is the overhead of the CSN LPF, is the maximum value among all the minimum tap length requirements for all the CSN LPF.It can be approximated by The derivation of ( 7) is shown in Appendix A.
The required number of complex multiplications per sample for IIR-RS-SCM-DBP is obtained as , The derivation of ( 8) is shown in Appendix B. For the LDBP XPolM scheme, the required number of complex multiplications is , where K CD,L and P CD,L = (1 + r)π|β 2 |NL span / T S,All 2 are the block size and the overhead for the linear compensation filter in the frequency domain using LDBP XPolM, respectively.T S,All is the sampling rate for all the subcarriers, which equals to T S /N SCM when compared to the sampling rate per subcarrier.The block size of NLC filter is K NL,L .P NL,L is the overhead of Gaussian LPF, which is expressed as , 3 =2 2 ln(2) / ( ) , and Δω 3dB is the 3 dB bandwidth [21].The derivation of ( 9) is shown in Appendix C.

Simulation setup and results
Three-channel Nyquist-WDM systems with DP-16QAM SCM signals were simulated using MATLAB and OptiSystem.For each channel, the symbol rate was 120 GBaud, and the number of subcarriers for SCM signals was 16 or 32, which was around the optimal subcarrier number [6,8,14].The subcarriers were processed by root-raised-cosine (RRC) pulse shaping with a roll-off factor of 0.1.In addition, the performance of the single carrier signals were also investigated for comparison.Figure 5 shows the simulation setup.The transmitted signals were first generated in MATLAB and then sent to the OptiSystem for transmission.The linewidths of the transmitter laser and local oscillator were both 100 kHz.Transmitted-side DSP is shown in [8].In the transmitter, after I-Q modulation and polarization multiplexing, the 3 WDM channels were multiplexed and launched into the fiber.The link consists of 20 spans, with each span containing 80 km standard single mode fibers (SMF-28e + ) and an inline EDFA of 5 dB noise figure.The nonlinear coefficient of fiber was 1.3 W -1 .km −1 and PMD was neglected.The output signals from the link were pre-amplified and filtered.After de-multiplexing, the signals were sent to the coherent receiver for coherent detection.Note that ideal digital to analog converters (DAC) and analog to digital converters (ADC) were assumed and the transceiver induced no extra filtering effects in our simulations.The central WDM channel was processed in the MATLAB for performance evaluation.The launch power was individually optimized for each transmission result of various system configurations to obtain the optimal Q-factor of each case in the following results.

Simulation results of RS-SCM-DBP
The Q-factor versus the number of interfering subcarriers involved in the CSN compensation is shown in Figs.6(a) and 6(b) for SCM channels with 16 subcarriers and 32 subcarriers, respectively.Q-factor is converted directly from the bit error rate (BER) [22].The performance of linear compensation (LC) in SC and SCM systems, denoted as LC SC and LC SCM, respectively, are also plotted for comparison.
For SCM systems with 16 subcarriers, when the step size is small, the difference in Qfactor for the RS-SCM-DBP scheme with different numbers of interfering subcarriers is within 0.2 dB.As the step size becomes larger, the difference in Q is smaller.In this case, the complexity can be saved by reducing the number of interfering subcarriers without inducing a significant performance loss.The reason, which was mentioned earlier, is that the CSN contributions of the remote interfering subcarriers (with large frequency spacing) to the probe subcarriers are small.For SCM systems with 32 subcarriers, the performances of RS-SCM-DBP with different step sizes are almost the same.The reason is that SCM systems with lower symbol rate per subcarrier mitigate the CD-induced waveform distortion even in a very large step size [14].Comparing Fig. 6(a) to Fig. 6(b), SCM systems with 16 subcarriers perform better than SCM systems with 32 subcarriers.In the following discussion, we will focus on SCM systems with 16 subcarriers.Figure 7 shows the performance of systems with different compensation schemes.For comparison, the LDBP XPolM in SCM systems and SC systems, and LDBP in SCM systems, denoted as LDBP XPolM SCM, LDBP XPolM SC and LDBP SCM, respectively, are investigated.In addition, the performances with LC for SCM systems and SC systems are also plotted.Even with only 4 interfering subcarriers, the loss of RS-SCM-DBP compared to SCM-DBP is within 0.16 dB.With 10 spans/step and 20 spans/step, the loss is less than 0.07 dB.It is observed that RS-SCM-DBP with only 4 interfering subcarriers achieves a better performance than LDBP XPolM SCM for all the step sizes shown in Fig. 7.The performance difference is around 0.2 dB when the step size is between 4 spans/step and 10 spans/step.Under the comparable performance, RS-SCM-DBP with 4 interfering subcarriers can reduce the number of steps by around 2-4 times compared to LDBP XPolM SCM.This is because the waveform of each subcarrier evolves much more slowly than the whole signal, leading to a small CD distortion and a more accurately modeling using RS-SCM-DBP [14].When compared with LC in SC systems and LC in SCM systems, RS-SCM-DBP with 4 interfering subcarriers implemented at 10 spans/step (2 step for the whole link) can increase the Q value by 0.8 dB and 0.3 dB, respectively.

Simulation results of IIR-RS-SCM-DBP
In this section, we evaluate the system performance with IIR filters applied in the CSN compensation for a further complexity reduction.The number of subcarriers investigated in this section is 16 as mentioned earlier.Figure 8(a) shows the Q-factor versus the number of the interfering subcarriers with IIR-RS-SCM-DBP operating at different step sizes.The variation in Q is smaller for IIR-RS-SCM-DBP with different number of interfering subcarriers compared to RS-SCM-DBP in Fig. 6(a).For 10 and 20 spans/step, IIR-RS-SCM-DBP with 4 interfering subcarriers can achieve a performance that is very close to the maximum.Figure 8(b) shows the difference of Q-factor between IIR-RS-SCM-DBP and RS-SCM-DBP as a function of the number of interfering subcarriers at different step sizes.The difference is within 0.08 dB.Moreover, with less than 10 interfering subcarriers the difference is within around 0.02 dB which suggests that the application of IIR filters is efficient.

Discussion of computational complexity
The computational complexities of SCM-DBP, RS-SCM-DBP, IIR-RS-SCM-DBP and LDBP XPolM have been analyzed in Section 3. In this section, we discuss the required complexities of the system configurations used in our early simulations.Note that the double side IIR filter used in the nonlinear compensation requires additional buffering [18].However, compared to the original frequency domain CSN LPF, which is based on large size FFTs/IFFTs, the required buffer size of the IIR filter can be much smaller.Since in the evaluated DBP schemes the power consumption of buffering is relatively small with respect to that of complex multiplications, we use the number of complex multipliers as the metric for complexity evaluation in this section.
In particular, the SCM systems with 16 subcarriers are considered, and RS-SCM-DBP and IIR-RS-SCM-DBP only use 4 interfering subcarriers.Note that for the LDBP XPolM scheme the required complexity is independent of the number of subcarriers [14].For the number of overhead samples of either CD compensation filters or NLC filters, they can be directly calculated from the equations provided in Section 3. The block sizes of all the FFT/IFFT's are set to the closest power of 2 which makes the overhead less than 10% [14].Table 1 shows the details of the complexity calculation for the DBP schemes with 10 spans/step.P CD and P NL is the overhead of the corresponding CDC filter and nonlinear filter, respectively.K CD and K NL is FFT/IFFT size of the CDC filter and nonlinear filter, respectively.Figure 10 shows the required number of complex multiplications per sample for various DBP schemes in SCM systems.When compared to SCM-DBP, RS-SCM-DBP and IIR-RS-SCM-DBP can reduce the complexity by 28%-40% and 44%-51% for the same step size, respectively.When compared to LDBP XPolM, RS-SCM-DBP and IIR-RS-SCM-DBP can reduce the complexity by 12%-19% and 30%-37% for the same step size, respectively.In addition to the complexity reduction, it has been shown in Fig. 9 that RS-SCM-DBP and IIR-RS-SCM-DBP achieve better performance than LDBP XPolM.The complexity reduction of RS-SCM-DBP and IIR-RS-SCM-DBP compared to LDBP XPolM is a result of two contributions.On one hand, RS-SCM-DBP and IIR-RS-SCM-DBP have less complex multiplications than LDBP XPolM at the CD compensation block due to the reduced accumulated CD per subcarrier.One the other hand, because of either less interfering subcarriers or the application of an IIR filter in the NLC block, RS-SCM-DBP and IIR-RS-SCM-DBP complexities are significantly reduced which results in a comparable complexity requirement when compared to LDBP XPolM.

Conclusion
In this paper, two modifications are proposed to reduce the complexity of SCM-DBP.One is to reduce the number of interfering subcarriers in the CSN compensation, denoted as RS-SCM-DBP.And the other is to use an IIR filter as a replacement for the frequency domain CSN filter, denoted as IIR-RS-SCM-DBP.We numerically demonstrate the performance of the proposed schemes in three-channel Nyquist WDM systems, with each channel delivering 120 GBaud DP-16QAM signals at a 20 × 80 km transmission distance.Compared to SCM systems and SC systems with linear compensation only, the proposed IIR-RS-SCM-DBP with 4 interfering subcarriers and 2 steps will improve system performance by 0.3 dB and 0.8 dB, respectively.Moreover, compared to the original SCM-DBP with 2 steps, IIR-RS-SCM-DBP experiences a 0.07 dB performance degradation while achieving a complexity reduction of 48%.

Appendix A: Complexity of RS-SCM-DBP
For RS-SCM-DBP, the required complexity for one linear step is the same as SCM-DBP [14].The expression is , where the parameters are specified in Section 3.
With respect to the complexity of the NLC block of the SCM-DBP scheme, if the exponential operation is assumed to be implemented using a lookup table, the number of complex multiplications per sample in one NLC section is [14] , The second term in Eq. ( 11) is the required complexity per sample for the CSN-induced phase noise and CSN-induced XPolM compensation, with K NL,R (log 2 K NL,R + N SCM -1)/(K NL,R -P NL,R ) used for evaluating the CSN-induced phase noise and 0.5K NL,R (log 2 K NL,R + N SCM -1)/(K NL,R -P NL,R ) used for evaluating the shared CSN-induced XPolM term.Note that (N SCM −1) in the denominator is the number of interfering subcarriers for each probe subcarrier.
If the RS-SCM-DBP is used with N RS interfering subcarriers, the required complex multiplications per sample in one NLC section is , As mentioned in Section 2, if the interfering subcarriers are symmetric about the probe subcarrier in the spectrum, the number of complex multiplications for CSN compensation for the probe subcarrier in the frequency domain can be halved.In our model we assume the number of the interfering subcarriers N RS is even.For (N SCM -N RS ) probe subcarriers located around the center of the spectrum, all the interfering subcarriers are symmetric about the probe subcarrier.Hence the complex multiplications for evaluating the CSN induced phase noise can be expressed as 2(N SCM -N RS )[K NL,R log 2 K NL,R +K NL,R N RS /2] for the two polarizations.In the expression, K NL,R log 2 K NL,R is for calculating the FFTs and IFFTs implemented per probe subcarrier, and K NL,R N RS /2 is for N RS interfering subcarriers multiplying their corresponding CSN LPF relative to the probe subcarrier in the frequency domain [14].However, for the remaining N RS probe subcarriers located around the edge of the spectrum, not all of the interfering subcarriers are symmetric about the probe subcarriers, leading to 2N RS [K NL,R log 2 K NL,R +K NL,R (3N RS /4+1/2)] complex multiplications for the two polarizations for calculating the CSN-induced phase noise.In total, the complex multiplications for the CSNinduced phase noise for the two polarizations can be expressed as In addition, for the CSN-induced XPolM term, the number of complex multiplications is half compared to CSN-induced phase noise, since the two polarizations share one XPolM term.So the number of complex multiplications for evaluating the CSN-induced phase noise and XPolM is 3N As a result, when taking into account the symmetrical property of CSN LPF, the required number of complex multiplications per sample in one NLC section for RS-SCM-DBP is 2 , In conclusion, for a symmetric DBP scheme with M steps, which contains (M+1) LC sections and M NLC sections [14,21], the required number of complex multiplications per sample is )

Appendix B: Complexity of IIR-RS-SCM-DBP
The complexity requirement for IIR-RS-SCM-DBP in one linear step is the same as that of SCM-DBP, as given in Eq. (10).For the NLC, compared with SCM-DBP, the difference comes from the complexity for evaluating the CSN-induced phase noise and XPolM.
With the help of an IIR filter, the CSN compensation can be implemented in the time domain, resulting in no FFT and IFFT in the NLC section.For the following discussion, we only consider the complexity when the first order Butterworth IIR filter is applied.
The transfer function for the first order Butterworth IIR filter can be expressed in the Z domain as H B (Z)=b(1+Z -1 )/(1+a 1 Z -1 ) [20], where b and a 1 are the constants.In addition, it is concluded from Eq. ( 4) that the spectrum of CSN LPF has Hermitian symmetry, which means the tap coefficients of IIR filter in the time domain are real values [20].When evaluating the multiplication between the tap and the sample (the sample is a complex value), the number of required complex multiplication is 0.5.Taking into account forward and backward filtering, the complex multiplications for employing one first order IIR filter per sample is 2. For N RS interfering subcarriers, N RS IIR filters are needed.Hence the required number of complex multiplications per sample for compensating CSN-induced phase noise is 2N RS .Since the two polarizations share one XPolM term, the complexity requirement for the CSN-induced XPolM compensation is half of that for the CSN-induced phase noise compensation.As a result, the number of complex multiplications per sample for evaluating the CSN induced phase noise and XPolM is 3N RS .
In conclusion, for the DBP link with M steps, the required number of complex multiplications per sample is Reference [14] provides the detailed analysis of the complexity for LDBP.For LDBP XPolM, the complex multiplications for one linear step is the same as that of LDBP, which is In the NLC section, similar to LDBP, a Gaussian LPF is used to remove the high frequency components in order to accurately evaluate the nonlinear noise.The block size of the NLC filter is K NL,L .1.5 complex multiplications per sample are first required in the time domain, with 1 multiplication for evaluating the intensity term and 0.5 for evaluating the shared XPolM term.In order to evaluate the nonlinear phase noise in the frequency domain, two FFTs, two IFFTs and 2K NL,L complex multiplications are required for the two polarizations, which is expressed as 2K NL,L (log 2 K NL,L +1).Taking into account the XPolM compensation, the number of complex multiplications is 3K NL,L (log 2 K NL,L +1) for the two polarizations in one nonlinear step in the frequency domain.After evaluating the phase noise and polarization crosstalk in the frequency domain, 2 complex multiplications per sample are needed to obtain the final result [13,14] as follows. 1 complex multiplication is needed for the signal multiplying the phase noise exponential term, and the other multiplication is needed for orthogonal polarization multiplying the XPolM term.In total, the number of complex multiplications per sample for one nonlinear step is , In conclusion, for a DBP link with M steps, the number of complex multiplications per sample is ,

Fig. 1 .
Fig. 1.Spectrum of an eight-subcarrier SCM signal with the leftmost subcarrier as the probe subcarrier.IS: interfering subcarrier.

Fig. 2 .
Fig. 2. Spectrum of various CSN LPFs for different interfering subcarriers relative to the probe subcarrier.

Fig. 3 .
Fig. 3.The locations of the interfering subcarriers in the RS-SCM-DBP scheme with (a) the fourth subcarrier and (b) the first subcarrier as the probe subcarrier.PS: probe subcarrier.

Figure 4 (
a) shows the comparison of IIR LPF and CSN LPF for different interfering subcarriers with respect to the probe subcarrier as shown in Fig.1.It can be seen that IIR LPF can be used to represent the CSN LPF to some extent.

Fig. 6 .
Fig. 6.The Q-factor versus the number of interfering subcarriers for SCM systems with (a) 16 subcarriers and (b) 32 subcarriers.

Fig. 7 .
Fig.7.The Q-factor versus the step length for 16-subcarrier SCM systems and SC systems with various compensation schemes at the optimal launch power.

Fig. 8 .
Fig. 8. (a) The Q-factor versus the number of interfering subcarriers using IIR-RS-SCM-DBP.(b) Difference of Q-factor between IIR-RS-SCM-DBP and RS-SCM-DBP versus the number of interfering subcarriers Figure 9 shows the performances of various compensation schemes including IIR-RS-SCM-DBP with 4 interfering subcarriers.IIR-RS-SCM-DBP has nearly the same performance as RS-SCM-DBP at all the step sizes.For 10 and 20 spans/step, the performance differences between IIR-RS-SCM-DBP and SCM-DBP are within 0.06 dB.Compared to LDBP XPolM SCM, IIR-RS-SCM-DBP can increase the step size by a factor of 2-4 at a comparable performance.Moreover, at 10 spans/step IIR-RS-SCM-DBP can increase the Q value by ~0.2 dB compared to LDBP XPolM SCM.The overall performance improvement of IIR-RS-SCM-DBP compared to SC systems with linear compensation only is ~0.8 dB, including 0.3 dB coming from nonlinear compensation and 0.5 dB improvement coming from the SCM scheme.

Fig. 9 .
Fig. 9. Various compensation schemes for 16-subcarrier SCM systems and SC systems at the optimized launch power.

Fig. 10 .
Fig. 10.The required number of complex multiplications per sample for various DBP schemes in 16-subcarrier SCM systems.