Abstract

In linear communication channels, spectral components (modes) defined by the Fourier transform of the signal propagate without interactions with each other. In certain nonlinear channels, such as the one modelled by the classical nonlinear Schrödinger equation, there are nonlinear modes (nonlinear signal spectrum) that also propagate without interacting with each other and without corresponding nonlinear cross talk, effectively, in a linear manner. Here, we describe in a constructive way how to introduce such nonlinear modes for a given input signal. We investigate the performance of the nonlinear inverse synthesis (NIS) method, in which the information is encoded directly onto the continuous part of the nonlinear signal spectrum. This transmission technique, combined with the appropriate distributed Raman amplification, can provide an effective eigenvalue division multiplexing with high spectral efficiency, thanks to highly suppressed channel cross talk. The proposed NIS approach can be integrated with any modulation formats. Here, we demonstrate numerically the feasibility of merging the NIS technique in a burst mode with high spectral efficiency methods, such as orthogonal frequency division multiplexing and Nyquist pulse shaping with advanced modulation formats (e.g., QPSK, 16QAM, and 64QAM), showing a performance improvement up to 4.5 dB, which is comparable to results achievable with multi-step per span digital back propagation.

© 2014 Optical Society of America

Full Article  |  PDF Article
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References

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  1. A. D. Ellis, Z. Jian, and D. Cotter, “Approaching the Non-Linear Shannon Limit,” J. Lightwave Technol. 28(4), 423–433 (2010).
    [Crossref]
  2. R. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity Limits of Optical Fiber Networks,” J. Lightwave Technol. 28(4), 662–701 (2010).
    [Crossref]
  3. E. Ip and J. M. Kahn, “Compensation of Dispersion and Nonlinear Impairments Using Digital Backpropagation,” J. Lightwave Technol. 26(20), 3416–3425 (2008).
    [Crossref]
  4. C. Xi, L. Xiang, S. Chandrasekhar, B. Zhu, and R. W. Tkach, “Experimental demonstration of fiber nonlinearity mitigation using digital phase conjugation,” in Optical Fiber Communication Conference and Exposition (OFC/NFOEC),2012, paper OTh3C.
  5. S. L. Jansen, D. Van den Borne, B. Spinnler, S. Calabro, H. Suche, P. M. Krummrich, W. Sohler, G. Khoe, and H. de Waardt, “Optical phase conjugation for ultra long-haul phase-shift-keyed transmission,” J. Lightwave Technol. 24(1), 54–64 (2006).
    [Crossref]
  6. D. M. Pepper and A. Yariv, “Compensation for phase distortions in nonlinear media by phase conjugation,” Opt. Lett. 5(2), 59–60 (1980).
    [Crossref] [PubMed]
  7. I. Phillips, M. Tan, M. F. Stephens, M. 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, San Francisco, California, 2014, paper M3C.1.
  8. S. Watanabe, S. Kaneko, and T. Chikama, “Long-Haul Fiber Transmission Using Optical Phase Conjugation,” Opt. Fiber Technol. 2(2), 169–178 (1996).
    [Crossref]
  9. 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(7), 560–568 (2013).
    [Crossref]
  10. G. P. Agrawal, Fiber-Optic Communication Systems, 4th ed. (Wyley, 2010).
  11. A. Hasegawa and Y. Kodama, Solitons in Optical Communications (Oxford University Press, 1996).
  12. V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).
  13. L. F. Mollenauer and J. P. Gordon, Solitons in Optical Fibers: Fundamentals and Applications (Academic Press, 2006).
  14. Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, 2003).
  15. A. Peleg, M. Chertkov, and I. Gabitov, “Inelastic interchannel collisions of pulses in optical fibers in the presence of third-order dispersion,” J. Opt. Soc. Am. B 21(1), 18–23 (2004).
    [Crossref]
  16. J. E. Prilepsky, S. A. Derevyanko, and S. K. Turitsyn, “Temporal Solitonic Crystals and Non-Hermitian Informational Lattices,” Phys. Rev. Lett. 108(18), 183902 (2012).
    [Crossref] [PubMed]
  17. O. V. Yushko and A. A. Redyuk, “Soliton communication lines based on spectrally efficient modulation formats,” Quantum Electron. 44(6), 606–611 (2014).
    [Crossref]
  18. M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, “The inverse scattering transform-Fourier analysis for nonlinear problems,” Stud. Appl. Math. 53, 249–315 (1974).
  19. V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevskii, Theory of Solitons. The Inverse Scattering Method. (Colsultants Bureau, New York, 1984).
  20. M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform. (SIAM, Philadelphia, 1981).
  21. A. R. Osborne, “The inverse scattering transform: tools for the nonlinear Fourier analysis and filtering of ocean surface waves,” Chaos Solitons Fractals 5(12), 2623–2637 (1995).
    [Crossref]
  22. A. S. Fokas and I. M. Gelfand, “Integrability of linear and nonlinear evolution equations and the associated nonlinear Fourier transforms,” Lett. Math. Phys. 32(3), 189–210 (1994).
    [Crossref]
  23. A. Hasegawa and T. Nyu, “Eigenvalue communication,” J. Lightwave Technol. 11(3), 395–399 (1993).
    [Crossref]
  24. E. G. Turitsyna and S. K. Turitsyn, “Digital signal processing based on inverse scattering transform,” Opt. Lett. 38(20), 4186–4188 (2013).
    [Crossref] [PubMed]
  25. H. Terauchi and A. Maruta, “Eigenvalue Modulated Optical Transmission System Based on Digital Coherent Technology,” in 18th OptoElectronics and Communications Conference held jointly with 2013 International Conference on Photonics in Switching (OECC/PS), IEICE,2013, paper WR2–5.
  26. M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, Part I: Mathematical tools,” IEEE Trans. Inf. Theory 60(7), 4312–4328 (2014).
    [Crossref]
  27. M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, Part II: Numerical methods,” IEEE Trans. Inf. Theory 60(7), 4329–4345 (2014).
    [Crossref]
  28. M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, Part III: Spectrum modulation,” IEEE Trans. Inf. Theory 60(7), 4346–4369 (2014).
    [Crossref]
  29. J. E. Prilepsky, S. A. Derevyanko, and S. K. Turitsyn, “Nonlinear spectral management: Linearization of the lossless fiber channel,” Opt. Express 21(20), 24344–24367 (2013).
    [Crossref] [PubMed]
  30. 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(1), 013901 (2014).
    [Crossref] [PubMed]
  31. S. Hari, F. Kschischang, and M. Yousefi, “Multi-eigenvalue communication via the nonlinear Fourier transform,” in Communications (QBSC),2014, 27th Biennial Symposium on, pp. 92–95.
    [Crossref]
  32. S. Chandrasekhar and L. Xiang, “OFDM Based Superchannel Transmission Technology,” J. Lightwave Technol. 30(24), 3816–3823 (2012).
    [Crossref]
  33. E. Giacoumidis, M. A. Jarajreh, S. Sygletos, S. T. Le, F. Farjady, A. Tsokanos, A. Hamié, E. Pincemin, Y. Jaouën, A. D. Ellis, and N. J. Doran, “Dual-polarization multi-band optical OFDM transmission and transceiver limitations for up to 500 Gb/s uncompensated long-haul links,” Opt. Express 22(9), 10975–10986 (2014).
    [Crossref] [PubMed]
  34. J. Armstrong, “OFDM for Optical Communications,” J. Lightwave Technol. 27(3), 189–204 (2009).
    [Crossref]
  35. S. B. Weinstein, “The history of orthogonal frequency-division multiplexing [history of communications],” IEEE Commun. Mag. 47(11), 26–35 (2009).
    [Crossref]
  36. W. Shieh and C. Athaudage, “Coherent optical orthogonal frequency division multiplexing,” Electron. Lett. 42(10), 587–588 (2006).
    [Crossref]
  37. S. T. Le, K. Blow, and S. Turitsyn, “Power pre-emphasis for suppression of FWM in coherent optical OFDM transmission,” Opt. Express 22(6), 7238–7248 (2014).
    [Crossref] [PubMed]
  38. D. Hillerkuss, R. Schmogrow, M. Meyer, S. Wolf, M. Jordan, P. Kleinow, N. Lindenmann, P. Schindler, A. Melikyan, X. Yang, S. Ben-Ezra, B. Nebendahl, M. Dreschmann, J. Meyer, F. Parmigiani, P. Petropoulos, B. Resan, A. Oehler, K. Weingarten, L. Altenhain, T. Ellermeyer, M. Moeller, M. Huebner, J. Becker, C. Koos, W. Freude, and J. Leuthold, “Single-Laser 32.5Tbit/s Nyquist WDM Transmission,” J. Opt. Commun. Netw. 4(10), 715–723 (2012).
    [Crossref]
  39. A. C. Newell, Solitons in mathematics and physics (SIAM Philadelphia, 1985).
  40. R. Bulirsch, Introduction to Numerical Analysis, 2nd ed. (Springer-Verlag, 1993).
  41. S. Burtsev, R. Camassa, and I. Timofeyev, “Numerical Algorithms for the Direct Spectral Transform with Applications to Nonlinear Schrödinger Type Systems,” J. Comput. Phys. 147(1), 166–186 (1998).
    [Crossref]
  42. G. Boffetta and A. R. Osborne, “Computation of the direct scattering transform for the nonlinear Schroedinger equation,” J. Comput. Phys. 102(2), 252–264 (1992).
    [Crossref]
  43. O. V. Belai, L. L. Frumin, E. V. Podivilov, and D. A. Shapiro, “Efficient numerical method of the fiber Bragg grating synthesis,” J. Opt. Soc. Am. B 24(7), 1451–1457 (2007).
    [Crossref]
  44. T. Kuusela, J. Hietarinta, K. Kokko, and R. Laiho, “Soliton experiments in a nonlinear electrical transmission line,” Eur. J. Phys. 8(1), 27–33 (1987).
    [Crossref]
  45. S. Zohar, “Toeplitz Matrix Inversion: The Algorithm of W. F. Trench,” Journal of the Association for Computing Machinery 16(4), 592–601 (1969).
    [Crossref]
  46. M. V. Barel, G. Heinig, and P. Kravanja, “A Stabilized Superfast Solver for Nonsymmetric Toeplitz Systems,” SIAM J. Matrix Anal. Appl. 23(2), 494–510 (2001).
    [Crossref]
  47. M. Stewart, “Fast algorithms for structured matrix computations” in Handbook of Linear Algebra (2nd edition) (Chapman & Hall, 2013), Chap. 62.
  48. A. Buryak, J. Bland-Hawthorn, and V. Steblina, “Comparison of Inverse Scattering Algorithms for Designing Ultrabroadband Fibre Bragg Gratings,” Opt. Express 17(3), 1995–2004 (2009).
    [Crossref] [PubMed]
  49. G. Xiao and K. Yashiro, “An Efficient Algorithm for Solving Zakharov–Shabat Inverse Scattering Problem,” IEEE Trans. Antenn. Propag. 50(6), 807–811 (2002).
    [Crossref]
  50. S. Wahls and H. V. Poor, “Introducing the fast nonlinear Fourier transform,” in Proceedings of International Conference on Acoustics, Speech, and Signal Processing (ICASSP 2013), IEEE, 2013, pp. 5780–5784.
    [Crossref]
  51. S. Wahls and H. V. Poor, “Fast Numerical Nonlinear Fourier Transforms,” submitted to IEEE Trans. Inf. Theory (2013), http://arxiv.org/abs/1402.1605 .
  52. S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electromagnetic waves,” Sov. Phys. JETP 38, 248–253 (1974).
  53. R. A. Shafik, S. Rahman, and R. Islam, “On the Extended Relationships Among EVM, BER and SNR as Performance Metrics,” in Electrical and Computer Engineering,2006. ICECE '06. International Conference.on, 2006, pp. 408–411.
    [Crossref]
  54. S. T. Le, K. J. Blow, V. K. Menzentsev, and S. K. Turitsyn, “Comparison of numerical bit error rate estimation methods in 112Gbs QPSK CO-OFDM transmission,” in Optical Communication (ECOC 2013), 39th European Conference and Exhibition on, 2013, paper P4.14.
    [Crossref]
  55. S. Kilmurray, T. Fehenberger, P. Bayvel, and R. I. Killey, “Comparison of the nonlinear transmission performance of quasi-Nyquist WDM and reduced guard interval OFDM,” Opt. Express 20(4), 4198–4205 (2012).
    [Crossref] [PubMed]
  56. Y. Lu, Y. Fang, B. Wu, K. Wang, W. Wan, F. Yu, L. Li, X. Shi, and Q. Xiong, “Experimental comparison of 32-Gbaud Electrical-OFDM and Nyquist-WDM transmission with 64GSa/s DAC,” in Optical Communication (ECOC 2013), 39th European Conference and Exhibition on (2013).
    [Crossref]

2014 (7)

O. V. Yushko and A. A. Redyuk, “Soliton communication lines based on spectrally efficient modulation formats,” Quantum Electron. 44(6), 606–611 (2014).
[Crossref]

M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, Part I: Mathematical tools,” IEEE Trans. Inf. Theory 60(7), 4312–4328 (2014).
[Crossref]

M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, Part II: Numerical methods,” IEEE Trans. Inf. Theory 60(7), 4329–4345 (2014).
[Crossref]

M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, Part III: Spectrum modulation,” IEEE Trans. Inf. Theory 60(7), 4346–4369 (2014).
[Crossref]

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(1), 013901 (2014).
[Crossref] [PubMed]

E. Giacoumidis, M. A. Jarajreh, S. Sygletos, S. T. Le, F. Farjady, A. Tsokanos, A. Hamié, E. Pincemin, Y. Jaouën, A. D. Ellis, and N. J. Doran, “Dual-polarization multi-band optical OFDM transmission and transceiver limitations for up to 500 Gb/s uncompensated long-haul links,” Opt. Express 22(9), 10975–10986 (2014).
[Crossref] [PubMed]

S. T. Le, K. Blow, and S. Turitsyn, “Power pre-emphasis for suppression of FWM in coherent optical OFDM transmission,” Opt. Express 22(6), 7238–7248 (2014).
[Crossref] [PubMed]

2013 (3)

2012 (4)

2010 (2)

2009 (3)

2008 (1)

2007 (1)

2006 (2)

2004 (1)

2002 (1)

G. Xiao and K. Yashiro, “An Efficient Algorithm for Solving Zakharov–Shabat Inverse Scattering Problem,” IEEE Trans. Antenn. Propag. 50(6), 807–811 (2002).
[Crossref]

2001 (1)

M. V. Barel, G. Heinig, and P. Kravanja, “A Stabilized Superfast Solver for Nonsymmetric Toeplitz Systems,” SIAM J. Matrix Anal. Appl. 23(2), 494–510 (2001).
[Crossref]

1998 (1)

S. Burtsev, R. Camassa, and I. Timofeyev, “Numerical Algorithms for the Direct Spectral Transform with Applications to Nonlinear Schrödinger Type Systems,” J. Comput. Phys. 147(1), 166–186 (1998).
[Crossref]

1996 (1)

S. Watanabe, S. Kaneko, and T. Chikama, “Long-Haul Fiber Transmission Using Optical Phase Conjugation,” Opt. Fiber Technol. 2(2), 169–178 (1996).
[Crossref]

1995 (1)

A. R. Osborne, “The inverse scattering transform: tools for the nonlinear Fourier analysis and filtering of ocean surface waves,” Chaos Solitons Fractals 5(12), 2623–2637 (1995).
[Crossref]

1994 (1)

A. S. Fokas and I. M. Gelfand, “Integrability of linear and nonlinear evolution equations and the associated nonlinear Fourier transforms,” Lett. Math. Phys. 32(3), 189–210 (1994).
[Crossref]

1993 (1)

A. Hasegawa and T. Nyu, “Eigenvalue communication,” J. Lightwave Technol. 11(3), 395–399 (1993).
[Crossref]

1992 (1)

G. Boffetta and A. R. Osborne, “Computation of the direct scattering transform for the nonlinear Schroedinger equation,” J. Comput. Phys. 102(2), 252–264 (1992).
[Crossref]

1987 (1)

T. Kuusela, J. Hietarinta, K. Kokko, and R. Laiho, “Soliton experiments in a nonlinear electrical transmission line,” Eur. J. Phys. 8(1), 27–33 (1987).
[Crossref]

1980 (1)

1974 (2)

M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, “The inverse scattering transform-Fourier analysis for nonlinear problems,” Stud. Appl. Math. 53, 249–315 (1974).

S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electromagnetic waves,” Sov. Phys. JETP 38, 248–253 (1974).

1972 (1)

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

1969 (1)

S. Zohar, “Toeplitz Matrix Inversion: The Algorithm of W. F. Trench,” Journal of the Association for Computing Machinery 16(4), 592–601 (1969).
[Crossref]

Ablowitz, M. J.

M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, “The inverse scattering transform-Fourier analysis for nonlinear problems,” Stud. Appl. Math. 53, 249–315 (1974).

Altenhain, L.

Armstrong, J.

Athaudage, C.

W. Shieh and C. Athaudage, “Coherent optical orthogonal frequency division multiplexing,” Electron. Lett. 42(10), 587–588 (2006).
[Crossref]

Barel, M. V.

M. V. Barel, G. Heinig, and P. Kravanja, “A Stabilized Superfast Solver for Nonsymmetric Toeplitz Systems,” SIAM J. Matrix Anal. Appl. 23(2), 494–510 (2001).
[Crossref]

Bayvel, P.

Becker, J.

Belai, O. V.

Ben-Ezra, S.

Bland-Hawthorn, J.

Blow, K.

Blow, K. J.

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(1), 013901 (2014).
[Crossref] [PubMed]

Boffetta, G.

G. Boffetta and A. R. Osborne, “Computation of the direct scattering transform for the nonlinear Schroedinger equation,” J. Comput. Phys. 102(2), 252–264 (1992).
[Crossref]

Burtsev, S.

S. Burtsev, R. Camassa, and I. Timofeyev, “Numerical Algorithms for the Direct Spectral Transform with Applications to Nonlinear Schrödinger Type Systems,” J. Comput. Phys. 147(1), 166–186 (1998).
[Crossref]

Buryak, A.

Calabro, S.

Camassa, R.

S. Burtsev, R. Camassa, and I. Timofeyev, “Numerical Algorithms for the Direct Spectral Transform with Applications to Nonlinear Schrödinger Type Systems,” J. Comput. Phys. 147(1), 166–186 (1998).
[Crossref]

Chandrasekhar, S.

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(7), 560–568 (2013).
[Crossref]

S. Chandrasekhar and L. Xiang, “OFDM Based Superchannel Transmission Technology,” J. Lightwave Technol. 30(24), 3816–3823 (2012).
[Crossref]

Chertkov, M.

Chikama, T.

S. Watanabe, S. Kaneko, and T. Chikama, “Long-Haul Fiber Transmission Using Optical Phase Conjugation,” Opt. Fiber Technol. 2(2), 169–178 (1996).
[Crossref]

Cotter, D.

de Waardt, H.

Derevyanko, S. A.

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(1), 013901 (2014).
[Crossref] [PubMed]

J. E. Prilepsky, S. A. Derevyanko, and S. K. Turitsyn, “Nonlinear spectral management: Linearization of the lossless fiber channel,” Opt. Express 21(20), 24344–24367 (2013).
[Crossref] [PubMed]

J. E. Prilepsky, S. A. Derevyanko, and S. K. Turitsyn, “Temporal Solitonic Crystals and Non-Hermitian Informational Lattices,” Phys. Rev. Lett. 108(18), 183902 (2012).
[Crossref] [PubMed]

Doran, N. J.

Dreschmann, M.

Ellermeyer, T.

Ellis, A. D.

Essiambre, R.

Farjady, F.

Fehenberger, T.

Fokas, A. S.

A. S. Fokas and I. M. Gelfand, “Integrability of linear and nonlinear evolution equations and the associated nonlinear Fourier transforms,” Lett. Math. Phys. 32(3), 189–210 (1994).
[Crossref]

Foschini, G. J.

Freude, W.

Frumin, L. L.

Gabitov, I.

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(1), 013901 (2014).
[Crossref] [PubMed]

A. Peleg, M. Chertkov, and I. Gabitov, “Inelastic interchannel collisions of pulses in optical fibers in the presence of third-order dispersion,” J. Opt. Soc. Am. B 21(1), 18–23 (2004).
[Crossref]

Gelfand, I. M.

A. S. Fokas and I. M. Gelfand, “Integrability of linear and nonlinear evolution equations and the associated nonlinear Fourier transforms,” Lett. Math. Phys. 32(3), 189–210 (1994).
[Crossref]

Giacoumidis, E.

Goebel, B.

Hamié, A.

Hasegawa, A.

A. Hasegawa and T. Nyu, “Eigenvalue communication,” J. Lightwave Technol. 11(3), 395–399 (1993).
[Crossref]

Heinig, G.

M. V. Barel, G. Heinig, and P. Kravanja, “A Stabilized Superfast Solver for Nonsymmetric Toeplitz Systems,” SIAM J. Matrix Anal. Appl. 23(2), 494–510 (2001).
[Crossref]

Hietarinta, J.

T. Kuusela, J. Hietarinta, K. Kokko, and R. Laiho, “Soliton experiments in a nonlinear electrical transmission line,” Eur. J. Phys. 8(1), 27–33 (1987).
[Crossref]

Hillerkuss, D.

Huebner, M.

Ip, E.

Jansen, S. L.

Jaouën, Y.

Jarajreh, M. A.

Jian, Z.

Jordan, M.

Kahn, J. M.

Kaneko, S.

S. Watanabe, S. Kaneko, and T. Chikama, “Long-Haul Fiber Transmission Using Optical Phase Conjugation,” Opt. Fiber Technol. 2(2), 169–178 (1996).
[Crossref]

Kaup, D. J.

M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, “The inverse scattering transform-Fourier analysis for nonlinear problems,” Stud. Appl. Math. 53, 249–315 (1974).

Khoe, G.

Killey, R. I.

Kilmurray, S.

Kleinow, P.

Kokko, K.

T. Kuusela, J. Hietarinta, K. Kokko, and R. Laiho, “Soliton experiments in a nonlinear electrical transmission line,” Eur. J. Phys. 8(1), 27–33 (1987).
[Crossref]

Koos, C.

Kramer, G.

Kravanja, P.

M. V. Barel, G. Heinig, and P. Kravanja, “A Stabilized Superfast Solver for Nonsymmetric Toeplitz Systems,” SIAM J. Matrix Anal. Appl. 23(2), 494–510 (2001).
[Crossref]

Krummrich, P. M.

Kschischang, F. R.

M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, Part III: Spectrum modulation,” IEEE Trans. Inf. Theory 60(7), 4346–4369 (2014).
[Crossref]

M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, Part I: Mathematical tools,” IEEE Trans. Inf. Theory 60(7), 4312–4328 (2014).
[Crossref]

M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, Part II: Numerical methods,” IEEE Trans. Inf. Theory 60(7), 4329–4345 (2014).
[Crossref]

Kuusela, T.

T. Kuusela, J. Hietarinta, K. Kokko, and R. Laiho, “Soliton experiments in a nonlinear electrical transmission line,” Eur. J. Phys. 8(1), 27–33 (1987).
[Crossref]

Laiho, R.

T. Kuusela, J. Hietarinta, K. Kokko, and R. Laiho, “Soliton experiments in a nonlinear electrical transmission line,” Eur. J. Phys. 8(1), 27–33 (1987).
[Crossref]

Le, S. T.

Leuthold, J.

Lindenmann, N.

Liu, A. R. C. X.

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(7), 560–568 (2013).
[Crossref]

Manakov, S. V.

S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electromagnetic waves,” Sov. Phys. JETP 38, 248–253 (1974).

Melikyan, A.

Meyer, J.

Meyer, M.

Moeller, M.

Nebendahl, B.

Newell, A. C.

M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, “The inverse scattering transform-Fourier analysis for nonlinear problems,” Stud. Appl. Math. 53, 249–315 (1974).

Nyu, T.

A. Hasegawa and T. Nyu, “Eigenvalue communication,” J. Lightwave Technol. 11(3), 395–399 (1993).
[Crossref]

Oehler, A.

Osborne, A. R.

A. R. Osborne, “The inverse scattering transform: tools for the nonlinear Fourier analysis and filtering of ocean surface waves,” Chaos Solitons Fractals 5(12), 2623–2637 (1995).
[Crossref]

G. Boffetta and A. R. Osborne, “Computation of the direct scattering transform for the nonlinear Schroedinger equation,” J. Comput. Phys. 102(2), 252–264 (1992).
[Crossref]

Parmigiani, F.

Peleg, A.

Pepper, D. M.

Petropoulos, P.

Pincemin, E.

Podivilov, E. V.

Poor, H. V.

S. Wahls and H. V. Poor, “Introducing the fast nonlinear Fourier transform,” in Proceedings of International Conference on Acoustics, Speech, and Signal Processing (ICASSP 2013), IEEE, 2013, pp. 5780–5784.
[Crossref]

Prilepsky, J. E.

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(1), 013901 (2014).
[Crossref] [PubMed]

J. E. Prilepsky, S. A. Derevyanko, and S. K. Turitsyn, “Nonlinear spectral management: Linearization of the lossless fiber channel,” Opt. Express 21(20), 24344–24367 (2013).
[Crossref] [PubMed]

J. E. Prilepsky, S. A. Derevyanko, and S. K. Turitsyn, “Temporal Solitonic Crystals and Non-Hermitian Informational Lattices,” Phys. Rev. Lett. 108(18), 183902 (2012).
[Crossref] [PubMed]

Redyuk, A. A.

O. V. Yushko and A. A. Redyuk, “Soliton communication lines based on spectrally efficient modulation formats,” Quantum Electron. 44(6), 606–611 (2014).
[Crossref]

Resan, B.

Schindler, P.

Schmogrow, R.

Segur, H.

M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, “The inverse scattering transform-Fourier analysis for nonlinear problems,” Stud. Appl. Math. 53, 249–315 (1974).

Shabat, A. B.

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

Shapiro, D. A.

Shieh, W.

W. Shieh and C. Athaudage, “Coherent optical orthogonal frequency division multiplexing,” Electron. Lett. 42(10), 587–588 (2006).
[Crossref]

Sohler, W.

Spinnler, B.

Steblina, V.

Suche, H.

Sygletos, S.

Timofeyev, I.

S. Burtsev, R. Camassa, and I. Timofeyev, “Numerical Algorithms for the Direct Spectral Transform with Applications to Nonlinear Schrödinger Type Systems,” J. Comput. Phys. 147(1), 166–186 (1998).
[Crossref]

Tkach, R. W.

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(7), 560–568 (2013).
[Crossref]

Tsokanos, A.

Turitsyn, S.

Turitsyn, S. K.

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(1), 013901 (2014).
[Crossref] [PubMed]

E. G. Turitsyna and S. K. Turitsyn, “Digital signal processing based on inverse scattering transform,” Opt. Lett. 38(20), 4186–4188 (2013).
[Crossref] [PubMed]

J. E. Prilepsky, S. A. Derevyanko, and S. K. Turitsyn, “Nonlinear spectral management: Linearization of the lossless fiber channel,” Opt. Express 21(20), 24344–24367 (2013).
[Crossref] [PubMed]

J. E. Prilepsky, S. A. Derevyanko, and S. K. Turitsyn, “Temporal Solitonic Crystals and Non-Hermitian Informational Lattices,” Phys. Rev. Lett. 108(18), 183902 (2012).
[Crossref] [PubMed]

Turitsyna, E. G.

Van den Borne, D.

Wahls, S.

S. Wahls and H. V. Poor, “Introducing the fast nonlinear Fourier transform,” in Proceedings of International Conference on Acoustics, Speech, and Signal Processing (ICASSP 2013), IEEE, 2013, pp. 5780–5784.
[Crossref]

Watanabe, S.

S. Watanabe, S. Kaneko, and T. Chikama, “Long-Haul Fiber Transmission Using Optical Phase Conjugation,” Opt. Fiber Technol. 2(2), 169–178 (1996).
[Crossref]

Weingarten, K.

Weinstein, S. B.

S. B. Weinstein, “The history of orthogonal frequency-division multiplexing [history of communications],” IEEE Commun. Mag. 47(11), 26–35 (2009).
[Crossref]

Winzer, P. J.

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(7), 560–568 (2013).
[Crossref]

R. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity Limits of Optical Fiber Networks,” J. Lightwave Technol. 28(4), 662–701 (2010).
[Crossref]

Wolf, S.

Xiang, L.

Xiao, G.

G. Xiao and K. Yashiro, “An Efficient Algorithm for Solving Zakharov–Shabat Inverse Scattering Problem,” IEEE Trans. Antenn. Propag. 50(6), 807–811 (2002).
[Crossref]

Yang, X.

Yariv, A.

Yashiro, K.

G. Xiao and K. Yashiro, “An Efficient Algorithm for Solving Zakharov–Shabat Inverse Scattering Problem,” IEEE Trans. Antenn. Propag. 50(6), 807–811 (2002).
[Crossref]

Yousefi, M. I.

M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, Part I: Mathematical tools,” IEEE Trans. Inf. Theory 60(7), 4312–4328 (2014).
[Crossref]

M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, Part III: Spectrum modulation,” IEEE Trans. Inf. Theory 60(7), 4346–4369 (2014).
[Crossref]

M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, Part II: Numerical methods,” IEEE Trans. Inf. Theory 60(7), 4329–4345 (2014).
[Crossref]

Yushko, O. V.

O. V. Yushko and A. A. Redyuk, “Soliton communication lines based on spectrally efficient modulation formats,” Quantum Electron. 44(6), 606–611 (2014).
[Crossref]

Zakharov, V. E.

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

Zohar, S.

S. Zohar, “Toeplitz Matrix Inversion: The Algorithm of W. F. Trench,” Journal of the Association for Computing Machinery 16(4), 592–601 (1969).
[Crossref]

Chaos Solitons Fractals (1)

A. R. Osborne, “The inverse scattering transform: tools for the nonlinear Fourier analysis and filtering of ocean surface waves,” Chaos Solitons Fractals 5(12), 2623–2637 (1995).
[Crossref]

Electron. Lett. (1)

W. Shieh and C. Athaudage, “Coherent optical orthogonal frequency division multiplexing,” Electron. Lett. 42(10), 587–588 (2006).
[Crossref]

Eur. J. Phys. (1)

T. Kuusela, J. Hietarinta, K. Kokko, and R. Laiho, “Soliton experiments in a nonlinear electrical transmission line,” Eur. J. Phys. 8(1), 27–33 (1987).
[Crossref]

IEEE Commun. Mag. (1)

S. B. Weinstein, “The history of orthogonal frequency-division multiplexing [history of communications],” IEEE Commun. Mag. 47(11), 26–35 (2009).
[Crossref]

IEEE Trans. Antenn. Propag. (1)

G. Xiao and K. Yashiro, “An Efficient Algorithm for Solving Zakharov–Shabat Inverse Scattering Problem,” IEEE Trans. Antenn. Propag. 50(6), 807–811 (2002).
[Crossref]

IEEE Trans. Inf. Theory (3)

M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, Part I: Mathematical tools,” IEEE Trans. Inf. Theory 60(7), 4312–4328 (2014).
[Crossref]

M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, Part II: Numerical methods,” IEEE Trans. Inf. Theory 60(7), 4329–4345 (2014).
[Crossref]

M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, Part III: Spectrum modulation,” IEEE Trans. Inf. Theory 60(7), 4346–4369 (2014).
[Crossref]

J. Comput. Phys. (2)

S. Burtsev, R. Camassa, and I. Timofeyev, “Numerical Algorithms for the Direct Spectral Transform with Applications to Nonlinear Schrödinger Type Systems,” J. Comput. Phys. 147(1), 166–186 (1998).
[Crossref]

G. Boffetta and A. R. Osborne, “Computation of the direct scattering transform for the nonlinear Schroedinger equation,” J. Comput. Phys. 102(2), 252–264 (1992).
[Crossref]

J. Lightwave Technol. (7)

J. Opt. Commun. Netw. (1)

J. Opt. Soc. Am. B (2)

Journal of the Association for Computing Machinery (1)

S. Zohar, “Toeplitz Matrix Inversion: The Algorithm of W. F. Trench,” Journal of the Association for Computing Machinery 16(4), 592–601 (1969).
[Crossref]

Lett. Math. Phys. (1)

A. S. Fokas and I. M. Gelfand, “Integrability of linear and nonlinear evolution equations and the associated nonlinear Fourier transforms,” Lett. Math. Phys. 32(3), 189–210 (1994).
[Crossref]

Nat. Photonics (1)

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(7), 560–568 (2013).
[Crossref]

Opt. Express (5)

Opt. Fiber Technol. (1)

S. Watanabe, S. Kaneko, and T. Chikama, “Long-Haul Fiber Transmission Using Optical Phase Conjugation,” Opt. Fiber Technol. 2(2), 169–178 (1996).
[Crossref]

Opt. Lett. (2)

Phys. Rev. Lett. (2)

J. E. Prilepsky, S. A. Derevyanko, and S. K. Turitsyn, “Temporal Solitonic Crystals and Non-Hermitian Informational Lattices,” Phys. Rev. Lett. 108(18), 183902 (2012).
[Crossref] [PubMed]

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(1), 013901 (2014).
[Crossref] [PubMed]

Quantum Electron. (1)

O. V. Yushko and A. A. Redyuk, “Soliton communication lines based on spectrally efficient modulation formats,” Quantum Electron. 44(6), 606–611 (2014).
[Crossref]

SIAM J. Matrix Anal. Appl. (1)

M. V. Barel, G. Heinig, and P. Kravanja, “A Stabilized Superfast Solver for Nonsymmetric Toeplitz Systems,” SIAM J. Matrix Anal. Appl. 23(2), 494–510 (2001).
[Crossref]

Sov. Phys. JETP (2)

S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electromagnetic waves,” Sov. Phys. JETP 38, 248–253 (1974).

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

Stud. Appl. Math. (1)

M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, “The inverse scattering transform-Fourier analysis for nonlinear problems,” Stud. Appl. Math. 53, 249–315 (1974).

Other (18)

V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevskii, Theory of Solitons. The Inverse Scattering Method. (Colsultants Bureau, New York, 1984).

M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform. (SIAM, Philadelphia, 1981).

I. Phillips, M. Tan, M. F. Stephens, M. 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, San Francisco, California, 2014, paper M3C.1.

G. P. Agrawal, Fiber-Optic Communication Systems, 4th ed. (Wyley, 2010).

A. Hasegawa and Y. Kodama, Solitons in Optical Communications (Oxford University Press, 1996).

L. F. Mollenauer and J. P. Gordon, Solitons in Optical Fibers: Fundamentals and Applications (Academic Press, 2006).

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, 2003).

S. Hari, F. Kschischang, and M. Yousefi, “Multi-eigenvalue communication via the nonlinear Fourier transform,” in Communications (QBSC),2014, 27th Biennial Symposium on, pp. 92–95.
[Crossref]

R. A. Shafik, S. Rahman, and R. Islam, “On the Extended Relationships Among EVM, BER and SNR as Performance Metrics,” in Electrical and Computer Engineering,2006. ICECE '06. International Conference.on, 2006, pp. 408–411.
[Crossref]

S. T. Le, K. J. Blow, V. K. Menzentsev, and S. K. Turitsyn, “Comparison of numerical bit error rate estimation methods in 112Gbs QPSK CO-OFDM transmission,” in Optical Communication (ECOC 2013), 39th European Conference and Exhibition on, 2013, paper P4.14.
[Crossref]

Y. Lu, Y. Fang, B. Wu, K. Wang, W. Wan, F. Yu, L. Li, X. Shi, and Q. Xiong, “Experimental comparison of 32-Gbaud Electrical-OFDM and Nyquist-WDM transmission with 64GSa/s DAC,” in Optical Communication (ECOC 2013), 39th European Conference and Exhibition on (2013).
[Crossref]

C. Xi, L. Xiang, S. Chandrasekhar, B. Zhu, and R. W. Tkach, “Experimental demonstration of fiber nonlinearity mitigation using digital phase conjugation,” in Optical Fiber Communication Conference and Exposition (OFC/NFOEC),2012, paper OTh3C.

S. Wahls and H. V. Poor, “Introducing the fast nonlinear Fourier transform,” in Proceedings of International Conference on Acoustics, Speech, and Signal Processing (ICASSP 2013), IEEE, 2013, pp. 5780–5784.
[Crossref]

S. Wahls and H. V. Poor, “Fast Numerical Nonlinear Fourier Transforms,” submitted to IEEE Trans. Inf. Theory (2013), http://arxiv.org/abs/1402.1605 .

M. Stewart, “Fast algorithms for structured matrix computations” in Handbook of Linear Algebra (2nd edition) (Chapman & Hall, 2013), Chap. 62.

H. Terauchi and A. Maruta, “Eigenvalue Modulated Optical Transmission System Based on Digital Coherent Technology,” in 18th OptoElectronics and Communications Conference held jointly with 2013 International Conference on Photonics in Switching (OECC/PS), IEICE,2013, paper WR2–5.

A. C. Newell, Solitons in mathematics and physics (SIAM Philadelphia, 1985).

R. Bulirsch, Introduction to Numerical Analysis, 2nd ed. (Springer-Verlag, 1993).

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Figures (13)

Fig. 1
Fig. 1 (a) Block diagram of NIS-based optical communication systems, (b, c) block diagrams of OFDM encoder and decoder, (d, e) block diagrams of Nyquist-shaped encoder and decoder.
Fig. 2
Fig. 2 (a) Continuous spectrum of rectangular pulse for A = 1, T2 - T1 = 1. (b) Continuous spectrum of rectangular pulse for A = 6, T2 - T1 = 1. (c) Mean square error as a function of the simulation time resolution (dt).
Fig. 3
Fig. 3 Comparison between the numerical and analytical results. (a) Numerical and analytical solution for q(t), dt = 0.01. (b) Error of numerical method as a function of t. (c) RMSE as a function of the time resolution dt.
Fig. 4
Fig. 4 Illustration of a burst mode transmission, in which neighbouring packets are separated by a guard time
Fig. 5
Fig. 5 Linear spectra of OFDM signals before and after BNFT, the launch power is 0dBm.
Fig. 6
Fig. 6 Q-factor as a function of the launch power for the (a) OFDM and (b) Nyquist-shaped NIS-based systems without the ASE noise.
Fig. 7
Fig. 7 Performance comparison of the 100-Gb/s QPSK OFDM systems with the NIS vs. the DBP methods for fiber nonlinearity compensation. The receiver filter bandwidth used was 40 GHz, the distance is 2000km.
Fig. 8
Fig. 8 Constellation diagrams at the optimum launch powers of the 100-Gb/s QPSK OFDM systems with and without the NIS and DBP methods for fiber compensation; (a) without NIS and DBP, (b) with the NIS method, (c) DBP with 10 steps/span, (d) DBP with 20 steps/span
Fig. 9
Fig. 9 Performance comparison of the 100-Gb/s QPSK Nyquist-shaped systems with the NIS and DBP methods for fiber nonlinearity compensation. The receiver filter bandwidth is 40 GHz, the distance is 2000km
Fig. 10
Fig. 10 Performance comparison of the 200-Gb/s 16QAM OFDM systems with the NIS and DBP methods for fiber nonlinearity compensation. The receiver filter bandwidth was 40 GHz, the distance is 2000kmand DBP, (b) DBP with 20 steps/span, (c) with the NIS method.
Fig. 11
Fig. 11 Constellation diagrams at the optimum launch powers of the 200-Gb/s 16QAM OFDM systems with and without the NIS and DBP methods for fiber compensation. (a) Without NIS
Fig. 12
Fig. 12 Performance comparison of the 300-Gb/s 64QAM OFDM systems with the NIS and DBP methods for fiber nonlinearity compensation. The receiver filter bandwidth was 40 GHz, the distance is 800km
Fig. 13
Fig. 13 Constellation diagrams at the optimum launch powers of the 300-Gb/s 64QAM OFDM systems with and without NIS and DBP methods for fiber compensation. (a) Without NIS and DBP, (b) DBP with 20 steps/span, (c) with the NIS method.

Equations (46)

Equations on this page are rendered with MathJax. Learn more.

S(ω)= s(t)exp(jωt)dt
r(ξ)| ξ=ω/2 = S(ω)
r(L,ξ)=r(ξ) e 2j ξ 2 L
S ¯ (ω)=r(L,ξ) e 2j ξ 2 L | ξ=ω/2
j q z β 2 2 q tt +γq | q | 2 =0
γ= n 2 ω 0 /c A eff
j q z + 1 2 q tt +q | q | 2 =0,
t T s t, z Z s z, q γ Z s q
d v 1 dt =q(t) v 2 jς v 1 , d v 2 dt = q ¯ (t) v 1 +jς v 2
Φ(t,ξ)= [ ϕ 1 , ϕ 2 ] T , Φ ˜ (t,ξ)= [ ϕ ¯ 2 , ϕ ¯ 1 ] T ,
Φ| t = [ e jξt ,0 ] T
Ψ(t,ξ)= [ ψ 1 , ψ 2 ] T , Ψ ˜ (t,ξ)= [ ψ ¯ 2 , ψ ¯ 1 ] T ,
Ψ| t+ = [ 0, e jξt ] T
{ Φ(t,ξ)=a(ξ) Ψ ˜ (t,ξ)+b(ξ)Ψ(t,ξ) Φ ˜ (t,ξ)= a ¯ (ξ)Ψ(t,ξ)+ b ¯ (ξ) Ψ ˜ (t,ξ)
r(ξ)= b ¯ (ξ)/a(ξ),
N(ω)=r(ξ)| ξ=ω/2
K(t,x)+F(t+x)+ t t K(t,λ) F ¯ (λ+σ)F(σ+x)dσdλ ,
F(t)= 1 2π + r(ξ) e jξt dξ
q(t)=2 lim xt0 K(t,x)
a(ξ)= lim t+ ϕ 1 (t,ξ) e jξt , b(ξ)= lim t+ ϕ 2 (t,ξ) e jξt
Φ( t n +Δt/2,ξ)=T( q n ,ξ)Φ( t n Δt/2,ξ),
T( q n ,ξ)=exp[ Δt( jξ q n q n * jξ ) ] =( cosh(kΔt)jξ k 1 sinh(kΔt) q n k 1 sinh(kΔt) q n * k 1 sinh(kΔt) cosh(kΔt)+jξ k 1 sinh(kΔt) ),
Φ( T 0 Δt/2,ξ)=Π(ξ)Φ( T 0 Δt/2,ξ), Π(ξ)= n=1 2M T( q n ,ξ)
Φ( T 0 Δt/2,ξ)=( 1 0 ) e jξ( T 0 Δt/2)
Φ( T 0 Δt/2,ξ)=( a(ξ) e jξ( T 0 Δt/2) b(ξ) e jξ( T 0 Δt/2) )=( Π 11 (ξ) Π 12 (ξ) Π 21 (ξ) Π 22 (ξ) )Φ( T 0 Δt/2,ξ) =( Π 11 (ξ) Π 12 (ξ) Π 21 (ξ) Π 22 (ξ) )( e jξ( T 0 Δt/2) 0 ),
a(ξ)= Π 11 (ξ) e 2jξ T 0 , b(ξ)= Π 21 (ξ) e jξΔt
a(ξ)= Π 11 (ξ) e jξ( T max T min ) , b(ξ)= Π 21 (ξ) e jξ( T max + T min Δt)
q(t)={ A, t[ T 1 , T 2 ] 0 otherwise
r(ξ)= A ¯ jξ e 2jξt ( 1 ξ 2 + | A | 2 jξ cot( ξ 2 + | A | 2 ( T 2 T 1 ) ) )
MSE= 1 A 1 N k=1 N | r exact ( ξ k ) r numeric ( ξ k ) | 2
A 1 (x,t)+ x F(t+y) A ¯ 2 (x,y)dy A 2 (x,t) x F(t+y) A ¯ 1 (x,y) dy=F(x+t), x>t
u(x,s)= A ¯ 1 (x,xs), v(x,τ)= A ¯ 2 (x,τx) ,
u(x,s)+ s 2x F ¯ (τs)v(x,τ)dτ=0, v(x,τ) 0 τ F(τs)u(x,s)ds=F(τ),
q(x)=2v(x,2x0)
s k =h(k1/2), k=1,2...m τ n =h(n1/2), n=1,2...m x m =mh/2, m=1,2...N
u n (m) =u( x m , τ n ), v n (m) =v( x m , τ n ), F n =F(nh)
u k (m) +h n=k m F ¯ nk v n (m) =0, v n (m) h k=1 m F nk u k (m) = F n
q (m) =2 v m (m)
G (m) ( u (m) v (m) )= b (m) ,
G (m) =( E (m) h F (m) h F (m) E (m) ).
F (m) =( F 0 0 0 ... 0 F 1 F 0 0 ... 0 F 2 F 1 F 0 ... 0 0 F m1 F m2 F m3 ... F 0 ),
G ˜ (m) =( E (N) h F (m) h F (m) E (m) ),
b ˜ (m) = [ 0,0,...0 N , F 0 , F 1 ,... F m1 ] T
F(t)=να e αt
q(t)= 4αυσ( σ1 ) ( σ1 ) 2 e 2σαt + υ 2 e 2σαt ,
N ASE =αLh f s K T

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