# Comparison of Nonlinear Compensation Techniques for 400-Gb/s Coherent Multi-Band OFDM Super-Channels

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## Abstract

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## Featured Application

**The results of the comparison study, between the well-known digital backpropagation nonlinear compensation method and the 3rd-order inverse Volterra series transfer function nonlinear equalizer, reveal the physical limits of the nonlinear equalizers when high bit rates are transmitted in long-haul optical communication systems, in the presence of increased nonlinear effects. In addition, the computational complexity and, by extension, the cost, introduced by each equalizer, are taken into account. The insight gained from this study could be valuable in the foreseeable future in order to indicate the nonlinear compensation method of choice for the next generation inter-data centers supporting bit rates as high as 400 Gb/s.**

## Abstract

## 1. Introduction

^{2}-factor improvement on the nonlinear tolerance relatively to the multistep-per-span DBP-SSF method when 2 samples per symbol (SpS) are used for the equalization process [14]. Even though the equalization performance of the 3rd-order Volterra-based NLEs can compete with the widely-used DBP-SSF method, the increasing need for reduced computational complexity has triggered more in-depth studies on this topic [15]. One of them is the numerical study of the performance of a simplified Volterra NLE, which showed reduction of the total computational complexity by a factor of ~3 compared to the full Volterra NLE [15], when tested on a long-haul 224 Gb/s dual-polarization (DP), 16-ary quadrature amplitude modulation (QAM) transmission system [15]. Then, a 3rd-order inverse Volterra series transfer function (IVSTF)-NLE was studied via simulations in [16], on a single-channel transmission system using a 256 Gb/s DP, 16QAM signal after 1024 km of an SSMF, revealing a 1 dB Q

^{2}-factor improvement and a 50% reduction in the computational complexity by lowering the processing rate. Later, the performance of a variant of the 3rd-order IVSTF-NLE was studied in a coherent orthogonal frequency division multiplexing (OFDM) single-channel system of the same total rate and transmission distance [17] showing a Q

^{2}-factor improvement of 2 dB, i.e., 1 dB higher compared to the study of [16].

## 2. Experimental Section

^{®}using a 1024-point fast Fourier transform (FFT)/inverse FFT (IFFT). There are 576 subcarriers, each carrying 16QAM. They are generated by a 15-GHz bandwidth Keysight

^{®}arbitrary waveform generator (AWG) operating at 64 Gsamples/s. The in-phase and quadrature components of the coherent OFDM signal are electrically amplified using two linear RF driver amplifiers (SHF807), which feed a quadrature modulator (CMZM). The first CMZM generates the odd channels while the second generates the even channels, in order to totally decorrelate the data carried by adjacent channels. The odd and even combs are combined together using a 3-dB polarization maintaining (PM) coupler. The polarization multiplexing is realized by dividing the signal into two paths, putting a delay of one-OFDM-symbol-duration on one of the paths, switching it to the perpendicular polarization, and combining them together, as shown in Figure 2. The one-OFDM-symbol-duration delay corresponds to 1152 samples (1024 FFT size samples and 128 CP samples). Then, the OFDM super-channel is combined with 58 wavelengths modulated at 100 Gb/s using DP-QPSK and spaced 50-GHz apart from each other. The transmission line is composed of 10 spans of 100 km of SSMF. A single-stage erbium doped fiber amplifiers (EDFAs), with 20 dB gain and ~4.5 dB noise figure, is applied at the end of each span and for all the 10 spans. At this point, we note that the linear and non-linear equalization schemes that are presented in this paper are not adaptive but feed forward schemes. Although they may require exact knowledge of the system’s parameters to operate, their operation is not affected by the amount of accumulated noise at the receiving end. On the other hand, their effectiveness in improving the system performance is affected by the noise added by the EDFA elements along the link and the nonlinear mixing between signal and noise. This additional non-linear noise effect cannot be treated by any equalization scheme at the receiver, but only with the use of distributed compensation methods, such as all optical regenerators or cascaded optical phase conjugation (OPC) elements [24].

^{6}bits to evaluate the bit error rate (BER) by error counting.

## 3. Simulation Setup

^{−1}km

^{−1}[27]. Considering the parameters of LEAF, we have α = 0.19 dB/km, D = 4 ps/nm/km, and γ = 1.5 W

^{−1}km

^{−1}[27]. The BER is calculated by error counting. A total number of 436,224 bits per band are used.

## 4. Results and Discussion

_{Nsteps}, where N

_{steps}is the number of steps per span. In both, experimental and simulation, studies we have only detected the 3rd-band of the central OFDM super-channel, since inner bands are more affected by inter-channel nonlinear impairments compared to the outer bands. To compare the performance of different NLEs, we use the Q

^{2}-factor as a figure of merit, which is related to BER by Q

^{2}(dB) = 20log

_{10}[√2 erfc

^{−1}(2BER)] [16]. We estimated the net Q

^{2}-factor improvement ($\u2206{Q}^{2}$) in dB which is obtained as follows: First, we estimated the maximum Q

^{2}-factor value when only linear equalization is applied (max.${Q}_{\mathrm{w}.\text{}\mathrm{LE}}^{2}$). Second, we applied both, linear and nonlinear, equalization and we evaluated the corresponding maximum Q

^{2}-factor value (max.${Q}_{\mathrm{w}.\text{}\mathrm{NLE}}^{2}$). We define as $\u2206{Q}^{2}={Q}_{\mathrm{w}.\text{}\mathrm{NLE}}^{2}-{Q}_{\mathrm{w}.\text{}\mathrm{LE}}^{2}$ the net Q

^{2}-factor improvement, thus, the higher the $\u2206{Q}^{2}$, the better the performance of the NLE. For both, experimental and simulation, results four trials were conducted and then the average was calculated and recorded as the final value.

#### 4.1. Experimental Results

^{2}-factor as a function of the launch power after 10 × 100 km of SSMF for single-channel transmission with linear compensation only, as well as with IVSTF or DBP-SSF

_{1,2,8}NLEs (scenario described in Figure 3a). The Q

^{2}-factor improvement is ~1 dB (i.e., BER = 3.2 × 10

^{−3}with linear equalization only (wo. NLE) and BER = 9.9 × 10

^{−4}with linear and nonlinear equalization (w. NLE)) for all NLEs, which agrees with the published results for the single-channel, single-carrier system in [16].

^{2}-factor improvements are ~0.44 dB (i.e., BER = 5.3 × 10

^{−3}wo. NLE and BER = 3.6 × 10

^{−3}w. NLE) for the 3rd-order IVSTF-NLE, ~0.5 dB for the DBP-SSF

_{1}NLE, and ~0.6 dB (i.e., BER = 5.3 × 10

^{−3}wo. NLE and BER = 3.2 × 10

^{−3}w. NLE) for the DBP-SSF

_{2,8}NLEs. We observe that, in this scenario, the performance of the DBP-SSF method saturates at two steps per span.

^{2}-factor vs. input power per band for a central 4-band OFDM super-channel surrounded by WDM DP QPSK channels (scenario described in Figure 3c). The measured Q

^{2}-factor improvements are ~0.33 dB (i.e., BER = 7.5 × 10

^{−3}wo. NLE and BER = 5.8 × 10

^{−3}w. NLE) for the 3rd-order IVSTF-NLE, ~0.28 dB for the DBP-SSF

_{1}NLE, ~0.35 dB (i.e., BER = 7.5 × 10

^{−3}wo. NLE and BER = 5.7 × 10

^{−3}w. NLE) by the DBP-SSF

_{2}NLE and ~0.4 dB (i.e., BER = 7.5 × 10

^{−3}wo. NLE and BER = 5.5 × 10

^{−3}w. NLE) by DBP-SSF

_{8}NLE.

_{2}NLE. Therefore, as the nonlinear phenomena become stronger, larger number of steps per span are required for the DBP-SSF method to reach its performance limit. On the contrary, the 3rd-order IVSTF-NLE, which operates with a single step per span, performs almost as good as the DBP-SSF

_{2}NLE, with the latter being twice more complex than the former. Finally, we emphasize that the primary focus of the experimental results is to compare and gain a better insight about the potential of each NLE in the presence of very strong nonlinear effects. Although the performance results of all NLEs is modest, especially in the worst-case scenario (see Figure 5c), they are indicative of the fact that the 3rd-order IVSTF-NLE can compete with significantly lower computational effort the heavily iterative DBP-SSF with multi-steps per span.

#### 4.2. Simulation Results

^{2}-factor evaluation as a function of the input power per band after propagation through 10 × 100 km of an SSMF, using 1.8 SpS for the equalization process. All NLEs were optimized, as we did for the experimental study, through their adjustable parameters [9,16]. The simulation results are shown in Figure 6. The measured Q

^{2}-factor improvements, compared to linear equalization, are ~0.32 dB (i.e., BER = 5.9 × 10

^{−3}wo. NLE and BER = 4.6 × 10

^{−3}w. NLE) for the 3rd-order IVSTF-NLE, ~0.3 dB (i.e., BER = 5.9 × 10

^{−3}wo. NLE and BER = 4.6 × 10

^{−3}w. NLE) for the DBP-SSF

_{1}NLE, and ~0.35 dB (i.e., BER =5.9 × 10

^{−3}wo. NLE and BER = 4.5 × 10

^{−3}w. NLE) for the DBP-SSF

_{2,8}NLEs. There is a very good agreement with the experimental results represented in Figure 5c. Thus, our model is a fair representation of the experimental setup.

^{−2}or Q

^{2}= 6.25 dB), the maximum reach is 1400 km with linear equalization only. After nonlinear compensation, the maximum reach was extended ~3% for the 3rd-order IVSTF-NLE, ~3% for the DBP-SSF

_{1}NLE, ~4.6% for the DBP-SSF

_{2}NLE and ~7% for the DBP-SSF

_{8}NLE, with respect to linear equalization. We observe that the DBP-SSF method slightly outperforms the 3rd-order IVSTF-NLE, especially if the number of steps per span increases.

_{1}NLE, ~6.3% for the DBP-SSF

_{2}NLE, and ~7.3% for the DBP-SSF

_{8}NLE compared to linear case. The DBP-SSF method with eight steps per span is once more the NLE with the best performance, compared to all the aforementioned NLEs, but at the vast expense of computational complexity, as shown in the following sub-section.

_{8}NLE while the former is almost eight times less complex.

#### 4.3. Evaluation of Computational Complexity

^{2}-factor provided by the various NLEs as a function of complexity, after transmission through SSMF and LEAF.

_{FFT}) for both the linear and the nonlinear branches, while in [16] each of the nonlinear branches operate with N

_{FFT}/2. We decided to use the same N

_{FFT}in both, the linear and the nonlinear, branches because it was observed, through trial and error, that only then the 3rd-order IVSTF-NLE exhibits its best performance in the worst case scenario (see Figure 3c) and after 1000 km of SSMF using only 1.8 SpS. We consider a radix-2 implementation of FFT. In this case, the number of real multiplications for every FFT/IFFT is equal to 2log

_{2}N

_{FFT}[16]. Therefore, the FFT and the IFFT at the input and at the output of the IVSTF-NLE, respectively, are realized with 4log

_{2}N

_{FFT}real multiplications [16]. The linear branch of the IVSTF-NLE, as depicted in Figure 1a, operates with 4 real multiplications.

_{2}N

_{FFT}real multiplications in the time domain. Then, 5 real multiplications per sample per polarization are required for multiplying the total power of x and y polarizations with jc (i.e., C

_{IVSTF}). The latter multiplication is common between the two polarizations, thus, only 2.5 multiplications are required per sample per polarization. An FFT follows, with 2log

_{2}N

_{FFT}real multiplications, for the compensation of the residual CD in the frequency domain, adding 4 more real multiplications. Totally, 4 + 4log

_{2}N

_{FFT}+ N

_{spans}× (4log

_{2}N

_{FFT}+ 10.5) real multiplications per sample per polarization are needed for the operation of the 3rd-order IVSTF-NLE. For the DBP-SSF method, the required number of real multiplications per polarization per sample is N

_{steps}× N

_{spans}× (4log

_{2}N

_{FFT}+ 10.5) [16].

_{FFT}, the lower the computational complexity per block. At this point, the key question is if there is an optimum N

_{FFT}, at which the performance limit of each NLE can be reached with a minimum computational complexity per block. To answer this question, we applied, right before each NLE, the method of overlap and save (OS) to divide the received, uncompensated signal into small FFT blocks. After the full compensation of each block (i.e., linear and nonlinear), we aggregated them back together, at the output of each NLE, to form the output signal. Then, we measured the Q

^{2}-factor as a function of the N

_{FFT}, after 14 × 100 km of SSMF, at the optimum input power per band (i.e., −3 dBm). We chose this transmission reach because it corresponds to the FEC limit without nonlinear compensation (i.e., Q

^{2}= 6.25 dB). Following a similar rationale, we repeated the simulation runs after 11 × 100 km of LEAF, at optimum input power per band equal to −6 dBm.

_{FFT}. Table 1 and Table 2 show the Q

^{2}-factor improvement, with respect to the FEC limit, provided by each NLE, at the optimum N

_{FFT}as well as the number of real multiplications required per sample per polarization for these improvements. After SSMF transmission (Table 1), we observe that the 3rd-order IVSTF-NLE is as good as the DBP-SSF

_{8}NLE but with the latter being almost eight times more complex. After LEAF transmission (Table 2), we observe that, in the presence of stronger nonlinear effects, the 3rd-order IVSTF-NLE is as good as the DBP-SSF

_{2}NLE with the latter requiring ~2 times more real multiplications, compared to the former. Finally, the DBP-SSF

_{8}NLE is only marginally better compared to the IVSTF-NLE.

_{1,2}NLEs might sound modest, we still draw useful conclusions considering the physical limits of the NLEs in the presence of increased nonlinearities.

## 5. Conclusions

_{1,2}NLEs, compared to the DBP-SSF

_{8}NLE and the 3rd-order IVSTF-NLE, are only marginal, they are still indicative of the robustness of each and every NLE in the presence of exacerbated nonlinear effects. Thus, the 3rd-order IVSTF-NLE could be a reasonable choice for compensating nonlinear impairments with lower computational complexity and, by extension, lower cost.

## Author Contributions

## Conflicts of Interest

## Appendix A

_{x}and A

_{y}denote the signal components in the two polarization states, while the sum (|A

_{x}|

^{2}+ |A

_{y}|

^{2}) is the total signal power. The variables t and z denote the time and distance axes, respectively, α is the attenuation coefficient, β

_{2}is the group velocity dispersion parameter, γ′ = (8/9)γ is the effective nonlinear coefficient, and γ is the nonlinear coefficient [28]. According to [16], the solution of (A1) and (A2) can be expended into Volterra series transfer function (VSTF) kernels up to the 3rd-order as follows:

_{x}(ω,z) and A

_{y}(ω,z) are the Fourier transforms of the signal components in the x and y polarization after distance z, while A

_{x}(ω) = A

_{x}(ω,0) and A

_{y}(ω) = A

_{y}(ω,0) are the optical signal spectra at the input of the fiber (i.e., z = 0). In addition, H

_{1}(ω,z) and H

_{3}(ω

_{1},ω

_{2},ω − ω

_{1}+ ω

_{2}) are the first- and third-order VSTF kernels, respectively. For an optically amplified transmission link with N spans, the mathematical expressions for the corresponding first- and third-order inverse VSTF (IVSTF) kernels are as follows [16]

_{1}− ω) × (ω

_{1}− ω

_{2}) is the spacing between the discrete frequencies in the sampled spectrum, L denotes the span length, and N is the total number of fiber spans. The last two equations (i.e., (A5) and (A6)) can be realized by the scheme shown in Figure 1a,b, respectively.

_{3,k}(ω

_{1},ω

_{2},ω − ω

_{1}+ ω

_{2}) with transfer function [16]

^{N−k}with transfer function [16]

## References

- Essiambre, R.-J.; Kramer, G.; Winzer, P.J.; Foschini, G.J.; Goebel, B. Capacity limits of optical fiber networks. J. Lightware Technol.
**2010**, 28, 667–701. [Google Scholar] [CrossRef] - Temprana, E.; Myslivets, E.; Kuo, P.-P.; Liu, L.; Ataie, V.; Alic, N.; Radic, S. Overcoming Kerr-induced capacity limit in optical fiber transmission. Science
**2015**, 384, 1445–1448. [Google Scholar] [CrossRef] [PubMed] - Cartledge, J.C.; Ellis, A.D.; Shiner, A.I.; Abd EL-Raman, A.I.; McCarthy, M.E.; Reimer, M.; Borowie, A.; Kashi, A. Signal processing techniques for reducing the impact of fiber nonlinearities on system performance. In Proceedings of the Optical Fiber Communication Conference and Exhibition (OFC), Anaheim, CA, USA, 20–24 March 2016. [Google Scholar]
- Liu, X.; Chaplyvy, R.; Winzer, P.J.; Tkach, R.W.; Chandrasekhar, S. Phase-conjugated twin waves for communication beyond the Kerr nonlinearity limit. Nat. Photonics
**2013**, 7, 560–568. [Google Scholar] [CrossRef] - Shiner, A.; Reimer, A.; Borowiec, A.; Oveis Gharan, S.; Gaubette, J.; Mehta, P.; Charlton, D.; Roberts, K.; O’Sullivan, M. Demonstration of an 8-dimensional format with reduced inter-channel nonlinearities in polarization multiplexed coherent systems. Opt. Express
**2014**, 22, 20366–20374. [Google Scholar] [CrossRef] [PubMed] - Tao, Z.; Dou, L.; Yan, W.; Li, L.; Hoshida, T.; Rasmussen, J.C. Multiplier-free intrachannel nonlinearity compensation algorithm operating at symbol rate. J. Lightware Technol.
**2011**, 29, 20366–20374. [Google Scholar] [CrossRef] - Liang, X.; Kumar, S. Multi-stage perturbation theory for compensating intra-channel nonlinear impairments in fiber-optic links. Opt. Express
**2014**, 22, 29733–29745. [Google Scholar] [CrossRef] [PubMed] - Gao, Y.; Cartledge, J.C.; Karar, A.S.; Yam, S.S.-H.; O’Sullivan, M.; Laperle, C.; Borowiec, A.; Roberts, K. Reducing the complexity of perturbation based nonlinearity pre-compensation using symmetric EDC and pulse shaping. Opt. Express
**2014**, 22, 1209–1219. [Google Scholar] [CrossRef] [PubMed] - Ip, E.; Kahn, J.M. Compensation of dispersion and nonlinear impairments using digital backpropagation. J. Lightware Technol.
**2008**, 26, 3416–3425. [Google Scholar] [CrossRef] - Napoli, A.; Maalej, Z.; Kuschnerov, M.; Rafique, D.; Timmers, E.; Spinnler, B.; Rahman, T.; Coelho, L.D.; Hanik, N. Reduced complexity digital back-propagation methods for optical communication systems. J. Lightware Technol.
**2014**, 32, 3416–3425. [Google Scholar] [CrossRef] - Maher, R.; Lavery, D.; Millar, A.; Alvarado, A.; Parsons, K.; Killey, R.; Bayvel, P. Reach enhancement of 100% for a DP-64QAM super-channel using MC-DBP. In Proceedings of the Optical Fiber Communication Conference and Exhibition (OFC), Los Angeles, CA, USA, 22–26 March 2015. [Google Scholar]
- Peddanarappagari, K.V.; Brandt-Pearce, M. Volterra series transfer function of single-mode fibers. J. Lightware Technol.
**1997**, 15, 2232–2241. [Google Scholar] [CrossRef] - Vannucci, A.; Serena, P.; Bononni, A. The RP method: A new tool for the iterative solution of the nonlinear Schrödinger equation. J. Lightware Technol.
**2002**, 20, 1102–1112. [Google Scholar] [CrossRef] - Guiomar, F.P.; Reis, J.D.; Teixeira, A.L.; Pinto, A.N. Mitigation on intra-channel nonlinearities using a frequency-domain Volterra series equalizer. Opt. Express
**2012**, 20, 1360–1369. [Google Scholar] [CrossRef] [PubMed] - Guiomar, F.P.; Pinto, A.N. Simplified Volterra series nonlinear equalizer for polarization multiplexed coherent optical systems. J. Lightware Technol.
**2013**, 31, 3879–3891. [Google Scholar] [CrossRef] - Liu, L.; Li, L.; Huang, Y.; Cui, K.; Xiong, Q.; Hauske, F.N.; Xie, C.; Cai, Y. Intrachannel nonlinearity compensation by inverse Volterra series transfer function. J. Lightware Technol.
**2012**, 21, 310–316. [Google Scholar] [CrossRef] - Giacoumidis, E.; Aldaya, I.; Jarajreh, M.A.; Tsokanos, A.; Le, S.T.; Farjady, F.; Jaouën, Y.; Ellis, A.D.; Doran, N.J. Volterra-based reconfigurable nonlinear equalizer for coherent OFDM. IEEE Photonics Technol. Lett.
**2014**, 26, 1383–1386. [Google Scholar] [CrossRef] - Gagni, M.; Guiomar, F.P.; Wabnitz, S.; Pinto, A.N. Simplified high-order Volterra series transfer function for optical transmission links. Opt. Express
**2017**, 25, 2446–2459. [Google Scholar] [CrossRef] [PubMed] - Amari, A.; Dobre, O.A.; Venkatesan, R. Fifth-order Volterra based equalizer for fiber nonlinearity compensation in Nyquist WDM superchannel system. In Proceedings of the International Conference on Transparent Optical Networks (ICTON), Girona, Spain, 2–6 July 2017. [Google Scholar]
- Klekamp, A.; Dischler, R.; Buchali, F. Transmission reach of optical OFDM superchannels with 10-600 Gb/s for transparent bit-rate adaptive networks. In Proceedings of the 37th European Conference and Exhibition on Optical Communication (ECOC), Geneva, Italy, 18–22 September 2011. [Google Scholar]
- Qiu, M.; Zhuge, Q.; Changon, M.; Gao, Y.; Xu, X.; Morsy-Osman, M.; Plant, D.V. Digital subcarrier multiplexing for fiber nonlinearity mitigation in coherent optical communication system. Opt. Express
**2014**, 22, 18770–18777. [Google Scholar] [CrossRef] [PubMed] - Zhuge, Q.; Chatelain, B.; Plant, D.V. Comparison of Intra-Channel Nonlinearity Tolerance between Reduced-Guard-Interval CO-OFDM Systems and Nyquist Single Carrier Systems. In Proceedings of the Optical Fiber Communication Conference and Exposition (OFC/NFOEC), Los Angeles, CA, USA, 4–8 March 2012. [Google Scholar]
- Bosco, G.; Carena, A.; Curri, V.; Poggiolini, P.; Forghieri, F. Performance Limits of Nyquist-WDM and CO-OFDM in High-Speed PM-QPSK Systems. IEEE Photonics Technol. Lett.
**2010**, 22, 1129–1131. [Google Scholar] [CrossRef] - Ellis, A.D.; Al Khateeb, M.A.Z.; McCarthy, M.E. Impact of optical phase conjugation on the nonlinear Shannon limit. J. Lightware Technol.
**2017**, 35, 792–798. [Google Scholar] [CrossRef] - Pincemin, E.; Song, M.; Karaki, J.; Zia-Chahabi, O.; Guillosou, T.; Grot, D.; Thouenon, G.; Betoule, C.; Clavier, R.; Poudoulec, A.; et al. Multi-channel OFDM transmission at 100 Gbps with sub-channel optical switching. J. Lightware Technol.
**2014**, 32, 2202–2219. [Google Scholar] [CrossRef] - Savory, S.J. Digital filters for coherent optical receivers. Opt. Express
**2008**, 16, 804–817. [Google Scholar] [CrossRef] [PubMed] - Wood, W.A.; Ten, S.; Roudas, I.; Sterlingov, P.M.; Kaliteevskiy, N.A.; Downie, J.D.; Rukosueva, M. Relative importance of optical fiber effective area and attenuation in span length optimization of ultra-long 100 Gbps PM-QPSK systems. In Proceedings of the SubOptic Conference, Paris, France, 22–25 April 2013. [Google Scholar]
- Agrawal, G.P. Fiber-Optic Communication System, 3rd ed.; Wiley-Interscience: New York, NY, USA, 2002; pp. 64–65. ISBN 0471215716. [Google Scholar]

**Figure 1.**(

**a**) Block diagram of the 3rd-order IVSTF-NLE; (

**b**) Operating principle of the kth nonlinear compensation branch of the 3rd-order IVSTF-NLE [16]. (Symbols: Inverse Fast Fourier Transform (IFFT), Linearly Equalized (LE), Non-Linearly Equalized (NLE), Fast Fourier Transform (FFT)).

**Figure 2.**Experimental setup. (Symbols: external cavity laser (ECL), polarization-maintaining (PM) coupler, polarization-maintaining erbium-doped fiber amplifier (PM EDFA), quadrature modulator (CMZM), arbitrary waveform generator (AWG), digital-to-analog converter (DAC), RF driver amplifier, variable optical attenuator (VOA), optical delay line (ODL), polarization beam combiner (PBC), dynamic gain equalizer (DGE), optical bandpass filter (OBPF), polarization beam splitter (PBS), analog-to-digital converter (ADC)).

**Figure 3.**Spectra of: (

**a**) a single-channel transmission of a single-band DP, 16-ary quadrature amplitude modulation (QAM) orthogonal frequency division multiplexing (OFDM) signal, (

**b**) Wavelength division multiplexing (WDM) transmission with a 2-band DP, 16QAM OFDM signal, (

**c**) WDM transmission with a 4-band, DP, 16QAM OFDM super-channel at the center of the WDM comb and (

**d**) the multi-band (MB) OFDM super-channel. The black arrow, in each case, indicates the band under measurement.

**Figure 4.**Simulation setup of a 4-band, single-polarization, OFDM super-channel using as a transmission link either an SSMF or a LEAF. (Abbreviations: pseudo-random bit sequence (PRBS), OFDM transmitter (OFDM Tx), digital-to-analog converter (DAC), standard single-mode fiber (SSMF), large effective area fiber (LEAF), erbium-doped fiber amplifier (EDFA), polarization-diversity coherent receiver (Polariz. Divers. CohRx), analog-to-digital converter (ADC), nonlinear equalizer (NLE), OFDM receiver (OFDM Rx), and bit-error-rate (BER).

**Figure 5.**Experimental results on Q

^{2}-factor vs. input power per band without/with IVSTF-NLE and DBP-SSF

_{1,2,8}in (

**a**) single-channel with a single OFDM band; (

**b**) a single-OFDM-band super-channel surrounded by WDM DP-QPSK channels; and (

**c**) a WDM, 4-band OFDM super-channel surrounded by WDM DP-QPSK channels after transmission through 10 × 100 km SSMF.

**Figure 6.**Simulation results on Q

^{2}-factor vs. input power per band without/with IVSTF-NLE and DBP-SSF

_{1,2,8}NLEs for the 3rd-band of the central OFDM in a WDM system with three wavelengths, 4-band DP-16QAM OFDM super-channels after 10 × 100 km of SSMF.

**Figure 7.**Maximum transmission reach at the FEC limit vs. launch power using IVSTF-NLE and DBP-SSF

_{1,2,8}NLEs for the 3rd band of the center OFDM super-channel in the simulation case, after signal transmission through (

**a**) SSMF and (

**b**) LEAF.

**Figure 8.**Q

^{2}-factor as a function of the N

_{FFT}, at the optimum input power per band, after (

**a**) 14 × 100 km of SSMF and (

**b**) 11 × 100 km of LEAF.

**Table 1.**Maximum Q

^{2}-factor and the corresponding number of real multiplications per sample per polarization required at the optimum N

_{FFT}, after 14 × 100 km of SSMF at optimum input power equal to −3 dBm.

NLE | Optimum N_{FFT} | Maximum Q^{2} (dB) | Maximum BER | Number of Real Multiplications |
---|---|---|---|---|

3rd-order IVSTF | 256 | 6.5 | 1.7 × 10^{−2} | 631 |

DBP-SSF_{1} | 256 | 6.4 | 1.8 × 10^{−2} | 595 |

DBP-SSF_{2} | 512 | 6.43 | 1.8 × 10^{−2} | 1302 |

DBP-SSF_{8} | 512 | 6.5 | 1.7 × 10^{−2} | 5208 |

_{FFT}), Inverse Volterra series transfer function (IVSTF), Digital backpropagation (DBP) based on split-step Fourier (DBP-SSF).

**Table 2.**Maximum Q

^{2}-factor and the corresponding number of real multiplications per sample per polarization required at the optimum N

_{FFT}, after 11 × 100 km of LEAF at optimum input power equal to −6 dBm.

NLE | Optimum N_{FFT} | Maximum Q^{2} (dB) | Maximum BER | Number of Real Multiplications |
---|---|---|---|---|

3rd-order IVSTF | 512 | 6.65 | 1.6 × 10^{−2} | 552 |

DBP-SSF_{1} | 512 | 6.6 | 1.6 × 10^{−2} | 512 |

DBP-SSF_{2} | 512 | 6.65 | 1.6 × 10^{−2} | 1023 |

DBP-SSF_{8} | 512 | 6.7 | 1.5 × 10^{−2} | 4092 |

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**MDPI and ACS Style**

Vgenopoulou, V.; Song, M.; Pincemin, E.; Jaouën, Y.; Sygletos, S.; Roudas, I.
Comparison of Nonlinear Compensation Techniques for 400-Gb/s Coherent Multi-Band OFDM Super-Channels. *Appl. Sci.* **2018**, *8*, 447.
https://doi.org/10.3390/app8030447

**AMA Style**

Vgenopoulou V, Song M, Pincemin E, Jaouën Y, Sygletos S, Roudas I.
Comparison of Nonlinear Compensation Techniques for 400-Gb/s Coherent Multi-Band OFDM Super-Channels. *Applied Sciences*. 2018; 8(3):447.
https://doi.org/10.3390/app8030447

**Chicago/Turabian Style**

Vgenopoulou, Vasiliki, Mengdi Song, Erwan Pincemin, Yves Jaouën, Stylianos Sygletos, and Ioannis Roudas.
2018. "Comparison of Nonlinear Compensation Techniques for 400-Gb/s Coherent Multi-Band OFDM Super-Channels" *Applied Sciences* 8, no. 3: 447.
https://doi.org/10.3390/app8030447