Abstract

We studied a simplified digital back propagation (DBP) scheme by including the correlation between neighboring signal samples. An analytical expression for calculating the correlation coefficients is derived based on a perturbation theory. In each propagation step, nonlinear distortion due to phase-dependent terms in the perturbative expansion are ignored which enhances the computational efficiency. The performance of the correlated DBP is evaluated by simulating a single-channel single-polarization fiber-optic system operating at 28 Gbaud, 32-quadrature amplitude modulation (32-QAM), and 40 × 80 km transmission distance. As compared to standard DBP, correlated DBP reduces the total number of propagation steps by a factor of 10 without performance penalty. Correlated DBP with only 2 steps per link provides about one dB improvement in Q-factor over linear compensation.

© 2015 Optical Society of America

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References

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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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2014 (6)

2013 (3)

J. Shao, S. Kumar, and X. Liang, “Digital back propagation with optimal step size for polarization multiplexed transmission,” IEEE Photon. Technol. Lett. 25(23), 2327–2330 (2013).
[Crossref]

Z. Tao, L. Dou, W. Yan, Y. Fan, L. Li, S. Oda, Y. Akiyama, H. Nakashima, T. Hoshida, and J. C. Rasmussen, “Complexity-reduced digital nonlinear compensation for coherent optical system,” Proc. SPIE 8647, 86470K (2013).

R. Dar, M. Feder, A. Mecozzi, and M. Shtaif, “Properties of nonlinear noise in long, dispersion-uncompensated fiber links,” Opt. Express 21(22), 25685–25699 (2013).
[Crossref] [PubMed]

2012 (3)

2011 (3)

2010 (2)

D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, and S. J. Savory, “Mitigation of fiber nonlinearity using a digital coherent receiver,” IEEE J. Sel. Top. Quantum Electron. 16(5), 1217–1226 (2010).
[Crossref]

L. B. Du and A. J. Lowery, “Improved single channel backpropagation for intra-channel fiber nonlinearity compensation in long-haul optical communication systems,” Opt. Express 18(16), 17075–17088 (2010).
[Crossref] [PubMed]

2009 (1)

2008 (2)

2005 (1)

2000 (1)

A. Mecozzi, C. B. Clausen, and M. Shtaif, “Analysis of intrachannel nonlinear effects in highly dispersed optical pulse transmission,” IEEE Photon. Technol. Lett. 12(4), 392–394 (2000).
[Crossref]

Akiyama, Y.

Z. Tao, L. Dou, W. Yan, Y. Fan, L. Li, S. Oda, Y. Akiyama, H. Nakashima, T. Hoshida, and J. C. Rasmussen, “Complexity-reduced digital nonlinear compensation for coherent optical system,” Proc. SPIE 8647, 86470K (2013).

Bayvel, P.

D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, and S. J. Savory, “Mitigation of fiber nonlinearity using a digital coherent receiver,” IEEE J. Sel. Top. Quantum Electron. 16(5), 1217–1226 (2010).
[Crossref]

Behrens, C.

D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, and S. J. Savory, “Mitigation of fiber nonlinearity using a digital coherent receiver,” IEEE J. Sel. Top. Quantum Electron. 16(5), 1217–1226 (2010).
[Crossref]

Cartledge, J. C.

Y. Gao, J. H. Ke, K. P. Zhong, J. C. Cartledge, and S. S.-H. Yam, “Assessment of intra-channel nonlinear compensation for 112 Gb/s dual polarization 16QAM systems,” J. Lightwave Technol. 30(24), 3902–3910 (2012).
[Crossref]

Y. Gao, J. C. Cartledge, A. S. Karar, and S. S.-H. Yam, “Reducing the complexity of nonlinearity pre-compensation using symmetric EDC and pulse shaping,” in 39th European Conference and Exhibition on Optical Communication (ECOC 2013), 1209–1219.
[Crossref]

Chen, X.

Chugtai, M. N.

Clausen, C. B.

A. Mecozzi, C. B. Clausen, and M. Shtaif, “Analysis of intrachannel nonlinear effects in highly dispersed optical pulse transmission,” IEEE Photon. Technol. Lett. 12(4), 392–394 (2000).
[Crossref]

Dar, R.

Dou, L.

Z. Tao, L. Dou, W. Yan, Y. Fan, L. Li, S. Oda, Y. Akiyama, H. Nakashima, T. Hoshida, and J. C. Rasmussen, “Complexity-reduced digital nonlinear compensation for coherent optical system,” Proc. SPIE 8647, 86470K (2013).

Z. Tao, L. Dou, W. Yan, L. Li, T. Hoshida, and J. C. Rasmussen, “Multiplier-free intrachannel nonlinearity compensating algorithm operating at symbol rate,” J. Lightwave Technol. 29(17), 2570–2576 (2011).
[Crossref]

Du, L. B.

Ellis, A. D.

Essiambre, R.-J.

Fan, Y.

Z. Tao, L. Dou, W. Yan, Y. Fan, L. Li, S. Oda, Y. Akiyama, H. Nakashima, T. Hoshida, and J. C. Rasmussen, “Complexity-reduced digital nonlinear compensation for coherent optical system,” Proc. SPIE 8647, 86470K (2013).

Feder, M.

Fischer, J. K.

Forzati, M.

Gao, Y.

Y. Gao, J. H. Ke, K. P. Zhong, J. C. Cartledge, and S. S.-H. Yam, “Assessment of intra-channel nonlinear compensation for 112 Gb/s dual polarization 16QAM systems,” J. Lightwave Technol. 30(24), 3902–3910 (2012).
[Crossref]

Y. Gao, J. C. Cartledge, A. S. Karar, and S. S.-H. Yam, “Reducing the complexity of nonlinearity pre-compensation using symmetric EDC and pulse shaping,” in 39th European Conference and Exhibition on Optical Communication (ECOC 2013), 1209–1219.
[Crossref]

Goldfarb, G.

Hellerbrand, S.

D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, and S. J. Savory, “Mitigation of fiber nonlinearity using a digital coherent receiver,” IEEE J. Sel. Top. Quantum Electron. 16(5), 1217–1226 (2010).
[Crossref]

Hoffmann, S.

Hoshida, T.

Z. Tao, L. Dou, W. Yan, Y. Fan, L. Li, S. Oda, Y. Akiyama, H. Nakashima, T. Hoshida, and J. C. Rasmussen, “Complexity-reduced digital nonlinear compensation for coherent optical system,” Proc. SPIE 8647, 86470K (2013).

Z. Tao, L. Dou, W. Yan, L. Li, T. Hoshida, and J. C. Rasmussen, “Multiplier-free intrachannel nonlinearity compensating algorithm operating at symbol rate,” J. Lightwave Technol. 29(17), 2570–2576 (2011).
[Crossref]

Ip, E.

Kahn, J. M.

Karar, A. S.

Y. Gao, J. C. Cartledge, A. S. Karar, and S. S.-H. Yam, “Reducing the complexity of nonlinearity pre-compensation using symmetric EDC and pulse shaping,” in 39th European Conference and Exhibition on Optical Communication (ECOC 2013), 1209–1219.
[Crossref]

Ke, J. H.

Killey, R. I.

D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, and S. J. Savory, “Mitigation of fiber nonlinearity using a digital coherent receiver,” IEEE J. Sel. Top. Quantum Electron. 16(5), 1217–1226 (2010).
[Crossref]

Kim, I.

Kumar, S.

J. Shao, X. Liang, and S. Kumar, “Comparison of split-step Fourier schemes for simulating fiber optic communication systems,” IEEE Photonics J. 6(4), 7200515 (2014).

S. N. Shahi, S. Kumar, and X. Liang, “Analytical modeling of cross-phase modulation in coherent fiber-optic system,” Opt. Express 22(2), 1426–1439 (2014).
[Crossref] [PubMed]

X. Liang and S. Kumar, “Analytical modeling of XPM in dispersion-managed coherent fiber-optic systems,” Opt. Express 22(9), 10579–10592 (2014).
[Crossref] [PubMed]

X. Liang, S. Kumar, J. Shao, M. Malekiha, and D. V. Plant, “Digital compensation of cross-phase modulation distortions using perturbation technique for dispersion-managed fiber-optic systems,” Opt. Express 22(17), 20634–20645 (2014).
[Crossref] [PubMed]

X. Liang and S. Kumar, “Multi-stage perturbation theory for compensating intra-channel nonlinear impairments in fiber-optic links,” Opt. Express 22(24), 29733–29745 (2014).
[Crossref] [PubMed]

J. Shao, S. Kumar, and X. Liang, “Digital back propagation with optimal step size for polarization multiplexed transmission,” IEEE Photon. Technol. Lett. 25(23), 2327–2330 (2013).
[Crossref]

S. Kumar, S. N. Shahi, and D. Yang, “Analytical modeling of a single channel nonlinear fiber optic system based on QPSK,” Opt. Express 20(25), 27740–27755 (2012).
[Crossref] [PubMed]

S. Kumar and D. Yang, “Second-order theory for self-phase modulation and cross-phase modulation in optical fibers,” J. Lightwave Technol. 23(6), 2073–2080 (2005).
[Crossref]

Li, G.

Li, L.

Z. Tao, L. Dou, W. Yan, Y. Fan, L. Li, S. Oda, Y. Akiyama, H. Nakashima, T. Hoshida, and J. C. Rasmussen, “Complexity-reduced digital nonlinear compensation for coherent optical system,” Proc. SPIE 8647, 86470K (2013).

Z. Tao, L. Dou, W. Yan, L. Li, T. Hoshida, and J. C. Rasmussen, “Multiplier-free intrachannel nonlinearity compensating algorithm operating at symbol rate,” J. Lightwave Technol. 29(17), 2570–2576 (2011).
[Crossref]

Li, X.

Liang, X.

Lowery, A. J.

Makovejs, S.

D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, and S. J. Savory, “Mitigation of fiber nonlinearity using a digital coherent receiver,” IEEE J. Sel. Top. Quantum Electron. 16(5), 1217–1226 (2010).
[Crossref]

Malekiha, M.

Mårtensson, J.

Mateo, E.

Mecozzi, A.

Millar, D. S.

D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, and S. J. Savory, “Mitigation of fiber nonlinearity using a digital coherent receiver,” IEEE J. Sel. Top. Quantum Electron. 16(5), 1217–1226 (2010).
[Crossref]

Molle, L.

Mussolin, M.

Nakashima, H.

Z. Tao, L. Dou, W. Yan, Y. Fan, L. Li, S. Oda, Y. Akiyama, H. Nakashima, T. Hoshida, and J. C. Rasmussen, “Complexity-reduced digital nonlinear compensation for coherent optical system,” Proc. SPIE 8647, 86470K (2013).

Noé, R.

Nölle, M.

Oda, S.

Z. Tao, L. Dou, W. Yan, Y. Fan, L. Li, S. Oda, Y. Akiyama, H. Nakashima, T. Hoshida, and J. C. Rasmussen, “Complexity-reduced digital nonlinear compensation for coherent optical system,” Proc. SPIE 8647, 86470K (2013).

Pfau, T.

Plant, D. V.

Rafique, D.

Rasmussen, J. C.

Z. Tao, L. Dou, W. Yan, Y. Fan, L. Li, S. Oda, Y. Akiyama, H. Nakashima, T. Hoshida, and J. C. Rasmussen, “Complexity-reduced digital nonlinear compensation for coherent optical system,” Proc. SPIE 8647, 86470K (2013).

Z. Tao, L. Dou, W. Yan, L. Li, T. Hoshida, and J. C. Rasmussen, “Multiplier-free intrachannel nonlinearity compensating algorithm operating at symbol rate,” J. Lightwave Technol. 29(17), 2570–2576 (2011).
[Crossref]

Savory, S. J.

D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, and S. J. Savory, “Mitigation of fiber nonlinearity using a digital coherent receiver,” IEEE J. Sel. Top. Quantum Electron. 16(5), 1217–1226 (2010).
[Crossref]

Schubert, C.

Shahi, S. N.

Shao, J.

J. Shao, X. Liang, and S. Kumar, “Comparison of split-step Fourier schemes for simulating fiber optic communication systems,” IEEE Photonics J. 6(4), 7200515 (2014).

X. Liang, S. Kumar, J. Shao, M. Malekiha, and D. V. Plant, “Digital compensation of cross-phase modulation distortions using perturbation technique for dispersion-managed fiber-optic systems,” Opt. Express 22(17), 20634–20645 (2014).
[Crossref] [PubMed]

J. Shao, S. Kumar, and X. Liang, “Digital back propagation with optimal step size for polarization multiplexed transmission,” IEEE Photon. Technol. Lett. 25(23), 2327–2330 (2013).
[Crossref]

Shtaif, M.

Tao, Z.

Z. Tao, L. Dou, W. Yan, Y. Fan, L. Li, S. Oda, Y. Akiyama, H. Nakashima, T. Hoshida, and J. C. Rasmussen, “Complexity-reduced digital nonlinear compensation for coherent optical system,” Proc. SPIE 8647, 86470K (2013).

Z. Tao, L. Dou, W. Yan, L. Li, T. Hoshida, and J. C. Rasmussen, “Multiplier-free intrachannel nonlinearity compensating algorithm operating at symbol rate,” J. Lightwave Technol. 29(17), 2570–2576 (2011).
[Crossref]

Yam, S. S.-H.

Y. Gao, J. H. Ke, K. P. Zhong, J. C. Cartledge, and S. S.-H. Yam, “Assessment of intra-channel nonlinear compensation for 112 Gb/s dual polarization 16QAM systems,” J. Lightwave Technol. 30(24), 3902–3910 (2012).
[Crossref]

Y. Gao, J. C. Cartledge, A. S. Karar, and S. S.-H. Yam, “Reducing the complexity of nonlinearity pre-compensation using symmetric EDC and pulse shaping,” in 39th European Conference and Exhibition on Optical Communication (ECOC 2013), 1209–1219.
[Crossref]

Yaman, F.

Yan, W.

Z. Tao, L. Dou, W. Yan, Y. Fan, L. Li, S. Oda, Y. Akiyama, H. Nakashima, T. Hoshida, and J. C. Rasmussen, “Complexity-reduced digital nonlinear compensation for coherent optical system,” Proc. SPIE 8647, 86470K (2013).

Z. Tao, L. Dou, W. Yan, L. Li, T. Hoshida, and J. C. Rasmussen, “Multiplier-free intrachannel nonlinearity compensating algorithm operating at symbol rate,” J. Lightwave Technol. 29(17), 2570–2576 (2011).
[Crossref]

Yang, D.

Zhong, K. P.

IEEE J. Sel. Top. Quantum Electron. (1)

D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, and S. J. Savory, “Mitigation of fiber nonlinearity using a digital coherent receiver,” IEEE J. Sel. Top. Quantum Electron. 16(5), 1217–1226 (2010).
[Crossref]

IEEE Photon. Technol. Lett. (2)

J. Shao, S. Kumar, and X. Liang, “Digital back propagation with optimal step size for polarization multiplexed transmission,” IEEE Photon. Technol. Lett. 25(23), 2327–2330 (2013).
[Crossref]

A. Mecozzi, C. B. Clausen, and M. Shtaif, “Analysis of intrachannel nonlinear effects in highly dispersed optical pulse transmission,” IEEE Photon. Technol. Lett. 12(4), 392–394 (2000).
[Crossref]

IEEE Photonics J. (1)

J. Shao, X. Liang, and S. Kumar, “Comparison of split-step Fourier schemes for simulating fiber optic communication systems,” IEEE Photonics J. 6(4), 7200515 (2014).

J. Lightwave Technol. (6)

Opt. Express (10)

L. B. Du and A. J. Lowery, “Improved single channel backpropagation for intra-channel fiber nonlinearity compensation in long-haul optical communication systems,” Opt. Express 18(16), 17075–17088 (2010).
[Crossref] [PubMed]

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

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

S. Kumar, S. N. Shahi, and D. Yang, “Analytical modeling of a single channel nonlinear fiber optic system based on QPSK,” Opt. Express 20(25), 27740–27755 (2012).
[Crossref] [PubMed]

D. Rafique, M. Mussolin, J. Mårtensson, M. Forzati, J. K. Fischer, L. Molle, M. Nölle, C. Schubert, and A. D. Ellis, “Polarization multiplexed 16QAM transmission employing modified digital back-propagation,” Opt. Express 19(26), B805–B810 (2011).
[Crossref] [PubMed]

S. N. Shahi, S. Kumar, and X. Liang, “Analytical modeling of cross-phase modulation in coherent fiber-optic system,” Opt. Express 22(2), 1426–1439 (2014).
[Crossref] [PubMed]

X. Liang and S. Kumar, “Analytical modeling of XPM in dispersion-managed coherent fiber-optic systems,” Opt. Express 22(9), 10579–10592 (2014).
[Crossref] [PubMed]

X. Liang, S. Kumar, J. Shao, M. Malekiha, and D. V. Plant, “Digital compensation of cross-phase modulation distortions using perturbation technique for dispersion-managed fiber-optic systems,” Opt. Express 22(17), 20634–20645 (2014).
[Crossref] [PubMed]

X. Liang and S. Kumar, “Multi-stage perturbation theory for compensating intra-channel nonlinear impairments in fiber-optic links,” Opt. Express 22(24), 29733–29745 (2014).
[Crossref] [PubMed]

R. Dar, M. Feder, A. Mecozzi, and M. Shtaif, “Properties of nonlinear noise in long, dispersion-uncompensated fiber links,” Opt. Express 21(22), 25685–25699 (2013).
[Crossref] [PubMed]

Opt. Lett. (1)

Proc. SPIE (1)

Z. Tao, L. Dou, W. Yan, Y. Fan, L. Li, S. Oda, Y. Akiyama, H. Nakashima, T. Hoshida, and J. C. Rasmussen, “Complexity-reduced digital nonlinear compensation for coherent optical system,” Proc. SPIE 8647, 86470K (2013).

Other (9)

M. Secondini, D. Marsella, and E. Forestieri, “Enhanced split-step Fourier method for digital backpropagation,” in 40th European Conference and Exhibition on Optical Communication (ECOC 2014), We.3.3.5.
[Crossref]

Y. Gao, J. C. Cartledge, A. S. Karar, and S. S.-H. Yam, “Reducing the complexity of nonlinearity pre-compensation using symmetric EDC and pulse shaping,” in 39th European Conference and Exhibition on Optical Communication (ECOC 2013), 1209–1219.
[Crossref]

Y. Gao, A. S. Karar, J. C. Cartledge, S. S.-H. Yam, M. O’Sullivan, C. Laperle, A. Borowiec, and K. Roberts, “Simplified nonlinearity pre-compensation using a modified summation criteria and non-uniform power profile,” in Optical Fiber Communication Conference, (Optical Society of America, 2014), paper Tu3A.6.
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W. Yan, Z. Tao, L. Dou, L. Li, S. Oda, T. Tanimura, T. Hoshida, and J. Rasmussen, “Low complexity digital perturbation back-propagation,” in European Conference and Exhibition on Optical Communication (ECOC 2011), Tu.3.A.2.
[Crossref]

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Figures (5)

Fig. 1
Fig. 1 Schematic of a single-channel fiber-optic system. Tx: transmitter, BPF: band-pass filter, DBP: digital back propagation, LPF: low-pass filter, CPR: carrier phase recovery.
Fig. 2
Fig. 2 Q-factor versus launch power. Transmission distance = 40 × 80 km.
Fig. 3
Fig. 3 Correlation coefficients calculated using perturbation theory for correlated DBP.
Fig. 4
Fig. 4 Q-factor vs. the number of neighboring weighting factors.
Fig. 5
Fig. 5 Q-factor improvement over dispersion compensation only versus computational cost. Nstp is the number of DBP steps for the 40 × 80km fiber-optic link.

Equations (36)

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q(z,t) z + α(z) 2 q(z,t)+i β 2 (z) 2 2 q(z,t) t 2 i γ 0 | q(z,t) | 2 q(z,t)=0,
i u(z,t) z β 2 (z) 2 2 u(z,t) t 2 =γ(z) | u(z,t) | 2 u(z,t),
q(z,t)= e w(z)/2 u(z,t),
D ^ =i β 2 (z) 2 2 t 2 ,
N ^ =i γ 0 e αz | u(z,t) | 2 .
u(Δz,t)=exp( Δz 2 D ^ )exp[ i γ 0 0 Δz e αz | u(z,t) | 2 dz ]exp( Δz 2 D ^ )u(0,t),
θ(t)=i γ 0 | u(Δz/2,t) | 2 L eff ,
L eff = 1exp( αΔz ) α .
u(0,t)= P n= N sym /2+1 N sym /2 d n p(0,tn T 0 ) ,
u(0,t)= P n= N smp /2+1 N smp /2 a n g(0,tn T s ) ,
a n = k= N sym /2+1 N sym /2 d k 1 T s p(0,tk T 0 ) g (0,tn T s )dt.
u(z,t)= u (0) (z,t)+ γ 0 u (1) (z,t)+ γ 0 2 u (2) (z,t)+...,
i u (1) (z,t) z β 2 (z) 2 2 u (1) (z,t) t 2 = e w(z) | u (0) (z,t) | 2 u (0) (z,t).
u (0) (z,t)= P n= N smp /2+1 N smp /2 a n g(z,tn T s ) ,
g(z,t)= P ^ (z){ g(0,t) },
P ^ (z)=exp[ i S(z) 2 2 t 2 ],
g(z,t)= T s 2π π/ T s π/ T s exp[ iS(z) ω 2 /2iωt ] dω.
u (0) (Δz/2,t)= P n= N smp /2+1 N smp /2 a n g(Δz/2,tn T s ) .
u (0) (Δz/2,t)= P n= N smp /2+1 N smp /2 b n g(0,tn T s ).
b k = n= N smp /2+1 N smp /2 a n v kn ,
v n = T s 2π π/ T s π/ T s exp[ iS(Δz/2) ω 2 /2iωn T s ] dω.
b m =IDFT{ DFT{ a n }×DFT{ v n } }.
Δu(Δz,t)= γ 0 u (1) (Δz,t) = γ 0 0 Δz P ^ (Δzs){ F(s,t) }ds ,
F(s,t)=i e w(s) | u (0) (s,t) | 2 u (0) (s,t).
Δ u (Δz,t)= P ^ (Δz/2){ Δu(Δz,t) } = γ 0 0 Δz P ^ (Δz/2s){ F(s,t) }ds .
Δ u (Δz,t)= γ 0 P k= N smp /2+1 N smp /2 b k (1) g(0,tk T s ) .
b k (1) = i T s P 0 Δz ds e w(s) dt | u (0) (s,t) | 2 u (0) (s,t) g (sΔz/2,tk T s ).
u (0) (s,t)= P n= N smp /2+1 N smp /2 b n g(sΔz/2,tn T s ) .
b k (1) =iP m= N nb N nb n= N nb N nb b m+k b n+k b m+n+k X mn ,
X mn = 1 T s 0 Δz ds e w(s) dt g ( s Δz 2 ,t ) g( s Δz 2 ,tm T s )g( s Δz 2 ,tn T s ) g ( s Δz 2 ,t(m+n) T s ).
b k = b k + γ 0 b k (1) b k exp( i ϕ nl,k ),
ϕ nl,k = γ 0 P( | b k | 2 X k +2 n=1 N nb [ ( | b n | 2 + | b n | 2 ) X n ] ),
X n X m=0,n = 1 T s 0 Δz ds e w(s) dt | g( s Δz 2 ,t ) | 2 | g( s Δz 2 ,tn T s ) | 2 ,
u( Δz/2,t )= n= N smp /2+1 N smp /2 b n g( 0,tn T s ) .
c n = n= N smp /2+1 N smp /2 b n v nm ,
u( Δz,t )= n= N smp /2+1 N smp /2 c n g( 0,tn T s ) .

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