W3A.4.pdf

OFC 2016 © OSA 2016

Experimental Comparison of Artificial Neural Network and Volterra based Nonlinear Equalization for CO-OFDM 1

2

3

4

2

2

E. Giacoumidis , S. T. Le , I. Aldaya , J. L. Wei , M. McCarthy , N. J. Doran , and B. J. Eggleton

1

1: Centre of Ultra High Bandwidth for Optical Systems (CUDOS), University of Sydney, Physics Road, NSW 2006, Sydney, Australia [email protected] 2: Aston Institute of Photonic Technologies, Aston University, Birmingham, B4 7ET, UK 3: Physics Institute, State University of Campinas, 777, Campinas, Brazil 4: ADVA Optical Networking SE, Campus Martinsried Fraunhoferstrasse 9a, Martinsried/Munich 82152, Germany.

Abstract: A novel artificial neural network (ANN)-based nonlinear equalizer (NLE) of low complexity is demonstrated for 40-Gb/s CO-OFDM at 2000 km, revealing ~1.5 dB enhancement in Q-factor compared to inverse Volterra-series transfer function based NLE. ©2016 Optical Society of America OCIS codes: (060.2330) Fiber optics communications; (060.1660) Coherent communications; (060.4080) Modulation

1. Introduction Endeavors to the Kerr nonlinearity limit have been performed by either inserting an optical phase conjugator (OPC) at the middle point of the link [1], or using electronic-based nonlinearity compensation (NLC) schemes such as digital back-propagation (DBP) [2], phase-conjugated twin-waves (PC-TW) [3], and nonlinear equalizers (NLE) based on the inverse Volterra-series transfer function (V-NLE) [4]. Unfortunately, OPC significantly reduces the flexibility in an optically routed network and requires both a symmetric chromatic dispersion (CD) map and power evolution. On the other hand, DBP is very complex, PC-TW halves the transmission capacity, and V-NLE shows marginal performance enhancement accompanied with a significant amount of floating-point operations. Recently, NLC based on transmitter-side DBP and employment of frequency reference carriers (FRC) was introduced in [5], which however required a sophisticated transmitter. Albeit the Kerr-mediate nonlinear process is deterministic, in multicarrier schemes like coherent optical OFDM (CO-OFDM) the resulting nonlinear interaction between subcarriers becomes so complicated that appears random due to its high peak-to-average power ratio (PAPR). This was confirmed in [6], in which NLC based on PC-subcarrier coding was hitlessly implemented in 8-QAM COOFDM due to inadequate compensation of inter-subcarrier interference (ICI) cross-phase modulation (XPM). Alternatively, nonlinearities could be mitigated using nonlinear decision classifiers such as artificial neural networks (ANN) [7] which learn the nonlinear impairments from observed data and build a probabilistic model of the impairment. ANN requires the transmission of an initial training sequence which might be high consuming in terms of required computational resources. Nonetheless, for a highly stable system, such as long-haul optical links, where CD and nonlinear effects do not change over time, ANN mapping could be accomplished once. In the simulation study in [7], 80-Gb/s ANN-NLE outperformed in terms of Q-factor to V-NLE by 1 dB at 1000 km. In this work, we experimentally demonstrate, for the first time, a novel ANN-NLE for 40-Gb/s 16-QAM COOFDM and compare it with the benchmark V-NLE. It is shown that ANN-NLE can reduce the nonlinearity penalty by ~2 dB at 2000 km. Compared to V-NLE, ANN-NLE outperforms by ~1.5 dB in Q-factor, concurrently providing a significant decrease in computational complexity. Numerical analysis indicates dependency of the nonlinearity penalty reduction on the length of the ANN training vector overhead (TVO) and the signal bit-rate.

(a) (b) Fig. 1. CO-OFDM receiver block diagram with (a) ANN-NLE, and (b) V-NLE [4]. LPF: low-pass filter; ADC: analog-to-digital converter; CP: cyclic prefix; (I)FFT: (inverse) fast-Fourier transform; MMSE: minimum mean-square error; HCD: nonlinear system chromatic dispersion.

W3A.4.pdf

OFC 2016 © OSA 2016

2. Proposed ANN and Volterra based NLEs for 16-QAM CO-OFDM In Fig. 1 (a), the ANN-NLE block-diagram for 16-QAM CO-OFDM is depicted which is similar to [7]. In summary, the ANN-NLE is comprised of k sub-neural networks (hidden layers), with each sub-network being associated to each subcarrier k and where s(k) is the training vector. The received symbols for each subcarrier x{k} are processed by the NLE neurons which are multiplied with the weight value for a given subcarrier/neuron wk,i, after which, the outputs of the different neurons are summed. In the training stage, the minimum mean-square error (MMSE) determines the error signal and updates the weights, which are iteratively updated until the desired error value is reached, thus indicating the optimum match between the sub-network output and the transmitted symbols. The error signal is given by (1)-(2): e(k) = s(k) − ŝ(k)

(1)

𝑠̂ (𝑘) = ∑𝑀 𝑖=1 𝑤𝑘,𝑖 𝜑𝑘,𝑖 (𝑠(𝑘))

(2)

E(n) = ‖S(n) − Ŝ(n)‖

2

(3)

where ŝ(k) is calculated in terms of a nonlinear activation function (NAF), 𝜑𝑘,𝑖 , performing the NLE. The NAF “f(.)” is a differential sigmoid function (SF) that solves the conflicting relationship between the boundedness and the differentiability of a complex function [7]. The SF is a “split” complex NAF, where two conventional real-valued functions process the I-Q components. The nonlinear Kerr-effect of a SSMF induces an approximately sigmoid-like power transmission function and therefore the SF-based NAF can tolerate the nonlinear spike or no-spike decision in spiking neurons. The number of neurons in every sub-neural network is equal to the number of points of the constellation, i.e. M in (2), which in the case of 16-QAM is 16. The ANN-NLE is based on the Riedmiller’s resilient-back propagation (RR-BP) algorithm performing an approximation to the global minimization achieved by the steepest descent. The training function updates the weights and bias values according to RR-BP, which minimizes the difference between the ANN output and the desired output by splitting the complex OFDM data in 2 real-valued data collections. The transfer functions for the hidden layer of the ANN-NLE are differentiable and similar to the hyperbolic tangent function. For the output layer, the linear function “purelin” is used. The block identified as MMSE in Fig. 1(a), represents the subsystem that implements RR-BP to find the weights that minimize the error vector. In (3) where 𝑆(𝑛) and 𝑆̂(𝑛) are the desired and calculated output vectors, respectively. The weights are updated according to the 5 steps described in [7] by applying the gradient descent on the cost function E(n) in order to reach a minimum. The block diagram of V-NLE is depicted in Fig. 1(b), which is similar to [7]. Compared to ANN, the V-NLE is placed after the ADCs to reduce DSP complexity by means of reducing the number of FFT/IFFT blocks. V-NLE inherits some of the features of the hybrid time-and-frequency domain implementation, such as non-frequency aliasing and simple implementation. From Fig. 1(b), it can be clearly identified that CD, i.e. (HCD)k, and the fiber nonlinearity are combated by the linear and nonlinear compensator tool, respectively. Very high-order Volterra kernels have not been considered here, thus offering ∼50% reduced computational complexity compared to single-step/span DBP. 3. Experimental transmission set-up, results, and theoretical analysis Fig. 2 depicts the experimental setup where an external cavity laser (ECL) with a linewidth of 100 KHz was modulated using a dual-parallel Mach-Zehnder modulator (DP-MZM) in IQ configuration. The DP-MZM modulator was fed with the I-Q components of the OFDM signal, which was generated offline. The transmission path at 1550.2 nm was a recirculating loop consisting of 20×100 km spans of Sterlite OH-LITE (E) fiber (attenuation of 18.9-19.5 dB/100 km) controlled by acousto-optic modulator (AOM). The loop switch was located in the mid-stage of the 1st Erbium-doped fiber amplifier (EDFA) and a gain-flattening filter (GFF) was placed in the mid-stage of the 3rd EDFA. The optimum launched optical power (LOP) was swept by controlling the output power of the EDFAs. At the receiver, the incoming signal was combined with another 100 KHz linewidth ECL acting as local oscillator (LO). After down-conversion, the baseband signal was sampled using a real-time oscilloscope operating at 80 GS/s and processed offline in MATLAB. 400 symbols were generated by 512-point inverse (I)FFT, 210 subcarriers were modulated with 16-QAM, while the rest were set to zero to reduce the PAPR. For ISI elimination due to linear dispersive effects, a CP of 2% was included.

Fig. 2. Experimental set-up of CO-OFDM equipped with ANN and V-NLE. ECL: external cavity laser, DSP: digital signal processing, AWG: arbitrary waveform generator, AOM: acousto-optic modulator, EDFA: Erbium-doped fiber amplifier, GFF: gain flatten filter, LO: local oscillator.

W3A.4.pdf

OFC 2016 © OSA 2016

The ANN training vector overhead was adjusted at 5% and by controlling the sampling rate we tailored the net signal bit-rate at ~40-Gb/s. The offline OFDM demodulator included both timing synchronization and frequency offset compensation, as well as I-Q imbalance compensation and CD compensation using an overlapped frequency domain equalizer employing the overlap-and-save method. The nonlinearity compensation capability was assessed based on Qfactor measurements averaging over 10 recorded traces (~106 bits), which was estimated from the BER obtained by error counting after hard-decision decoding. The Q-factor is related to BER by: Q = 20log10[√2𝑒𝑟𝑓𝑐 −1 (2𝐵𝐸𝑅)]. For 16QAM, a BER of 3×10-3 results in a required Q-factor of 8.7 dB. In Fig. 3 (a), the Q-factor against the LOP is plotted for a 40Gb/s CO-OFDM at 2000 km, with ANN-NLE, V-NLE and without (w/o) NLE. Nonlinearity reduction of ~2 dB at optimum LOP (FEC-limit) of 4 dBm is observed with 1.5 dB difference compared to V-NLE. ANN-NLE outperforms V-NLE due to the knack of reducing stochastic ICI–XPM/FWM. In Fig. 3(b), by simulation and using the same parameters as the experiment, we employed different lengths of ANN TVO at 5%, 10%, 15%, and 35% resulting in training lengths of 20, 40, 60, 140 symbols, respectively, and compared to V-NLE at different bit-rates at 1000 km of transmission (10 spans×100 km). It is shown that ANN-NLE for a soft-decision FEC-limit of 25% (Q = 6.25 dB) can reduce the nonlinearity penalty by ~4 dB at 70-Gb/s, with a minimum TVO of 10%, improving the Q-factor up to ~2 dB compared to V-NLE. V-NLE w/o NLE

10

Q-factor (dB)

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ANN-NLE TVO = 35% TVO = 15% TVO = 10% TVO = 5%

6 4 2 0

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65 70 75 Signal bit-rate (Gb/s)

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(a) (b) (c) Fig. 3. (a) Q-factor vs. launched power of ANN- and V-NLEs for 40-Gb/s experimental CO-OFDM at 2000 km. (b) Q-factor vs. bit-rate of ANN-NLE for numerical CO-OFDM at 1000 km for different training vector overhead (TVO). (c) Computational complexity comparison between ANN- and V-NLEs: Blue bars represent the computational complexity of V-NLE for different subcarrier number, NSC, and number of spans, Nspan, whereas red bars are for ANN-NLE considering various NSC, and bits per subcarrier, Nbits.

ANN- and V-NLEs are compared in Fig. 3(b) in terms of computational complexity. The numbers of floatingpoint real-valued operations required by V-NLE is given by NVolterra=(Nspan+1)8NSCNoverlog2 (NSCNover)+(20Nspan−6)NSCNover+16(Nspan+1). Where Nspan is the number of spans, NSC the subcarrier number, and Nover is the oversampling factor. On the other hand, the number of operations required to perform ANN-NLE is NANN=2NSC(2Nbits+1), where Nbits is the number of bits coded in each data subcarrier. The complexity of both NLEs increases as NSC does in a radically different way. Due to their different structure, the numbers of operations required by ANN- and V-NLEs does not depend on the same set of parameters. In particular, the computational complexity of V-NLE depends on Nspan but not on Nbits, while ANN-NLE does not depend on the link-related parameters but on Nbits since it is sensitive to the number of points in the constellation. Fig. 3(b) shows a quantitative comparison terms of NSC for different system parameters. For V-NLE, a Nover of 4 has been set and the Nspan has been varied (1, 5, and 10), whereas for ANN-NLE, Nbits has been swept (1, 2, and 4). It is shown for all considered NSC values, ANN-NLE outperforms to V-NLE in terms of computational complexity. Even when comparing the best-case scenario of V-NLE, i.e. only 1 span, to the worst-case scenario of ANN, i.e. 4 Nbits, the latter always outperforms. This difference increases accordingly to the number of spans, which is the case of long-haul networks. 4. Conclusions A novel ANN-NLE was experimentally demonstrated for 40-Gb/s 16-QAM CO-OFDM, reporting ~2 dB of nonlinearity penalty reduction at 2000 km. ANN-NLE outperformed the benchmark V-NLE by ~1.5 dB indicating also a significant reduction in computational complexity. Acknowledgement: This work was partly supported by the Centre of Excellence (CUDOS/IPOS, CE110001018) and Laureate Fellowship programs (FL120100029) of the Australian Research Council.

6. References [1] I.D. Phillips et al, OFC 2014, paper M3C.1. [5] E. Temprana et al, Science 348(6242), 1445–1448 (2015). [2] G. Gao et al, Photon. Technol. Lett. 25(8), 717–720 (2013). [6] S.T. Le et al, J. Lightw. Technol. 33(11), 2206–2212 (2015). [3] X. Liu et al, Nature Photon. 7(7), 560–568 (2013). [7] M.A. Jarajreh et al, Photon. Technol. Lett. 27(4), 387–390 (2015). [4] E. Giacoumidis et al, Photon. Technol. Lett. 26(14), 1383–1386 (2014).

OFC 2016 © OSA 2016

Experimental Comparison of Artificial Neural Network and Volterra based Nonlinear Equalization for CO-OFDM 1

2

3

4

2

2

E. Giacoumidis , S. T. Le , I. Aldaya , J. L. Wei , M. McCarthy , N. J. Doran , and B. J. Eggleton

1

1: Centre of Ultra High Bandwidth for Optical Systems (CUDOS), University of Sydney, Physics Road, NSW 2006, Sydney, Australia [email protected] 2: Aston Institute of Photonic Technologies, Aston University, Birmingham, B4 7ET, UK 3: Physics Institute, State University of Campinas, 777, Campinas, Brazil 4: ADVA Optical Networking SE, Campus Martinsried Fraunhoferstrasse 9a, Martinsried/Munich 82152, Germany.

Abstract: A novel artificial neural network (ANN)-based nonlinear equalizer (NLE) of low complexity is demonstrated for 40-Gb/s CO-OFDM at 2000 km, revealing ~1.5 dB enhancement in Q-factor compared to inverse Volterra-series transfer function based NLE. ©2016 Optical Society of America OCIS codes: (060.2330) Fiber optics communications; (060.1660) Coherent communications; (060.4080) Modulation

1. Introduction Endeavors to the Kerr nonlinearity limit have been performed by either inserting an optical phase conjugator (OPC) at the middle point of the link [1], or using electronic-based nonlinearity compensation (NLC) schemes such as digital back-propagation (DBP) [2], phase-conjugated twin-waves (PC-TW) [3], and nonlinear equalizers (NLE) based on the inverse Volterra-series transfer function (V-NLE) [4]. Unfortunately, OPC significantly reduces the flexibility in an optically routed network and requires both a symmetric chromatic dispersion (CD) map and power evolution. On the other hand, DBP is very complex, PC-TW halves the transmission capacity, and V-NLE shows marginal performance enhancement accompanied with a significant amount of floating-point operations. Recently, NLC based on transmitter-side DBP and employment of frequency reference carriers (FRC) was introduced in [5], which however required a sophisticated transmitter. Albeit the Kerr-mediate nonlinear process is deterministic, in multicarrier schemes like coherent optical OFDM (CO-OFDM) the resulting nonlinear interaction between subcarriers becomes so complicated that appears random due to its high peak-to-average power ratio (PAPR). This was confirmed in [6], in which NLC based on PC-subcarrier coding was hitlessly implemented in 8-QAM COOFDM due to inadequate compensation of inter-subcarrier interference (ICI) cross-phase modulation (XPM). Alternatively, nonlinearities could be mitigated using nonlinear decision classifiers such as artificial neural networks (ANN) [7] which learn the nonlinear impairments from observed data and build a probabilistic model of the impairment. ANN requires the transmission of an initial training sequence which might be high consuming in terms of required computational resources. Nonetheless, for a highly stable system, such as long-haul optical links, where CD and nonlinear effects do not change over time, ANN mapping could be accomplished once. In the simulation study in [7], 80-Gb/s ANN-NLE outperformed in terms of Q-factor to V-NLE by 1 dB at 1000 km. In this work, we experimentally demonstrate, for the first time, a novel ANN-NLE for 40-Gb/s 16-QAM COOFDM and compare it with the benchmark V-NLE. It is shown that ANN-NLE can reduce the nonlinearity penalty by ~2 dB at 2000 km. Compared to V-NLE, ANN-NLE outperforms by ~1.5 dB in Q-factor, concurrently providing a significant decrease in computational complexity. Numerical analysis indicates dependency of the nonlinearity penalty reduction on the length of the ANN training vector overhead (TVO) and the signal bit-rate.

(a) (b) Fig. 1. CO-OFDM receiver block diagram with (a) ANN-NLE, and (b) V-NLE [4]. LPF: low-pass filter; ADC: analog-to-digital converter; CP: cyclic prefix; (I)FFT: (inverse) fast-Fourier transform; MMSE: minimum mean-square error; HCD: nonlinear system chromatic dispersion.

W3A.4.pdf

OFC 2016 © OSA 2016

2. Proposed ANN and Volterra based NLEs for 16-QAM CO-OFDM In Fig. 1 (a), the ANN-NLE block-diagram for 16-QAM CO-OFDM is depicted which is similar to [7]. In summary, the ANN-NLE is comprised of k sub-neural networks (hidden layers), with each sub-network being associated to each subcarrier k and where s(k) is the training vector. The received symbols for each subcarrier x{k} are processed by the NLE neurons which are multiplied with the weight value for a given subcarrier/neuron wk,i, after which, the outputs of the different neurons are summed. In the training stage, the minimum mean-square error (MMSE) determines the error signal and updates the weights, which are iteratively updated until the desired error value is reached, thus indicating the optimum match between the sub-network output and the transmitted symbols. The error signal is given by (1)-(2): e(k) = s(k) − ŝ(k)

(1)

𝑠̂ (𝑘) = ∑𝑀 𝑖=1 𝑤𝑘,𝑖 𝜑𝑘,𝑖 (𝑠(𝑘))

(2)

E(n) = ‖S(n) − Ŝ(n)‖

2

(3)

where ŝ(k) is calculated in terms of a nonlinear activation function (NAF), 𝜑𝑘,𝑖 , performing the NLE. The NAF “f(.)” is a differential sigmoid function (SF) that solves the conflicting relationship between the boundedness and the differentiability of a complex function [7]. The SF is a “split” complex NAF, where two conventional real-valued functions process the I-Q components. The nonlinear Kerr-effect of a SSMF induces an approximately sigmoid-like power transmission function and therefore the SF-based NAF can tolerate the nonlinear spike or no-spike decision in spiking neurons. The number of neurons in every sub-neural network is equal to the number of points of the constellation, i.e. M in (2), which in the case of 16-QAM is 16. The ANN-NLE is based on the Riedmiller’s resilient-back propagation (RR-BP) algorithm performing an approximation to the global minimization achieved by the steepest descent. The training function updates the weights and bias values according to RR-BP, which minimizes the difference between the ANN output and the desired output by splitting the complex OFDM data in 2 real-valued data collections. The transfer functions for the hidden layer of the ANN-NLE are differentiable and similar to the hyperbolic tangent function. For the output layer, the linear function “purelin” is used. The block identified as MMSE in Fig. 1(a), represents the subsystem that implements RR-BP to find the weights that minimize the error vector. In (3) where 𝑆(𝑛) and 𝑆̂(𝑛) are the desired and calculated output vectors, respectively. The weights are updated according to the 5 steps described in [7] by applying the gradient descent on the cost function E(n) in order to reach a minimum. The block diagram of V-NLE is depicted in Fig. 1(b), which is similar to [7]. Compared to ANN, the V-NLE is placed after the ADCs to reduce DSP complexity by means of reducing the number of FFT/IFFT blocks. V-NLE inherits some of the features of the hybrid time-and-frequency domain implementation, such as non-frequency aliasing and simple implementation. From Fig. 1(b), it can be clearly identified that CD, i.e. (HCD)k, and the fiber nonlinearity are combated by the linear and nonlinear compensator tool, respectively. Very high-order Volterra kernels have not been considered here, thus offering ∼50% reduced computational complexity compared to single-step/span DBP. 3. Experimental transmission set-up, results, and theoretical analysis Fig. 2 depicts the experimental setup where an external cavity laser (ECL) with a linewidth of 100 KHz was modulated using a dual-parallel Mach-Zehnder modulator (DP-MZM) in IQ configuration. The DP-MZM modulator was fed with the I-Q components of the OFDM signal, which was generated offline. The transmission path at 1550.2 nm was a recirculating loop consisting of 20×100 km spans of Sterlite OH-LITE (E) fiber (attenuation of 18.9-19.5 dB/100 km) controlled by acousto-optic modulator (AOM). The loop switch was located in the mid-stage of the 1st Erbium-doped fiber amplifier (EDFA) and a gain-flattening filter (GFF) was placed in the mid-stage of the 3rd EDFA. The optimum launched optical power (LOP) was swept by controlling the output power of the EDFAs. At the receiver, the incoming signal was combined with another 100 KHz linewidth ECL acting as local oscillator (LO). After down-conversion, the baseband signal was sampled using a real-time oscilloscope operating at 80 GS/s and processed offline in MATLAB. 400 symbols were generated by 512-point inverse (I)FFT, 210 subcarriers were modulated with 16-QAM, while the rest were set to zero to reduce the PAPR. For ISI elimination due to linear dispersive effects, a CP of 2% was included.

Fig. 2. Experimental set-up of CO-OFDM equipped with ANN and V-NLE. ECL: external cavity laser, DSP: digital signal processing, AWG: arbitrary waveform generator, AOM: acousto-optic modulator, EDFA: Erbium-doped fiber amplifier, GFF: gain flatten filter, LO: local oscillator.

W3A.4.pdf

OFC 2016 © OSA 2016

The ANN training vector overhead was adjusted at 5% and by controlling the sampling rate we tailored the net signal bit-rate at ~40-Gb/s. The offline OFDM demodulator included both timing synchronization and frequency offset compensation, as well as I-Q imbalance compensation and CD compensation using an overlapped frequency domain equalizer employing the overlap-and-save method. The nonlinearity compensation capability was assessed based on Qfactor measurements averaging over 10 recorded traces (~106 bits), which was estimated from the BER obtained by error counting after hard-decision decoding. The Q-factor is related to BER by: Q = 20log10[√2𝑒𝑟𝑓𝑐 −1 (2𝐵𝐸𝑅)]. For 16QAM, a BER of 3×10-3 results in a required Q-factor of 8.7 dB. In Fig. 3 (a), the Q-factor against the LOP is plotted for a 40Gb/s CO-OFDM at 2000 km, with ANN-NLE, V-NLE and without (w/o) NLE. Nonlinearity reduction of ~2 dB at optimum LOP (FEC-limit) of 4 dBm is observed with 1.5 dB difference compared to V-NLE. ANN-NLE outperforms V-NLE due to the knack of reducing stochastic ICI–XPM/FWM. In Fig. 3(b), by simulation and using the same parameters as the experiment, we employed different lengths of ANN TVO at 5%, 10%, 15%, and 35% resulting in training lengths of 20, 40, 60, 140 symbols, respectively, and compared to V-NLE at different bit-rates at 1000 km of transmission (10 spans×100 km). It is shown that ANN-NLE for a soft-decision FEC-limit of 25% (Q = 6.25 dB) can reduce the nonlinearity penalty by ~4 dB at 70-Gb/s, with a minimum TVO of 10%, improving the Q-factor up to ~2 dB compared to V-NLE. V-NLE w/o NLE

10

Q-factor (dB)

8

ANN-NLE TVO = 35% TVO = 15% TVO = 10% TVO = 5%

6 4 2 0

60

65 70 75 Signal bit-rate (Gb/s)

80

(a) (b) (c) Fig. 3. (a) Q-factor vs. launched power of ANN- and V-NLEs for 40-Gb/s experimental CO-OFDM at 2000 km. (b) Q-factor vs. bit-rate of ANN-NLE for numerical CO-OFDM at 1000 km for different training vector overhead (TVO). (c) Computational complexity comparison between ANN- and V-NLEs: Blue bars represent the computational complexity of V-NLE for different subcarrier number, NSC, and number of spans, Nspan, whereas red bars are for ANN-NLE considering various NSC, and bits per subcarrier, Nbits.

ANN- and V-NLEs are compared in Fig. 3(b) in terms of computational complexity. The numbers of floatingpoint real-valued operations required by V-NLE is given by NVolterra=(Nspan+1)8NSCNoverlog2 (NSCNover)+(20Nspan−6)NSCNover+16(Nspan+1). Where Nspan is the number of spans, NSC the subcarrier number, and Nover is the oversampling factor. On the other hand, the number of operations required to perform ANN-NLE is NANN=2NSC(2Nbits+1), where Nbits is the number of bits coded in each data subcarrier. The complexity of both NLEs increases as NSC does in a radically different way. Due to their different structure, the numbers of operations required by ANN- and V-NLEs does not depend on the same set of parameters. In particular, the computational complexity of V-NLE depends on Nspan but not on Nbits, while ANN-NLE does not depend on the link-related parameters but on Nbits since it is sensitive to the number of points in the constellation. Fig. 3(b) shows a quantitative comparison terms of NSC for different system parameters. For V-NLE, a Nover of 4 has been set and the Nspan has been varied (1, 5, and 10), whereas for ANN-NLE, Nbits has been swept (1, 2, and 4). It is shown for all considered NSC values, ANN-NLE outperforms to V-NLE in terms of computational complexity. Even when comparing the best-case scenario of V-NLE, i.e. only 1 span, to the worst-case scenario of ANN, i.e. 4 Nbits, the latter always outperforms. This difference increases accordingly to the number of spans, which is the case of long-haul networks. 4. Conclusions A novel ANN-NLE was experimentally demonstrated for 40-Gb/s 16-QAM CO-OFDM, reporting ~2 dB of nonlinearity penalty reduction at 2000 km. ANN-NLE outperformed the benchmark V-NLE by ~1.5 dB indicating also a significant reduction in computational complexity. Acknowledgement: This work was partly supported by the Centre of Excellence (CUDOS/IPOS, CE110001018) and Laureate Fellowship programs (FL120100029) of the Australian Research Council.

6. References [1] I.D. Phillips et al, OFC 2014, paper M3C.1. [5] E. Temprana et al, Science 348(6242), 1445–1448 (2015). [2] G. Gao et al, Photon. Technol. Lett. 25(8), 717–720 (2013). [6] S.T. Le et al, J. Lightw. Technol. 33(11), 2206–2212 (2015). [3] X. Liu et al, Nature Photon. 7(7), 560–568 (2013). [7] M.A. Jarajreh et al, Photon. Technol. Lett. 27(4), 387–390 (2015). [4] E. Giacoumidis et al, Photon. Technol. Lett. 26(14), 1383–1386 (2014).