Abstract

A recursive perturbation theory to model the fiber-optic system is developed. Using this perturbation theory, a multi-stage compensation technique to mitigate the intra-channel nonlinear impairments is investigated. The technique is validated by numerical simulations of a single-polarization single-channel fiber-optic system operating at 28 Gbaud, 32-quadrature amplitude modulation (32-QAM), and 40 × 80 km transmission distance. It is found that, with 2 samples per symbol, the multi-stage scheme with eight compensation stages increases the Q-factor as compared with linear compensation by 4.5 dB; as compared with single-stage compensation, the computational complexity is reduced by a factor of 1.3 and the required memory for storing perturbation coefficients is decreased by a factor of 13.

© 2014 Optical Society of America

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References

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  1. E. Ip and J. M. Kahn, “Compensation of dispersion and nonlinear impairments using digital backpropagation,” J. Lightwave Technol. 26(20), 3416–3425 (2008).
    [Crossref]
  2. 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), 880–888 (2008).
    [Crossref] [PubMed]
  3. L. B. Du and A. J. Lowery, “Improved single channel backpropagation for intra-channel fiber nonlinearity compensation in long-haul optical communication systems,” Opt. Express 18(16), 17075–17088 (2010).
    [Crossref] [PubMed]
  4. D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, 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]
  5. D. Rafique and A. D. Ellis, “Impact of signal-ASE four-wave mixing on the effectiveness of digital back-propagation in 112 Gb/s PM-QPSK systems,” Opt. Express 19(4), 3449–3454 (2011).
    [Crossref] [PubMed]
  6. Y. Gao, J. H. Ke, K. P. Zhong, J. C. Cartledge, and S. S.-H. Yam, “Assessment of intra-channel nonlinear compensation for 112 Gb/s dual polarization 16QAM systems,” J. Lightwave Technol. 30(24), 3902–3910 (2012).
    [Crossref]
  7. J. Shao, S. Kumar, and X. Liang, “Digital back propagation with optimal step size for polarization multiplexed transmission,” IEEE Photon. Technol. Lett. 25(23), 2327–2330 (2013).
    [Crossref]
  8. S. Kumar and M. J. Deen, Fiber Optic Communications: Fundamentals and Applications, (Wiley, 2014), Chapters. 10 and 11.
  9. Z. Tao, L. Dou, W. Yan, L. Li, T. Hoshida, and J. C. Rasmussen, “Multiplier-free intrachannel nonlinearity compensating algorithm operating at symbol rate,” J. Lightwave Technol. 29(17), 2570–2576 (2011).
    [Crossref]
  10. Y. Gao, J. C. Cartledge, A. S. Karar, and S. S.-H. Yam, “Reducing the complexity of nonlinearity pre-compensation using symmetric EDC and pulse shaping,” in 39th European Conference and Exhibition on Optical Communication (ECOC 2013), Pd3.E.5.
    [Crossref]
  11. Y. Gao, J. C. Cartledge, A. S. Karar, S. S.-H. Yam, M. O’Sullivan, C. Laperle, A. Borowiec, and K. Roberts, “Reducing the complexity of perturbation based nonlinearity pre-compensation using symmetric EDC and pulse shaping,” Opt. Express 22(2), 1209–1219 (2014).
    [Crossref] [PubMed]
  12. Y. Fan, L. Dou, Z. Tao, T. Hoshida, and J. C. Rasmussen, “A high performance nonlinear compensation algorithm with reduced complexity based on XPM model,” in Optical Fiber Communication Conference, (Optical Society of America, 2014), paper Th2A.8.
    [Crossref]
  13. Y. Gao, A. S. Karar, J. C. Cartledge, S. S.-H. Yam, M. O’Sullivan, C. Laperle, A. Borowiec, and K. Roberts, “Simplified nonlinearity pre-compensation using a modified summation criteria and non-uniform power profile,” in Optical Fiber Communication Conference, (Optical Society of America, 2014), paper Tu3A.6.
    [Crossref]
  14. T. Oyama, H. Nakashima, S. Oda, T. Yamauchi, Z. Tao, T. Hoshida, and J. C. Rasmussen, “Robust and efficient receiver-side compensation method for intra-channel nonlinear effects,” in Optical Fiber Communication Conference, (Optical Society of America, 2014), paper Tu3A.3.
    [Crossref]
  15. Y. Gao, A. S. Karar, J. C. Cartledge, S. S.-H. Yam, M. O’Sullivan, C. Laperle, A. Borowiec, and K. Roberts, “Joint pre-compensation and selective post-compensation for fiber nonlinearities,” IEEE Photon. Technol. Lett. 26(17), 1746–1749 (2014).
    [Crossref]
  16. A. Mecozzi, C. B. Clausen, and M. Shtaif, “Analysis of intrachannel nonlinear effects in highly dispersed optical pulse transmission,” IEEE Photon. Technol. Lett. 12(4), 392–394 (2000).
    [Crossref]
  17. A. Mecozzi and R.-J. Essiambre, “Nonlinear shannon limit in pseudolinear coherent systems,” J. Lightwave Technol. 30(12), 2010–2024 (2011).
    [Crossref]
  18. R.-J. Essiambre, G. Raybon, and B. Mikkelsen, “Pseudo-linear transmission of high-speed TDM signals: 40 and 160 Gb/s,” in Optical Fiber Telecommunication IVB, I. P. Kaminow and T. Li, ed. (Academic Press, 2002).
  19. A. Vannucci, P. Serena, and A. Bononi, “The RP method: A new tool for the iterative solution of the nonlinear Schrödinger equation,” J. Lightwave Technol. 20(7), 1102–1112 (2002).
    [Crossref]
  20. S. Kumar and D. Yang, “Second-order theory for self-phase modulation and cross-phase modulation in optical fibers,” J. Lightwave Technol. 23(6), 2073–2080 (2005).
    [Crossref]
  21. D. Yang and S. Kumar, “Intra-channel four-wave mixing impairments in dispersion-managed coherent fiber-optic systems based on binary phase-shift keying,” J. Lightwave Technol. 27(14), 2916–2923 (2009).
    [Crossref]
  22. S. Kumar, “Analysis of nonlinear phase noise in coherent fiber-optic systems based on phase shift keying,” J. Lightwave Technol. 27(21), 4722–4733 (2009).
    [Crossref]
  23. A. Mecozzi, “A unified theory of intrachannel nonlinearity in pseudolinear transmission,” in Impact of Nonlinearities on Fiber Optic Communications, S. Kumar, ed. (Springer, 2011).
  24. R. Dar, M. Feder, A. Mecozzi, and M. Shtaif, “Properties of nonlinear noise in long, dispersion-uncompensated fiber links,” Opt. Express 21(22), 25685–25699 (2013).
    [Crossref] [PubMed]
  25. S. N. Shahi, S. Kumar, and X. Liang, “Analytical modeling of cross-phase modulation in coherent fiber-optic system,” Opt. Express 22(2), 1426–1439 (2014).
    [Crossref] [PubMed]
  26. X. Liang and S. Kumar, “Analytical modeling of XPM in dispersion-managed coherent fiber-optic systems,” Opt. Express 22(9), 10579–10592 (2014).
    [Crossref] [PubMed]
  27. X. Liang, S. Kumar, J. Shao, M. Malekiha, and D. V. Plant, “Digital compensation of cross-phase modulation distortions using perturbation technique for dispersion-managed fiber-optic systems,” Opt. Express 22(17), 20634–20645 (2014).
    [Crossref] [PubMed]
  28. M. Secondini, D. Marsella, and E. Forestieri, “Enhanced split-step Fourier method for digital backpropagation,” in 40th European Conference and Exhibition on Optical Communication (ECOC 2014), We.3.3.5.
  29. Y. Fan, L. Dou, Z. Tao, L. Lei, S. Oda, T. Hoshida, and J. C. Rasmussen, “Modulation format dependent phase noise caused by intra-channel nonlinearity,” in 38th European Conference and Exhibition on Optical Communication (ECOC 2012), We.2.C.3.
    [Crossref]
  30. T. Pfau, S. Hoffmann, and R. Noé, “Hardware-efficient coherent digital receiver concept with feedforward carrier recovery for M-QAM constellations,” J. Lightwave Technol. 27(8), 989–999 (2009).
    [Crossref]

2014 (5)

2013 (2)

J. Shao, S. Kumar, and X. Liang, “Digital back propagation with optimal step size for polarization multiplexed transmission,” IEEE Photon. Technol. Lett. 25(23), 2327–2330 (2013).
[Crossref]

R. Dar, M. Feder, A. Mecozzi, and M. Shtaif, “Properties of nonlinear noise in long, dispersion-uncompensated fiber links,” Opt. Express 21(22), 25685–25699 (2013).
[Crossref] [PubMed]

2012 (1)

2011 (3)

2010 (2)

D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, 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]

L. B. Du and A. J. Lowery, “Improved single channel backpropagation for intra-channel fiber nonlinearity compensation in long-haul optical communication systems,” Opt. Express 18(16), 17075–17088 (2010).
[Crossref] [PubMed]

2009 (3)

2008 (2)

2005 (1)

2002 (1)

2000 (1)

A. Mecozzi, C. B. Clausen, and M. Shtaif, “Analysis of intrachannel nonlinear effects in highly dispersed optical pulse transmission,” IEEE Photon. Technol. Lett. 12(4), 392–394 (2000).
[Crossref]

Bayvel, P.

D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, 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]

Behrens, C.

D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, 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]

Bononi, A.

Borowiec, A.

Y. Gao, A. S. Karar, J. C. Cartledge, S. S.-H. Yam, M. O’Sullivan, C. Laperle, A. Borowiec, and K. Roberts, “Joint pre-compensation and selective post-compensation for fiber nonlinearities,” IEEE Photon. Technol. Lett. 26(17), 1746–1749 (2014).
[Crossref]

Y. Gao, J. C. Cartledge, A. S. Karar, S. S.-H. Yam, M. O’Sullivan, C. Laperle, A. Borowiec, and K. Roberts, “Reducing the complexity of perturbation based nonlinearity pre-compensation using symmetric EDC and pulse shaping,” Opt. Express 22(2), 1209–1219 (2014).
[Crossref] [PubMed]

Cartledge, J. C.

Chen, X.

Clausen, C. B.

A. Mecozzi, C. B. Clausen, and M. Shtaif, “Analysis of intrachannel nonlinear effects in highly dispersed optical pulse transmission,” IEEE Photon. Technol. Lett. 12(4), 392–394 (2000).
[Crossref]

Dar, R.

Dou, L.

Du, L. B.

Ellis, A. D.

Essiambre, R.-J.

A. Mecozzi and R.-J. Essiambre, “Nonlinear shannon limit in pseudolinear coherent systems,” J. Lightwave Technol. 30(12), 2010–2024 (2011).
[Crossref]

Feder, M.

Gao, Y.

Goldfarb, G.

Hellerbrand, S.

D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, 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]

Hoffmann, S.

Hoshida, T.

Ip, E.

Kahn, J. M.

Karar, A. S.

Y. Gao, A. S. Karar, J. C. Cartledge, S. S.-H. Yam, M. O’Sullivan, C. Laperle, A. Borowiec, and K. Roberts, “Joint pre-compensation and selective post-compensation for fiber nonlinearities,” IEEE Photon. Technol. Lett. 26(17), 1746–1749 (2014).
[Crossref]

Y. Gao, J. C. Cartledge, A. S. Karar, S. S.-H. Yam, M. O’Sullivan, C. Laperle, A. Borowiec, and K. Roberts, “Reducing the complexity of perturbation based nonlinearity pre-compensation using symmetric EDC and pulse shaping,” Opt. Express 22(2), 1209–1219 (2014).
[Crossref] [PubMed]

Ke, J. H.

Killey, R. I.

D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, 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]

Kim, I.

Kumar, S.

Laperle, C.

Y. Gao, A. S. Karar, J. C. Cartledge, S. S.-H. Yam, M. O’Sullivan, C. Laperle, A. Borowiec, and K. Roberts, “Joint pre-compensation and selective post-compensation for fiber nonlinearities,” IEEE Photon. Technol. Lett. 26(17), 1746–1749 (2014).
[Crossref]

Y. Gao, J. C. Cartledge, A. S. Karar, S. S.-H. Yam, M. O’Sullivan, C. Laperle, A. Borowiec, and K. Roberts, “Reducing the complexity of perturbation based nonlinearity pre-compensation using symmetric EDC and pulse shaping,” Opt. Express 22(2), 1209–1219 (2014).
[Crossref] [PubMed]

Li, G.

Li, L.

Li, X.

Liang, X.

Lowery, A. J.

Makovejs, S.

D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, 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]

Malekiha, M.

Mateo, E.

Mecozzi, A.

R. Dar, M. Feder, A. Mecozzi, and M. Shtaif, “Properties of nonlinear noise in long, dispersion-uncompensated fiber links,” Opt. Express 21(22), 25685–25699 (2013).
[Crossref] [PubMed]

A. Mecozzi and R.-J. Essiambre, “Nonlinear shannon limit in pseudolinear coherent systems,” J. Lightwave Technol. 30(12), 2010–2024 (2011).
[Crossref]

A. Mecozzi, C. B. Clausen, and M. Shtaif, “Analysis of intrachannel nonlinear effects in highly dispersed optical pulse transmission,” IEEE Photon. Technol. Lett. 12(4), 392–394 (2000).
[Crossref]

Millar, D. S.

D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, 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]

Noé, R.

O’Sullivan, M.

Y. Gao, J. C. Cartledge, A. S. Karar, S. S.-H. Yam, M. O’Sullivan, C. Laperle, A. Borowiec, and K. Roberts, “Reducing the complexity of perturbation based nonlinearity pre-compensation using symmetric EDC and pulse shaping,” Opt. Express 22(2), 1209–1219 (2014).
[Crossref] [PubMed]

Y. Gao, A. S. Karar, J. C. Cartledge, S. S.-H. Yam, M. O’Sullivan, C. Laperle, A. Borowiec, and K. Roberts, “Joint pre-compensation and selective post-compensation for fiber nonlinearities,” IEEE Photon. Technol. Lett. 26(17), 1746–1749 (2014).
[Crossref]

Pfau, T.

Plant, D. V.

Rafique, D.

Rasmussen, J. C.

Roberts, K.

Y. Gao, J. C. Cartledge, A. S. Karar, S. S.-H. Yam, M. O’Sullivan, C. Laperle, A. Borowiec, and K. Roberts, “Reducing the complexity of perturbation based nonlinearity pre-compensation using symmetric EDC and pulse shaping,” Opt. Express 22(2), 1209–1219 (2014).
[Crossref] [PubMed]

Y. Gao, A. S. Karar, J. C. Cartledge, S. S.-H. Yam, M. O’Sullivan, C. Laperle, A. Borowiec, and K. Roberts, “Joint pre-compensation and selective post-compensation for fiber nonlinearities,” IEEE Photon. Technol. Lett. 26(17), 1746–1749 (2014).
[Crossref]

Savory, S. J.

D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, 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]

Serena, P.

Shahi, S. N.

Shao, J.

X. Liang, S. Kumar, J. Shao, M. Malekiha, and D. V. Plant, “Digital compensation of cross-phase modulation distortions using perturbation technique for dispersion-managed fiber-optic systems,” Opt. Express 22(17), 20634–20645 (2014).
[Crossref] [PubMed]

J. Shao, S. Kumar, and X. Liang, “Digital back propagation with optimal step size for polarization multiplexed transmission,” IEEE Photon. Technol. Lett. 25(23), 2327–2330 (2013).
[Crossref]

Shtaif, M.

R. Dar, M. Feder, A. Mecozzi, and M. Shtaif, “Properties of nonlinear noise in long, dispersion-uncompensated fiber links,” Opt. Express 21(22), 25685–25699 (2013).
[Crossref] [PubMed]

A. Mecozzi, C. B. Clausen, and M. Shtaif, “Analysis of intrachannel nonlinear effects in highly dispersed optical pulse transmission,” IEEE Photon. Technol. Lett. 12(4), 392–394 (2000).
[Crossref]

Tao, Z.

Vannucci, A.

Yam, S. S.-H.

Yaman, F.

Yan, W.

Yang, D.

Zhong, K. P.

IEEE J. Sel. Top. Quantum Electron. (1)

D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, 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]

IEEE Photon. Technol. Lett. (3)

J. Shao, S. Kumar, and X. Liang, “Digital back propagation with optimal step size for polarization multiplexed transmission,” IEEE Photon. Technol. Lett. 25(23), 2327–2330 (2013).
[Crossref]

Y. Gao, A. S. Karar, J. C. Cartledge, S. S.-H. Yam, M. O’Sullivan, C. Laperle, A. Borowiec, and K. Roberts, “Joint pre-compensation and selective post-compensation for fiber nonlinearities,” IEEE Photon. Technol. Lett. 26(17), 1746–1749 (2014).
[Crossref]

A. Mecozzi, C. B. Clausen, and M. Shtaif, “Analysis of intrachannel nonlinear effects in highly dispersed optical pulse transmission,” IEEE Photon. Technol. Lett. 12(4), 392–394 (2000).
[Crossref]

J. Lightwave Technol. (9)

A. Mecozzi and R.-J. Essiambre, “Nonlinear shannon limit in pseudolinear coherent systems,” J. Lightwave Technol. 30(12), 2010–2024 (2011).
[Crossref]

Z. Tao, L. Dou, W. Yan, L. Li, T. Hoshida, and J. C. Rasmussen, “Multiplier-free intrachannel nonlinearity compensating algorithm operating at symbol rate,” J. Lightwave Technol. 29(17), 2570–2576 (2011).
[Crossref]

E. Ip and J. M. Kahn, “Compensation of dispersion and nonlinear impairments using digital backpropagation,” J. Lightwave Technol. 26(20), 3416–3425 (2008).
[Crossref]

A. Vannucci, P. Serena, and A. Bononi, “The RP method: A new tool for the iterative solution of the nonlinear Schrödinger equation,” J. Lightwave Technol. 20(7), 1102–1112 (2002).
[Crossref]

S. Kumar and D. Yang, “Second-order theory for self-phase modulation and cross-phase modulation in optical fibers,” J. Lightwave Technol. 23(6), 2073–2080 (2005).
[Crossref]

D. Yang and S. Kumar, “Intra-channel four-wave mixing impairments in dispersion-managed coherent fiber-optic systems based on binary phase-shift keying,” J. Lightwave Technol. 27(14), 2916–2923 (2009).
[Crossref]

S. Kumar, “Analysis of nonlinear phase noise in coherent fiber-optic systems based on phase shift keying,” J. Lightwave Technol. 27(21), 4722–4733 (2009).
[Crossref]

Y. Gao, J. H. Ke, K. P. Zhong, J. C. Cartledge, and S. S.-H. Yam, “Assessment of intra-channel nonlinear compensation for 112 Gb/s dual polarization 16QAM systems,” J. Lightwave Technol. 30(24), 3902–3910 (2012).
[Crossref]

T. Pfau, S. Hoffmann, and R. Noé, “Hardware-efficient coherent digital receiver concept with feedforward carrier recovery for M-QAM constellations,” J. Lightwave Technol. 27(8), 989–999 (2009).
[Crossref]

Opt. Express (8)

R. Dar, M. Feder, A. Mecozzi, and M. Shtaif, “Properties of nonlinear noise in long, dispersion-uncompensated fiber links,” Opt. Express 21(22), 25685–25699 (2013).
[Crossref] [PubMed]

S. N. Shahi, S. Kumar, and X. Liang, “Analytical modeling of cross-phase modulation in coherent fiber-optic system,” Opt. Express 22(2), 1426–1439 (2014).
[Crossref] [PubMed]

X. Liang and S. Kumar, “Analytical modeling of XPM in dispersion-managed coherent fiber-optic systems,” Opt. Express 22(9), 10579–10592 (2014).
[Crossref] [PubMed]

X. Liang, S. Kumar, J. Shao, M. Malekiha, and D. V. Plant, “Digital compensation of cross-phase modulation distortions using perturbation technique for dispersion-managed fiber-optic systems,” Opt. Express 22(17), 20634–20645 (2014).
[Crossref] [PubMed]

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), 880–888 (2008).
[Crossref] [PubMed]

L. B. Du and A. J. Lowery, “Improved single channel backpropagation for intra-channel fiber nonlinearity compensation in long-haul optical communication systems,” Opt. Express 18(16), 17075–17088 (2010).
[Crossref] [PubMed]

D. Rafique and A. D. Ellis, “Impact of signal-ASE four-wave mixing on the effectiveness of digital back-propagation in 112 Gb/s PM-QPSK systems,” Opt. Express 19(4), 3449–3454 (2011).
[Crossref] [PubMed]

Y. Gao, J. C. Cartledge, A. S. Karar, S. S.-H. Yam, M. O’Sullivan, C. Laperle, A. Borowiec, and K. Roberts, “Reducing the complexity of perturbation based nonlinearity pre-compensation using symmetric EDC and pulse shaping,” Opt. Express 22(2), 1209–1219 (2014).
[Crossref] [PubMed]

Other (9)

Y. Fan, L. Dou, Z. Tao, T. Hoshida, and J. C. Rasmussen, “A high performance nonlinear compensation algorithm with reduced complexity based on XPM model,” in Optical Fiber Communication Conference, (Optical Society of America, 2014), paper Th2A.8.
[Crossref]

Y. Gao, A. S. Karar, J. C. Cartledge, S. S.-H. Yam, M. O’Sullivan, C. Laperle, A. Borowiec, and K. Roberts, “Simplified nonlinearity pre-compensation using a modified summation criteria and non-uniform power profile,” in Optical Fiber Communication Conference, (Optical Society of America, 2014), paper Tu3A.6.
[Crossref]

T. Oyama, H. Nakashima, S. Oda, T. Yamauchi, Z. Tao, T. Hoshida, and J. C. Rasmussen, “Robust and efficient receiver-side compensation method for intra-channel nonlinear effects,” in Optical Fiber Communication Conference, (Optical Society of America, 2014), paper Tu3A.3.
[Crossref]

R.-J. Essiambre, G. Raybon, and B. Mikkelsen, “Pseudo-linear transmission of high-speed TDM signals: 40 and 160 Gb/s,” in Optical Fiber Telecommunication IVB, I. P. Kaminow and T. Li, ed. (Academic Press, 2002).

Y. Gao, J. C. Cartledge, A. S. Karar, and S. S.-H. Yam, “Reducing the complexity of nonlinearity pre-compensation using symmetric EDC and pulse shaping,” in 39th European Conference and Exhibition on Optical Communication (ECOC 2013), Pd3.E.5.
[Crossref]

S. Kumar and M. J. Deen, Fiber Optic Communications: Fundamentals and Applications, (Wiley, 2014), Chapters. 10 and 11.

M. Secondini, D. Marsella, and E. Forestieri, “Enhanced split-step Fourier method for digital backpropagation,” in 40th European Conference and Exhibition on Optical Communication (ECOC 2014), We.3.3.5.

Y. Fan, L. Dou, Z. Tao, L. Lei, S. Oda, T. Hoshida, and J. C. Rasmussen, “Modulation format dependent phase noise caused by intra-channel nonlinearity,” in 38th European Conference and Exhibition on Optical Communication (ECOC 2012), We.2.C.3.
[Crossref]

A. Mecozzi, “A unified theory of intrachannel nonlinearity in pseudolinear transmission,” in Impact of Nonlinearities on Fiber Optic Communications, S. Kumar, ed. (Springer, 2011).

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Figures (8)

Fig. 1
Fig. 1 Comparisons of the additive perturbation model and the additive-multiplicative perturbation model (40-span SSMFS, power = 2dBm): (a) output signals vs. time, (b) signal square error vs. time, and (c) signal power vs. number of perturbation stages.
Fig. 2
Fig. 2 Schematic of a single-channel fiber-optic system with multi-stage perturbation-based compensation. Tx: transmitter, BPF: band-pass filter, LPF: low-pass filter, CPR: carrier phase recovery.
Fig. 3
Fig. 3 Perturbation coefficient matrices, 20log10|Xmn|: (a) Nstg = 40, (b) Nstg = 8, (c) Nstg = 1.
Fig. 4
Fig. 4 Signal constellations: (a) input signal, (b) compensated signal using the additive model, (c) compensated signal using the additive-multiplicative model.
Fig. 5
Fig. 5 Q-factor versus (a) launch power and (b) truncation threshold, 1 sample/symbol is used.
Fig. 6
Fig. 6 Q-factor versus (a) launch power and (b) truncation threshold, 2 samples/symbol are used.
Fig. 7
Fig. 7 Comparisons of multi-stage perturbation-based compensation scheme of different number of stages: (a) compensation performance, (b) computational complexity.
Fig. 8
Fig. 8 (a) Q-factor gain vs. truncation threshold, (b) number of complex multiplications per symbol vs. number of perturbation stages for given Q-factor gains. (2 samples per symbol are used. Reference Q-factor is 6.2 dB.)

Tables (3)

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Table 1 Q-factor improvement (ΔQ), number of complex multiplications per symbol (Mc), and required memory for storing Xmn. Reference Q-factor = 6.2 dB.

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Table 2 Comparison of scheme complexity to achieve ΔQ = 2.0 dB. Reference Q-factor = 6.2 dB.

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Table 3 Comparison of scheme complexity to achieve ΔQ = 3.0 dB. Reference Q-factor = 6.2 dB.

Equations (32)

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q z + α 2 q+i β 2 2 2 q T 2 i γ 0 | q | 2 q=0,
q(z,T)= e w(z)/2 u(z,T),
i u z β 2 2 2 u T 2 =γ | u | 2 u,
u(0,T)= P n= N sym N sym d n p(0,Tn T 0 ) ,
u(0,T)= P n=N/2+1 N/2 a n g(0,Tn T s ) ,
u= u (0) + γ 0 u (1) + γ 0 2 u (2) +...,
u (0) (z,T)= P n a n g(z,Tn T s ),
g(z,T)= T s 2π π/ T s π/ T s exp[ iS(z) ω 2 /2iωT ] dω,
i u (1) z β 2 2 2 u (1) T 2 = e w(z) | u (0) | 2 u (0) .
u ˜ (1) z i β 2 2 ω 2 u ˜ (1) = G ˜ (z,ω),
u ˜ (1) (z,ω)= 0 z G ˜ (s,ω) exp{ i[ S(z)S(s) ] ω 2 /2 }ds.
Δu(z,T)= γ 0 u (1) (z,T)= γ 0 F 1 { u ˜ (1) (z,ω) },
Δu(z,T)= γ 0 P n a n (1) g(z,Tn T s ) ,
a n (1) = Δu(z,T) T s γ 0 P g (z,Tn T s ) dT, = i T s P 0 z ds e w(s) dT g (s,Tn T s ) | u (0) (s,T) | 2 u (0) (s,T).
a j (1) =iP m=K/2 K/2 n=K/2 K/2 a m+j a n+j a m+n+j X mn ,
X mn = 1 T s 0 z ds e w(s) dT g (s,T) g(s,Tm T s )g(s,Tn T s ) g (s,T(m+n) T s ),
u(L,T)= P n a n g(L,Tn T s ),
u(L,T)= P m b m g(0,Tm T s ),
b m =u(L,T=m T s )/ P = n a n g(L,(mn) T s ).
c mn =g(L,(mn) T s ).
c n = T s 2π π/ T s π/ T s exp[ iS(L) ω 2 /2iωn T s ] dω.
b m = n a n c mn .
b n =DFT{ IDFT{ a ' n }×IDFT{ c n } },
a n=0 (1) =iP a 0 [ | a 0 | 2 X m=0,n=0 +2 n0 | a n | 2 X m=0,n ]+iP m0 n0 a m a n a m+n X mn ,
ϕ nl = γ 0 P( | a 0 | 2 X m=0,n=0 +2 n0 | a n | 2 X m=0,n ),
Δ a IFWM =i γ 0 P m0 n0 a m a n a m+n X mn ,
a n=0 = a 0 + γ 0 a n=0 (1)
= a 0 ( 1+i ϕ nl )+Δ a IFWM
a 0 exp( i ϕ nl )+Δ a IFWM .
h n =DFT{ IDFT{ r n }×IDFT{ c n } },
c n = T s 2π π/ T s π/ T s exp[ iS(L) ω 2 /2iωn T s ] dω,
v n =( h n Δ a IFWM )exp( i ϕ nl ),

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