Abstract

In this article, for the first time, a full-spectrum periodic nonlinear Fourier transform (NFT)-based communication system with the inverse transformation at the transmitter performed by using the solution of Riemann-Hilbert problem (RHP), is proposed and studied. The entire control over the nonlinear spectrum rendered by our technique, where we operate with two qualitatively different components of this spectrum represented, correspondingly, in terms of the main spectrum and the phases, allows us to design a time-domain signal tailored to the characteristics of the transmission channel. In the heart of our system is the RHP-based signal processing utilised to generate the time-domain signal from the modulated nonlinear spectrum. This type of NFT processing leads to a computational complexity that scales linearly with the number of time-domain samples, and we can process signal samples in parallel. In this article, we suggest the way of getting an exactly periodic signal through the correctly formulated RHP, and present evidence of the analogy between band-limited (in ordinary Fourier sense) signals and finite-band (in RHP sense) signals. Also, for the first time, we explain how to modulate the phases of individual periodic nonlinear modes. The performance of our transmission system is evaluated through numerical simulations in terms of bit error rate and Q $^2$ -factor dependencies on the transmission distance and power, and the results demonstrate the good potential of the approach.

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2020 (1)

M. Pankratova, A. Vasylchenkova, S. A. Derevyanko, N. B. Chichkov, and J. E. Prilepsky, “Signal-noise interaction in optical fiber communication systems employing nonlinear frequency division multiplexing,” Phys. Rev. Applied, 2020.

2019 (5)

W. Q. Zhang, “Correlated eigenvalues of multi-soliton optical communications,” Scientific Rep., vol. 9, no. 1, 2019, Art. no. .

X. Yangzhang, V. Aref, S. T. Le, H. Buelow, D. Lavery, and P. Bayvel, “Dual-polarization non-linear frequency-division multiplexed transmission with $ b$-modulation,” J. Lightw. Technol., vol. 37, no. 6, pp. 1570–1578, 2019.

S. Medvedev, I. Vaseva, I. Chekhovskoy, and M. Fedoruk, “Numerical algorithm with fourth-order accuracy for the direct Zakharov-Shabat problem,” Opt. Lett., vol. 44, no. 9, pp. 2264–2267, 2019.

J.-W. Goossens, H. Hafermann, and Y. Jaouën, “Data transmission based on exact inverse periodic nonlinear Fourier transform, Part I: Theory,” 2019, arXiv:1911.12614.

J.-W. Goossens, H. Hafermann, and Y. Jaouën, “Data transmission based on exact inverse periodic nonlinear Fourier transform, Part II: Waveform design and experiment,” 2019, arXiv:1911.12615.

2018 (1)

M. Kamalian, A. Vasylchenkova, D. Shepelsky, J. Prilepsky, and S. Turitsyn, “Periodic nonlinear Fourier transform communication solving the Riemann-Hilbert problem,” J. Lightw. Technol., vol. 36, no. 24, pp. 5714–5727, 2018.

2017 (7)

M. Kamalian, J. Prilepsky, S. Le, and S. Turitsyn, “On the design of NFT-based communication systems with lumped amplification,” J. Lightw. Technol., vol. 35, no. 24, pp. 5464–5472, 2017.

V. Kotlyarov and D. Shepelsky, “Planar unmodular Baker-Akhiezer function for the nonlinear Schrödinger equation,” Ann. Math. Sci. Appl., vol. 2, no. 2, pp. 343–384, 2017.

S. Civelli, E. Forestieri, and M. Secondini, “Why noise and dispersion may seriously hamper nonlinear frequency-division multiplexing,” IEEE Photon. Technol. Lett., vol. 29, no. 16, pp. 1332–1335,  2017.

S. Turitsyn, “Nonlinear Fourier transform for optical data processing and transmission: Advances and perspectives,” Optica, vol. 4, no. 3, pp. 307–322, 2017.

S. Le, V. Aref, and H. Buelow, “Nonlinear signal multiplexing for communication beyond the Kerr nonlinearity limit,” Nature Photon., vol. 11, no. 9, pp. 570–576, 2017.

I. T. Lima, T. D. DeMenezes, V. S. Grigoryan, M. O'sullivan, and C. R. Menyuk, “Nonlinear compensation in optical communications systems with normal dispersion fibers using the nonlinear Fourier transform,” J. Lightw. Technol., vol. 35, no. 23, pp. 5056–5068, 2017.

T. Gui, C. Lu, A. P. T. Lau, and P. Wai, “High-order modulation on a single discrete eigenvalue for optical communications based on nonlinear Fourier transform,” Opt. Express, vol. 25, no. 17, pp. 20 286–20 297, 2017.

2016 (4)

S. Derevyanko, J. Prilepsky, and S. Turitsyn, “Capacity estimates for optical transmission based on the nonlinear Fourier transform,” Nature Commun., vol. 7, 2016, Art. no. .

M. Kamalian, J. Prilepsky, S. Le, and S. Turitsyn, “Periodic nonlinear Fourier transform for fiber-optic communications, Part I: Theory and numerical methods,” Opt. Express, vol. 24, no. 16, pp. 18 353–18 369, 2016.

S. Wahls and V. Vaibhav, “Fast inverse nonlinear Fourier transforms for continuous spectra of Zakharov-Shabat type,” pp. 1–5, 2016, arXiv:1607.01305.

M. Kamalian, J. Prilepsky, S. Le, and S. Turitsyn, “Periodic nonlinear Fourier transform for fiber-optic communications, part II: Eigenvalue communication,” Opt. Express, vol. 24, no. 16, pp. 18 370–18 381, 2016.

2015 (1)

S. Wahls and V. Poor, “Fast numerical nonlinear Fourier transforms,” IEEE Trans. Inf. Theory, vol. 61, no. 12, pp. 6957–6974,  2015.

2014 (2)

M. Yousefi and F. Kschischang, “Information transmission using the nonlinear Fourier transform, Part I: Mathematical tools,” IEEE Trans. Inf. Theory, vol. 60, no. 7, pp. 4312–4328,  2014.

M. Yousefi and F. Kschischang, “Information transmission using the nonlinear Fourier transform, Part III: Spectrum modulation,” IEEE Trans. Inf. Theory, vol. 60, no. 7, pp. 4346–4369,  2014.

2012 (1)

S. Olver, “A general framework for solving Riemann–Hilbert problems numerically,” Numerische Mathematik, vol. 122, no. 2, pp. 305–340, 2012.

2011 (1)

A. Arico, G. Rodriguez, and S. Seatzu, “Numerical solution of the nonlinear Schrödinger equation, starting from the scattering data,” Calcolo, vol. 48, no. 1, pp. 75–88, 2011.

1993 (1)

A. Hasegawa and T. Nyu, “Eigenvalue communication,” J. Lightw. Technol., vol. 11, no. 3, pp. 395–399, 1993.

1988 (1)

E. Tracy and H. Chen, “Nonlinear self-modulation: An exactly solvable model,” Physical Rev. A, vol. 37, no. 3, pp. 815–839, 1988.

1972 (1)

V. Zakharov and A. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP, vol. 34, no. 1, pp. 62–69, 1972.

Agrawal, G.

G. Agrawal, Nonlinear Fiber Optics. New York, NY, USA: Springer, 2000.

Aref, V.

X. Yangzhang, V. Aref, S. T. Le, H. Buelow, D. Lavery, and P. Bayvel, “Dual-polarization non-linear frequency-division multiplexed transmission with $ b$-modulation,” J. Lightw. Technol., vol. 37, no. 6, pp. 1570–1578, 2019.

S. Le, V. Aref, and H. Buelow, “Nonlinear signal multiplexing for communication beyond the Kerr nonlinearity limit,” Nature Photon., vol. 11, no. 9, pp. 570–576, 2017.

Arico, A.

A. Arico, G. Rodriguez, and S. Seatzu, “Numerical solution of the nonlinear Schrödinger equation, starting from the scattering data,” Calcolo, vol. 48, no. 1, pp. 75–88, 2011.

Barletti, L.

S. Civelli, L. Barletti, and M. Secondini, “Numerical methods for the inverse nonlinear fourier transform,” in Proc. Tyrrhenian Int. Workshop Digit. Commun., 2015, pp. 13–16.

Bayvel, P.

X. Yangzhang, V. Aref, S. T. Le, H. Buelow, D. Lavery, and P. Bayvel, “Dual-polarization non-linear frequency-division multiplexed transmission with $ b$-modulation,” J. Lightw. Technol., vol. 37, no. 6, pp. 1570–1578, 2019.

Belokolos, E.

E. Belokolos, A. Bobenko, V. Enol'skii, A. Its, and V. Matveev, Algebro-Geometric Approach to Nonlinear Integrable Equations. New York, NY, USA: Springer, 1994.

Bobenko, A.

E. Belokolos, A. Bobenko, V. Enol'skii, A. Its, and V. Matveev, Algebro-Geometric Approach to Nonlinear Integrable Equations. New York, NY, USA: Springer, 1994.

Buelow, H.

X. Yangzhang, V. Aref, S. T. Le, H. Buelow, D. Lavery, and P. Bayvel, “Dual-polarization non-linear frequency-division multiplexed transmission with $ b$-modulation,” J. Lightw. Technol., vol. 37, no. 6, pp. 1570–1578, 2019.

S. Le, V. Aref, and H. Buelow, “Nonlinear signal multiplexing for communication beyond the Kerr nonlinearity limit,” Nature Photon., vol. 11, no. 9, pp. 570–576, 2017.

Chekhovskoy, I.

Chen, H.

E. Tracy and H. Chen, “Nonlinear self-modulation: An exactly solvable model,” Physical Rev. A, vol. 37, no. 3, pp. 815–839, 1988.

Chichkov, N. B.

M. Pankratova, A. Vasylchenkova, S. A. Derevyanko, N. B. Chichkov, and J. E. Prilepsky, “Signal-noise interaction in optical fiber communication systems employing nonlinear frequency division multiplexing,” Phys. Rev. Applied, 2020.

Chimmalgi, S.

S. Wahls, S. Chimmalgi, and P. J. Prins, “Wiener-Hopf method for b-modulation,” in Proc. IEEE Opt. Fiber Commun. Conf. Exhib., 2019, Paper W2A.50.

S. Wahls, S. Chimmalgi, and P. J. Prins, “Wiener-Hopf method for b-modulation,” in Proc. Opt. Fiber Commun. Conf., 2019, pp. W2A–50.

S. Chimmalgi, P. J. Prins, and S. Wahls, “Nonlinear Fourier transform algorithm using a higher order exponential integrator,” in Proc. Adv. Photon., 2018, Paper SpM4G.5.

Civelli, S.

S. Civelli, E. Forestieri, and M. Secondini, “Why noise and dispersion may seriously hamper nonlinear frequency-division multiplexing,” IEEE Photon. Technol. Lett., vol. 29, no. 16, pp. 1332–1335,  2017.

S. Civelli, L. Barletti, and M. Secondini, “Numerical methods for the inverse nonlinear fourier transform,” in Proc. Tyrrhenian Int. Workshop Digit. Commun., 2015, pp. 13–16.

DeMenezes, T. D.

I. T. Lima, T. D. DeMenezes, V. S. Grigoryan, M. O'sullivan, and C. R. Menyuk, “Nonlinear compensation in optical communications systems with normal dispersion fibers using the nonlinear Fourier transform,” J. Lightw. Technol., vol. 35, no. 23, pp. 5056–5068, 2017.

Derevyanko, S.

S. Derevyanko, J. Prilepsky, and S. Turitsyn, “Capacity estimates for optical transmission based on the nonlinear Fourier transform,” Nature Commun., vol. 7, 2016, Art. no. .

M. Kamalian, J. Prilepsky, S. Derevyanko, S. Le, and S. Turitsyn, “Nonlinear fourier based spectral filtering,” in Proc. IEEE Lasers Electro-Opt., Conf., 2017, Paper JTh2A.135.

Derevyanko, S. A.

M. Pankratova, A. Vasylchenkova, S. A. Derevyanko, N. B. Chichkov, and J. E. Prilepsky, “Signal-noise interaction in optical fiber communication systems employing nonlinear frequency division multiplexing,” Phys. Rev. Applied, 2020.

Enol'skii, V.

E. Belokolos, A. Bobenko, V. Enol'skii, A. Its, and V. Matveev, Algebro-Geometric Approach to Nonlinear Integrable Equations. New York, NY, USA: Springer, 1994.

Fedoruk, M.

Forestieri, E.

S. Civelli, E. Forestieri, and M. Secondini, “Why noise and dispersion may seriously hamper nonlinear frequency-division multiplexing,” IEEE Photon. Technol. Lett., vol. 29, no. 16, pp. 1332–1335,  2017.

Goossens, J.-W.

J.-W. Goossens, H. Hafermann, and Y. Jaouën, “Data transmission based on exact inverse periodic nonlinear Fourier transform, Part I: Theory,” 2019, arXiv:1911.12614.

J.-W. Goossens, H. Hafermann, and Y. Jaouën, “Data transmission based on exact inverse periodic nonlinear Fourier transform, Part II: Waveform design and experiment,” 2019, arXiv:1911.12615.

J.-W. Goossens, Y. Jaouën, and H. Hafermann, “Experimental demonstration of data transmission based on the exact inverse periodic nonlinear Fourier transform,” in Proc. Opt. Fiber Commun. Conf., 2019, pp. M1I.6.

Grigoryan, V. S.

I. T. Lima, T. D. DeMenezes, V. S. Grigoryan, M. O'sullivan, and C. R. Menyuk, “Nonlinear compensation in optical communications systems with normal dispersion fibers using the nonlinear Fourier transform,” J. Lightw. Technol., vol. 35, no. 23, pp. 5056–5068, 2017.

Gui, T.

T. Gui, C. Lu, A. P. T. Lau, and P. Wai, “High-order modulation on a single discrete eigenvalue for optical communications based on nonlinear Fourier transform,” Opt. Express, vol. 25, no. 17, pp. 20 286–20 297, 2017.

Hafermann, H.

J.-W. Goossens, H. Hafermann, and Y. Jaouën, “Data transmission based on exact inverse periodic nonlinear Fourier transform, Part II: Waveform design and experiment,” 2019, arXiv:1911.12615.

J.-W. Goossens, H. Hafermann, and Y. Jaouën, “Data transmission based on exact inverse periodic nonlinear Fourier transform, Part I: Theory,” 2019, arXiv:1911.12614.

J.-W. Goossens, Y. Jaouën, and H. Hafermann, “Experimental demonstration of data transmission based on the exact inverse periodic nonlinear Fourier transform,” in Proc. Opt. Fiber Commun. Conf., 2019, pp. M1I.6.

Hasegawa, A.

A. Hasegawa and T. Nyu, “Eigenvalue communication,” J. Lightw. Technol., vol. 11, no. 3, pp. 395–399, 1993.

Its, A.

E. Belokolos, A. Bobenko, V. Enol'skii, A. Its, and V. Matveev, Algebro-Geometric Approach to Nonlinear Integrable Equations. New York, NY, USA: Springer, 1994.

Jaouën, Y.

J.-W. Goossens, H. Hafermann, and Y. Jaouën, “Data transmission based on exact inverse periodic nonlinear Fourier transform, Part I: Theory,” 2019, arXiv:1911.12614.

J.-W. Goossens, H. Hafermann, and Y. Jaouën, “Data transmission based on exact inverse periodic nonlinear Fourier transform, Part II: Waveform design and experiment,” 2019, arXiv:1911.12615.

J.-W. Goossens, Y. Jaouën, and H. Hafermann, “Experimental demonstration of data transmission based on the exact inverse periodic nonlinear Fourier transform,” in Proc. Opt. Fiber Commun. Conf., 2019, pp. M1I.6.

Kamalian, M.

M. Kamalian, A. Vasylchenkova, D. Shepelsky, J. Prilepsky, and S. Turitsyn, “Periodic nonlinear Fourier transform communication solving the Riemann-Hilbert problem,” J. Lightw. Technol., vol. 36, no. 24, pp. 5714–5727, 2018.

M. Kamalian, J. Prilepsky, S. Le, and S. Turitsyn, “On the design of NFT-based communication systems with lumped amplification,” J. Lightw. Technol., vol. 35, no. 24, pp. 5464–5472, 2017.

M. Kamalian, J. Prilepsky, S. Le, and S. Turitsyn, “Periodic nonlinear Fourier transform for fiber-optic communications, part II: Eigenvalue communication,” Opt. Express, vol. 24, no. 16, pp. 18 370–18 381, 2016.

M. Kamalian, J. Prilepsky, S. Le, and S. Turitsyn, “Periodic nonlinear Fourier transform for fiber-optic communications, Part I: Theory and numerical methods,” Opt. Express, vol. 24, no. 16, pp. 18 353–18 369, 2016.

M. Kamalian, A. Vasylchenkova, J. Prilepsky, D. Shepelsky, and S. Turitsyn, “Communication system based on periodic nonlinear fourier transform with exact inverse transformation,” in Proc. ECOC 44nd Eur. Conf. Opt. Commun., 2018, Paper Tu3A.2.

M. Kamalian, S. Le, J. Prilepsky, and S. Turitsyn, “Statistical analysis of a communication system based on the periodic nonlinear Fourier transform,” in Proc. Australian Conf. Opt. Fibre Technol., 2016, pp. ATh1C–4.

M. Kamalian, J. Prilepsky, A. Vasylchenkova, D. Shepelsky, and S. Turitsyn, “Methods of nonlinear Fourier-based optical transmission with periodically-extended signals,” in Proc. IEEE Int. Conf. Sci. Elect. Eng. Isr., 2018, pp. 1–5.

M. Kamalian, D. Shepelsky, A. Vasylchenkova, J. Prilepsky, and S. Turitsyn, “Communication system using periodic nonlinear Fourier transform based on Riemann-Hilbert problem,” in Proc. ECOC 44nd Eur. Conf. Opt. Commun., 2018, Paper Tu3A.3.

O. Kotlyar, M. Pankratova, M. Kamalian, A. Vasylchenkova, J. E. Prilepsky, and S. K. Turitsyn, “Unsupervised and supervised machine learning for performance improvement of NFT optical transmission,” in Proc. IEEE Brit. Irish Conf. Opt. Photon., 2018, pp. 1–4.

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V. Kotlyarov and D. Shepelsky, “Planar unmodular Baker-Akhiezer function for the nonlinear Schrödinger equation,” Ann. Math. Sci. Appl., vol. 2, no. 2, pp. 343–384, 2017.

Calcolo (1)

A. Arico, G. Rodriguez, and S. Seatzu, “Numerical solution of the nonlinear Schrödinger equation, starting from the scattering data,” Calcolo, vol. 48, no. 1, pp. 75–88, 2011.

IEEE Photon. Technol. Lett. (1)

S. Civelli, E. Forestieri, and M. Secondini, “Why noise and dispersion may seriously hamper nonlinear frequency-division multiplexing,” IEEE Photon. Technol. Lett., vol. 29, no. 16, pp. 1332–1335,  2017.

IEEE Trans. Inf. Theory (3)

S. Wahls and V. Poor, “Fast numerical nonlinear Fourier transforms,” IEEE Trans. Inf. Theory, vol. 61, no. 12, pp. 6957–6974,  2015.

M. Yousefi and F. Kschischang, “Information transmission using the nonlinear Fourier transform, Part I: Mathematical tools,” IEEE Trans. Inf. Theory, vol. 60, no. 7, pp. 4312–4328,  2014.

M. Yousefi and F. Kschischang, “Information transmission using the nonlinear Fourier transform, Part III: Spectrum modulation,” IEEE Trans. Inf. Theory, vol. 60, no. 7, pp. 4346–4369,  2014.

Int. Math. Res. Notices (1)

K. T.-R. McLaughlin and P. V. Nabelek, “A Riemann–Hilbert problem approach to periodic infinite gap Hill's operators and the Korteweg–de Vries equation,” Int. Math. Res. Notices, to be published, doi: .
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M. Kamalian, J. Prilepsky, A. Vasylchenkova, D. Shepelsky, and S. Turitsyn, “Methods of nonlinear Fourier-based optical transmission with periodically-extended signals,” in Proc. IEEE Int. Conf. Sci. Elect. Eng. Isr., 2018, pp. 1–5.

M. K. Kopae, J. E. Prilepsky, S. T. Le, and S. K. Turitsyn, “Optical communication based on the periodic nonlinear Fourier transform signal processing,” in Proc. IEEE 6th Int. Conf. Photon., 2016, pp. 1–3.

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O. Kotlyar, M. Pankratova, M. Kamalian, A. Vasylchenkova, J. E. Prilepsky, and S. K. Turitsyn, “Unsupervised and supervised machine learning for performance improvement of NFT optical transmission,” in Proc. IEEE Brit. Irish Conf. Opt. Photon., 2018, pp. 1–4.

M. Kamalian-Kopae, A. Vasylchenkova, O. Kotlyar, M. Pankratova, J. Prilepsky, and S. Turitsyn, “Artificial neural network-based equaliser in the nonlinear Fourier domain for fibre-optic communication applications,” in Proc. Conf. Lasers Electro-Opt. Europe Eur. Quantum Electron. Conf., 2019, Paper ci_1_4.

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