Abstract

Fiber-optic communication systems are nowadays facing serious challenges due to the fast growing demand on capacity from various new applications and services. It is now well recognized that nonlinear effects limit the spectral efficiency and transmission reach of modern fiber-optic communications. Nonlinearity compensation is therefore widely believed to be of paramount importance for increasing the capacity of future optical networks. Recently, there has been steadily growing interest in the application of a powerful mathematical tool—the nonlinear Fourier transform (NFT)—in the development of fundamentally novel nonlinearity mitigation tools for fiber-optic channels. It has been recognized that, within this paradigm, the nonlinear crosstalk due to the Kerr effect is effectively absent, and fiber nonlinearity due to the Kerr effect can enter as a constructive element rather than a degrading factor. The novelty and the mathematical complexity of the NFT, the versatility of the proposed system designs, and the lack of a unified vision of an optimal NFT-type communication system, however, constitute significant difficulties for communication researchers. In this paper, we therefore survey the existing approaches in a common framework and review the progress in this area with a focus on practical implementation aspects. First, an overview of existing key algorithms for the efficacious computation of the direct and inverse NFT is given, and the issues of accuracy and numerical complexity are elucidated. We then describe different approaches for the utilization of the NFT in practical transmission schemes. After that we discuss the differences, advantages, and challenges of various recently emerged system designs employing the NFT, as well as the spectral efficiency estimates available up-to-date. With many practical implementation aspects still being open, our mini-review is aimed at helping researchers assess the perspectives, understand the bottlenecks, and envision the development paths in the upcoming NFT-based transmission technologies.

© 2017 Optical Society of America

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D. J. Richardson, “New optical fibres for high-capacity optical communications,” Philos. Trans. R. Soc. A 374, 20140441 (2016).
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P. Bayvel, R. Maher, T. Xu, G. Liga, N. A. Shevchenko, D. Lavery, A. Alvarado, and R. I. Killey, “Maximizing the optical network capacity,” Philos. Trans. R. Soc. A 374, 20140440 (2016).
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D. Rafique, “Fiber nonlinearity compensation: commercial applications and complexity analysis,” J. Lightwave Technol. 34, 544–553 (2016).
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S. T. Le, J. E. Prilepsky, P. Rosa, J. D. Ania-Castanon, and S. K. Turitsyn, “Nonlinear inverse synthesis for optical links with distributed Raman amplification,” J. Lightwave Technol. 34, 1778–1786 (2016).
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S. T. Le, I. D. Philips, J. E. Prilepsky, P. Harper, A. D. Ellis, and S. K. Turitsyn, “Demonstration of nonlinear inverse synthesis transmission over transoceanic distances,” J. Lightwave Technol. 34, 2459–2466 (2016).
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M. Kamalian, J. E. Prilepsky, S. T. Le, and S. K. Turitsyn, “Periodic nonlinear Fourier transform for fiber-optic communications, Part I: theory and numerical methods,” Opt. Express 24, 18353–18369 (2016).
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M. Kamalian, J. E. Prilepsky, S. T. Le, and S. K. Turitsyn, “Periodic nonlinear Fourier transform for fiber-optic communications, Part II: eigenvalue communication,” Opt. Express 24, 18370–18381 (2016).
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C. Swierczewski and B. Deconinck, “Computing Riemann theta functions in Sage with applications,” Math. Comput. Simul. 127, 263–272 (2016).
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S. Hari and F. R. Kschischang, “Bi-directional algorithm for computing discrete spectral amplitudes in the NFT,” J. Lightwave Technol. 34, 3529–3537 (2016).
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S. Hari, M. I. Yousefi, and F. R. Kschischang, “Multieigenvalue communication,” J. Lightwave Technol. 34, 3110–3117 (2016).
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E. Agrell, A. Alvarado, and F. R. Kschishang, “Implications of information theory in optical fibre communications,” Philos. Trans. R. Soc. A 374, 20140438 (2016).
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S. A. Derevyanko, J. E. Prilepsky, and S. K. Turitsyn, “Capacity estimates for optical transmission based on the nonlinear Fourier transform,” Nat. Commun. 7, 12710 (2016).
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2015 (8)

J. L. Aurentz, T. Mach, R. Vandebril, and D. S. Watkins, “Fast and backward stable computation of roots of polynomials,” SIAM J. Matrix Anal. Appl. 36, 942–973 (2015).
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L. L. Frumin, O. V. Belai, E. V. Podivilov, and D. A. Shapiro, “Efficient numerical method for solving the direct Zakharov-Shabat scattering problem,” J. Opt. Soc. Am. B 32, 290–295 (2015).
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S. Wahls and H. V. Poor, “Fast numerical nonlinear Fourier transforms,” IEEE Trans. Inf. Theory 61, 6957–6974 (2015).
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H. Bülow, “Experimental demonstration of optical signal detection using nonlinear Fourier transform,” J. Lightwave Technol. 33, 1433–1439 (2015).
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S. T. Le, J. E. Prilepsky, and S. K. Turitsyn, “Nonlinear inverse synthesis technique for optical links with lumped amplification,” Opt. Express 23, 8317–8328 (2015).
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P. J. Winzer, “Scaling optical fiber networks: challenges and solutions,” Opt. Photon. News 26(3), 28–35 (2015).

2014 (7)

P. J. Winzer, “Spatial multiplexing in fiber optics: the 10X scaling of metro/core capacities,” Bell Labs Tech. J. 19, 22–30 (2014).

L. B. Du, D. Rafique, A. Napoli, B. Spinnler, A. D. Ellis, M. Kuschnerov, and A. J. Lawery, “Digital fiber nonlinearity compensation: toward 1-Tb/s transport,” IEEE Signal Process. Mag. 31(2), 46–56 (2014).
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M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, Part I: mathematical tools,” IEEE Trans. Inf. Theory 60, 4312–4328 (2014).
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M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, Part II: numerical methods,” IEEE Trans. Inf. Theory 60, 4329–4345 (2014).
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M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, Part III: spectrum modulation,” IEEE Trans. Inf. Theory 60, 4346–4369 (2014).
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J. E. Prilepsky, S. A. Derevyanko, K. J. Blow, I. Gabitov, and S. K. Turitsyn, “Nonlinear inverse synthesis and eigenvalue division multiplexing in optical fiber channels,” Phys. Rev. Lett. 113, 013901 (2014).
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S. T. Le, J. E. Prilepsky, and S. K. Turitsyn, “Nonlinear inverse synthesis for high spectral efficiency transmission in optical fibers,” Opt. Express 22, 26720–26741 (2014).
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2013 (4)

J. E. Prilepsky, S. A. Derevyanko, and S. K. Turitsyn, “Nonlinear spectral management: linearization of the lossless fiber channel,” Opt. Express 21, 24344–24367 (2013).
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E. G. Turitsyna and S. K. Turitsyn, “Digital signal processing based on inverse scattering transform,” Opt. Lett. 38, 4186–4188 (2013).
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A. R. C. X. Liu, P. J. Winzer, R. W. Tkach, and S. Chandrasekhar, “Phase-conjugated twin waves for communication beyond the Kerr nonlinearity limit,” Nat. Photonics 7, 560–568 (2013).
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D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7, 354–362 (2013).
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2012 (2)

T. Trogdon and S. Olver, “Numerical inverse scattering for the focusing and defocusing nonlinear Schrödinger equations,” Proc. R. Soc. London A 469, 20120330 (2012).
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P. Boito, Y. Eidelman, L. Gemignami, and I. Gohberg, “Implicit QR with compression,” Indag. Math. 23, 733–761 (2012).
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2011 (3)

A. Arico, G. Rodriguez, and S. Seatzu, “Numerical solution of the nonlinear Schrödinger equation, starting from the scattering data,” Calcolo 48, 75–88 (2011).
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S. A. Derevyanko, “Appearance of bound states in random potentials with applications to soliton theory,” Phys. Rev. E 84, 016601 (2011).
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R. I. Killey and C. Behrens, “Shannon’s theory in nonlinear systems,” J. Mod. Opt. 58, 1–10 (2011).
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J. D. Ania-Castañón, V. Karalekas, P. Harper, and S. K. Turitsyn, “Simultaneous spatial and spectral transparency in ultralong fiber lasers,” Phys. Rev. Lett. 101, 123903 (2008).
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S. K. Turitsyn and S. A. Derevyanko, “Soliton-based discriminator of noncoherent optical pulses,” Phys. Rev. A 78, 063819 (2008).
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S. A. Derevyanko and J. E. Prilepsky, “Random input problem for the nonlinear Schrödinger equation,” Phys. Rev. E 78, 046610 (2008).
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S. A. Derevyanko, “Design of a flat-top fiber Bragg filter via quasi-random modulation of the refractive index,” Opt. Lett. 33, 2404–2406 (2008).
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O. V. Belai, L. L. Frumin, E. V. Podivilov, O. Y. Schwarz, and D. A. Shapiro, “Finite Bragg grating synthesis by numerical solution of Hermitian Gel’fand-Levitan-Marchenko equations,” J. Opt. Soc. Am. B 23, 2040–2045 (2006).
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S. A. Derevyanko, J. E. Prilepsky, and D. A. Yakushev, “Statistics of a noise-driven Manakov soliton,” J. Phys. A 39, 1297–1309 (2006).

J. D. Ania-Castañón, T. J. Ellingham, R. Ibbotson, X. Chen, L. Zhang, and S. K. Turitsyn, “Ultralong Raman fibre lasers as virtually lossless optical media,” Phys. Rev. Lett. 96, 023902 (2006).
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S. L. Jansen, D. Van den Borne, B. Spinnler, S. Calabro, H. Suche, P. M. Krummrich, G.-D. Khoe, and H. de Waardt, “Optical phase conjugation for ultra long-haul phase-shift-keyed transmission,” J. Lightwave Technol. 24, 54–64 (2006).
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J. D. Ania-Castañón, “Quasi-lossless transmission using second-order Raman amplification and fibre Bragg gratings,” Opt. Express 12, 4372–4377 (2004).
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A. Rosenthal and M. Horowitz, “Inverse scattering algorithm for reconstructing strongly reflecting fiber Bragg gratings,” IEEE J. Quantum Electron. 39, 1018–1026 (2003).
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S. T. Le, J. E. Prilepsky, P. Rosa, J. D. Ania-Castanon, and S. K. Turitsyn, “Nonlinear inverse synthesis for optical links with distributed Raman amplification,” J. Lightwave Technol. 34, 1778–1786 (2016).
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J. D. Ania-Castañón, T. J. Ellingham, R. Ibbotson, X. Chen, L. Zhang, and S. K. Turitsyn, “Ultralong Raman fibre lasers as virtually lossless optical media,” Phys. Rev. Lett. 96, 023902 (2006).
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J. E. Prilepsky, S. A. Derevyanko, and S. K. Turitsyn, “Nonlinear spectral management: linearization of the lossless fiber channel,” Opt. Express 21, 24344–24367 (2013).
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N. A. Shevchenko, S. A. Derevyanko, J. E. Prilepsky, A. Alvarado, P. Bavel, and S. K. Turitsyn, “A lower bound on the capacity of the noncentral chi channel with applications to soliton amplitude modulation,” arXiv:1609.02318 (2016).

S. T. Le, J. E. Prilepsky, M. Kamalian, P. Rosa, M. Tan, J. D. Ania-Castanon, P. Harper, and S. K. Turitsyn, “Modified nonlinear inverse synthesis for optical links with distributed Raman amplification,” in 41st European Conference on Optical Communications (ECOC), Valencia, Spain, 2015, paper Tu 1.1.3.

S. T. Le, I. D. Philips, J. E. Prilepsky, M. Kamalian, A. D. Ellis, P. Harper, and S. K. Turitsyn, “Achievable information rate of nonlinear inverse synthesis based 16QAM OFDM transmission,” in 42nd European Conference on Optical Communications (ECOC), Germany, 2016, paper Th.2.PS2.SC5.

S. T. Le, S. Wahls, D. Lavery, J. E. Prilepsky, and S. K. Turitsyn, “Reduced complexity nonlinear inverse synthesis for nonlinearity compensation in optical fiber links,” in Proceedings of Conference on Lasers and Electro-Optics/Europe and the European Quantum Electronics Conference (CLEO/Europe-EQEC), Munich, Germany, 2015, paper CI_3_2.

M. Kamalian, J. E. Prilepsky, S. T. Le, and S. K. Turitsyn, “Periodic nonlinear Fourier transform based transmissions with high order QAM formats,” in European Conference on Optical Communications (ECOC), Germany, 2016, pp. 1–3.

M. Kamalian Kopae, J. E. Prilepsky, S. T. Le, and S. K. Turitsyn, “Optical communication based on the periodic nonlinear Fourier transform signal processing,” in IEEE 6th International Conference on Photonics (ICP), Kuching, Malaysia, 2016, pp. 1–3.

M. Kamalian Kopae, J. E. Prilepsky, S. T. Le, and S. Turitsyn, “Periodic nonlinear Fourier transform based optical communication systems in a band-limited regime,” in Advanced Photonics (IPR, NOMA, Sensors, Networks, SPPCom, SOF), Vancouver, Canada, 2016, paper JTu4A.34.

N. A. Shevchenko, J. E. Prilepsky, S. A. Derevyanko, A. Alvarado, P. Bavel, and S. K. Turitsyn, “A lower bound on the per soliton capacity of the nonlinear optical fibre channel,” in IEEE Information Theory Workshop (ITW), Jeju Island, Korea, 2015, pp. 104–108.

S. Wahls, S. T. Le, J. E. Prilepsky, H. V. Poor, and S. K. Turitsyn, “Digital backpropagation in the nonlinear Fourier domain,” in IEEE 16th International Workshop in Signal Processing Advances in Wireless Communications (SPAWC), Stockholm, Sweden, 2015, pp. 445–449.

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N. A. Shevchenko, S. A. Derevyanko, J. E. Prilepsky, A. Alvarado, P. Bavel, and S. K. Turitsyn, “A lower bound on the capacity of the noncentral chi channel with applications to soliton amplitude modulation,” arXiv:1609.02318 (2016).

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S. T. Le, J. E. Prilepsky, M. Kamalian, P. Rosa, M. Tan, J. D. Ania-Castanon, P. Harper, and S. K. Turitsyn, “Modified nonlinear inverse synthesis for optical links with distributed Raman amplification,” in 41st European Conference on Optical Communications (ECOC), Valencia, Spain, 2015, paper Tu 1.1.3.

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M. Kamalian Kopae, J. E. Prilepsky, S. T. Le, and S. Turitsyn, “Periodic nonlinear Fourier transform based optical communication systems in a band-limited regime,” in Advanced Photonics (IPR, NOMA, Sensors, Networks, SPPCom, SOF), Vancouver, Canada, 2016, paper JTu4A.34.

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S. T. Le, J. E. Prilepsky, P. Rosa, J. D. Ania-Castanon, and S. K. Turitsyn, “Nonlinear inverse synthesis for optical links with distributed Raman amplification,” J. Lightwave Technol. 34, 1778–1786 (2016).
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S. T. Le, I. D. Philips, J. E. Prilepsky, P. Harper, A. D. Ellis, and S. K. Turitsyn, “Demonstration of nonlinear inverse synthesis transmission over transoceanic distances,” J. Lightwave Technol. 34, 2459–2466 (2016).
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J. E. Prilepsky, S. A. Derevyanko, K. J. Blow, I. Gabitov, and S. K. Turitsyn, “Nonlinear inverse synthesis and eigenvalue division multiplexing in optical fiber channels,” Phys. Rev. Lett. 113, 013901 (2014).
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J. E. Prilepsky, S. A. Derevyanko, and S. K. Turitsyn, “Nonlinear spectral management: linearization of the lossless fiber channel,” Opt. Express 21, 24344–24367 (2013).
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E. G. Turitsyna and S. K. Turitsyn, “Digital signal processing based on inverse scattering transform,” Opt. Lett. 38, 4186–4188 (2013).
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J. D. Ania-Castañón, V. Karalekas, P. Harper, and S. K. Turitsyn, “Simultaneous spatial and spectral transparency in ultralong fiber lasers,” Phys. Rev. Lett. 101, 123903 (2008).
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S. K. Turitsyn and S. A. Derevyanko, “Soliton-based discriminator of noncoherent optical pulses,” Phys. Rev. A 78, 063819 (2008).
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J. E. Prilepsky, S. A. Derevyanko, and S. K. Turitsyn, “Conversion of a chirped Gaussian pulse to a soliton or a bound multisoliton state in quasi-lossless and lossy optical fiber spans,” J. Opt. Soc. Am. B 24, 1254–1261 (2007).
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J. D. Ania-Castañón, T. J. Ellingham, R. Ibbotson, X. Chen, L. Zhang, and S. K. Turitsyn, “Ultralong Raman fibre lasers as virtually lossless optical media,” Phys. Rev. Lett. 96, 023902 (2006).
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S. T. Le, I. D. Philips, J. E. Prilepsky, M. Kamalian, A. D. Ellis, P. Harper, and S. K. Turitsyn, “Equalization-enhanced phase noise in nonlinear inverse synthesis transmissions,” in European Conference on Optical Communications (ECOC), Germany, 2016, paper Tu.3.B.2.

N. A. Shevchenko, S. A. Derevyanko, J. E. Prilepsky, A. Alvarado, P. Bavel, and S. K. Turitsyn, “A lower bound on the capacity of the noncentral chi channel with applications to soliton amplitude modulation,” arXiv:1609.02318 (2016).

J. E. Prilepsky and S. K. Turitsyn, “Eigenvalue communications in nonlinear fiber channels,” in Odyssey of Light in Nonlinear Optical Fibers: Theory and Applications, K. Porsezian and R. Ganapathy, eds. (CRC Press, 2015) Chap. 18, pp. 459–490.

S. Wahls, S. T. Le, J. E. Prilepsky, H. V. Poor, and S. K. Turitsyn, “Digital backpropagation in the nonlinear Fourier domain,” in IEEE 16th International Workshop in Signal Processing Advances in Wireless Communications (SPAWC), Stockholm, Sweden, 2015, pp. 445–449.

I. D. Phillips, M. Tan, M. F. C. Stephens, M. E. McCarthy, E. Giacoumidis, S. Sygletos, P. Rosa, S. Fabbri, S. T. Le, T. Kanesan, S. K. Turitsyn, N. J. Doran, P. Harper, and A. D. Ellis, “Exceeding the Nonlinear-Shannon limit using Raman laser based amplification and optical phase conjugation,” in Optical Fiber Communication Conference (OFC), Los Angeles, CA, 2014, paper M3C.1.

S. T. Le, J. E. Prilepsky, M. Kamalian, P. Rosa, M. Tan, J. D. Ania-Castanon, P. Harper, and S. K. Turitsyn, “Modified nonlinear inverse synthesis for optical links with distributed Raman amplification,” in 41st European Conference on Optical Communications (ECOC), Valencia, Spain, 2015, paper Tu 1.1.3.

M. Kamalian Kopae, J. E. Prilepsky, S. T. Le, and S. K. Turitsyn, “Optical communication based on the periodic nonlinear Fourier transform signal processing,” in IEEE 6th International Conference on Photonics (ICP), Kuching, Malaysia, 2016, pp. 1–3.

N. A. Shevchenko, J. E. Prilepsky, S. A. Derevyanko, A. Alvarado, P. Bavel, and S. K. Turitsyn, “A lower bound on the per soliton capacity of the nonlinear optical fibre channel,” in IEEE Information Theory Workshop (ITW), Jeju Island, Korea, 2015, pp. 104–108.

S. T. Le, S. Wahls, D. Lavery, J. E. Prilepsky, and S. K. Turitsyn, “Reduced complexity nonlinear inverse synthesis for nonlinearity compensation in optical fiber links,” in Proceedings of Conference on Lasers and Electro-Optics/Europe and the European Quantum Electronics Conference (CLEO/Europe-EQEC), Munich, Germany, 2015, paper CI_3_2.

S. T. Le, I. D. Philips, J. E. Prilepsky, M. Kamalian, A. D. Ellis, P. Harper, and S. K. Turitsyn, “Achievable information rate of nonlinear inverse synthesis based 16QAM OFDM transmission,” in 42nd European Conference on Optical Communications (ECOC), Germany, 2016, paper Th.2.PS2.SC5.

M. Kamalian, J. E. Prilepsky, S. T. Le, and S. K. Turitsyn, “Periodic nonlinear Fourier transform based transmissions with high order QAM formats,” in European Conference on Optical Communications (ECOC), Germany, 2016, pp. 1–3.

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Supplementary Material (1)

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Figures (4)

Fig. 1.
Fig. 1.

Exemplary NF spectrum (anomalous dispersion case), containing solitons (discrete eigenvalues) and continuous nonlinear spectrum (depicted on the real axis ξ).

Fig. 2.
Fig. 2.

Burst mode for the window in vanishing signal processing (ordinary NFT) and the processing window for the periodic signal with cyclic extension (PNFT).

Fig. 3.
Fig. 3.

Diagram of the currently proposed and studied NFT-based methods.

Fig. 4.
Fig. 4.

Basic designs of NFT-based transmission systems, including transmission in the NFT domain, DBP with the use of NFT operations, and the hybrid method.

Equations (33)

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iqz±122qt2+|q|2q=0.
iqzβ222qt2+γ|q|2q=ig(z)q+η(z,t),
E[η(z,t)η¯(z,t)]=2Dδ(tt)δ(zz),
t/Tst,z/Zsz,q/P0q,
dv1dt=q(z0,t)v2iζv1,dv2dt=q¯(z0,t)v1+iζv2
ϕ1(t,ζ)=eiζt+o(1),ϕ2(t,ζ)=o(1)for  t,
ψ1(t,ζ)=o(1),ψ2(t,ζ)=eiζt+o(1)for  t+.
[ϕ1ϕ2]=a(ζ)[ψ˜1ψ˜2]+b(ζ)[ψ1ψ2],
[ϕ˜1ϕ˜2]=a˜(ζ)[ψ1ψ2]+b˜(ζ)[ψ˜1ψ˜2].
a(ζ)=limtϕ1(t,ζ)eiζt,b(ζ)=limtϕ2(t,ζ)eiζt.
l(ξ)=b¯(ξ)/a(ξ),r(ξ)=b(ξ)/a(ξ),ξR.
ln=[b(ζn)a(ζn)]1,rn=b(ζn)/a(ζn),
Σl={l(ξ),[ζn,ln]n=1N},Σr={r(ξ),[ζn,rn]n=1N},
l(ξ,z)=l(ξ,z0)e2iξ2(zz0),ln(ξ,z)=ln(ξ,z0)e2iζn2(zz0),r(ξ,z)=r(ξ,z0)e2iξ2(zz0),rn(ξ,z)=rn(ξ,z0)e2iζn2(zz0).
ε1r¯ε(ξ),ε1lε(ξ)q(ω)|ω=2ξwhen  ε0,
K¯1(τ,τ)+τdyL(τ+y)K2(τ,y)=0,K¯2(τ,τ)+L(τ+τ)+τdyL(τ+y)K1(τ,y)=0
Lsol(τ)=inlneiζnτ,Lrad(τ)=12πdξl(ξ)eiξτ
M={ζm|Δ(ζm)=±1,dΔdζ|ζ=ζm0}.
q(z,t)=q(0,0)Θ(W|τ)Θ(W+|τ)eik0ziω0t,
Θ(W|τ)=mZgexp(2πimTW+πimTτm).
qm=q(z0,T1+ϵ(m1)),m=1,,M,
[ϕ1(T1+(m+0.5)ϵ,ζ)ϕ2(T1+(m+0.5)ϵ,ζ)]=Tm[ϕ1(T1+(m0.5)ϵ,ζ)ϕ2(T1+(m0.5)ϵ,ζ)],
where  Tm(ζ)=expm(ϵ[iζqmq¯miζ]).
[a(ζ)b(ζ)]diag(eiζ(T2+0.5ϵ),eiζ(T2+0.5ϵ))TM(ζ)××T1(ζ)eiζ(T10.5ϵ).
Tm(ζ)=11±ϵ2|qm|2[Zϵqmϵq¯mZ1],Z=eiζϵ.
u(τ,τ)=K1(τ,ττ),w(τ,τ)=K¯2(τ,ττ).
u(τ,y)y2τL¯(τy)w(τ,τ)dτ=0,w(τ,τ)+0τL(τy)u(τ,y)dy+L(τ)=0,
Tm(ζ)=Sm(Z)/dm(Z),Z=eiλϵ,
Sm(Z)=11+ϵ2|qm|2[Z2ϵqmZϵq¯mZ1],dm(Z)=Z.
[a(ζ)b(ζ)]diag(eiζ(T2+0.5ϵ),eiζ(T2+0.5ϵ))S(Z)d(Z)eiζ(T10.5ϵ).
S(Z)=k=02M1S(k)Zk,d(Z)=k=02M1d(k)Zk.
S(Z)=SM(Z)[SM1(Z)[SM2(Z)[[S1(Z)]]]].
Zm=ei(λ1+mδϵ)=eiλ1ϵ(eδϵ)m=awm.