Abstract

We introduce an efficient and accurate nonlinear compensator (NLC) for digital back-propagation (DBP) of coherent optical OFDM receivers, based on a factorization procedure for the Volterra Series Transfer Function (VSTF) with 3N degrees of freedom for N frequency samples. The O(N2) nonlinear compensation complexity of generic Volterra evaluation (normalized per-subcarrier) is reduced to 28 + 6logN. Our analysis and simulations indicate that this NLC system outperforms previous VSTF-based non-linear compensation methods. Compared to a most recent VSTF-based method, the new method incurs 52% extra computational complexity in return for improved nonlinear tolerance of ~2 dB for the particular analyzed link.

© 2013 OSA

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  1. 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. Express16(2), 880–888 (2008).
    [CrossRef] [PubMed]
  2. E. Ip and J. M. Kahn, “Compensation of dispersion and nonlinear impairments using digital backpropagation,” J. Lightwave Technol.26(20), 3416–3425 (2008).
    [CrossRef]
  3. E. Ip, N. Bai, and T. Wang, “Complexity versus performance tradeoff for fiber nonlinearity compensation using frequency-shaped, multi-subband backpropagation,” in OFC/NFOEC - Conference on Optical Fiber Communication and the National Fiber Optic Engineers Conference, (2011).
    [CrossRef]
  4. E. F. Mateo, F. Yaman, T. Wang, and G. Li, “Nonlinearity compensation using digital backward propagation,” J. Lightwave Technol.1, 10–11 (2011).
  5. D. Rafique, M. Mussolin, M. Forzati, J. Mårtensson, M. N. Chugtai, and A. D. Ellis, “Compensation of intra-channel nonlinear fibre impairments using simplified digital back-propagation algorithm,” Opt. Express19(10), 9453–9460 (2011).
    [CrossRef] [PubMed]
  6. M. Schetzen, The Volterra and Wiener Theories of Nonlinear Systems (John Wiley and Sons Inc., 1980).
  7. A. Bononi, P. Serena, N. Rossi, E. Grellier, and F. Vacondio, “Modeling nonlinearity in coherent intrachannel-four-wave-mixing,” Opt. Express20, 3416–3425 (2012).
    [CrossRef] [PubMed]
  8. A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of nonlinear propagation effects in uncompensated optical coherent transmission links,” J. Lightwave Technol.30(10), 1524–1539 (2012).
    [CrossRef]
  9. X. Chen and W. Shieh, “Closed-form expressions for nonlinear transmission performance of densely spaced coherent optical OFDM systems,” Opt. Express18(18), 19039–19054 (2010).
    [CrossRef] [PubMed]
  10. A. Mecozzi, R. Essiambre, and S. Member, “Nonlinear Shannon limit in pseudolinear coherent systems,” J. Lightwave Technol.30(12), 2011–2024 (2012).
    [CrossRef]
  11. C. Xia and W. Rosenkranz, “Nonlinear electrical equalization for different modulation formats with optical filtering,” J. Lightwave Technol.25(4), 996–1001 (2007).
    [CrossRef]
  12. Z. Pan, C. Benoit, M. Chagnon, and D. V. Plant, “Volterra filtering for nonlinearity impairment mitigation in DP-16QAM and DP-QPSK Fiber Optic Communication Systems,” in OFC/NFOEC - Conference on Optical Fiber Communication and the National Fiber Optic Engineers Conference, (2011).
    [CrossRef]
  13. K. Peddanarappagari and M. Brandt-Pearce, “Volterra Series transfer function of single-mode fibers,” J. Lightwave Technol.15(12), 2232–2241 (1997).
    [CrossRef]
  14. B. Xu, M. Brandt-pearce, and S. Member, “Comparison of FWM- and XPM-induced crosstalk using the Volterra series transfer function method,” J. Lightwave Technol.21(1), 40–53 (2003).
    [CrossRef]
  15. T. Freckmann, C. V. Gonzalez, and J. M. R.-C. Crespo, “Joint electronic dispersion compensation for DQPSK,” in OFC/NFOEC - Conference on Optical Fiber Communication and the National Fiber Optic Engineers Conference, (2008).
    [CrossRef]
  16. M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express16(20), 15777–15810 (2008).
    [CrossRef] [PubMed]
  17. R. Weidenfeld, M. Nazarathy, R. Noe, and I. Shpantzer, “Volterra nonlinear compensation of 112 Gb/s ultra-long-haul coherent optical OFDM based on frequency-shaped decision feedback,” in European Conference of Optical Communication (ECOC), 1–2, (2009).
  18. R. Weidenfeld, M. Nazarathy, R. Noe, and I. Shpantzer, “Volterra nonlinear compensation of 100G coherent OFDM with baud-rate ADC, tolerable complexity and low intra-channel FWM/XPM error propagation,” in OFC/NFOEC - Conference on Optical Fiber Communication and the National Fiber Optic Engineers Conference, (2010).
    [CrossRef]
  19. S. Kumar, Impact of Nonlinearities on Fiber Optic Communications (Springer, 2011), Chap. 3.
  20. L. B. Du and A. J. Lowery, “Improved nonlinearity precompensation for long-haul high-data-rate transmission using coherent optical OFDM,” Opt. Express16(24), 19920–19925 (2008).
    [CrossRef] [PubMed]
  21. J. D. Reis and A. L. Teixeira, “Unveiling nonlinear effects in dense coherent optical WDM systems with Volterra series,” Opt. Express18(8), 8660–8670 (2010).
    [CrossRef] [PubMed]
  22. F. P. Guiomar, J. D. Reis, S. Member, A. L. Teixeira, A. N. Pinto, and S. Member, “Digital Postcompensation Using Volterra Series Transfer Function,” Photon. Technol. Lett.23(19), 1412–1414 (2011).
    [CrossRef]
  23. F. P. Guiomar, J. D. Reis, A. L. Teixeira, and A. N. Pinto, “Mitigation of intra-channel nonlinearities using a frequency-domain Volterra series equalizer,” Opt. Express20(2), 1360–1369 (2012).
    [CrossRef] [PubMed]
  24. H.-M. Chin, F. Marco, and M. Jonas, “Volterra Based Nonlinear Compensation on 224 Gb/s PolMux-16QAM Optical Fibre Link,” in OFC/NFOEC - Conference on Optical Fiber Communication and the National Fiber Optic Engineers Conference, (2012).
    [CrossRef]
  25. L. Liu, L. Li, Y. Huang, K. Cui, Q. Xiong, F. N. Hauske, C. Xie, and Y. Cai, “Electronic Nonlinearity Compensation of 256Gb / s PDM- 16QAM Based on Inverse Volterra Transfer Function,” in European Conference of Optical Communication (ECOC), 25–27, (2011).
  26. 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. Lightwave Technol.30(3), 310–316 (2012).
    [CrossRef]
  27. G. Shulkind and M. Nazarathy, “Estimating the Volterra series transfer function over coherent optical OFDM for efficient monitoring of the fiber channel nonlinearity,” Opt. Express20(27), 29035–29062 (2012).
    [CrossRef] [PubMed]
  28. G. Shulkind and M. Nazarathy, “An analytical study of the improved nonlinear tolerance of DFT-spread OFDM and its unitary-spread OFDM generalization,” Opt. Express20(23), 25884–25901 (2012).
    [CrossRef] [PubMed]

2012 (7)

2011 (3)

E. F. Mateo, F. Yaman, T. Wang, and G. Li, “Nonlinearity compensation using digital backward propagation,” J. Lightwave Technol.1, 10–11 (2011).

D. Rafique, M. Mussolin, M. Forzati, J. Mårtensson, M. N. Chugtai, and A. D. Ellis, “Compensation of intra-channel nonlinear fibre impairments using simplified digital back-propagation algorithm,” Opt. Express19(10), 9453–9460 (2011).
[CrossRef] [PubMed]

F. P. Guiomar, J. D. Reis, S. Member, A. L. Teixeira, A. N. Pinto, and S. Member, “Digital Postcompensation Using Volterra Series Transfer Function,” Photon. Technol. Lett.23(19), 1412–1414 (2011).
[CrossRef]

2010 (2)

2008 (4)

2007 (1)

2003 (1)

1997 (1)

K. Peddanarappagari and M. Brandt-Pearce, “Volterra Series transfer function of single-mode fibers,” J. Lightwave Technol.15(12), 2232–2241 (1997).
[CrossRef]

Bononi, A.

A. Bononi, P. Serena, N. Rossi, E. Grellier, and F. Vacondio, “Modeling nonlinearity in coherent intrachannel-four-wave-mixing,” Opt. Express20, 3416–3425 (2012).
[CrossRef] [PubMed]

Bosco, G.

Brandt-pearce, M.

B. Xu, M. Brandt-pearce, and S. Member, “Comparison of FWM- and XPM-induced crosstalk using the Volterra series transfer function method,” J. Lightwave Technol.21(1), 40–53 (2003).
[CrossRef]

K. Peddanarappagari and M. Brandt-Pearce, “Volterra Series transfer function of single-mode fibers,” J. Lightwave Technol.15(12), 2232–2241 (1997).
[CrossRef]

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. Lightwave Technol.30(3), 310–316 (2012).
[CrossRef]

L. Liu, L. Li, Y. Huang, K. Cui, Q. Xiong, F. N. Hauske, C. Xie, and Y. Cai, “Electronic Nonlinearity Compensation of 256Gb / s PDM- 16QAM Based on Inverse Volterra Transfer Function,” in European Conference of Optical Communication (ECOC), 25–27, (2011).

Carena, A.

Chen, X.

Cho, P.

Chugtai, M. N.

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. Lightwave Technol.30(3), 310–316 (2012).
[CrossRef]

L. Liu, L. Li, Y. Huang, K. Cui, Q. Xiong, F. N. Hauske, C. Xie, and Y. Cai, “Electronic Nonlinearity Compensation of 256Gb / s PDM- 16QAM Based on Inverse Volterra Transfer Function,” in European Conference of Optical Communication (ECOC), 25–27, (2011).

Curri, V.

Du, L. B.

Ellis, A. D.

Essiambre, R.

Forghieri, F.

Forzati, M.

Goldfarb, G.

Grellier, E.

A. Bononi, P. Serena, N. Rossi, E. Grellier, and F. Vacondio, “Modeling nonlinearity in coherent intrachannel-four-wave-mixing,” Opt. Express20, 3416–3425 (2012).
[CrossRef] [PubMed]

Guiomar, F. P.

F. P. Guiomar, J. D. Reis, A. L. Teixeira, and A. N. Pinto, “Mitigation of intra-channel nonlinearities using a frequency-domain Volterra series equalizer,” Opt. Express20(2), 1360–1369 (2012).
[CrossRef] [PubMed]

F. P. Guiomar, J. D. Reis, S. Member, A. L. Teixeira, A. N. Pinto, and S. Member, “Digital Postcompensation Using Volterra Series Transfer Function,” Photon. Technol. Lett.23(19), 1412–1414 (2011).
[CrossRef]

Hauske, F. N.

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. Lightwave Technol.30(3), 310–316 (2012).
[CrossRef]

L. Liu, L. Li, Y. Huang, K. Cui, Q. Xiong, F. N. Hauske, C. Xie, and Y. Cai, “Electronic Nonlinearity Compensation of 256Gb / s PDM- 16QAM Based on Inverse Volterra Transfer Function,” in European Conference of Optical Communication (ECOC), 25–27, (2011).

Huang, 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. Lightwave Technol.30(3), 310–316 (2012).
[CrossRef]

L. Liu, L. Li, Y. Huang, K. Cui, Q. Xiong, F. N. Hauske, C. Xie, and Y. Cai, “Electronic Nonlinearity Compensation of 256Gb / s PDM- 16QAM Based on Inverse Volterra Transfer Function,” in European Conference of Optical Communication (ECOC), 25–27, (2011).

Ip, E.

Kahn, J. M.

Karagodsky, V.

Khurgin, J.

Kim, I.

Li, G.

Li, L.

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. Lightwave Technol.30(3), 310–316 (2012).
[CrossRef]

L. Liu, L. Li, Y. Huang, K. Cui, Q. Xiong, F. N. Hauske, C. Xie, and Y. Cai, “Electronic Nonlinearity Compensation of 256Gb / s PDM- 16QAM Based on Inverse Volterra Transfer Function,” in European Conference of Optical Communication (ECOC), 25–27, (2011).

Li, X.

Liu, L.

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. Lightwave Technol.30(3), 310–316 (2012).
[CrossRef]

L. Liu, L. Li, Y. Huang, K. Cui, Q. Xiong, F. N. Hauske, C. Xie, and Y. Cai, “Electronic Nonlinearity Compensation of 256Gb / s PDM- 16QAM Based on Inverse Volterra Transfer Function,” in European Conference of Optical Communication (ECOC), 25–27, (2011).

Lowery, A. J.

Mårtensson, J.

Mateo, E.

Mateo, E. F.

E. F. Mateo, F. Yaman, T. Wang, and G. Li, “Nonlinearity compensation using digital backward propagation,” J. Lightwave Technol.1, 10–11 (2011).

Mecozzi, A.

Meiman, Y.

Member, S.

A. Mecozzi, R. Essiambre, and S. Member, “Nonlinear Shannon limit in pseudolinear coherent systems,” J. Lightwave Technol.30(12), 2011–2024 (2012).
[CrossRef]

F. P. Guiomar, J. D. Reis, S. Member, A. L. Teixeira, A. N. Pinto, and S. Member, “Digital Postcompensation Using Volterra Series Transfer Function,” Photon. Technol. Lett.23(19), 1412–1414 (2011).
[CrossRef]

F. P. Guiomar, J. D. Reis, S. Member, A. L. Teixeira, A. N. Pinto, and S. Member, “Digital Postcompensation Using Volterra Series Transfer Function,” Photon. Technol. Lett.23(19), 1412–1414 (2011).
[CrossRef]

B. Xu, M. Brandt-pearce, and S. Member, “Comparison of FWM- and XPM-induced crosstalk using the Volterra series transfer function method,” J. Lightwave Technol.21(1), 40–53 (2003).
[CrossRef]

Mussolin, M.

Nazarathy, M.

Noe, R.

M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express16(20), 15777–15810 (2008).
[CrossRef] [PubMed]

R. Weidenfeld, M. Nazarathy, R. Noe, and I. Shpantzer, “Volterra nonlinear compensation of 112 Gb/s ultra-long-haul coherent optical OFDM based on frequency-shaped decision feedback,” in European Conference of Optical Communication (ECOC), 1–2, (2009).

Peddanarappagari, K.

K. Peddanarappagari and M. Brandt-Pearce, “Volterra Series transfer function of single-mode fibers,” J. Lightwave Technol.15(12), 2232–2241 (1997).
[CrossRef]

Pinto, A. N.

F. P. Guiomar, J. D. Reis, A. L. Teixeira, and A. N. Pinto, “Mitigation of intra-channel nonlinearities using a frequency-domain Volterra series equalizer,” Opt. Express20(2), 1360–1369 (2012).
[CrossRef] [PubMed]

F. P. Guiomar, J. D. Reis, S. Member, A. L. Teixeira, A. N. Pinto, and S. Member, “Digital Postcompensation Using Volterra Series Transfer Function,” Photon. Technol. Lett.23(19), 1412–1414 (2011).
[CrossRef]

Poggiolini, P.

Rafique, D.

Reis, J. D.

Rosenkranz, W.

Rossi, N.

A. Bononi, P. Serena, N. Rossi, E. Grellier, and F. Vacondio, “Modeling nonlinearity in coherent intrachannel-four-wave-mixing,” Opt. Express20, 3416–3425 (2012).
[CrossRef] [PubMed]

Serena, P.

A. Bononi, P. Serena, N. Rossi, E. Grellier, and F. Vacondio, “Modeling nonlinearity in coherent intrachannel-four-wave-mixing,” Opt. Express20, 3416–3425 (2012).
[CrossRef] [PubMed]

Shieh, W.

Shpantzer, I.

M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express16(20), 15777–15810 (2008).
[CrossRef] [PubMed]

R. Weidenfeld, M. Nazarathy, R. Noe, and I. Shpantzer, “Volterra nonlinear compensation of 112 Gb/s ultra-long-haul coherent optical OFDM based on frequency-shaped decision feedback,” in European Conference of Optical Communication (ECOC), 1–2, (2009).

Shulkind, G.

Teixeira, A. L.

Vacondio, F.

A. Bononi, P. Serena, N. Rossi, E. Grellier, and F. Vacondio, “Modeling nonlinearity in coherent intrachannel-four-wave-mixing,” Opt. Express20, 3416–3425 (2012).
[CrossRef] [PubMed]

Wang, T.

E. F. Mateo, F. Yaman, T. Wang, and G. Li, “Nonlinearity compensation using digital backward propagation,” J. Lightwave Technol.1, 10–11 (2011).

Weidenfeld, R.

M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express16(20), 15777–15810 (2008).
[CrossRef] [PubMed]

R. Weidenfeld, M. Nazarathy, R. Noe, and I. Shpantzer, “Volterra nonlinear compensation of 112 Gb/s ultra-long-haul coherent optical OFDM based on frequency-shaped decision feedback,” in European Conference of Optical Communication (ECOC), 1–2, (2009).

Xia, C.

Xie, C.

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. Lightwave Technol.30(3), 310–316 (2012).
[CrossRef]

L. Liu, L. Li, Y. Huang, K. Cui, Q. Xiong, F. N. Hauske, C. Xie, and Y. Cai, “Electronic Nonlinearity Compensation of 256Gb / s PDM- 16QAM Based on Inverse Volterra Transfer Function,” in European Conference of Optical Communication (ECOC), 25–27, (2011).

Xiong, Q.

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. Lightwave Technol.30(3), 310–316 (2012).
[CrossRef]

L. Liu, L. Li, Y. Huang, K. Cui, Q. Xiong, F. N. Hauske, C. Xie, and Y. Cai, “Electronic Nonlinearity Compensation of 256Gb / s PDM- 16QAM Based on Inverse Volterra Transfer Function,” in European Conference of Optical Communication (ECOC), 25–27, (2011).

Xu, B.

Yaman, F.

J. Lightwave Technol. (8)

Opt. Express (10)

G. Shulkind and M. Nazarathy, “Estimating the Volterra series transfer function over coherent optical OFDM for efficient monitoring of the fiber channel nonlinearity,” Opt. Express20(27), 29035–29062 (2012).
[CrossRef] [PubMed]

G. Shulkind and M. Nazarathy, “An analytical study of the improved nonlinear tolerance of DFT-spread OFDM and its unitary-spread OFDM generalization,” Opt. Express20(23), 25884–25901 (2012).
[CrossRef] [PubMed]

L. B. Du and A. J. Lowery, “Improved nonlinearity precompensation for long-haul high-data-rate transmission using coherent optical OFDM,” Opt. Express16(24), 19920–19925 (2008).
[CrossRef] [PubMed]

J. D. Reis and A. L. Teixeira, “Unveiling nonlinear effects in dense coherent optical WDM systems with Volterra series,” Opt. Express18(8), 8660–8670 (2010).
[CrossRef] [PubMed]

F. P. Guiomar, J. D. Reis, A. L. Teixeira, and A. N. Pinto, “Mitigation of intra-channel nonlinearities using a frequency-domain Volterra series equalizer,” Opt. Express20(2), 1360–1369 (2012).
[CrossRef] [PubMed]

M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express16(20), 15777–15810 (2008).
[CrossRef] [PubMed]

X. Chen and W. Shieh, “Closed-form expressions for nonlinear transmission performance of densely spaced coherent optical OFDM systems,” Opt. Express18(18), 19039–19054 (2010).
[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. Express16(2), 880–888 (2008).
[CrossRef] [PubMed]

A. Bononi, P. Serena, N. Rossi, E. Grellier, and F. Vacondio, “Modeling nonlinearity in coherent intrachannel-four-wave-mixing,” Opt. Express20, 3416–3425 (2012).
[CrossRef] [PubMed]

D. Rafique, M. Mussolin, M. Forzati, J. Mårtensson, M. N. Chugtai, and A. D. Ellis, “Compensation of intra-channel nonlinear fibre impairments using simplified digital back-propagation algorithm,” Opt. Express19(10), 9453–9460 (2011).
[CrossRef] [PubMed]

Photon. Technol. Lett. (1)

F. P. Guiomar, J. D. Reis, S. Member, A. L. Teixeira, A. N. Pinto, and S. Member, “Digital Postcompensation Using Volterra Series Transfer Function,” Photon. Technol. Lett.23(19), 1412–1414 (2011).
[CrossRef]

Other (9)

H.-M. Chin, F. Marco, and M. Jonas, “Volterra Based Nonlinear Compensation on 224 Gb/s PolMux-16QAM Optical Fibre Link,” in OFC/NFOEC - Conference on Optical Fiber Communication and the National Fiber Optic Engineers Conference, (2012).
[CrossRef]

L. Liu, L. Li, Y. Huang, K. Cui, Q. Xiong, F. N. Hauske, C. Xie, and Y. Cai, “Electronic Nonlinearity Compensation of 256Gb / s PDM- 16QAM Based on Inverse Volterra Transfer Function,” in European Conference of Optical Communication (ECOC), 25–27, (2011).

M. Schetzen, The Volterra and Wiener Theories of Nonlinear Systems (John Wiley and Sons Inc., 1980).

E. Ip, N. Bai, and T. Wang, “Complexity versus performance tradeoff for fiber nonlinearity compensation using frequency-shaped, multi-subband backpropagation,” in OFC/NFOEC - Conference on Optical Fiber Communication and the National Fiber Optic Engineers Conference, (2011).
[CrossRef]

R. Weidenfeld, M. Nazarathy, R. Noe, and I. Shpantzer, “Volterra nonlinear compensation of 112 Gb/s ultra-long-haul coherent optical OFDM based on frequency-shaped decision feedback,” in European Conference of Optical Communication (ECOC), 1–2, (2009).

R. Weidenfeld, M. Nazarathy, R. Noe, and I. Shpantzer, “Volterra nonlinear compensation of 100G coherent OFDM with baud-rate ADC, tolerable complexity and low intra-channel FWM/XPM error propagation,” in OFC/NFOEC - Conference on Optical Fiber Communication and the National Fiber Optic Engineers Conference, (2010).
[CrossRef]

S. Kumar, Impact of Nonlinearities on Fiber Optic Communications (Springer, 2011), Chap. 3.

T. Freckmann, C. V. Gonzalez, and J. M. R.-C. Crespo, “Joint electronic dispersion compensation for DQPSK,” in OFC/NFOEC - Conference on Optical Fiber Communication and the National Fiber Optic Engineers Conference, (2008).
[CrossRef]

Z. Pan, C. Benoit, M. Chagnon, and D. V. Plant, “Volterra filtering for nonlinearity impairment mitigation in DP-16QAM and DP-QPSK Fiber Optic Communication Systems,” in OFC/NFOEC - Conference on Optical Fiber Communication and the National Fiber Optic Engineers Conference, (2011).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

OFDM receiver structure with Multi-span Volterra-based DBP nonlinear compensator.

Fig. 2
Fig. 2

Alternative FWM emulator designs. (a). Based on factorized VSTF with 4N DOFs. (b) Based on factorized VSTF with 3N DOFs accounting for VSTF symmetries.

Fig. 3
Fig. 3

Multi-span link emulator, based on concatenating processing segments each of which features a FWM emulator realized as in Fig. 2.

Fig. 4
Fig. 4

Nonlinear emulator verification scheme. The FWM emulator synthesizes the FWM nonlinear interference by processing the transmitted symbols and its output is subtracted from the actually received NL distorted signal, as generated by the cascade of the OFDM Tx, an SSF simulation of the channel and the OFDM Rx.

Fig. 5
Fig. 5

Nonlinear MER (ratio of the average signal and NLI powers) vs. frequency-domain tone index following NLCs, based on either factorized or full VSTF. The lowest curve corresponds to the uncompensated NLI MER. The transmitted constellation is 16-QAM, with an average power of 0dBm. An ASE free link was assumed to determine the net NL performance of the proposed compensation scheme. The fiber link parameters are: BW=25GHz,N=64,α=0.2dB/Km,γ=1.3/W/km,D=17ps/nm/km,L=6×100km. The number of accumulated received symbols in the MER averaging was 64,000, corresponding to 1000 frames x 64symbols/frame.

Fig. 6
Fig. 6

Digital back-propagation nonlinear compensation topology.

Fig. 7
Fig. 7

Compensation performance (Nonlinear MER vs. OFDM tone index) for a link with parameters as in the caption of Fig. 5. The top curve is the full VS-DBP (which would require O(N2) complexity per sample; the next curve, ~1 dB down is our factorized VS-DBP with substantially reduced O(logN) complexity per sample per span; ~2 dB down we have the parallel-combined frequency-flat VS-DBP [26] and at the bottom we have the uncompensated link. The number of received symbols was 10240 symbols (160frames*64symbols/frame), which sufficed for good convergence of the MER averaging process.

Fig. 8
Fig. 8

Frequency-flat parallel NLC akin to [26]. (a). Top-level block diagram. The top path transfer function represents the end-to-end linear filtering effect of CD. The common structure of the Nspan parallel paths is detailed in (b). The the s-th NL comp module generates the NLI contribution of the s-th fiber span (s = 1,2,…,Nspan) at the link output.

Fig. 9
Fig. 9

Conceptual comparison of topological decompositions of our F-VSTF NLC DBP vs. the VSTF based system of [26]. (a). A fiber link with three spans. (b). Parallel decomposition of the link and its NL emulator as realized in [26]. (c) The topology of our F-VSTF based emulator (d).Trellis-like decomposition of the serial NL emulator of (c) in terms of inter-span NL interaction terms, explicitly showing all possible routes between emulator end-points.

Tables (1)

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Appendix Table 1 – Glossary

Equations (30)

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r i =[ A i + R i FWM (A) ] H i CD+XPM
R i FWM (A)= ( j,k )S[i] H i;j,k A j A k A j+ki *
H i;j,k =j 0 L γ( z ) G P ( z )exp( jΔ β i;j,k z )dz G P ( z )exp( 0 z α( z )d z );Δ β i;j,k ( 2πΔv ) 2 ( ji )( ki ) β 2
H i;j,k H ^ j 1 H ^ k 2 H ^ j+ki 3* H ^ i 4
ε 2 i | R i FWM R ^ i FWM | 2 = i | j,kS[ i ] ( H i;j,k H ^ j 1 H ^ k 2 H ^ j+ki 3* H ^ i 4 ) A j A k A j+ki | 2
R ^ i FWM H ^ i 4 j,kS[i] A j A k A j+ki * H ^ j 1 H ^ k 2 H ^ j+ki 3*
argmin H ^ 1 , H ^ 2 , H ^ 3 , H ^ 4 i j,kS[i] | H i;j,k H ^ j 1 H ^ k 2 H ^ j+ki 3* H ^ i 4 |
H i;j,k H ^ j 1 H ^ k 1 H ^ j+ki 3* H ^ i 4
A ^ k ( H s ) A k H ^ k s ; s=1,2,3
a ( H s ) [ t ]IDF T 4N { A ^ 1 ( H s ) ,..., A ^ N/2 ( H s ) ,0,...,0, A ^ N/2+1 ( H s ) ,..., A ^ 0 ( H s ) }
a ( H s ) [ t ] k A ^ k ( H s ) e j 2π 4N kt
r NL [t] a ( H 1 ) [t] a ( H 2 ) [t] a ( H 3 )* [t]
r NL [ t ] k 1 k 2 k 3 A ^ k 1 ( H 1 ) A ^ k 2 ( H 2 ) A ^ k 3 ( H 3 )* e j 2π 4N ( k 1 + k 2 k 3 )t
R ˜ i NL H ^ i 4 ( k 1 , k 2 ) S ˜ [i] A ^ k 1 ( H 1 ) A ^ k 2 ( H 2 ) A ^ k 1 + k 2 i ( H 3 )* = H ^ i 4 ( j,k ) S ˜ [i] A j A k A j+ki * H ^ j 1 H ^ k 2 H ^ j+ki 3*
S ˜ [i]={ ( j,k ) |1jN,1kN,1j+kiN }= =S[i]{ ( j=i,ki ) }{ ( ji,k=i ) }{ ( j=i,k=i ) }
δ R ˜ i NLCXPM R ˜ i NL R i FWM A i H ^ i 1 [ 2 k | A k | 2 H ^ k 2 H ^ k 3* | A i | 2 H ^ i 2 H ^ i 3* ] H ^ i 4
R i FWM R ˜ i NL δ R ˜ i NLCXPM
R i FWM = r i / H i CD+XPM A i
H i;j,k = N span H i;j,k oneSpan F i;j,k H i;j,k oneSpan jγ 1 e α L s e jΔ β i;j,k L s jΔ β i;j,k +α ; F i;j,k 1 N span s=0 N span 1 e js L s Δ β i;j,k
H i;j,k =( e +j L s Δ β i;j,k H i;j,k oneSpan ) s=0 N span 1 e js L s Δ β i;j,k = H i;j,k oneSpan s=1 N span e js L s Δ β i;j,k
H i;j,k oneSpan = e +j L s Δ β i;j,k H i;j,k oneSpan .
e +j L s Δ β i;j,k = e 1 2 jδ i 2 e 1 2 jδ j 2 e 1 2 jδ k 2 e 1 2 jδ ( j+ki ) 2 ;δ β 2 L s ( 2πΔν ) 2
{2N,2 1 2 4Nlog4N,24N+ 1 2 4Nlog4N+N,4N}
C F-VSTF = N span ( 28N+6NlogN ) 1 N = N span ( 28+6logN )
C parallel combinedfreq.flat-VS-DBP = 1 N [ N direct_path + N span ( 2N pre/post_dispersion + 2 1 2 4Nlog4N+24N NL ) ] =1+ N span ( 18+4logN )
argmin | H ^ | { i ( j,k )S[i] | ln| H i;j,k |ln| H ^ j 1 |ln| H ^ k 2 |ln| H ^ j+ki 3 |ln| H ^ i 4 | | 2 }
argmin H ^ { i ( j,k )S[i] | H i;j,k H ^ j 1 H ^ k 2 + H ^ j+ki 3 H ^ i 4 | 2 }
{ λ,γ }{ ln| H ^ |, H ^ } H ^ =exp( λ+jγ ) [ A 4mag ] s,0N+ j s 1, [ A 4mag ] s,1N+ k s 1, [ A 4mag ] s,2N+ j s + k s i s 1, [ A 4mag ] s,3N+ i s 1, [ A 4ang ] s,0N+ j s 1, [ A 4ang ] s,1N+ k s 1, [ A 4ang ] s,2N+ j s + k s i s 1, [ A 4ang ] s,3N+ i s 1 B mag [ | H i 1 ; j 1 , k 1 |,| H i 2 ; j 2 , k 2 |,... ] T , B ang [ H i 1 ; j 1 , k 1 , H i 2 ; j 2 , k 2 ,... ] T
λ opt = argmin λ B mag A mag λ 2 ; γ opt = argmin γ B ang A ang γ 2
{ λ opt , γ opt }={ [ A 4mag ] B mag , [ A 4ang ] B ang } H ^ opt =exp( λ opt +j γ opt )

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