Abstract

The computational complexity and system bit-error-rate (BER) performance of four types of neural-network-based nonlinear equalizers are analyzed for a 50-Gb/s pulse amplitude modulation (PAM)-4 direct-detection (DD) optical link. The four types are feedforward neural networks (F-NN), radial basis function neural networks (RBF-NN), auto-regressive recurrent neural networks (AR-RNN) and layer-recurrent neural networks (L-RNN). Numerical results show that, for a fixed BER threshold, the AR-RNN-based equalizers have the lowest computational complexity. Amongst all the nonlinear NN-based equalizers with the same number of inputs and hidden neurons, F-NN-based equalizers have the lowest computational complexity while the AR-RNN-based equalizers exhibit the best BER performance. Compared with F-NN or RNN, RBF-NN tends to require more hidden neurons with the increase of the number of inputs, making it not suitable for long fiber transmission distance. We also demonstrate that only a few tens of multiplications per symbol are needed for NN-based equalizers to guarantee a good BER performance. This relatively low computational complexity signifies that various NN-based equalizers can be potentially implemented in real time. More broadly, this paper provides guidelines for selecting a suitable NN-based equalizer based on BER and computational complexity requirements.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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[Crossref]

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[Crossref]

2015 (3)

2014 (2)

2012 (1)

2008 (1)

Q. Liu and J. Wang, “A one-layer recurrent neural network with a discontinuous hard-limiting activation function for quadratic programming,” IEEE Trans. Neural Netw. 19(4), 558–570 (2008).
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J. Park and I. W. Sandberg, “Universal approximation using radial-basis-function networks,” Neural Comput. 3(2), 246–257 (1991).
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F. N. Khan, K. Zhong, X. Zhou, W. H. Al-Arashi, C. Yu, C. Lu, and A. P. T. Lau, “Joint OSNR monitoring and modulation format identification in digital coherent receivers using deep neural networks,” Opt. Express 25(15), 17767–17776 (2017).
[Crossref]

F. N. Khan, K. Zhong, W. H. Al-Arashi, C. Yu, C. Lu, and A. P. T. Lau, “Modulation format identification in coherent receivers using deep machine learning,” IEEE Photonics Technol. Lett. 28(17), 1886–1889 (2016).
[Crossref]

Anthapadmanabhan, N. P.

Balemarthy, K.

Bengio, Y.

X. Glorot and Y. Bengio, “Understanding the difficulty of training deep feedforward neural networks,” in Proceedings of the thirteenth international conference on artificial intelligence and statistics (2010), pp. 249–256.

Bigo, S.

J. Estaran, R. Rios-Mueller, M. A. Mestre, F. Jorge, H. Mardoyan, A. Konczykowska, J.-Y. Dupuy, and S. Bigo, “Artificial neural networks for linear and non-linear impairment mitigation in high-baudrate IMDD systems,” in European Conference on Optical Communication (2016), pp. M.2.B.2.

Borowiec, A.

Campos, L. A.

M. Xu, J. Zhang, H. Zhang, Z. Jia, J. Wang, L. Cheng, L. A. Campos, and C. Knittle, “Multi-stage machine learning enhanced DSP for DP-64QAM coherent optical transmission systems,” in Optical Fiber Communication Conference (Optical Society of America, 2019), paper M2H.1.

Cartledge, J. C.

Chang, W. F.

C. Y. Chuang, W. F. Chang, C. C. Wei, C. J. Ho, C. Y. Huang, J. W. Shi, L. Henrickson, Y. K. Chen, and J. Chen, “Sparse volterra nonlinear equalizer by employing pruning algorithm for high-speed PAM-4 850-nm VCSEL optical interconnect,” in Optical Fiber Communication Conference (Optical Society of America, 2019), paper M1F.2.

Charlton, D. W.

Chen, J.

C. Y. Chuang, W. F. Chang, C. C. Wei, C. J. Ho, C. Y. Huang, J. W. Shi, L. Henrickson, Y. K. Chen, and J. Chen, “Sparse volterra nonlinear equalizer by employing pruning algorithm for high-speed PAM-4 850-nm VCSEL optical interconnect,” in Optical Fiber Communication Conference (Optical Society of America, 2019), paper M1F.2.

C. Y. Chuang, C. C. Wei, T. C. Lin, K. L. Chi, L. C. Liu, J. W. Shi, Y. K. Chen, and J. Chen, “Employing deep neural network for high speed 4-PAM optical interconnect,” in European Conference on Optical Communication (2017), pp. W.2.D.2.

C. Y. Chuang, L. C. Liu, C. C. Wei, J. J. Liu, L. Henrickson, W. J. Huang, C. L. Wang, Y. K. Chen, and J. Chen, “Convolutional neural network based nonlinear classifier for 112-Gbps high speed optical link,” in Optical Fiber Communication Conference (Optical Society of America, 2018), paper W2A.43.

Chen, W.

Chen, Y. K.

C. Y. Chuang, C. C. Wei, T. C. Lin, K. L. Chi, L. C. Liu, J. W. Shi, Y. K. Chen, and J. Chen, “Employing deep neural network for high speed 4-PAM optical interconnect,” in European Conference on Optical Communication (2017), pp. W.2.D.2.

C. Y. Chuang, W. F. Chang, C. C. Wei, C. J. Ho, C. Y. Huang, J. W. Shi, L. Henrickson, Y. K. Chen, and J. Chen, “Sparse volterra nonlinear equalizer by employing pruning algorithm for high-speed PAM-4 850-nm VCSEL optical interconnect,” in Optical Fiber Communication Conference (Optical Society of America, 2019), paper M1F.2.

C. Y. Chuang, L. C. Liu, C. C. Wei, J. J. Liu, L. Henrickson, W. J. Huang, C. L. Wang, Y. K. Chen, and J. Chen, “Convolutional neural network based nonlinear classifier for 112-Gbps high speed optical link,” in Optical Fiber Communication Conference (Optical Society of America, 2018), paper W2A.43.

Cheng, L.

M. Xu, J. Zhang, H. Zhang, Z. Jia, J. Wang, L. Cheng, L. A. Campos, and C. Knittle, “Multi-stage machine learning enhanced DSP for DP-64QAM coherent optical transmission systems,” in Optical Fiber Communication Conference (Optical Society of America, 2019), paper M2H.1.

Chi, K. L.

C. Y. Chuang, C. C. Wei, T. C. Lin, K. L. Chi, L. C. Liu, J. W. Shi, Y. K. Chen, and J. Chen, “Employing deep neural network for high speed 4-PAM optical interconnect,” in European Conference on Optical Communication (2017), pp. W.2.D.2.

Chuang, C. Y.

C. Y. Chuang, C. C. Wei, T. C. Lin, K. L. Chi, L. C. Liu, J. W. Shi, Y. K. Chen, and J. Chen, “Employing deep neural network for high speed 4-PAM optical interconnect,” in European Conference on Optical Communication (2017), pp. W.2.D.2.

C. Y. Chuang, W. F. Chang, C. C. Wei, C. J. Ho, C. Y. Huang, J. W. Shi, L. Henrickson, Y. K. Chen, and J. Chen, “Sparse volterra nonlinear equalizer by employing pruning algorithm for high-speed PAM-4 850-nm VCSEL optical interconnect,” in Optical Fiber Communication Conference (Optical Society of America, 2019), paper M1F.2.

C. Y. Chuang, L. C. Liu, C. C. Wei, J. J. Liu, L. Henrickson, W. J. Huang, C. L. Wang, Y. K. Chen, and J. Chen, “Convolutional neural network based nonlinear classifier for 112-Gbps high speed optical link,” in Optical Fiber Communication Conference (Optical Society of America, 2018), paper W2A.43.

Compernolle, L. V.

T. Nguyen, S. Mhatli, E. Giacoumidis, L. V. Compernolle, M. Wuilpart, and P. Megret, “Fiber nonlinearity equalizer based on support vector classification for coherent optical OFDM,” IEEE Photonics J. 8(2), 1–9 (2016).
[Crossref]

Cui, Y.

D. Wang, M. Zhang, Z. Li, Y. Cui, J. Liu, Y. Yang, and H. Wang, “Nonlinear decision boundary created by a machine learning-based classifier to mitigate nonlinear phase noise,” in European Conference on Optical Communication (2015), pp. P.3.1.

Deng, L.

Dupuy, J.-Y.

J. Estaran, R. Rios-Mueller, M. A. Mestre, F. Jorge, H. Mardoyan, A. Konczykowska, J.-Y. Dupuy, and S. Bigo, “Artificial neural networks for linear and non-linear impairment mitigation in high-baudrate IMDD systems,” in European Conference on Optical Communication (2016), pp. M.2.B.2.

Estaran, J.

J. Estaran, R. Rios-Mueller, M. A. Mestre, F. Jorge, H. Mardoyan, A. Konczykowska, J.-Y. Dupuy, and S. Bigo, “Artificial neural networks for linear and non-linear impairment mitigation in high-baudrate IMDD systems,” in European Conference on Optical Communication (2016), pp. M.2.B.2.

Etemad, S. A.

Feng, H.

C. Ye, D. Zhang, X. Huang, H. Feng, and K. Zhang, “Demonstration of 50Gbps IM/DD PAM4 PON over 10 GHz class optics using neural network based nonlinear equalization,” in European Conference on Optical Communication (2017), pp. W.2.B.4.

C. Ye, D. Zhang, X. Hu, X. Huang, H. Feng, and K. Zhang, “Recurrent neural network (RNN) based end-to-end nonlinear management for symmetrical 50Gbps NRZ PON with 29dB+ loss budget,” in European Conference on Optical Communication (2018), pp. Mo4B.3.

Flunkert, V.

D. Salinas, V. Flunkert, and J. Gasthaus, “DeepAR: Probabilistic forecasting with autoregressive recurrent networks,” https://arxiv.org/abs/1704.04110 .

Fu, S.

Fukumoto, Y.

T. Kyono, Y. Otsuka, Y. Fukumoto, S. Owaki, and M. Nakamura, “Computational-complexity comparison of artificial neural network and volterra series transfer function for optical nonlinearity compensation with time- and frequency-domain dispersion equalization,” in European Conference on Optical Communication (2018), pp. P.2.

Gao, F.

Z. Yang, F. Gao, S. Fu, X. Li, L. Deng, Z. He, M. Tang, and D. Liu, “Radial basis function neural network enabled C-band 4 × 50 Gbs PAM-4 transmission over 80 km SSMF,” Opt. Lett. 43(15), 3542–3545 (2018).
[Crossref]

M. Luo, F. Gao, X. Li, Z. He, and S. Fu, “Transmission of 4×50-Gbs PAM-4 signal over 80-km single mode fiber using neural network,” in Optical Fiber Communication Conference (Optical Society of America, 2018), paper M2F.2.

Gao, M.

Gao, Y.

K. Zhong, X. Zhou, T. Gui, L. Tao, Y. Gao, W. Chen, J. Man, L. Zeng, A. P. T. Lau, and C. Lu, “Experimental Study of PAM-4, CAP-16, and DMT for 100 Gb/s Short Reach Optical Transmission Systems,” Opt. Express 23(2), 1176–1189 (2015).
[Crossref]

Y. Gao, J. C. Cartledge, S. S.-H. Yam, A. Rezania, and Y. Matsui, “112 Gb/s PAM-4 using a directly modulated laser with linear pre-compensation and nonlinear post-compensation,” in European Conference on Optical Communication (2016), pp. M.2.C.2.

Gasthaus, J.

D. Salinas, V. Flunkert, and J. Gasthaus, “DeepAR: Probabilistic forecasting with autoregressive recurrent networks,” https://arxiv.org/abs/1704.04110 .

Gazula, D.

Ghiasi, A.

Giacoumidis, E.

T. Nguyen, S. Mhatli, E. Giacoumidis, L. V. Compernolle, M. Wuilpart, and P. Megret, “Fiber nonlinearity equalizer based on support vector classification for coherent optical OFDM,” IEEE Photonics J. 8(2), 1–9 (2016).
[Crossref]

Glorot, X.

X. Glorot and Y. Bengio, “Understanding the difficulty of training deep feedforward neural networks,” in Proceedings of the thirteenth international conference on artificial intelligence and statistics (2010), pp. 249–256.

Graham, L. A.

Guan, K.

Guenter, J. K.

Gui, T.

He, Z.

Z. Yang, F. Gao, S. Fu, X. Li, L. Deng, Z. He, M. Tang, and D. Liu, “Radial basis function neural network enabled C-band 4 × 50 Gbs PAM-4 transmission over 80 km SSMF,” Opt. Lett. 43(15), 3542–3545 (2018).
[Crossref]

M. Luo, F. Gao, X. Li, Z. He, and S. Fu, “Transmission of 4×50-Gbs PAM-4 signal over 80-km single mode fiber using neural network,” in Optical Fiber Communication Conference (Optical Society of America, 2018), paper M2F.2.

Henrickson, L.

C. Y. Chuang, W. F. Chang, C. C. Wei, C. J. Ho, C. Y. Huang, J. W. Shi, L. Henrickson, Y. K. Chen, and J. Chen, “Sparse volterra nonlinear equalizer by employing pruning algorithm for high-speed PAM-4 850-nm VCSEL optical interconnect,” in Optical Fiber Communication Conference (Optical Society of America, 2019), paper M1F.2.

C. Y. Chuang, L. C. Liu, C. C. Wei, J. J. Liu, L. Henrickson, W. J. Huang, C. L. Wang, Y. K. Chen, and J. Chen, “Convolutional neural network based nonlinear classifier for 112-Gbps high speed optical link,” in Optical Fiber Communication Conference (Optical Society of America, 2018), paper W2A.43.

Hinton, K. J.

Ho, C. J.

C. Y. Chuang, W. F. Chang, C. C. Wei, C. J. Ho, C. Y. Huang, J. W. Shi, L. Henrickson, Y. K. Chen, and J. Chen, “Sparse volterra nonlinear equalizer by employing pruning algorithm for high-speed PAM-4 850-nm VCSEL optical interconnect,” in Optical Fiber Communication Conference (Optical Society of America, 2019), paper M1F.2.

Hu, R.

C. Yang, R. Hu, M. Luo, Q. Yang, C. Li, H. Li, and S. Yu, “IM/DD-based 112-Gb/s/lambda PAM-4 transmission using 18-Gbps DML,” IEEE Photonics J. 8(3), 1–7 (2016).
[Crossref]

Hu, X.

C. Ye, D. Zhang, X. Hu, X. Huang, H. Feng, and K. Zhang, “Recurrent neural network (RNN) based end-to-end nonlinear management for symmetrical 50Gbps NRZ PON with 29dB+ loss budget,” in European Conference on Optical Communication (2018), pp. Mo4B.3.

Huang, C. Y.

C. Y. Chuang, W. F. Chang, C. C. Wei, C. J. Ho, C. Y. Huang, J. W. Shi, L. Henrickson, Y. K. Chen, and J. Chen, “Sparse volterra nonlinear equalizer by employing pruning algorithm for high-speed PAM-4 850-nm VCSEL optical interconnect,” in Optical Fiber Communication Conference (Optical Society of America, 2019), paper M1F.2.

Huang, W. J.

C. Y. Chuang, L. C. Liu, C. C. Wei, J. J. Liu, L. Henrickson, W. J. Huang, C. L. Wang, Y. K. Chen, and J. Chen, “Convolutional neural network based nonlinear classifier for 112-Gbps high speed optical link,” in Optical Fiber Communication Conference (Optical Society of America, 2018), paper W2A.43.

Huang, X.

C. Ye, D. Zhang, X. Hu, X. Huang, H. Feng, and K. Zhang, “Recurrent neural network (RNN) based end-to-end nonlinear management for symmetrical 50Gbps NRZ PON with 29dB+ loss budget,” in European Conference on Optical Communication (2018), pp. Mo4B.3.

C. Ye, D. Zhang, X. Huang, H. Feng, and K. Zhang, “Demonstration of 50Gbps IM/DD PAM4 PON over 10 GHz class optics using neural network based nonlinear equalization,” in European Conference on Optical Communication (2017), pp. W.2.B.4.

Jia, Z.

M. Xu, J. Zhang, H. Zhang, Z. Jia, J. Wang, L. Cheng, L. A. Campos, and C. Knittle, “Multi-stage machine learning enhanced DSP for DP-64QAM coherent optical transmission systems,” in Optical Fiber Communication Conference (Optical Society of America, 2019), paper M2H.1.

Johnson, R. H.

Jorge, F.

J. Estaran, R. Rios-Mueller, M. A. Mestre, F. Jorge, H. Mardoyan, A. Konczykowska, J.-Y. Dupuy, and S. Bigo, “Artificial neural networks for linear and non-linear impairment mitigation in high-baudrate IMDD systems,” in European Conference on Optical Communication (2016), pp. M.2.B.2.

Karar, A. S.

Kashi, A. S.

Khan, F. N.

King, J.

Knittle, C.

M. Xu, J. Zhang, H. Zhang, Z. Jia, J. Wang, L. Cheng, L. A. Campos, and C. Knittle, “Multi-stage machine learning enhanced DSP for DP-64QAM coherent optical transmission systems,” in Optical Fiber Communication Conference (Optical Society of America, 2019), paper M2H.1.

Kocot, C.

Konczykowska, A.

J. Estaran, R. Rios-Mueller, M. A. Mestre, F. Jorge, H. Mardoyan, A. Konczykowska, J.-Y. Dupuy, and S. Bigo, “Artificial neural networks for linear and non-linear impairment mitigation in high-baudrate IMDD systems,” in European Conference on Optical Communication (2016), pp. M.2.B.2.

Kyono, T.

T. Kyono, Y. Otsuka, Y. Fukumoto, S. Owaki, and M. Nakamura, “Computational-complexity comparison of artificial neural network and volterra series transfer function for optical nonlinearity compensation with time- and frequency-domain dispersion equalization,” in European Conference on Optical Communication (2018), pp. P.2.

Landry, G. D.

Laperle, C.

Lau, A. P. T.

Li, C.

C. Yang, R. Hu, M. Luo, Q. Yang, C. Li, H. Li, and S. Yu, “IM/DD-based 112-Gb/s/lambda PAM-4 transmission using 18-Gbps DML,” IEEE Photonics J. 8(3), 1–7 (2016).
[Crossref]

Li, H.

C. Yang, R. Hu, M. Luo, Q. Yang, C. Li, H. Li, and S. Yu, “IM/DD-based 112-Gb/s/lambda PAM-4 transmission using 18-Gbps DML,” IEEE Photonics J. 8(3), 1–7 (2016).
[Crossref]

Li, J.

Li, X.

Li, Z.

D. Wang, M. Zhang, Z. Li, Y. Cui, J. Liu, Y. Yang, and H. Wang, “Nonlinear decision boundary created by a machine learning-based classifier to mitigate nonlinear phase noise,” in European Conference on Optical Communication (2015), pp. P.3.1.

Lin, T. C.

C. Y. Chuang, C. C. Wei, T. C. Lin, K. L. Chi, L. C. Liu, J. W. Shi, Y. K. Chen, and J. Chen, “Employing deep neural network for high speed 4-PAM optical interconnect,” in European Conference on Optical Communication (2017), pp. W.2.D.2.

Liu, D.

Liu, J.

D. Wang, M. Zhang, Z. Li, Y. Cui, J. Liu, Y. Yang, and H. Wang, “Nonlinear decision boundary created by a machine learning-based classifier to mitigate nonlinear phase noise,” in European Conference on Optical Communication (2015), pp. P.3.1.

Liu, J. J.

C. Y. Chuang, L. C. Liu, C. C. Wei, J. J. Liu, L. Henrickson, W. J. Huang, C. L. Wang, Y. K. Chen, and J. Chen, “Convolutional neural network based nonlinear classifier for 112-Gbps high speed optical link,” in Optical Fiber Communication Conference (Optical Society of America, 2018), paper W2A.43.

Liu, L. C.

C. Y. Chuang, L. C. Liu, C. C. Wei, J. J. Liu, L. Henrickson, W. J. Huang, C. L. Wang, Y. K. Chen, and J. Chen, “Convolutional neural network based nonlinear classifier for 112-Gbps high speed optical link,” in Optical Fiber Communication Conference (Optical Society of America, 2018), paper W2A.43.

C. Y. Chuang, C. C. Wei, T. C. Lin, K. L. Chi, L. C. Liu, J. W. Shi, Y. K. Chen, and J. Chen, “Employing deep neural network for high speed 4-PAM optical interconnect,” in European Conference on Optical Communication (2017), pp. W.2.D.2.

Liu, Q.

Q. Liu and J. Wang, “A one-layer recurrent neural network with a discontinuous hard-limiting activation function for quadratic programming,” IEEE Trans. Neural Netw. 19(4), 558–570 (2008).
[Crossref]

Lu, C.

Luo, M.

Z. Wan, J. Li, L. Shu, M. Luo, X. Li, S. Fu, and K. Xu, “Nonlinear equalization based on pruned artificial neural networks for 112-Gbs SSB-PAM4 transmission over 80-km SSMF,” Opt. Express 26(8), 10631–10642 (2018).
[Crossref]

C. Yang, R. Hu, M. Luo, Q. Yang, C. Li, H. Li, and S. Yu, “IM/DD-based 112-Gb/s/lambda PAM-4 transmission using 18-Gbps DML,” IEEE Photonics J. 8(3), 1–7 (2016).
[Crossref]

M. Luo, F. Gao, X. Li, Z. He, and S. Fu, “Transmission of 4×50-Gbs PAM-4 signal over 80-km single mode fiber using neural network,” in Optical Fiber Communication Conference (Optical Society of America, 2018), paper M2F.2.

Lyubomirsky, I.

Ma, Y.

MacInnes, A. N.

Man, J.

Manton, J. H.

Z. Xu, C. Sun, J. H. Manton, and W. Shieh, “Computational complexity analysis of neural network-based nonlinear equalization for short reach direct detection systems,” in Asia Communications and Photonics Conference (2019), paper T2G.4.

Mardoyan, H.

J. Estaran, R. Rios-Mueller, M. A. Mestre, F. Jorge, H. Mardoyan, A. Konczykowska, J.-Y. Dupuy, and S. Bigo, “Artificial neural networks for linear and non-linear impairment mitigation in high-baudrate IMDD systems,” in European Conference on Optical Communication (2016), pp. M.2.B.2.

Matsui, Y.

Y. Gao, J. C. Cartledge, S. S.-H. Yam, A. Rezania, and Y. Matsui, “112 Gb/s PAM-4 using a directly modulated laser with linear pre-compensation and nonlinear post-compensation,” in European Conference on Optical Communication (2016), pp. M.2.C.2.

Megret, P.

T. Nguyen, S. Mhatli, E. Giacoumidis, L. V. Compernolle, M. Wuilpart, and P. Megret, “Fiber nonlinearity equalizer based on support vector classification for coherent optical OFDM,” IEEE Photonics J. 8(2), 1–9 (2016).
[Crossref]

Mestre, M. A.

J. Estaran, R. Rios-Mueller, M. A. Mestre, F. Jorge, H. Mardoyan, A. Konczykowska, J.-Y. Dupuy, and S. Bigo, “Artificial neural networks for linear and non-linear impairment mitigation in high-baudrate IMDD systems,” in European Conference on Optical Communication (2016), pp. M.2.B.2.

Mhatli, S.

T. Nguyen, S. Mhatli, E. Giacoumidis, L. V. Compernolle, M. Wuilpart, and P. Megret, “Fiber nonlinearity equalizer based on support vector classification for coherent optical OFDM,” IEEE Photonics J. 8(2), 1–9 (2016).
[Crossref]

Nakamura, M.

T. Kyono, Y. Otsuka, Y. Fukumoto, S. Owaki, and M. Nakamura, “Computational-complexity comparison of artificial neural network and volterra series transfer function for optical nonlinearity compensation with time- and frequency-domain dispersion equalization,” in European Conference on Optical Communication (2018), pp. P.2.

Nguyen, T.

T. Nguyen, S. Mhatli, E. Giacoumidis, L. V. Compernolle, M. Wuilpart, and P. Megret, “Fiber nonlinearity equalizer based on support vector classification for coherent optical OFDM,” IEEE Photonics J. 8(2), 1–9 (2016).
[Crossref]

O’Sullivan, M.

Otsuka, Y.

T. Kyono, Y. Otsuka, Y. Fukumoto, S. Owaki, and M. Nakamura, “Computational-complexity comparison of artificial neural network and volterra series transfer function for optical nonlinearity compensation with time- and frequency-domain dispersion equalization,” in European Conference on Optical Communication (2018), pp. P.2.

Owaki, S.

T. Kyono, Y. Otsuka, Y. Fukumoto, S. Owaki, and M. Nakamura, “Computational-complexity comparison of artificial neural network and volterra series transfer function for optical nonlinearity compensation with time- and frequency-domain dispersion equalization,” in European Conference on Optical Communication (2018), pp. P.2.

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J. Park and I. W. Sandberg, “Universal approximation using radial-basis-function networks,” Neural Comput. 3(2), 246–257 (1991).
[Crossref]

Pillai, B. S. G.

Reza, A. G.

A. G. Reza and J.-K. K. Rhee, “Nonlinear equalizer based on neural networks for PAM-4 signal transmission using DML,” IEEE Photonics Technol. Lett. 30(15), 1416–1419 (2018).
[Crossref]

Rezania, A.

Y. Gao, J. C. Cartledge, S. S.-H. Yam, A. Rezania, and Y. Matsui, “112 Gb/s PAM-4 using a directly modulated laser with linear pre-compensation and nonlinear post-compensation,” in European Conference on Optical Communication (2016), pp. M.2.C.2.

Rhee, J.-K. K.

A. G. Reza and J.-K. K. Rhee, “Nonlinear equalizer based on neural networks for PAM-4 signal transmission using DML,” IEEE Photonics Technol. Lett. 30(15), 1416–1419 (2018).
[Crossref]

Rios-Mueller, R.

J. Estaran, R. Rios-Mueller, M. A. Mestre, F. Jorge, H. Mardoyan, A. Konczykowska, J.-Y. Dupuy, and S. Bigo, “Artificial neural networks for linear and non-linear impairment mitigation in high-baudrate IMDD systems,” in European Conference on Optical Communication (2016), pp. M.2.B.2.

Salinas, D.

D. Salinas, V. Flunkert, and J. Gasthaus, “DeepAR: Probabilistic forecasting with autoregressive recurrent networks,” https://arxiv.org/abs/1704.04110 .

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J. Park and I. W. Sandberg, “Universal approximation using radial-basis-function networks,” Neural Comput. 3(2), 246–257 (1991).
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Sedighi, B.

Shaw, E. M.

Shen, G.

Shi, J. W.

C. Y. Chuang, C. C. Wei, T. C. Lin, K. L. Chi, L. C. Liu, J. W. Shi, Y. K. Chen, and J. Chen, “Employing deep neural network for high speed 4-PAM optical interconnect,” in European Conference on Optical Communication (2017), pp. W.2.D.2.

C. Y. Chuang, W. F. Chang, C. C. Wei, C. J. Ho, C. Y. Huang, J. W. Shi, L. Henrickson, Y. K. Chen, and J. Chen, “Sparse volterra nonlinear equalizer by employing pruning algorithm for high-speed PAM-4 850-nm VCSEL optical interconnect,” in Optical Fiber Communication Conference (Optical Society of America, 2019), paper M1F.2.

Shieh, W.

B. S. G. Pillai, B. Sedighi, K. Guan, N. P. Anthapadmanabhan, W. Shieh, K. J. Hinton, and R. S. Tucker, “End-to-end energy modeling and analysis of long-haul coherent transmission systems,” J. Lightwave Technol. 32(18), 3093–3111 (2014).
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Z. Xu, C. Sun, J. H. Manton, and W. Shieh, “Computational complexity analysis of neural network-based nonlinear equalization for short reach direct detection systems,” in Asia Communications and Photonics Conference (2019), paper T2G.4.

Shu, L.

Shubochkin, R.

Sun, C.

Z. Xu, C. Sun, J. H. Manton, and W. Shieh, “Computational complexity analysis of neural network-based nonlinear equalization for short reach direct detection systems,” in Asia Communications and Photonics Conference (2019), paper T2G.4.

Tang, F.

Tang, M.

Tao, L.

Tatum, J. A.

Tucker, R. S.

Vaidya, D.

Wan, Z.

Wang, C. L.

C. Y. Chuang, L. C. Liu, C. C. Wei, J. J. Liu, L. Henrickson, W. J. Huang, C. L. Wang, Y. K. Chen, and J. Chen, “Convolutional neural network based nonlinear classifier for 112-Gbps high speed optical link,” in Optical Fiber Communication Conference (Optical Society of America, 2018), paper W2A.43.

Wang, D.

D. Wang, M. Zhang, Z. Li, Y. Cui, J. Liu, Y. Yang, and H. Wang, “Nonlinear decision boundary created by a machine learning-based classifier to mitigate nonlinear phase noise,” in European Conference on Optical Communication (2015), pp. P.3.1.

Wang, H.

D. Wang, M. Zhang, Z. Li, Y. Cui, J. Liu, Y. Yang, and H. Wang, “Nonlinear decision boundary created by a machine learning-based classifier to mitigate nonlinear phase noise,” in European Conference on Optical Communication (2015), pp. P.3.1.

Wang, J.

Q. Liu and J. Wang, “A one-layer recurrent neural network with a discontinuous hard-limiting activation function for quadratic programming,” IEEE Trans. Neural Netw. 19(4), 558–570 (2008).
[Crossref]

M. Xu, J. Zhang, H. Zhang, Z. Jia, J. Wang, L. Cheng, L. A. Campos, and C. Knittle, “Multi-stage machine learning enhanced DSP for DP-64QAM coherent optical transmission systems,” in Optical Fiber Communication Conference (Optical Society of America, 2019), paper M2H.1.

Wei, C. C.

C. Y. Chuang, L. C. Liu, C. C. Wei, J. J. Liu, L. Henrickson, W. J. Huang, C. L. Wang, Y. K. Chen, and J. Chen, “Convolutional neural network based nonlinear classifier for 112-Gbps high speed optical link,” in Optical Fiber Communication Conference (Optical Society of America, 2018), paper W2A.43.

C. Y. Chuang, W. F. Chang, C. C. Wei, C. J. Ho, C. Y. Huang, J. W. Shi, L. Henrickson, Y. K. Chen, and J. Chen, “Sparse volterra nonlinear equalizer by employing pruning algorithm for high-speed PAM-4 850-nm VCSEL optical interconnect,” in Optical Fiber Communication Conference (Optical Society of America, 2019), paper M1F.2.

C. Y. Chuang, C. C. Wei, T. C. Lin, K. L. Chi, L. C. Liu, J. W. Shi, Y. K. Chen, and J. Chen, “Employing deep neural network for high speed 4-PAM optical interconnect,” in European Conference on Optical Communication (2017), pp. W.2.D.2.

Wuilpart, M.

T. Nguyen, S. Mhatli, E. Giacoumidis, L. V. Compernolle, M. Wuilpart, and P. Megret, “Fiber nonlinearity equalizer based on support vector classification for coherent optical OFDM,” IEEE Photonics J. 8(2), 1–9 (2016).
[Crossref]

Xu, K.

Xu, M.

M. Xu, J. Zhang, H. Zhang, Z. Jia, J. Wang, L. Cheng, L. A. Campos, and C. Knittle, “Multi-stage machine learning enhanced DSP for DP-64QAM coherent optical transmission systems,” in Optical Fiber Communication Conference (Optical Society of America, 2019), paper M2H.1.

Xu, Z.

Z. Xu, C. Sun, J. H. Manton, and W. Shieh, “Computational complexity analysis of neural network-based nonlinear equalization for short reach direct detection systems,” in Asia Communications and Photonics Conference (2019), paper T2G.4.

Yam, S. S.-H.

Y. Gao, J. C. Cartledge, S. S.-H. Yam, A. Rezania, and Y. Matsui, “112 Gb/s PAM-4 using a directly modulated laser with linear pre-compensation and nonlinear post-compensation,” in European Conference on Optical Communication (2016), pp. M.2.C.2.

Yan, M.

Yang, C.

C. Yang, R. Hu, M. Luo, Q. Yang, C. Li, H. Li, and S. Yu, “IM/DD-based 112-Gb/s/lambda PAM-4 transmission using 18-Gbps DML,” IEEE Photonics J. 8(3), 1–7 (2016).
[Crossref]

Yang, Q.

C. Yang, R. Hu, M. Luo, Q. Yang, C. Li, H. Li, and S. Yu, “IM/DD-based 112-Gb/s/lambda PAM-4 transmission using 18-Gbps DML,” IEEE Photonics J. 8(3), 1–7 (2016).
[Crossref]

Yang, Y.

D. Wang, M. Zhang, Z. Li, Y. Cui, J. Liu, Y. Yang, and H. Wang, “Nonlinear decision boundary created by a machine learning-based classifier to mitigate nonlinear phase noise,” in European Conference on Optical Communication (2015), pp. P.3.1.

Yang, Z.

Ye, C.

C. Ye, D. Zhang, X. Hu, X. Huang, H. Feng, and K. Zhang, “Recurrent neural network (RNN) based end-to-end nonlinear management for symmetrical 50Gbps NRZ PON with 29dB+ loss budget,” in European Conference on Optical Communication (2018), pp. Mo4B.3.

C. Ye, D. Zhang, X. Huang, H. Feng, and K. Zhang, “Demonstration of 50Gbps IM/DD PAM4 PON over 10 GHz class optics using neural network based nonlinear equalization,” in European Conference on Optical Communication (2017), pp. W.2.B.4.

Yu, C.

F. N. Khan, K. Zhong, X. Zhou, W. H. Al-Arashi, C. Yu, C. Lu, and A. P. T. Lau, “Joint OSNR monitoring and modulation format identification in digital coherent receivers using deep neural networks,” Opt. Express 25(15), 17767–17776 (2017).
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F. N. Khan, K. Zhong, W. H. Al-Arashi, C. Yu, C. Lu, and A. P. T. Lau, “Modulation format identification in coherent receivers using deep machine learning,” IEEE Photonics Technol. Lett. 28(17), 1886–1889 (2016).
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Yu, S.

C. Yang, R. Hu, M. Luo, Q. Yang, C. Li, H. Li, and S. Yu, “IM/DD-based 112-Gb/s/lambda PAM-4 transmission using 18-Gbps DML,” IEEE Photonics J. 8(3), 1–7 (2016).
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Zeng, L.

Zhang, D.

C. Ye, D. Zhang, X. Huang, H. Feng, and K. Zhang, “Demonstration of 50Gbps IM/DD PAM4 PON over 10 GHz class optics using neural network based nonlinear equalization,” in European Conference on Optical Communication (2017), pp. W.2.B.4.

C. Ye, D. Zhang, X. Hu, X. Huang, H. Feng, and K. Zhang, “Recurrent neural network (RNN) based end-to-end nonlinear management for symmetrical 50Gbps NRZ PON with 29dB+ loss budget,” in European Conference on Optical Communication (2018), pp. Mo4B.3.

Zhang, H.

M. Xu, J. Zhang, H. Zhang, Z. Jia, J. Wang, L. Cheng, L. A. Campos, and C. Knittle, “Multi-stage machine learning enhanced DSP for DP-64QAM coherent optical transmission systems,” in Optical Fiber Communication Conference (Optical Society of America, 2019), paper M2H.1.

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J. Zhang, W. Chen, M. Gao, Y. Ma, Y. Zhao, W. Chen, and G. Shen, “Intelligent adaptive coherent optical receiver based on convolutional neural network and clustering algorithm,” Opt. Express 26(14), 18684–18698 (2018).
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M. Xu, J. Zhang, H. Zhang, Z. Jia, J. Wang, L. Cheng, L. A. Campos, and C. Knittle, “Multi-stage machine learning enhanced DSP for DP-64QAM coherent optical transmission systems,” in Optical Fiber Communication Conference (Optical Society of America, 2019), paper M2H.1.

Zhang, K.

C. Ye, D. Zhang, X. Hu, X. Huang, H. Feng, and K. Zhang, “Recurrent neural network (RNN) based end-to-end nonlinear management for symmetrical 50Gbps NRZ PON with 29dB+ loss budget,” in European Conference on Optical Communication (2018), pp. Mo4B.3.

C. Ye, D. Zhang, X. Huang, H. Feng, and K. Zhang, “Demonstration of 50Gbps IM/DD PAM4 PON over 10 GHz class optics using neural network based nonlinear equalization,” in European Conference on Optical Communication (2017), pp. W.2.B.4.

Zhang, M.

D. Wang, M. Zhang, Z. Li, Y. Cui, J. Liu, Y. Yang, and H. Wang, “Nonlinear decision boundary created by a machine learning-based classifier to mitigate nonlinear phase noise,” in European Conference on Optical Communication (2015), pp. P.3.1.

Zhao, Y.

Zhong, K.

Zhou, X.

Zhou, Y.

Zhuge, Q.

IEEE Photonics J. (2)

C. Yang, R. Hu, M. Luo, Q. Yang, C. Li, H. Li, and S. Yu, “IM/DD-based 112-Gb/s/lambda PAM-4 transmission using 18-Gbps DML,” IEEE Photonics J. 8(3), 1–7 (2016).
[Crossref]

T. Nguyen, S. Mhatli, E. Giacoumidis, L. V. Compernolle, M. Wuilpart, and P. Megret, “Fiber nonlinearity equalizer based on support vector classification for coherent optical OFDM,” IEEE Photonics J. 8(2), 1–9 (2016).
[Crossref]

IEEE Photonics Technol. Lett. (2)

F. N. Khan, K. Zhong, W. H. Al-Arashi, C. Yu, C. Lu, and A. P. T. Lau, “Modulation format identification in coherent receivers using deep machine learning,” IEEE Photonics Technol. Lett. 28(17), 1886–1889 (2016).
[Crossref]

A. G. Reza and J.-K. K. Rhee, “Nonlinear equalizer based on neural networks for PAM-4 signal transmission using DML,” IEEE Photonics Technol. Lett. 30(15), 1416–1419 (2018).
[Crossref]

IEEE Trans. Neural Netw. (1)

Q. Liu and J. Wang, “A one-layer recurrent neural network with a discontinuous hard-limiting activation function for quadratic programming,” IEEE Trans. Neural Netw. 19(4), 558–570 (2008).
[Crossref]

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Neural Comput. (1)

J. Park and I. W. Sandberg, “Universal approximation using radial-basis-function networks,” Neural Comput. 3(2), 246–257 (1991).
[Crossref]

Opt. Express (6)

Opt. Lett. (1)

Other (15)

D. Salinas, V. Flunkert, and J. Gasthaus, “DeepAR: Probabilistic forecasting with autoregressive recurrent networks,” https://arxiv.org/abs/1704.04110 .

C. Ye, D. Zhang, X. Huang, H. Feng, and K. Zhang, “Demonstration of 50Gbps IM/DD PAM4 PON over 10 GHz class optics using neural network based nonlinear equalization,” in European Conference on Optical Communication (2017), pp. W.2.B.4.

Cisco, “Cisco global cloud index: forecast and methodology, 2016-2021 white paper,” (Cisco, 2018). https://www.cisco.com/c/en/us/solutions/collateral/service-provider/global-cloud-index-gci/white-paper-c11-738085.html .

T. Kyono, Y. Otsuka, Y. Fukumoto, S. Owaki, and M. Nakamura, “Computational-complexity comparison of artificial neural network and volterra series transfer function for optical nonlinearity compensation with time- and frequency-domain dispersion equalization,” in European Conference on Optical Communication (2018), pp. P.2.

Z. Xu, C. Sun, J. H. Manton, and W. Shieh, “Computational complexity analysis of neural network-based nonlinear equalization for short reach direct detection systems,” in Asia Communications and Photonics Conference (2019), paper T2G.4.

X. Glorot and Y. Bengio, “Understanding the difficulty of training deep feedforward neural networks,” in Proceedings of the thirteenth international conference on artificial intelligence and statistics (2010), pp. 249–256.

C. Ye, D. Zhang, X. Hu, X. Huang, H. Feng, and K. Zhang, “Recurrent neural network (RNN) based end-to-end nonlinear management for symmetrical 50Gbps NRZ PON with 29dB+ loss budget,” in European Conference on Optical Communication (2018), pp. Mo4B.3.

C. Y. Chuang, L. C. Liu, C. C. Wei, J. J. Liu, L. Henrickson, W. J. Huang, C. L. Wang, Y. K. Chen, and J. Chen, “Convolutional neural network based nonlinear classifier for 112-Gbps high speed optical link,” in Optical Fiber Communication Conference (Optical Society of America, 2018), paper W2A.43.

M. Xu, J. Zhang, H. Zhang, Z. Jia, J. Wang, L. Cheng, L. A. Campos, and C. Knittle, “Multi-stage machine learning enhanced DSP for DP-64QAM coherent optical transmission systems,” in Optical Fiber Communication Conference (Optical Society of America, 2019), paper M2H.1.

D. Wang, M. Zhang, Z. Li, Y. Cui, J. Liu, Y. Yang, and H. Wang, “Nonlinear decision boundary created by a machine learning-based classifier to mitigate nonlinear phase noise,” in European Conference on Optical Communication (2015), pp. P.3.1.

Y. Gao, J. C. Cartledge, S. S.-H. Yam, A. Rezania, and Y. Matsui, “112 Gb/s PAM-4 using a directly modulated laser with linear pre-compensation and nonlinear post-compensation,” in European Conference on Optical Communication (2016), pp. M.2.C.2.

C. Y. Chuang, W. F. Chang, C. C. Wei, C. J. Ho, C. Y. Huang, J. W. Shi, L. Henrickson, Y. K. Chen, and J. Chen, “Sparse volterra nonlinear equalizer by employing pruning algorithm for high-speed PAM-4 850-nm VCSEL optical interconnect,” in Optical Fiber Communication Conference (Optical Society of America, 2019), paper M1F.2.

M. Luo, F. Gao, X. Li, Z. He, and S. Fu, “Transmission of 4×50-Gbs PAM-4 signal over 80-km single mode fiber using neural network,” in Optical Fiber Communication Conference (Optical Society of America, 2018), paper M2F.2.

C. Y. Chuang, C. C. Wei, T. C. Lin, K. L. Chi, L. C. Liu, J. W. Shi, Y. K. Chen, and J. Chen, “Employing deep neural network for high speed 4-PAM optical interconnect,” in European Conference on Optical Communication (2017), pp. W.2.D.2.

J. Estaran, R. Rios-Mueller, M. A. Mestre, F. Jorge, H. Mardoyan, A. Konczykowska, J.-Y. Dupuy, and S. Bigo, “Artificial neural networks for linear and non-linear impairment mitigation in high-baudrate IMDD systems,” in European Conference on Optical Communication (2016), pp. M.2.B.2.

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Figures (8)

Fig. 1.
Fig. 1. Schematic of a 2-layer F-NN/RBF-NN [29,30].
Fig. 2.
Fig. 2. Schematic of a 2-layer AR-RNN [31].
Fig. 3.
Fig. 3. Schematic of a 2-layer L-RNN [32].
Fig. 4.
Fig. 4. System setup of a 50-Gb/s PAM4 direct detection optical link.
Fig. 5.
Fig. 5. (a)/(b) BER/${N_{mul}}$ contour map for F-NN based equalizer; (c), (d) RBF-NN based equalizer; (e), (f) L-RNN based equalizer; (g), (h) AR-RNN based equalizer.
Fig. 6.
Fig. 6. (a) Minimum ${N_{mul}}$ required for 4 types of NNs under different BER, and (b) minimum ${N_{mul}}$ for hard-decision/soft-decision BER thresholds.
Fig. 7.
Fig. 7. (a) BER versus ROP using different NN-based equalizers, and (b) ${N_{mul}}$ of different NNs with (5,4) or (11,7).
Fig. 8.
Fig. 8. BER versus fiber length using different NN-based equalizers.

Equations (9)

Equations on this page are rendered with MathJax. Learn more.

y = f [ 2 ] ( w [ 2 ] f [ 1 ] ( w [ 1 ] X + b [ 1 ] ) + b [ 2 ] ) ,
N m u l _ F N N = ( n [ 0 ] + 1 ) n [ 1 ] .
H i [ 1 ] = f [ 1 ] ( b i [ 1 ] ( w i [ 1 ] ) T X ) ,   i = 1 , 2 , n [ 1 ] ,
y = f [ 2 ] ( w [ 2 ] H [ 1 ] + b [ 2 ] ) .
N m u l _ R B F N N = ( n [ 0 ] + 2 ) n [ 1 ] .
y = f [ 2 ] ( w [ 2 ] f [ 1 ] ( [ w [ 1 ] , w d ] [ X Y d ] + b [ 1 ] ) + b [ 2 ] ) ,
N m u l _ A R R N N = ( n [ 0 ] + k + 1 ) n [ 1 ] .
y = f [ 2 ] ( w [ 2 ] f [ 1 ] ( [ w [ 1 ] , w h ] [ X H h ] + b [ 1 ] ) + b [ 2 ] ) .
N m u l _ L R N N = ( n [ 0 ] + n [ 1 ] + 1 ) n [ 1 ] .

Metrics