Skip to main content
SpringerLink
Log in

Single-step digital backpropagation for nonlinearity mitigation

  • Published:
Photonic Network Communications Aims and scope Submit manuscript

Abstract

Nonlinearity mitigation based on the enhanced split-step Fourier method (ESSFM) for the implementation of low-complexity digital backpropagation (DBP) is investigated and experimentally demonstrated. After reviewing the main computational aspects of DBP and of the conventional split-step Fourier method (SSFM), the ESSFM for dual-polarization signals is introduced. Computational complexity, latency, and power consumption of DBP based on the SSFM and ESSFM algorithms are estimated and compared. Effective low-complexity nonlinearity mitigation in a 112 Gb/s polarization-multiplexed QPSK system is experimentally demonstrated by using a single-step DBP based on the ESSFM. The proposed DBP implementation requires only a single step of the ESSFM algorithm to achieve a transmission distance of 3200 km over a dispersion-unmanaged link. In comparison, a conventional DBP implementation requires 20 steps of the SSFM algorithm to achieve the same performance. An analysis of the computational complexity and structure of the two algorithms reveals that the overall complexity and power consumption of DBP are reduced by a factor of 16 with respect to a conventional implementation, while the computation time is reduced by a factor of 20. Similar complexity reductions can be obtained at longer distances if higher error probabilities are acceptable. The results indicate that the proposed algorithm enables a practical and effective implementation of DBP in real-time optical receivers, with only a moderate increase in the computational complexity, power consumption, and latency with respect to a simple feed-forward equalizer for bulk dispersion compensation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

Notes

  1. We consider the classical Cooley–Tukey radix-2 FFT algorithm [5] and assume that each complex multiplication requires four real multiplications and two real additions. Though slightly more efficient implementations are possible, this provides a reasonable indication of the required operations. Moreover, we assume that all fixed quantities (e.g., \(\gamma \Delta zc_{i}\) or \(\exp (-j2\pi ^{2}\beta _{2}f_{k}^{2}\Delta z)\)) are precalculated, and that the complex exponential in (5) is evaluated by using a lookup table.

  2. This approach can be employed even in a real system, as the optimization can be done off-line when designing the link. A more practical (and possibly accurate) approach is that of minimizing the MSE between the output samples (after DBP, equalization, and phase noise/frequency offset compensation) and the transmitted symbols, as suggested in [28]. This, however, needs some care to handle possible interactions with the convergence of the butterfly equalizer and is left to a future investigation.

  3. When the total number of spans is not an exact multiple of the number of spans per step, a final shorter step is considered to account for the remaining spans.

References

  1. Asif, R., Lin, C.Y., Holtmannspoetter, M., Schmauss, B.: Optimized digital backward propagation for phase modulated signals in mixed-optical fiber transmission link. Opt. Express 18(22), 22796–22807 (2010)

    Article  Google Scholar 

  2. Ciaramella, E., Forestieri, E.: Analytical approximation of nonlinear distortions. IEEE Photonics Technol. Lett. 17(1), 91–93 (2005)

    Article  Google Scholar 

  3. Colavolpe, G., Foggi, T., Forestieri, E., Secondini, M.: Impact of phase noise and compensation techniques in coherent optical systems. J. Lightwave Technol. 29(18), 2790–2800 (2011)

    Article  Google Scholar 

  4. Cugini, F., Paolucci, F., Meloni, G., Berrettini, G., Secondini, M., Fresi, F., Sambo, N., Poti, L., Castoldi, P.: Push-pull defragmentation without traffic disruption in flexible grid optical networks. J. Lightwave Technol. 31(1), 125–133 (2013)

    Article  Google Scholar 

  5. Cooley, J.W., Tukey, J.W.: An algorithm for the machine calculation of complex Fourier series. Math. Comp. 19, 297–301 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  6. Du, L.B., Lowery, A.J.: Improved single channel backpropagation for intra-channel fiber nonlinearity compensation in long-haul optical communication systems. Opt. Express 18(16), 17075–17088 (2010)

    Article  Google Scholar 

  7. Forestieri, E., Secondini, M.: Solving the nonlinear Schrödinger equation. In: Forestieri, E. (ed.) Optical Communication Theory and Techniques, pp. 3–11. Springer, New York (2004)

    Google Scholar 

  8. Goldfarb, G., Li, G.: Efficient backward-propagation using wavelet-based filtering for fiber backward-propagation. Opt. Express 17(11), 8815–8821 (2009)

    Article  Google Scholar 

  9. Hardin, R.H., Tappert, F.D.: Applications of the split-step Fourier method to the numerical solution of nonlinear and variable coefficient wave equations. SIAM Rev. 15, 423 (1973)

    Google Scholar 

  10. Ip, E., Bai, N., Wang, T.: Complexity versus performance tradeoff for fiber nonlinearity compensation using frequency-shaped, multi-subband backpropagation. In: Optical Fiber Communications Conference (OFC), p. OThF4 (2011)

  11. Ip, E., Kahn, J.M.: Compensation of dispersion and nonlinear impairments using digital backpropagation. J. Lightwave Technol. 26(20), 3416–3425 (2008)

    Article  Google Scholar 

  12. Irukulapati, N.V., Marsella, D., Johannisson, P., Secondini, M., Wymeersch, H., Agrell, E., Forestieri, E.: On maximum likelihood sequence detectors for single-channel coherent optical communications. In: European Conference on Optical Communications (ECOC), p. P.3.19 (2014)

  13. Irukulapati, N.V., Wymeersch, H., Johannisson, P., Agrell, E.: Stochastic digital backpropagation. IEEE Trans. Commun. 62(11), 3956–3968 (2014)

    Article  Google Scholar 

  14. Kuschnerov, M., Hauske, F.N., Piyawanno, K., Spinnler, B., Alfiad, M.S., Napoli, A., Lankl, B.: DSP for coherent single-carrier receivers. J. Lightwave Technol. 27(16), 3614–3622 (2009)

    Article  Google Scholar 

  15. Li, L., Tao, Z., Dou, L., Yan, W., Oda, S., Tanimura, T., Hoshida, T., Rasmussen, J.C.: Implementation efficient nonlinear equalizer based on correlated digital backpropagation. In: Optical Fiber Communications Conference (OFC), p. OWW3 (2011)

  16. Li, X., Chen, X., Goldfarb, G., Mateo, E., Kim, I., Yaman, F., Li, G.: Electronic post-compensation of WDM transmission impairments using coherent detection and digital signal processing. Opt. Express 16(2), 880–888 (2008)

    Article  Google Scholar 

  17. Liang, X., Kumar, S.: Correlated digital back propagation based on perturbation theory. Opt. Express 23(11), 14655–14665 (2015)

    Article  Google Scholar 

  18. Maher, R., Xu, T., Galdino, L., Sato, M., Alvarado, A., Shi, K., Savory, S.J., Thomsen, B.C., Killey, R.I., Bayvel, P.: Spectrally shaped DP-16QAM super-channel transmission with multi-channel digital back-propagation. Sci. Rep. 5, 8214 (2015). doi:10.1038/srep08214

  19. Marsella, D., Secondini, M., Forestieri, E.: Maximum likelihood sequence detection for mitigating nonlinear effects. J. Lightwave Technol. 32(5), 908–916 (2014)

    Article  Google Scholar 

  20. Menyuk, C., Marks, B.: Interaction of polarization mode dispersion and nonlinearity in optical fiber transmission systems. J. Lightwave Technol. 24(7), 2806–2826 (2006)

    Article  Google Scholar 

  21. Napoli, A., Maalej, Z., Sleiffer, V.A., Kuschnerov, M., Rafique, D., Timmers, E., Spinnler, B., Rahman, T., Coelho, L.D., Hanik, N.: Reduced complexity digital back-propagation methods for optical communication systems. J. Lightwave Technol. 32(7), 1351–1362 (2014)

    Article  Google Scholar 

  22. Oppenheim, A.V., Schafer, R.W.: Discrete-Time Signal Processing. Prentice Hall, Upper Saddle River (1999)

    MATH  Google Scholar 

  23. Peddanarappagari, K.V., Brandt-Pearce, M.: Volterra series transfer function of single-mode fibers. J. Lightwave Technol. 15, 2232–2241 (1997)

  24. Rafique, D., Mussolin, M., Forzati, M., Mårtensson, J., Chugtai, M.N., Ellis, A.D.: Compensation of intra-channel nonlinear fibre impairments using simplified digital back-propagation algorithm. Opt. Express 19(10), 9453–9460 (2011)

    Article  Google Scholar 

  25. Secondini, M., Forestieri, E.: Analytical fiber-optic channel model in the presence of cross-phase modulation. IEEE Photonics Technol. Lett. 24(22), 2016–2019 (2012)

    Article  Google Scholar 

  26. Secondini, M., Forestieri, E.: On XPM mitigation in WDM fiber-optic systems. IEEE Photonics Technol. Lett. 26(22), 2252–2255 (2014)

    Article  Google Scholar 

  27. Secondini, M., Forestieri, E., Prati, G.: Achievable information rate in nonlinear WDM fiber-optic systems with arbitrary modulation formats and dispersion maps. J. Lightwave Technol. 31(23), 3839–3852 (2013)

    Article  Google Scholar 

  28. Secondini, M., Marsella, D., Forestieri, E.: Enhanced split-step fourier method for digital backpropagation. In: Proceedings of the European Conference on Optical Communications (ECOC), p. We.3.3.5 (2014)

  29. Secondini, M., Rommel, S., Fresi, F., Forestieri, E., Meloni, G., Potì, L.: Coherent 100G nonlinear compensation with single-step digital backpropagation. In: Optical Network Design and Modeling (ONDM), 2015, pp. 63–67. IEEE (2015)

  30. Stockham, T.G., Jr.: High-speed convolution and correlation. In: Proceedings of the 1966 Spring Joint Computer Conference, AFIPS, vol. 28, pp. 229–233 (1966)

  31. Taha, T.R., Ablowitz, M.J.: Analytical and numerical aspects of certain nonlinear evolution equation, II, numerical, nonlinear Schroedinger equation. J. Comput. Phys. 5, 203–230 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  32. Taylor, M.G.: Coherent detection method using DSP for demodulation of signal and subsequent equalization of propagation impairments. IEEE Photonics Technol. Lett. 16(2), 674–676 (2004)

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

    Article  Google Scholar 

Download references

Acknowledgments

This work was supported in part by the Italian MIUR under the FIRB project COTONE and by the EU FP-7 GÉANT project COFFEE.

Author information

Authors and Affiliations

  1. TeCIP Institute, Scuola Superiore Sant’Anna, 56124, Pisa, Italy

    Marco Secondini, Francesco Fresi & Enrico Forestieri

  2. Department of Photonics Engineering, Technical University of Denmark, 2800, Kgs. Lyngby, Denmark

    Simon Rommel

  3. National Laboratory of Photonics Networks, CNIT, 56124, Pisa, Italy

    Gianluca Meloni & Luca Potì

Authors
  1. Marco Secondini
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Simon Rommel
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Gianluca Meloni
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Francesco Fresi
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Enrico Forestieri
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Luca Potì
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marco Secondini.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Secondini, M., Rommel, S., Meloni, G. et al. Single-step digital backpropagation for nonlinearity mitigation. Photon Netw Commun 31, 493–502 (2016). https://doi.org/10.1007/s11107-015-0586-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11107-015-0586-z

Keywords

Search

Navigation