Abstract

We propose a method to improve the performance of the nonlinear Fourier transform (NFT)-based optical transmission system by applying the neural network post-processing of the nonlinear spectrum at the receiver. We demonstrate through numerical modeling about one order of magnitude bit error rate improvement and compare this method with machine learning processing based on the classification of the received symbols. The proposed approach also offers a way to improve numerical accuracy of the inverse NFT; therefore, it can find a range of applications beyond optical communications.

Published by The Optical Society under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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  1. V. E. Zakharov and A. B. Shabat, Soviet Phys. JETP 34, 62 (1972).
  2. A. Hasegawa and T. Nyu, J. Lightwave Technol. 11, 395 (1993).
    [Crossref]
  3. J. E. Prilepsky, S. A. Derevyanko, K. J. Blow, I. Gabitov, and S. K. Turitsyn, Phys. Rev. Lett. 113, 013901 (2014).
    [Crossref]
  4. M. Yousefi and F. Kschischang, IEEE Trans. Inf. Theory 60, 4312 (2014).
    [Crossref]
  5. S. K. Turitsyn, J. E. Prilepsky, S. T. Le, S. Wahls, L. L. Frumin, M. Kamalian, and S. A. Derevyanko, Optica 4, 307 (2017).
    [Crossref]
  6. E. G. Turitsyna and S. K. Turitsyn, Opt. Lett. 38, 4186 (2013).
    [Crossref]
  7. S. Civelli, E. Forestieri, and M. Secondini, IEEE Photon. Technol. Lett. 29, 1332 (2017).
    [Crossref]
  8. S. A. Derevyanko, J. E. Prilepsky, and S. K. Turitsyn, Nat. Commun. 7, 307 (2016).
    [Crossref]
  9. M. Pankratova, A. Vasylchenkova, S. A. Derevyanko, N. B. Chichkov, and J. E. Prilepsky, Phys. Rev. Appl 13, 054021 (2020).
    [Crossref]
  10. F. Musumeci, C. Rottondi, A. Nag, I. Macaluso, D. Zibar, M. Ruffini, and M. Tornatore, IEEE Commun. Surv. Tutorials 21, 1383 (2019).
    [Crossref]
  11. F. N. Khan, Q. Fan, C. Lu, and A. P. T. Lau, J. Lightwave Technol. 37, 493 (2019).
    [Crossref]
  12. J. Mata, I. de Miguel, R. J. Durán, N. Merayo, S. K. Singh, A. Jukan, and M. Chamania, Opt. Switch. Netw. 28, 43 (2018).
    [Crossref]
  13. D. Zibar, F. Da Ros, G. Brajato, and U. C. de Moura, Opt. Photon. News 31, 34 (2020).
    [Crossref]
  14. R. T. Jones, S. Gaiarin, M. P. Yankov, and D. Zibar, IEEE Photon. Technol. Lett. 30, 1079 (2018).
    [Crossref]
  15. O. Kotlyar, M. Pankratova, M. Kamalian, A. Vasylchenkova, J. E. Prilepsky, and S. K. Turitsyn, in 2018 IEEE British and Irish Conference on Optics and Photonics (BICOP) (2018).
  16. O. Kotlyar, M. Kamalian, M. Pankratova, J. E. Prilepsky, and S. K. Turitsyn, in European Conference on Optical Communication (ECOC) (2019).
  17. M. Kamalian-Kopae, A. Vasylchenkova, O. Kotlyar, M. Pankratova, J. Prilepsky, and S. Turitsyn, in 2019 Conference on Lasers and Electro-Optics Europe and European Quantum Electronics Conference (CLEO/Europe-EQEC) (IEEE, 2019).
  18. S. T. Le, V. Aref, and H. Buelow, J. Lightwave Technol. 36, 1296 (2018).
    [Crossref]
  19. S. Wahls, in 2017 European Conference on Optical Communication (ECOC) (2017).
  20. R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, J. Lightwave Technol. 28, 662 (2010).
    [Crossref]
  21. S. Wahls, S. Chimmalgi, and P. J. Prins, J. Open Source Software 3, 597 (2018).
    [Crossref]
  22. T. Dozat, in International Conference on Learning Representations (ICLR) (2016).

2020 (2)

M. Pankratova, A. Vasylchenkova, S. A. Derevyanko, N. B. Chichkov, and J. E. Prilepsky, Phys. Rev. Appl 13, 054021 (2020).
[Crossref]

D. Zibar, F. Da Ros, G. Brajato, and U. C. de Moura, Opt. Photon. News 31, 34 (2020).
[Crossref]

2019 (2)

F. Musumeci, C. Rottondi, A. Nag, I. Macaluso, D. Zibar, M. Ruffini, and M. Tornatore, IEEE Commun. Surv. Tutorials 21, 1383 (2019).
[Crossref]

F. N. Khan, Q. Fan, C. Lu, and A. P. T. Lau, J. Lightwave Technol. 37, 493 (2019).
[Crossref]

2018 (4)

J. Mata, I. de Miguel, R. J. Durán, N. Merayo, S. K. Singh, A. Jukan, and M. Chamania, Opt. Switch. Netw. 28, 43 (2018).
[Crossref]

R. T. Jones, S. Gaiarin, M. P. Yankov, and D. Zibar, IEEE Photon. Technol. Lett. 30, 1079 (2018).
[Crossref]

S. T. Le, V. Aref, and H. Buelow, J. Lightwave Technol. 36, 1296 (2018).
[Crossref]

S. Wahls, S. Chimmalgi, and P. J. Prins, J. Open Source Software 3, 597 (2018).
[Crossref]

2017 (2)

2016 (1)

S. A. Derevyanko, J. E. Prilepsky, and S. K. Turitsyn, Nat. Commun. 7, 307 (2016).
[Crossref]

2014 (2)

J. E. Prilepsky, S. A. Derevyanko, K. J. Blow, I. Gabitov, and S. K. Turitsyn, Phys. Rev. Lett. 113, 013901 (2014).
[Crossref]

M. Yousefi and F. Kschischang, IEEE Trans. Inf. Theory 60, 4312 (2014).
[Crossref]

2013 (1)

2010 (1)

1993 (1)

A. Hasegawa and T. Nyu, J. Lightwave Technol. 11, 395 (1993).
[Crossref]

1972 (1)

V. E. Zakharov and A. B. Shabat, Soviet Phys. JETP 34, 62 (1972).

Aref, V.

Blow, K. J.

J. E. Prilepsky, S. A. Derevyanko, K. J. Blow, I. Gabitov, and S. K. Turitsyn, Phys. Rev. Lett. 113, 013901 (2014).
[Crossref]

Brajato, G.

D. Zibar, F. Da Ros, G. Brajato, and U. C. de Moura, Opt. Photon. News 31, 34 (2020).
[Crossref]

Buelow, H.

Chamania, M.

J. Mata, I. de Miguel, R. J. Durán, N. Merayo, S. K. Singh, A. Jukan, and M. Chamania, Opt. Switch. Netw. 28, 43 (2018).
[Crossref]

Chichkov, N. B.

M. Pankratova, A. Vasylchenkova, S. A. Derevyanko, N. B. Chichkov, and J. E. Prilepsky, Phys. Rev. Appl 13, 054021 (2020).
[Crossref]

Chimmalgi, S.

S. Wahls, S. Chimmalgi, and P. J. Prins, J. Open Source Software 3, 597 (2018).
[Crossref]

Civelli, S.

S. Civelli, E. Forestieri, and M. Secondini, IEEE Photon. Technol. Lett. 29, 1332 (2017).
[Crossref]

Da Ros, F.

D. Zibar, F. Da Ros, G. Brajato, and U. C. de Moura, Opt. Photon. News 31, 34 (2020).
[Crossref]

de Miguel, I.

J. Mata, I. de Miguel, R. J. Durán, N. Merayo, S. K. Singh, A. Jukan, and M. Chamania, Opt. Switch. Netw. 28, 43 (2018).
[Crossref]

de Moura, U. C.

D. Zibar, F. Da Ros, G. Brajato, and U. C. de Moura, Opt. Photon. News 31, 34 (2020).
[Crossref]

Derevyanko, S. A.

M. Pankratova, A. Vasylchenkova, S. A. Derevyanko, N. B. Chichkov, and J. E. Prilepsky, Phys. Rev. Appl 13, 054021 (2020).
[Crossref]

S. K. Turitsyn, J. E. Prilepsky, S. T. Le, S. Wahls, L. L. Frumin, M. Kamalian, and S. A. Derevyanko, Optica 4, 307 (2017).
[Crossref]

S. A. Derevyanko, J. E. Prilepsky, and S. K. Turitsyn, Nat. Commun. 7, 307 (2016).
[Crossref]

J. E. Prilepsky, S. A. Derevyanko, K. J. Blow, I. Gabitov, and S. K. Turitsyn, Phys. Rev. Lett. 113, 013901 (2014).
[Crossref]

Dozat, T.

T. Dozat, in International Conference on Learning Representations (ICLR) (2016).

Durán, R. J.

J. Mata, I. de Miguel, R. J. Durán, N. Merayo, S. K. Singh, A. Jukan, and M. Chamania, Opt. Switch. Netw. 28, 43 (2018).
[Crossref]

Essiambre, R.-J.

Fan, Q.

Forestieri, E.

S. Civelli, E. Forestieri, and M. Secondini, IEEE Photon. Technol. Lett. 29, 1332 (2017).
[Crossref]

Foschini, G. J.

Frumin, L. L.

Gabitov, I.

J. E. Prilepsky, S. A. Derevyanko, K. J. Blow, I. Gabitov, and S. K. Turitsyn, Phys. Rev. Lett. 113, 013901 (2014).
[Crossref]

Gaiarin, S.

R. T. Jones, S. Gaiarin, M. P. Yankov, and D. Zibar, IEEE Photon. Technol. Lett. 30, 1079 (2018).
[Crossref]

Goebel, B.

Hasegawa, A.

A. Hasegawa and T. Nyu, J. Lightwave Technol. 11, 395 (1993).
[Crossref]

Jones, R. T.

R. T. Jones, S. Gaiarin, M. P. Yankov, and D. Zibar, IEEE Photon. Technol. Lett. 30, 1079 (2018).
[Crossref]

Jukan, A.

J. Mata, I. de Miguel, R. J. Durán, N. Merayo, S. K. Singh, A. Jukan, and M. Chamania, Opt. Switch. Netw. 28, 43 (2018).
[Crossref]

Kamalian, M.

S. K. Turitsyn, J. E. Prilepsky, S. T. Le, S. Wahls, L. L. Frumin, M. Kamalian, and S. A. Derevyanko, Optica 4, 307 (2017).
[Crossref]

O. Kotlyar, M. Pankratova, M. Kamalian, A. Vasylchenkova, J. E. Prilepsky, and S. K. Turitsyn, in 2018 IEEE British and Irish Conference on Optics and Photonics (BICOP) (2018).

O. Kotlyar, M. Kamalian, M. Pankratova, J. E. Prilepsky, and S. K. Turitsyn, in European Conference on Optical Communication (ECOC) (2019).

Kamalian-Kopae, M.

M. Kamalian-Kopae, A. Vasylchenkova, O. Kotlyar, M. Pankratova, J. Prilepsky, and S. Turitsyn, in 2019 Conference on Lasers and Electro-Optics Europe and European Quantum Electronics Conference (CLEO/Europe-EQEC) (IEEE, 2019).

Khan, F. N.

Kotlyar, O.

O. Kotlyar, M. Pankratova, M. Kamalian, A. Vasylchenkova, J. E. Prilepsky, and S. K. Turitsyn, in 2018 IEEE British and Irish Conference on Optics and Photonics (BICOP) (2018).

O. Kotlyar, M. Kamalian, M. Pankratova, J. E. Prilepsky, and S. K. Turitsyn, in European Conference on Optical Communication (ECOC) (2019).

M. Kamalian-Kopae, A. Vasylchenkova, O. Kotlyar, M. Pankratova, J. Prilepsky, and S. Turitsyn, in 2019 Conference on Lasers and Electro-Optics Europe and European Quantum Electronics Conference (CLEO/Europe-EQEC) (IEEE, 2019).

Kramer, G.

Kschischang, F.

M. Yousefi and F. Kschischang, IEEE Trans. Inf. Theory 60, 4312 (2014).
[Crossref]

Lau, A. P. T.

Le, S. T.

Lu, C.

Macaluso, I.

F. Musumeci, C. Rottondi, A. Nag, I. Macaluso, D. Zibar, M. Ruffini, and M. Tornatore, IEEE Commun. Surv. Tutorials 21, 1383 (2019).
[Crossref]

Mata, J.

J. Mata, I. de Miguel, R. J. Durán, N. Merayo, S. K. Singh, A. Jukan, and M. Chamania, Opt. Switch. Netw. 28, 43 (2018).
[Crossref]

Merayo, N.

J. Mata, I. de Miguel, R. J. Durán, N. Merayo, S. K. Singh, A. Jukan, and M. Chamania, Opt. Switch. Netw. 28, 43 (2018).
[Crossref]

Musumeci, F.

F. Musumeci, C. Rottondi, A. Nag, I. Macaluso, D. Zibar, M. Ruffini, and M. Tornatore, IEEE Commun. Surv. Tutorials 21, 1383 (2019).
[Crossref]

Nag, A.

F. Musumeci, C. Rottondi, A. Nag, I. Macaluso, D. Zibar, M. Ruffini, and M. Tornatore, IEEE Commun. Surv. Tutorials 21, 1383 (2019).
[Crossref]

Nyu, T.

A. Hasegawa and T. Nyu, J. Lightwave Technol. 11, 395 (1993).
[Crossref]

Pankratova, M.

M. Pankratova, A. Vasylchenkova, S. A. Derevyanko, N. B. Chichkov, and J. E. Prilepsky, Phys. Rev. Appl 13, 054021 (2020).
[Crossref]

O. Kotlyar, M. Pankratova, M. Kamalian, A. Vasylchenkova, J. E. Prilepsky, and S. K. Turitsyn, in 2018 IEEE British and Irish Conference on Optics and Photonics (BICOP) (2018).

O. Kotlyar, M. Kamalian, M. Pankratova, J. E. Prilepsky, and S. K. Turitsyn, in European Conference on Optical Communication (ECOC) (2019).

M. Kamalian-Kopae, A. Vasylchenkova, O. Kotlyar, M. Pankratova, J. Prilepsky, and S. Turitsyn, in 2019 Conference on Lasers and Electro-Optics Europe and European Quantum Electronics Conference (CLEO/Europe-EQEC) (IEEE, 2019).

Prilepsky, J.

M. Kamalian-Kopae, A. Vasylchenkova, O. Kotlyar, M. Pankratova, J. Prilepsky, and S. Turitsyn, in 2019 Conference on Lasers and Electro-Optics Europe and European Quantum Electronics Conference (CLEO/Europe-EQEC) (IEEE, 2019).

Prilepsky, J. E.

M. Pankratova, A. Vasylchenkova, S. A. Derevyanko, N. B. Chichkov, and J. E. Prilepsky, Phys. Rev. Appl 13, 054021 (2020).
[Crossref]

S. K. Turitsyn, J. E. Prilepsky, S. T. Le, S. Wahls, L. L. Frumin, M. Kamalian, and S. A. Derevyanko, Optica 4, 307 (2017).
[Crossref]

S. A. Derevyanko, J. E. Prilepsky, and S. K. Turitsyn, Nat. Commun. 7, 307 (2016).
[Crossref]

J. E. Prilepsky, S. A. Derevyanko, K. J. Blow, I. Gabitov, and S. K. Turitsyn, Phys. Rev. Lett. 113, 013901 (2014).
[Crossref]

O. Kotlyar, M. Kamalian, M. Pankratova, J. E. Prilepsky, and S. K. Turitsyn, in European Conference on Optical Communication (ECOC) (2019).

O. Kotlyar, M. Pankratova, M. Kamalian, A. Vasylchenkova, J. E. Prilepsky, and S. K. Turitsyn, in 2018 IEEE British and Irish Conference on Optics and Photonics (BICOP) (2018).

Prins, P. J.

S. Wahls, S. Chimmalgi, and P. J. Prins, J. Open Source Software 3, 597 (2018).
[Crossref]

Rottondi, C.

F. Musumeci, C. Rottondi, A. Nag, I. Macaluso, D. Zibar, M. Ruffini, and M. Tornatore, IEEE Commun. Surv. Tutorials 21, 1383 (2019).
[Crossref]

Ruffini, M.

F. Musumeci, C. Rottondi, A. Nag, I. Macaluso, D. Zibar, M. Ruffini, and M. Tornatore, IEEE Commun. Surv. Tutorials 21, 1383 (2019).
[Crossref]

Secondini, M.

S. Civelli, E. Forestieri, and M. Secondini, IEEE Photon. Technol. Lett. 29, 1332 (2017).
[Crossref]

Shabat, A. B.

V. E. Zakharov and A. B. Shabat, Soviet Phys. JETP 34, 62 (1972).

Singh, S. K.

J. Mata, I. de Miguel, R. J. Durán, N. Merayo, S. K. Singh, A. Jukan, and M. Chamania, Opt. Switch. Netw. 28, 43 (2018).
[Crossref]

Tornatore, M.

F. Musumeci, C. Rottondi, A. Nag, I. Macaluso, D. Zibar, M. Ruffini, and M. Tornatore, IEEE Commun. Surv. Tutorials 21, 1383 (2019).
[Crossref]

Turitsyn, S.

M. Kamalian-Kopae, A. Vasylchenkova, O. Kotlyar, M. Pankratova, J. Prilepsky, and S. Turitsyn, in 2019 Conference on Lasers and Electro-Optics Europe and European Quantum Electronics Conference (CLEO/Europe-EQEC) (IEEE, 2019).

Turitsyn, S. K.

S. K. Turitsyn, J. E. Prilepsky, S. T. Le, S. Wahls, L. L. Frumin, M. Kamalian, and S. A. Derevyanko, Optica 4, 307 (2017).
[Crossref]

S. A. Derevyanko, J. E. Prilepsky, and S. K. Turitsyn, Nat. Commun. 7, 307 (2016).
[Crossref]

J. E. Prilepsky, S. A. Derevyanko, K. J. Blow, I. Gabitov, and S. K. Turitsyn, Phys. Rev. Lett. 113, 013901 (2014).
[Crossref]

E. G. Turitsyna and S. K. Turitsyn, Opt. Lett. 38, 4186 (2013).
[Crossref]

O. Kotlyar, M. Pankratova, M. Kamalian, A. Vasylchenkova, J. E. Prilepsky, and S. K. Turitsyn, in 2018 IEEE British and Irish Conference on Optics and Photonics (BICOP) (2018).

O. Kotlyar, M. Kamalian, M. Pankratova, J. E. Prilepsky, and S. K. Turitsyn, in European Conference on Optical Communication (ECOC) (2019).

Turitsyna, E. G.

Vasylchenkova, A.

M. Pankratova, A. Vasylchenkova, S. A. Derevyanko, N. B. Chichkov, and J. E. Prilepsky, Phys. Rev. Appl 13, 054021 (2020).
[Crossref]

M. Kamalian-Kopae, A. Vasylchenkova, O. Kotlyar, M. Pankratova, J. Prilepsky, and S. Turitsyn, in 2019 Conference on Lasers and Electro-Optics Europe and European Quantum Electronics Conference (CLEO/Europe-EQEC) (IEEE, 2019).

O. Kotlyar, M. Pankratova, M. Kamalian, A. Vasylchenkova, J. E. Prilepsky, and S. K. Turitsyn, in 2018 IEEE British and Irish Conference on Optics and Photonics (BICOP) (2018).

Wahls, S.

S. Wahls, S. Chimmalgi, and P. J. Prins, J. Open Source Software 3, 597 (2018).
[Crossref]

S. K. Turitsyn, J. E. Prilepsky, S. T. Le, S. Wahls, L. L. Frumin, M. Kamalian, and S. A. Derevyanko, Optica 4, 307 (2017).
[Crossref]

S. Wahls, in 2017 European Conference on Optical Communication (ECOC) (2017).

Winzer, P. J.

Yankov, M. P.

R. T. Jones, S. Gaiarin, M. P. Yankov, and D. Zibar, IEEE Photon. Technol. Lett. 30, 1079 (2018).
[Crossref]

Yousefi, M.

M. Yousefi and F. Kschischang, IEEE Trans. Inf. Theory 60, 4312 (2014).
[Crossref]

Zakharov, V. E.

V. E. Zakharov and A. B. Shabat, Soviet Phys. JETP 34, 62 (1972).

Zibar, D.

D. Zibar, F. Da Ros, G. Brajato, and U. C. de Moura, Opt. Photon. News 31, 34 (2020).
[Crossref]

F. Musumeci, C. Rottondi, A. Nag, I. Macaluso, D. Zibar, M. Ruffini, and M. Tornatore, IEEE Commun. Surv. Tutorials 21, 1383 (2019).
[Crossref]

R. T. Jones, S. Gaiarin, M. P. Yankov, and D. Zibar, IEEE Photon. Technol. Lett. 30, 1079 (2018).
[Crossref]

IEEE Commun. Surv. Tutorials (1)

F. Musumeci, C. Rottondi, A. Nag, I. Macaluso, D. Zibar, M. Ruffini, and M. Tornatore, IEEE Commun. Surv. Tutorials 21, 1383 (2019).
[Crossref]

IEEE Photon. Technol. Lett. (2)

R. T. Jones, S. Gaiarin, M. P. Yankov, and D. Zibar, IEEE Photon. Technol. Lett. 30, 1079 (2018).
[Crossref]

S. Civelli, E. Forestieri, and M. Secondini, IEEE Photon. Technol. Lett. 29, 1332 (2017).
[Crossref]

IEEE Trans. Inf. Theory (1)

M. Yousefi and F. Kschischang, IEEE Trans. Inf. Theory 60, 4312 (2014).
[Crossref]

J. Lightwave Technol. (4)

J. Open Source Software (1)

S. Wahls, S. Chimmalgi, and P. J. Prins, J. Open Source Software 3, 597 (2018).
[Crossref]

Nat. Commun. (1)

S. A. Derevyanko, J. E. Prilepsky, and S. K. Turitsyn, Nat. Commun. 7, 307 (2016).
[Crossref]

Opt. Lett. (1)

Opt. Photon. News (1)

D. Zibar, F. Da Ros, G. Brajato, and U. C. de Moura, Opt. Photon. News 31, 34 (2020).
[Crossref]

Opt. Switch. Netw. (1)

J. Mata, I. de Miguel, R. J. Durán, N. Merayo, S. K. Singh, A. Jukan, and M. Chamania, Opt. Switch. Netw. 28, 43 (2018).
[Crossref]

Optica (1)

Phys. Rev. Appl (1)

M. Pankratova, A. Vasylchenkova, S. A. Derevyanko, N. B. Chichkov, and J. E. Prilepsky, Phys. Rev. Appl 13, 054021 (2020).
[Crossref]

Phys. Rev. Lett. (1)

J. E. Prilepsky, S. A. Derevyanko, K. J. Blow, I. Gabitov, and S. K. Turitsyn, Phys. Rev. Lett. 113, 013901 (2014).
[Crossref]

Soviet Phys. JETP (1)

V. E. Zakharov and A. B. Shabat, Soviet Phys. JETP 34, 62 (1972).

Other (5)

S. Wahls, in 2017 European Conference on Optical Communication (ECOC) (2017).

O. Kotlyar, M. Pankratova, M. Kamalian, A. Vasylchenkova, J. E. Prilepsky, and S. K. Turitsyn, in 2018 IEEE British and Irish Conference on Optics and Photonics (BICOP) (2018).

O. Kotlyar, M. Kamalian, M. Pankratova, J. E. Prilepsky, and S. K. Turitsyn, in European Conference on Optical Communication (ECOC) (2019).

M. Kamalian-Kopae, A. Vasylchenkova, O. Kotlyar, M. Pankratova, J. Prilepsky, and S. Turitsyn, in 2019 Conference on Lasers and Electro-Optics Europe and European Quantum Electronics Conference (CLEO/Europe-EQEC) (IEEE, 2019).

T. Dozat, in International Conference on Learning Representations (ICLR) (2016).

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Figures (5)

Fig. 1.
Fig. 1. Schematics of the NFT-based communication systems with two strategies of the receiver-side processing involving ML methods, marked with yellow and purple lines piercing the respective receiver processing blocks.
Fig. 2.
Fig. 2. Schematics of the spectra equalizer implemented as the feed-forward NN with three fully connected hidden layers. As an input to the NN, we take a sample from received nonlinear spectra (blue circle) and its neighbors (gray circles) from each side. The output of the NN is an “equalized” sample.
Fig. 3.
Fig. 3. BER versus power dependence for propagation distance 1000 km, using hard decision (purple curves), classification of received symbols by NN (yellow curves), or applying NN for equalization of the received spectrum (blue curves) for two data rates of 63 Gbit/s (solid lines) and 74 Gbit/s (dashed lines). Horizontal black dashed line represents HD FEC threshold.
Fig. 4.
Fig. 4. BER as a function of the number of neighbors (taps) used in the NN equalization for propagation distance of 1000 km, where the power was taken in the vicinity of optimal value in $-18\; {\rm dBm}$ .
Fig. 5.
Fig. 5. Received constellations without (a) and with (b) ML-based spectrum equalization processing for the launch power of $-18.5\; {\rm dBm}$ of 74 Gbit/s transmission.

Equations (2)

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i q z + 1 2 2 q t 2 + | q | 2 q = η .
r ( ξ , z = 0 ) = k = 63 64 c k s i n c ( 2 ξ k ) ,

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