Abstract

We investigate the application of dynamic deep neural networks for nonlinear equalization in long haul transmission systems. Through extensive numerical analysis we identify their optimum dimensions and calculate their computational complexity as a function of system length. Performing comparison with traditional back-propagation based nonlinear compensation of 2 steps-per-span and 2 samples-per-symbol, we demonstrate equivalent mitigation performance at significantly lower computational cost.

© 2018 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]
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    [Crossref]
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2017 (2)

D. Wang, M. Zhang, Z. Li, C. Song, M. Fu, J. Li, and X. Chen, “System impairment compensation in coherent optical communications by using a bio-inspired detector based on artificial neural network and genetic algorithm,” Opt. Commun. 399, 1–12 (2017).
[Crossref]

M. Gagni, F. Guiomar, S. Wabnitz, and A. Pinto, “Simplified high-order Volterra series transfer function for optical transmission links,” Opt. Express 25(3), 2446–2459 (2017).
[Crossref]

2016 (5)

D. Rafique, “Fiber” Nonlinearity Compensation: Commercial Applications and Complexity Analysis,” J. Lightw. Technol. 34(2), 544–553 (2016).
[Crossref]

D. Wang, M. Zhang, M. Fu, Z. Cai, Z. Li, H. Han, Y. Cui, and B. Luo, “Nonlinearity Mitigation Using a Machine Learning Detector Based on k-Nearest Neighbors,” IEEE Photon. Technol. Lett. 28(19), 2102–2105 (2016).
[Crossref]

D. Wang, M. Zhang, Z. Cai, Y. Cui, Z. Li, H. Han, M. Fu, and B. Luo, “Combatting nonlinear phase noise in coherent optical systems with an optimized decision processor based on machine learning,” Opt. Commun. 369, 199–208 (2016).
[Crossref]

T. Nguyen, S. Mhatli, E. Giacoumidis, L.V. Compernolle, M. Wuilpart, and P. Megret, “Fiber onlinearity Equalizer Based on Support Vector Classification for Coherent Optical OFDM,” IEEE Photon. J. 8(2), 7802009 (2016).
[Crossref]

S. T. Ahmad and K. P. Kumar, “Radial Basis Function Neural Network Nonlinear Equalizer for 16-QAM Coherent Optical OFDM,” IEEE Photon. Technol. Lett. 28(22), 2507–2510 (2016).
[Crossref]

2015 (3)

M. A. Jarajreh, E. Giacoumidis, I. Aldaya, S. T. Thai, A. Tsokanos, Z. Ghassenmlooy, and N. J. Doran, “Artificial Neural Network Nonlinear Equalizer for Coherent Optical OFDM,” IEEE Photon. Technol. Lett. 27(4), 387–390 (2015).
[Crossref]

E. Temprana, E.B. Myslivets, P.-P. Kuo, V. Ataie, N. Alic, and S. Radic, “Overcoming Kerr-induced capacity limit in optical fiber transmission,” Science 348(6242), 1445–1448 (2015).
[Crossref] [PubMed]

E. Giacoumidis, S. Le, M. Ghanbarisabagh, M. McCarthy, I. Aldaya, S. Mhatli, M. Jarajreh, P. Haigh, N. Doran, A. Ellis, and B. Eggleton, “Fiber nonlinearity-induced penalty reduction in CO-OFDM by ANN-based nonlinear equalization,” Opt. Lett. 40(21), 5113–5116 (2015).
[Crossref] [PubMed]

2014 (2)

A. Napoli, Z. Maalej, V. Sleiffer, M. Kuschnerov, D. Rafique, E. Timmers, B. Spinnler, T. Rahman, L. Coelho, and N. Hanik, “Reduced Complexity Digital Back-Propagation Methods for Optical Communication Systems,” J. Lightw. Technol. 32(7), 1351–1362 (2014).
[Crossref]

L. B. Du, D. Rafique, A. Napoli, B. Spinnler, A. D. Ellis, M. Kuschnerov, and A. J. Lowery, “Digital Fiber” Nonlinearity Compensation: Toward 1-Tb/s transport,” IEEE Commun. Mag. 31(2), 46–56 (2014).

2013 (2)

M. Li, S. Yu, J. Yang, Z. Chen, Y. Han, and W. Gu, “Nonparameter Nonlinear Phase Noise Mitigation by Using M-ary Support Vector Machine for Coherent Optical Systems,” IEEE Photon. J. 5(6), 7800312 (2013).
[Crossref]

S. Li, B. Liu, B. Chen, and Y. Lou, “Neural network based mobile phone localization using Bluetooth connectivity,” Neural Comput. Appl. 23(3–4), 667–675 (2013).
[Crossref]

2012 (3)

2011 (1)

2010 (4)

B. Spinnler, “Equalizer Design and Complexity for Digital Coherent Receivers,” IEEE J. Sel. Top. Quantum Electron. 16(5), 1180–1192 (2010).
[Crossref]

E. Ip, “Nonlinear Compensation Using Backpropagation for Polarization-Multiplexed Transmission,” J. Lightw. Technol 28(6), 939–951 (2010).
[Crossref]

R. Tkach, “Scaling optical communications for next decade and beyond,” Bell Labs Tech. J. 14(4), 3–9 (2010).
[Crossref]

D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Payvel, and S. J. Savory, “Mitigation of Fiber” Nonlinearity Using a Digital Coherent Receiver,” IEEE J. Sel. Top. Quantum Electron. 16(5), 1217–1226 (2010).
[Crossref]

2009 (1)

S. Rajbhandari, Z. Ghassemlooy, and M. Angelova, “Effective Denoising and Adaptive Equalization of Indoor Optical Wireless Channel With Artificial Light Using the Discrete Wavelet Transform and Artificial Neural Network,” J. Lightw. Technol. 27(20), 4493–4500 (2009).
[Crossref]

2008 (2)

E. Ip and J. M. Kahn, “Compensation of Dispersion and Nonlinear Impairments Using Digital Backpropagation,” J. Lightw. Technol. 26(20), 3416–3425 (2008).
[Crossref]

X. Li, X. Chen, G. Goldfarb, E. Mateo, I. Kim, F. Yaman, and G. Li, “Electronic post-compensation of WDM transmission impairments using coherent detection and digital signal processing,” Opt. Express 16(2), 881–888 (2008).

1948 (1)

C. E. Shannon, “A mathematical theory of communication,” Bell Labs Tech. J. 27(3), 379–423 (1948).
[Crossref]

Ahmad, S. T.

S. T. Ahmad and K. P. Kumar, “Radial Basis Function Neural Network Nonlinear Equalizer for 16-QAM Coherent Optical OFDM,” IEEE Photon. Technol. Lett. 28(22), 2507–2510 (2016).
[Crossref]

Aldaya, I.

M. A. Jarajreh, E. Giacoumidis, I. Aldaya, S. T. Thai, A. Tsokanos, Z. Ghassenmlooy, and N. J. Doran, “Artificial Neural Network Nonlinear Equalizer for Coherent Optical OFDM,” IEEE Photon. Technol. Lett. 27(4), 387–390 (2015).
[Crossref]

E. Giacoumidis, S. Le, M. Ghanbarisabagh, M. McCarthy, I. Aldaya, S. Mhatli, M. Jarajreh, P. Haigh, N. Doran, A. Ellis, and B. Eggleton, “Fiber nonlinearity-induced penalty reduction in CO-OFDM by ANN-based nonlinear equalization,” Opt. Lett. 40(21), 5113–5116 (2015).
[Crossref] [PubMed]

E. Giacoumidis, I. Aldaya, J. L. Wei, C. Sanchez, H. Mrabet, and L. P. Barry, “Affinity propagation clustering for blind nonlinearity compensation in coherent optical OFDM,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (online) (Optical Society of America, 2018), paper STh1C.5.
[Crossref]

Alic, N.

E. Temprana, E.B. Myslivets, P.-P. Kuo, V. Ataie, N. Alic, and S. Radic, “Overcoming Kerr-induced capacity limit in optical fiber transmission,” Science 348(6242), 1445–1448 (2015).
[Crossref] [PubMed]

Angelova, M.

S. Rajbhandari, Z. Ghassemlooy, and M. Angelova, “Effective Denoising and Adaptive Equalization of Indoor Optical Wireless Channel With Artificial Light Using the Discrete Wavelet Transform and Artificial Neural Network,” J. Lightw. Technol. 27(20), 4493–4500 (2009).
[Crossref]

Arlunno, V.

Ataie, V.

E. Temprana, E.B. Myslivets, P.-P. Kuo, V. Ataie, N. Alic, and S. Radic, “Overcoming Kerr-induced capacity limit in optical fiber transmission,” Science 348(6242), 1445–1448 (2015).
[Crossref] [PubMed]

Barry, L. P.

E. Giacoumidis, I. Aldaya, J. L. Wei, C. Sanchez, H. Mrabet, and L. P. Barry, “Affinity propagation clustering for blind nonlinearity compensation in coherent optical OFDM,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (online) (Optical Society of America, 2018), paper STh1C.5.
[Crossref]

Behrens, C.

D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Payvel, and S. J. Savory, “Mitigation of Fiber” Nonlinearity Using a Digital Coherent Receiver,” IEEE J. Sel. Top. Quantum Electron. 16(5), 1217–1226 (2010).
[Crossref]

Bigo, S.

J. Estaran, R. Rios-Muller, 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 IM/DD Systems,” in Proceedings of European Conference on Optical Communication (ECOC) (2016), paper M.2.B.2.

Borkowski, R.

Braun, H.

M. Riedmiller and H. Braun, “A direct adaptive method for faster backpropagation learning: the RPROP algorithm,” in Proceedings of the International Conference on Neural Networks (IEEE, 1993), pp. 586–591.
[Crossref]

Caballero, A.

Cai, Y.

L. Liu, L. Li, Y. Huang, K. Cui, Q. Xiong, F. N. Hauske, C. Xie, and Y. Cai, “Intrachannel” Nonlinearity Compensation by Inverse Volterra Series Transfer Function,” J. Lightw. Technol. 30(3), 310–316 (2012).
[Crossref]

Cai, Z.

D. Wang, M. Zhang, M. Fu, Z. Cai, Z. Li, H. Han, Y. Cui, and B. Luo, “Nonlinearity Mitigation Using a Machine Learning Detector Based on k-Nearest Neighbors,” IEEE Photon. Technol. Lett. 28(19), 2102–2105 (2016).
[Crossref]

D. Wang, M. Zhang, Z. Cai, Y. Cui, Z. Li, H. Han, M. Fu, and B. Luo, “Combatting nonlinear phase noise in coherent optical systems with an optimized decision processor based on machine learning,” Opt. Commun. 369, 199–208 (2016).
[Crossref]

Chen, B.

S. Li, B. Liu, B. Chen, and Y. Lou, “Neural network based mobile phone localization using Bluetooth connectivity,” Neural Comput. Appl. 23(3–4), 667–675 (2013).
[Crossref]

Chen, X.

D. Wang, M. Zhang, Z. Li, C. Song, M. Fu, J. Li, and X. Chen, “System impairment compensation in coherent optical communications by using a bio-inspired detector based on artificial neural network and genetic algorithm,” Opt. Commun. 399, 1–12 (2017).
[Crossref]

X. Li, X. Chen, G. Goldfarb, E. Mateo, I. Kim, F. Yaman, and G. Li, “Electronic post-compensation of WDM transmission impairments using coherent detection and digital signal processing,” Opt. Express 16(2), 881–888 (2008).

Chen, Z.

M. Li, S. Yu, J. Yang, Z. Chen, Y. Han, and W. Gu, “Nonparameter Nonlinear Phase Noise Mitigation by Using M-ary Support Vector Machine for Coherent Optical Systems,” IEEE Photon. J. 5(6), 7800312 (2013).
[Crossref]

Chugtai, M. N.

Coelho, L.

A. Napoli, Z. Maalej, V. Sleiffer, M. Kuschnerov, D. Rafique, E. Timmers, B. Spinnler, T. Rahman, L. Coelho, and N. Hanik, “Reduced Complexity Digital Back-Propagation Methods for Optical Communication Systems,” J. Lightw. Technol. 32(7), 1351–1362 (2014).
[Crossref]

Compernolle, L.V.

T. Nguyen, S. Mhatli, E. Giacoumidis, L.V. Compernolle, M. Wuilpart, and P. Megret, “Fiber onlinearity Equalizer Based on Support Vector Classification for Coherent Optical OFDM,” IEEE Photon. J. 8(2), 7802009 (2016).
[Crossref]

Cui, K.

L. Liu, L. Li, Y. Huang, K. Cui, Q. Xiong, F. N. Hauske, C. Xie, and Y. Cai, “Intrachannel” Nonlinearity Compensation by Inverse Volterra Series Transfer Function,” J. Lightw. Technol. 30(3), 310–316 (2012).
[Crossref]

Cui, Y.

D. Wang, M. Zhang, M. Fu, Z. Cai, Z. Li, H. Han, Y. Cui, and B. Luo, “Nonlinearity Mitigation Using a Machine Learning Detector Based on k-Nearest Neighbors,” IEEE Photon. Technol. Lett. 28(19), 2102–2105 (2016).
[Crossref]

D. Wang, M. Zhang, Z. Cai, Y. Cui, Z. Li, H. Han, M. Fu, and B. Luo, “Combatting nonlinear phase noise in coherent optical systems with an optimized decision processor based on machine learning,” Opt. Commun. 369, 199–208 (2016).
[Crossref]

Doran, N.

Doran, N. J.

M. A. Jarajreh, E. Giacoumidis, I. Aldaya, S. T. Thai, A. Tsokanos, Z. Ghassenmlooy, and N. J. Doran, “Artificial Neural Network Nonlinear Equalizer for Coherent Optical OFDM,” IEEE Photon. Technol. Lett. 27(4), 387–390 (2015).
[Crossref]

Du, L. B.

L. B. Du, D. Rafique, A. Napoli, B. Spinnler, A. D. Ellis, M. Kuschnerov, and A. J. Lowery, “Digital Fiber” Nonlinearity Compensation: Toward 1-Tb/s transport,” IEEE Commun. Mag. 31(2), 46–56 (2014).

Duan, C.

Dupuy, J.-Y.

J. Estaran, R. Rios-Muller, 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 IM/DD Systems,” in Proceedings of European Conference on Optical Communication (ECOC) (2016), paper M.2.B.2.

Eggleton, B.

Ellis, A.

Ellis, A. D.

L. B. Du, D. Rafique, A. Napoli, B. Spinnler, A. D. Ellis, M. Kuschnerov, and A. J. Lowery, “Digital Fiber” Nonlinearity Compensation: Toward 1-Tb/s transport,” IEEE Commun. Mag. 31(2), 46–56 (2014).

D. Rafique, M. Mussolin, M. Forzati, J. Martensson, M. N. Chugtai, and A. D. Ellis, “Compensation of intra-channel nonlinear fibre impairments using simplified digital back-propagation algorithm,” Opt. Express 19(10), 9453–9460 (2011).
[Crossref] [PubMed]

Estaran, J.

J. Estaran, R. Rios-Muller, 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 IM/DD Systems,” in Proceedings of European Conference on Optical Communication (ECOC) (2016), paper M.2.B.2.

Forzati, M.

Franceschi, N.

Fu, M.

D. Wang, M. Zhang, Z. Li, C. Song, M. Fu, J. Li, and X. Chen, “System impairment compensation in coherent optical communications by using a bio-inspired detector based on artificial neural network and genetic algorithm,” Opt. Commun. 399, 1–12 (2017).
[Crossref]

D. Wang, M. Zhang, Z. Cai, Y. Cui, Z. Li, H. Han, M. Fu, and B. Luo, “Combatting nonlinear phase noise in coherent optical systems with an optimized decision processor based on machine learning,” Opt. Commun. 369, 199–208 (2016).
[Crossref]

D. Wang, M. Zhang, M. Fu, Z. Cai, Z. Li, H. Han, Y. Cui, and B. Luo, “Nonlinearity Mitigation Using a Machine Learning Detector Based on k-Nearest Neighbors,” IEEE Photon. Technol. Lett. 28(19), 2102–2105 (2016).
[Crossref]

Gagni, M.

Ghanbarisabagh, M.

Ghassemlooy, Z.

S. Rajbhandari, Z. Ghassemlooy, and M. Angelova, “Effective Denoising and Adaptive Equalization of Indoor Optical Wireless Channel With Artificial Light Using the Discrete Wavelet Transform and Artificial Neural Network,” J. Lightw. Technol. 27(20), 4493–4500 (2009).
[Crossref]

Ghassenmlooy, Z.

M. A. Jarajreh, E. Giacoumidis, I. Aldaya, S. T. Thai, A. Tsokanos, Z. Ghassenmlooy, and N. J. Doran, “Artificial Neural Network Nonlinear Equalizer for Coherent Optical OFDM,” IEEE Photon. Technol. Lett. 27(4), 387–390 (2015).
[Crossref]

Giacoumidis, E.

T. Nguyen, S. Mhatli, E. Giacoumidis, L.V. Compernolle, M. Wuilpart, and P. Megret, “Fiber onlinearity Equalizer Based on Support Vector Classification for Coherent Optical OFDM,” IEEE Photon. J. 8(2), 7802009 (2016).
[Crossref]

M. A. Jarajreh, E. Giacoumidis, I. Aldaya, S. T. Thai, A. Tsokanos, Z. Ghassenmlooy, and N. J. Doran, “Artificial Neural Network Nonlinear Equalizer for Coherent Optical OFDM,” IEEE Photon. Technol. Lett. 27(4), 387–390 (2015).
[Crossref]

E. Giacoumidis, S. Le, M. Ghanbarisabagh, M. McCarthy, I. Aldaya, S. Mhatli, M. Jarajreh, P. Haigh, N. Doran, A. Ellis, and B. Eggleton, “Fiber nonlinearity-induced penalty reduction in CO-OFDM by ANN-based nonlinear equalization,” Opt. Lett. 40(21), 5113–5116 (2015).
[Crossref] [PubMed]

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Jarajreh, M. A.

M. A. Jarajreh, E. Giacoumidis, I. Aldaya, S. T. Thai, A. Tsokanos, Z. Ghassenmlooy, and N. J. Doran, “Artificial Neural Network Nonlinear Equalizer for Coherent Optical OFDM,” IEEE Photon. Technol. Lett. 27(4), 387–390 (2015).
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A. Napoli, Z. Maalej, V. Sleiffer, M. Kuschnerov, D. Rafique, E. Timmers, B. Spinnler, T. Rahman, L. Coelho, and N. Hanik, “Reduced Complexity Digital Back-Propagation Methods for Optical Communication Systems,” J. Lightw. Technol. 32(7), 1351–1362 (2014).
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Mateo, E.

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T. Nguyen, S. Mhatli, E. Giacoumidis, L.V. Compernolle, M. Wuilpart, and P. Megret, “Fiber onlinearity Equalizer Based on Support Vector Classification for Coherent Optical OFDM,” IEEE Photon. J. 8(2), 7802009 (2016).
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A. Napoli, Z. Maalej, V. Sleiffer, M. Kuschnerov, D. Rafique, E. Timmers, B. Spinnler, T. Rahman, L. Coelho, and N. Hanik, “Reduced Complexity Digital Back-Propagation Methods for Optical Communication Systems,” J. Lightw. Technol. 32(7), 1351–1362 (2014).
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A. Napoli, Z. Maalej, V. Sleiffer, M. Kuschnerov, D. Rafique, E. Timmers, B. Spinnler, T. Rahman, L. Coelho, and N. Hanik, “Reduced Complexity Digital Back-Propagation Methods for Optical Communication Systems,” J. Lightw. Technol. 32(7), 1351–1362 (2014).
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Figures (8)

Fig. 1
Fig. 1 Scheme of simulated transmission link.
Fig. 2
Fig. 2 Deep neural network architecture. The size of the input layer (Li) is ni, and the two hidden layers (L1, L2) and the output layer (Lo) have n1, n2 and no neurons, respectively
Fig. 3
Fig. 3 (a) BER as a function of number of delay taps Ndel for different number of spans; (b) number of required delay taps Ndel a function of number of spans in transmission links. Span length equals 100 km.
Fig. 4
Fig. 4 (a) BER as a function of launch power for different number of neurons per layer; (b) BER as a function of launch power for different number of hidden layers.
Fig. 5
Fig. 5 (a) Q2-factor as a function of number of spans for single channel transmission; (b) Q2-factor as a function of number of spans for 5-channel Nyquist-WDM transmission. Results correspond to the middle channel.
Fig. 6
Fig. 6 The dependency of the number of performed epochs on number of spans.
Fig. 7
Fig. 7 Dependency of complexity for DBP and (FDE+dNN) based nonlinear equalization schemes as a function of the number of transmitted symbols Nps.
Fig. 8
Fig. 8 Complexity of DBP and (FDE+dNN) based nonlinear equalizers as a function of transmission length

Equations (10)

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

C L i n D B P = N ( log 2 N + 1 ) n s ( N N D + 1 ) log 2 M ,
C D B P = 4 N S p a n N S t p S p ( N ( log 2 N + 1 ) n s ( N N D + 1 ) log 2 M + n s ) .
C F D E = 4 N ( log 2 N + 1 ) n s ( N N D + 1 ) log 2 M .
Δ ω i , j ( t ) = { Δ i , j ( t ) , if E ( t ) ω i , j > 0 , + Δ i , j ( t ) , if E ( t ) ω i , j < 0 , 0 , else ,
Δ i , j ( t ) = { η + Δ i , j ( t 1 ) , if E ( t 1 ) ω i , j E ( t ) ω i , j > 0 , η Δ i , j ( t 1 ) , if E ( t 1 ) ω i , j E ( t ) ω i , j < 0 , Δ i , j ( t 1 ) , else ,
E ω i , j = x i δ j = x i ( x j t j ) ,
E ω i , j = x i δ j = x i ( l L o δ l ω j , l ) ( 1 + x j ) ( 1 x j ) .
C d N N t r a i n = N e p [ ( N t s + N v s ) ( n i n 1 + n 1 n 2 + n 2 n o ) + 3 n o n 2 + n o n 2 n 1 + 5 n 2 n 1 + n 2 n 1 n i + 5 n 1 n i ] ( N p s + N t s + N v s ) log 2 M ,
C d N N p r e d i c t = N p s ( n i n 1 + n 1 n 2 + n 2 n o ) ( N p s + N t s + N v s ) log 2 M .
C F D E + d N N = C d N N t r a i n + C d N N p r e d i c t + C F D E .

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