Abstract

The recently proposed b-modulation method for nonlinear Fourier transform-based fiber-optic transmission offers explicit control over the duration of the generated pulses and therewith solves a longstanding practical problem. The currently used b-modulation however suffers from a fundamental energy barrier. There is a limit to the energy of the pulses, in normalized units, that can be generated. In this paper, we discuss how the energy barrier can be shifted by proper design of the carrier waveform and the modulation alphabet. In an experiment, it is found that the improved b-modulator achieves both a higher Q-factor and a further reach than a comparable conventional b-modulator. Furthermore, it performs significantly better than conventional approaches that modulate the reflection coefficient.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. E. G. Turitsyna and S. K. Turitsyn, “Digital signal processing based on inverse scattering transform,” Opt. Lett. 38(20), 4186–4188 (2013).
    [Crossref] [PubMed]
  2. 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]
  3. 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]
  4. 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]
  5. 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]
  6. 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]
  7. S. K. Turitsyn, J. E. Prilepsky, S. T. Le, S. Wahls, L. L. Frumin, M. Kamalian, and S. A. Derevyanko, “Nonlinear Fourier transform for optical data processing and transmission: advances and perspectives,” Optica 4(3), 307–322 (2017).
    [Crossref]
  8. 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(4), 249–315 (1974).
    [Crossref]
  9. A. Hasegawa and T. Nyu, “Eigenvalue communication,” J. Lightwave Technol. 11(3), 395–399 (1993).
    [Crossref]
  10. T. Gui, C. Lu, A. P. T. Lau, and P. K. A. Wai, “High-order modulation on a single discrete eigenvalue for optical communications based on nonlinear Fourier transform,” Opt. Express 25(17), 20286–20297 (2017).
    [Crossref] [PubMed]
  11. S. T. Le and H. Buelow, “64×0.5Gbaud nonlinear frequency division multiplexed transmissions with high order modulation formats,” J. Lightwave Technol. 35(17), 3692–3698 (2017).
    [Crossref]
  12. V. Aref, S. T. Le, and H. Buelow, “Demonstration of fully nonlinear spectrum modulated system in the highly nonlinear optical transmission regime,” inProceedings of European Conference on Optical Communication (Institute of Electrical and Electronics Engineers, 2016), post deadline paper.
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    [Crossref]
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    [Crossref] [PubMed]
  15. 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(16), 18370–18381 (2016).
    [Crossref] [PubMed]
  16. S. Wahls, “Generation of time-limited signals in the nonlinear Fourier domain via b-modulation,” inProceedings of European Conference on Optical Communication (Institute of Electrical and Electronics Engineers, 2017), paper W.3.C.6.
    [Crossref]
  17. S. T. Le, K. Schuh, F. Buchali, and H. Buelow, “100 Gbps b-modulated nonlinear frequency division multiplexed transmission,” in Optical Fiber Communication Conference (Optical Society of America, 2018), paper W1G.6.
    [Crossref]
  18. X. Yangzhang, V. Aref, S. T. Le, H. Buelow, and P. Bayvel, “400 Gbps dual-polarisation non-linear frequency-division multiplexed transmission with b-modulation,” Preprint arXiv:1806.10367 [eess.SP] (2018).
  19. X. Yangzhang, M. I. Yousefi, A. Alvarado, D. Lavery, and P. Bayvel, “Nonlinear frequency-division multiplexing in the focusing regime,” in Optical Fiber Communication Conference (Optical Society of America, 2017), paper Tu3D.1.
    [Crossref]
  20. K. Duda, T. P. Zielinski, and S. H. Barczentewicz, “Perfectly flat-top and equiripple flat-top cosine windows,” IEEE Trans. Instrum. Meas. 65(7), 1558–1567 (2016).
    [Crossref]
  21. T. Gui, T. H. Chan, C. Lu, A. P. T. Lau, and P. K. A. Wai, “Alternative decoding methods for optical communications based on nonlinear Fourier transform,” J. Lightwave Technol. 35(9), 1542–1550 (2017).
    [Crossref]
  22. J. K. Brenne and J. Skaar, “Design of grating-assisted codirectional couplers with discrete inverse-scattering algorithms,” J. Lightwave Technol. 21(1), 254–263 (2003).
    [Crossref]
  23. S. T. Le, V. Aref, and H. Buelow, “Combating the Kerr-nonlinearity limit with nonlinear signal multiplexing,” in Signal Processing in Photonic Communications (Optical Society of America, 2018), paper SpM4G. 3.
    [Crossref]
  24. J. C. Portinari, “An inverse scattering transform for potentials of compact support,” J. Math. Phys. 19(10), 2100–2102 (1978).
    [Crossref]
  25. T. Aktosun, “Inverse scattering on the line with incomplete scattering data,” Contemp. Math. 362, 362 (2004).
  26. S. Wahls, S. Chimmalgi, and P. J. Prins, “FNFT: a software library for computing nonlinear Fourier transforms,” J. Open Source Software 3(23), 597 1–11 (2018).
  27. S. Wahls and V. Vaibhav, “Fast inverse nonlinear Fourier transforms for continuous spectra of Zakharov-Shabat type,” Withdrawn Preprint arXiv:1607.01305v2 [cs.IT] (2016).
  28. P. J. Prins and S. Wahls, “Higher order exponential splittings for the fast non-linear Fourier transform of the Korteweg-de Vries Equation,” in International Conference on Acoustics, Speech and Signal Processing (Institute of Electrical and Electronics Engineers, 2018), paper SPTM-P9.4.

2018 (1)

S. Wahls, S. Chimmalgi, and P. J. Prins, “FNFT: a software library for computing nonlinear Fourier transforms,” J. Open Source Software 3(23), 597 1–11 (2018).

2017 (4)

2016 (3)

2015 (1)

S. Wahls and H. V. Poor, “Fast numerical nonlinear Fourier transforms,” IEEE Trans. Inf. Theory 61(12), 6957–6974 (2015).
[Crossref]

2014 (4)

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]

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]

2013 (2)

2004 (1)

T. Aktosun, “Inverse scattering on the line with incomplete scattering data,” Contemp. Math. 362, 362 (2004).

2003 (1)

1993 (1)

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

1978 (1)

J. C. Portinari, “An inverse scattering transform for potentials of compact support,” J. Math. Phys. 19(10), 2100–2102 (1978).
[Crossref]

1974 (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(4), 249–315 (1974).
[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(4), 249–315 (1974).
[Crossref]

Aktosun, T.

T. Aktosun, “Inverse scattering on the line with incomplete scattering data,” Contemp. Math. 362, 362 (2004).

Alvarado, A.

X. Yangzhang, M. I. Yousefi, A. Alvarado, D. Lavery, and P. Bayvel, “Nonlinear frequency-division multiplexing in the focusing regime,” in Optical Fiber Communication Conference (Optical Society of America, 2017), paper Tu3D.1.
[Crossref]

Barczentewicz, S. H.

K. Duda, T. P. Zielinski, and S. H. Barczentewicz, “Perfectly flat-top and equiripple flat-top cosine windows,” IEEE Trans. Instrum. Meas. 65(7), 1558–1567 (2016).
[Crossref]

Bayvel, P.

X. Yangzhang, M. I. Yousefi, A. Alvarado, D. Lavery, and P. Bayvel, “Nonlinear frequency-division multiplexing in the focusing regime,” in Optical Fiber Communication Conference (Optical Society of America, 2017), paper Tu3D.1.
[Crossref]

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]

Brenne, J. K.

Buchali, F.

S. T. Le, K. Schuh, F. Buchali, and H. Buelow, “100 Gbps b-modulated nonlinear frequency division multiplexed transmission,” in Optical Fiber Communication Conference (Optical Society of America, 2018), paper W1G.6.
[Crossref]

Buelow, H.

S. T. Le and H. Buelow, “64×0.5Gbaud nonlinear frequency division multiplexed transmissions with high order modulation formats,” J. Lightwave Technol. 35(17), 3692–3698 (2017).
[Crossref]

S. T. Le, K. Schuh, F. Buchali, and H. Buelow, “100 Gbps b-modulated nonlinear frequency division multiplexed transmission,” in Optical Fiber Communication Conference (Optical Society of America, 2018), paper W1G.6.
[Crossref]

Chan, T. H.

Chimmalgi, S.

S. Wahls, S. Chimmalgi, and P. J. Prins, “FNFT: a software library for computing nonlinear Fourier transforms,” J. Open Source Software 3(23), 597 1–11 (2018).

Derevyanko, S. A.

Duda, K.

K. Duda, T. P. Zielinski, and S. H. Barczentewicz, “Perfectly flat-top and equiripple flat-top cosine windows,” IEEE Trans. Instrum. Meas. 65(7), 1558–1567 (2016).
[Crossref]

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]

Gui, T.

Hasegawa, A.

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

Kamalian, M.

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(4), 249–315 (1974).
[Crossref]

Kschischang, F. R.

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 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]

Lau, A. P. T.

Lavery, D.

X. Yangzhang, M. I. Yousefi, A. Alvarado, D. Lavery, and P. Bayvel, “Nonlinear frequency-division multiplexing in the focusing regime,” in Optical Fiber Communication Conference (Optical Society of America, 2017), paper Tu3D.1.
[Crossref]

Le, S. T.

Lu, C.

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(4), 249–315 (1974).
[Crossref]

Nyu, T.

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

Poor, H. V.

S. Wahls and H. V. Poor, “Fast numerical nonlinear Fourier transforms,” IEEE Trans. Inf. Theory 61(12), 6957–6974 (2015).
[Crossref]

Portinari, J. C.

J. C. Portinari, “An inverse scattering transform for potentials of compact support,” J. Math. Phys. 19(10), 2100–2102 (1978).
[Crossref]

Prilepsky, J. E.

Prins, P. J.

S. Wahls, S. Chimmalgi, and P. J. Prins, “FNFT: a software library for computing nonlinear Fourier transforms,” J. Open Source Software 3(23), 597 1–11 (2018).

Schuh, K.

S. T. Le, K. Schuh, F. Buchali, and H. Buelow, “100 Gbps b-modulated nonlinear frequency division multiplexed transmission,” in Optical Fiber Communication Conference (Optical Society of America, 2018), paper W1G.6.
[Crossref]

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(4), 249–315 (1974).
[Crossref]

Skaar, J.

Turitsyn, S. K.

Turitsyna, E. G.

Wahls, S.

S. Wahls, S. Chimmalgi, and P. J. Prins, “FNFT: a software library for computing nonlinear Fourier transforms,” J. Open Source Software 3(23), 597 1–11 (2018).

S. K. Turitsyn, J. E. Prilepsky, S. T. Le, S. Wahls, L. L. Frumin, M. Kamalian, and S. A. Derevyanko, “Nonlinear Fourier transform for optical data processing and transmission: advances and perspectives,” Optica 4(3), 307–322 (2017).
[Crossref]

S. Wahls and H. V. Poor, “Fast numerical nonlinear Fourier transforms,” IEEE Trans. Inf. Theory 61(12), 6957–6974 (2015).
[Crossref]

S. Wahls, “Generation of time-limited signals in the nonlinear Fourier domain via b-modulation,” inProceedings of European Conference on Optical Communication (Institute of Electrical and Electronics Engineers, 2017), paper W.3.C.6.
[Crossref]

Wai, P. K. A.

Yangzhang, X.

X. Yangzhang, M. I. Yousefi, A. Alvarado, D. Lavery, and P. Bayvel, “Nonlinear frequency-division multiplexing in the focusing regime,” in Optical Fiber Communication Conference (Optical Society of America, 2017), paper Tu3D.1.
[Crossref]

Yousefi, M. I.

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]

X. Yangzhang, M. I. Yousefi, A. Alvarado, D. Lavery, and P. Bayvel, “Nonlinear frequency-division multiplexing in the focusing regime,” in Optical Fiber Communication Conference (Optical Society of America, 2017), paper Tu3D.1.
[Crossref]

Zielinski, T. P.

K. Duda, T. P. Zielinski, and S. H. Barczentewicz, “Perfectly flat-top and equiripple flat-top cosine windows,” IEEE Trans. Instrum. Meas. 65(7), 1558–1567 (2016).
[Crossref]

Contemp. Math. (1)

T. Aktosun, “Inverse scattering on the line with incomplete scattering data,” Contemp. Math. 362, 362 (2004).

IEEE Trans. Inf. Theory (4)

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]

S. Wahls and H. V. Poor, “Fast numerical nonlinear Fourier transforms,” IEEE Trans. Inf. Theory 61(12), 6957–6974 (2015).
[Crossref]

IEEE Trans. Instrum. Meas. (1)

K. Duda, T. P. Zielinski, and S. H. Barczentewicz, “Perfectly flat-top and equiripple flat-top cosine windows,” IEEE Trans. Instrum. Meas. 65(7), 1558–1567 (2016).
[Crossref]

J. Lightwave Technol. (4)

J. Math. Phys. (1)

J. C. Portinari, “An inverse scattering transform for potentials of compact support,” J. Math. Phys. 19(10), 2100–2102 (1978).
[Crossref]

J. Open Source Software (1)

S. Wahls, S. Chimmalgi, and P. J. Prins, “FNFT: a software library for computing nonlinear Fourier transforms,” J. Open Source Software 3(23), 597 1–11 (2018).

Opt. Express (4)

Opt. Lett. (1)

Optica (1)

Phys. Rev. Lett. (1)

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]

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(4), 249–315 (1974).
[Crossref]

Other (8)

S. Wahls, “Generation of time-limited signals in the nonlinear Fourier domain via b-modulation,” inProceedings of European Conference on Optical Communication (Institute of Electrical and Electronics Engineers, 2017), paper W.3.C.6.
[Crossref]

S. T. Le, K. Schuh, F. Buchali, and H. Buelow, “100 Gbps b-modulated nonlinear frequency division multiplexed transmission,” in Optical Fiber Communication Conference (Optical Society of America, 2018), paper W1G.6.
[Crossref]

X. Yangzhang, V. Aref, S. T. Le, H. Buelow, and P. Bayvel, “400 Gbps dual-polarisation non-linear frequency-division multiplexed transmission with b-modulation,” Preprint arXiv:1806.10367 [eess.SP] (2018).

X. Yangzhang, M. I. Yousefi, A. Alvarado, D. Lavery, and P. Bayvel, “Nonlinear frequency-division multiplexing in the focusing regime,” in Optical Fiber Communication Conference (Optical Society of America, 2017), paper Tu3D.1.
[Crossref]

S. T. Le, V. Aref, and H. Buelow, “Combating the Kerr-nonlinearity limit with nonlinear signal multiplexing,” in Signal Processing in Photonic Communications (Optical Society of America, 2018), paper SpM4G. 3.
[Crossref]

V. Aref, S. T. Le, and H. Buelow, “Demonstration of fully nonlinear spectrum modulated system in the highly nonlinear optical transmission regime,” inProceedings of European Conference on Optical Communication (Institute of Electrical and Electronics Engineers, 2016), post deadline paper.

S. Wahls and V. Vaibhav, “Fast inverse nonlinear Fourier transforms for continuous spectra of Zakharov-Shabat type,” Withdrawn Preprint arXiv:1607.01305v2 [cs.IT] (2016).

P. J. Prins and S. Wahls, “Higher order exponential splittings for the fast non-linear Fourier transform of the Korteweg-de Vries Equation,” in International Conference on Acoustics, Speech and Signal Processing (Institute of Electrical and Electronics Engineers, 2018), paper SPTM-P9.4.

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Figures (5)

Fig. 1
Fig. 1 The shape of the carrier ψ ( ξ ) and its inverse Fourier transform Ψ ( τ ) for T = 4.5 .
Fig. 2
Fig. 2 (a) Constellation shape of 16 QAM and its shaped version with E d = 4 ; (b) Time domain shape of the fiber inputs | q ( t ) | generated by the improved b-modulator, original b-modulator, two conventional modulators by q ^ 1 / 2   with same information; (c) 99.9% durations and bandwidths of fiber inputs q ( t ) by the four modulation methods; (d) Fourier transform spectra for the four fiber inputs q ( t ) .
Fig. 3
Fig. 3 For improved b-modulated, original b-modulated, q ^ 1 / 2   modulated and FDM systems in B2B scenario (a) BER vs SNR at an average energy of E d = 4 ; (b) Q factor as function of average energy of E d under SNR = 8dB.
Fig. 4
Fig. 4 DSP structure and experimental setup. AWG: arbitrary waveform generator; OBPF: optical band-pass filter; PC: polarization controller.
Fig. 5
Fig. 5 Q-factor as function of average power at 640 km for improved b-modulated, original b-modulated, q ^ 1 / 2 modulated and linear FDM systems.

Tables (1)

Tables Icon

Table 1 The value of coefficients a m

Equations (26)

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d dt [ ϕ 1 ( t ; λ ) ϕ 2 ( t ; λ ) ] = [ j λ q ( t ) q * ( t ) j λ ] [ ϕ 1 ( t ; λ ) ϕ 2 ( t ; λ ) ] ,   [ a ( t ; λ ) b ( t ; λ ) ] : = [ e j λ t ϕ 1 ( t ; λ ) e j λ t ϕ 2 ( t ; λ ) ] t [ 1 0 ] ,
j u z = 2 u t 2 + 2 u | u | 2 ,           u = u ( z , t ) ,
a z ( ξ ) = a 0 ( ξ ) ,     b z ( ξ ) = e 4 j ξ 2 z b 0 ( ξ ) .
q ^ 1 ( ξ ) : = A u ( ξ ) o r q ^ 2 ( ξ ) : = e A 2 | u ( ξ ) | 2 1 e j < u ( ξ ) ,
u ( ξ ) : = n = N N s n ψ ( ξ n ξ s ) .
b ( ξ ) = Au ( ξ ) ,                 with   A   and   u ( ξ )   as defined above .
q ( t ) = 0      for  t [ T 2 , T 2 ]
Ψ ( τ ) : = ψ ( ξ ) e j τ ξ d ξ 2 π
E = 1 π log ( 1 | b ( ξ ) | 2 ) d ξ .
| b ( ξ ) | < 1             f o r   a l l   ξ .
MCE [ ψ ] : = lim A ( 1 / sup ξ | ψ ( ξ ) | ) 1 π log ( 1 A 2 | ψ ( ξ ) | 2 ) d ξ
ψ e x a m p l e ( ξ ) = { 1 ξ 2 n ,     i f   | ξ | 1   0 ,                                   i f     | ξ | > 1 ,       n { 1 , 2 , 3 , } .
MCE [ ψ e x a m p l e ] = lim A 1 1 π 1 1 log ( 1 A 2 ( 1 ξ 2 n ) ) d ξ = 1 π 1 1 log ξ 2 n d ξ = 4 n π < .
ψ r e c t ( ξ ) = { 1 ,     i f   | ξ | 1 0 ,     i f   | ξ | > 1  
MCE [ ψ r e c t ] = lim A 1 1 π 1 1 log ( 1 A 2 ) d ξ = 2 π lim A 1 log ( 1 A 2 ) = .
E = | q ( t ) | 2 d t = 1 π log ( 1 + | q ^ 1 ( ξ ) | 2 ) d ξ = 1 π log ( 1 + A 2 | ψ ( ξ ) | 2 ) d ξ .
Ψ flat top ( τ ) = 1 T m = 0 15 a m r ( τ T ) cos ( π τ T m ) ,       r ( τ ) = { 1 ,     i f   | τ | 1 0 ,     o t h e r w i s e     ,
ψ f l a t   t o p ( ξ ) = m = 0 15 a m ( sin c ( ξ T π m ) + sin c ( ξ T π + m ) ) .
E [ s n ] = 1 π log ( 1 A 2 | s n | 2 | ψ ( ξ ) 2 | ) d ξ .
A shaped = { a 1 shaped , , a M shaped }
b ( ξ ) = m = N N s n shaped Ψ ( ξ n ξ s ) ,     where     s n shaped :=a m ( n ) shaped A shaped .
a m shaped : = γ m a m ,         where       γ m > 0.
E [ a m shaped ] : = 1 π log ( 1 γ m | a m | 2 | ψ ( ξ ) | 2 ) d ξ = | a m | 2 ( | a 1 | 2 + + | a M | 2 ) / M E d ,
E [ s n s h a p e d ] / E [ s k s h a p e d ] = | s n | 2 / | s k | 2 .
E [ a 1 shaped ] + + E [ a M shaped ] M = E d   .
a ( λ k ) = 0 b ( λ ) b * ( λ * ) = 1.

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