Abstract

A digital compensation scheme based on a perturbation theory for mitigation of cross-phase modulation (XPM) distortions is developed for dispersion-managed fiber-optic communication systems. It is a receiver-side scheme that uses a hard-decision unit to estimate data for the calculation of XPM fields using the perturbation technique. The intra-channel nonlinear distortions are removed by intra-channel digital backward propagation (DBP) based on split-step Fourier scheme before the hard-decision unit. The perturbation technique is shown to be effective in mitigating XPM distortions. However, wrong estimations in the hard-decision unit result in performance degradation. A hard-decision correction method is proposed to correct the wrong estimations. Numerical simulations show that the hybrid compensation scheme with DBP for dispersion and intra-channel nonlinear impairments compensation and the perturbation technique for XPM compensation brings up to 3.7 dBQ and 1.7 dBQ improvements as compared with the schemes of linear compensation only and intra-channel DBP, respectively. The perturbation technique for XPM compensation requires only one-stage (or two-stage when hard-decision correction is applied) compensation and symbol-rate signal processing.

© 2014 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Multi-stage perturbation theory for compensating intra-channel nonlinear impairments in fiber-optic links

Xiaojun Liang and Shiva Kumar
Opt. Express 22(24) 29733-29745 (2014)

Advanced perturbation technique for digital backward propagation in WDM systems

Lian Xiang, Paul Harper, and Xiaoping Zhang
Opt. Express 21(11) 13607-13616 (2013)

References

  • View by:
  • |
  • |
  • |

  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]
  2. S. Kumar, J. Mauro, S. Raghavan, and D. Chowdhury, “Intrachannel nonlinear penalties in dispersion-managed transmission systems,” IEEE J. Sel. Top. Quantum Electron. 8(3), 626–631 (2002).
    [Crossref]
  3. R.-J. Essiambre, G. Raybon, and B. Mikkelsen, “Pseudo-linear transmission of high-speed TDM signals: 40 and 160 Gb/s,” in Optical Fiber Telecommunication IVB, I. P. Kaminow and T. Li, ed. (Academic Press, 2002).
  4. D. Marcuse, A. R. Chraplyvy, and R. W. Tkach, “Dependence of cross-phase modulation on channel number in fiber WDM systems,” J. Lightwave Technol. 12(5), 885–890 (1994).
    [Crossref]
  5. R. Hui, Y. Wang, K. Demarest, and C. Allen, “Frequency response of cross-phase modulation in multispan WDM optical fiber systems,” IEEE Photon. Technol. Lett. 10(9), 1271–1273 (1998).
    [Crossref]
  6. A. Cartaxo, “Cross-phase modulation in intensity modulation-direct detection WDM systems with multiple optical amplifiers and dispersion compensators,” J. Lightwave Technol. 17(2), 178–190 (1999).
    [Crossref]
  7. A. Mecozzi and R. Essiambre, “Nonlinear Shannon limit in pseudolinear coherent systems,” J. Lightwave Technol. 30(12), 2011–2024 (2012).
    [Crossref]
  8. 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]
  9. E. Ip and J. M. Kahn, “Compensation of dispersion and nonlinear impairments using digital backpropagation,” J. Lightwave Technol. 26(20), 3416–3425 (2008).
    [Crossref]
  10. J. Shao and S. Kumar, “Optical backpropagation for fiber-optic communications using optical phase conjugation at the receiver,” Opt. Lett. 37(15), 3012–3014 (2012).
    [Crossref] [PubMed]
  11. S. Kumar and J. Shao, “Optical back propagation with optimal step size for fiber optic transmission systems,” IEEE Photon. Technol. Lett. 25(5), 523–526 (2013).
    [Crossref]
  12. 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]
  13. T. Oyama, H. Nakashima, S. Oda, T. Yamauchi, Z. Tao, T. Hoshida, and J. C. Rasmussen, “Robust and efficient receiver-side compensation method for intra-channel nonlinear effects,” in Optical Fiber Communication Conference, (Optical Society of America, 2014), paper Tu3A.3.
    [Crossref]
  14. Y. Fan, L. Dou, Z. Tao, T. Hoshida, and J. C. Rasmussen, “A high performance nonlinear compensation algorithm with reduced complexity based on XPM model,” in Optical Fiber Communication Conference, (Optical Society of America, 2014), paper Th2A.8.
    [Crossref]
  15. 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), 1–3.
    [Crossref]
  16. 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.
    [Crossref]
  17. E. Mateo, L. Zhu, and G. Li, “Impact of XPM and FWM on the digital implementation of impairment compensation for WDM transmission using backward propagation,” Opt. Express 16(20), 16124–16137 (2008).
    [Crossref] [PubMed]
  18. E. F. Mateo, X. Zhou, and G. Li, “Improved digital backward propagation for the compensation of inter-channel nonlinear effects in polarization-multiplexed WDM systems,” Opt. Express 19(2), 570–583 (2011).
    [Crossref] [PubMed]
  19. 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]
  20. 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]
  21. 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]
  22. X. Liu and S. Chandrasekhar, “Superchannel for next-generation optical networks,” in Optical Fiber Communication Conference,(Optical Society of America, 2014), paper W1H.5.
    [Crossref]
  23. X. Liu, S. Chandrasekhar, and P. J. Winzer, “Digital signal processing techniques enabling multi-Tb/s superchannel transmission: an overview of recent advances in DSP-enabled superchannels,” IEEE Signal Process. Mag. 31(2), 16–24 (2014).
    [Crossref]
  24. T. Zeng, “Superchannel transmission system based on multi-channel equalization,” Opt. Express 21(12), 14799–14807 (2013).
    [Crossref] [PubMed]
  25. S. F. Boys, “Electronic wave functions. I. A general method of calculation for the stationary states of any molecular system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 200(1063), 542–554 (1950).
    [Crossref]
  26. T. Pfau, S. Hoffmann, and R. Noé, “Hardware-efficient coherent digital receiver concept with feedforward carrier recovery for M-QAM constellations,” J. Lightwave Technol. 27(8), 989–999 (2009).
    [Crossref]

2014 (2)

X. Liu, S. Chandrasekhar, and P. J. Winzer, “Digital signal processing techniques enabling multi-Tb/s superchannel transmission: an overview of recent advances in DSP-enabled superchannels,” IEEE Signal Process. Mag. 31(2), 16–24 (2014).
[Crossref]

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]

2013 (3)

2012 (2)

2011 (2)

2009 (1)

2008 (3)

2005 (1)

2002 (1)

S. Kumar, J. Mauro, S. Raghavan, and D. Chowdhury, “Intrachannel nonlinear penalties in dispersion-managed transmission systems,” IEEE J. Sel. Top. Quantum Electron. 8(3), 626–631 (2002).
[Crossref]

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]

1999 (1)

1998 (1)

R. Hui, Y. Wang, K. Demarest, and C. Allen, “Frequency response of cross-phase modulation in multispan WDM optical fiber systems,” IEEE Photon. Technol. Lett. 10(9), 1271–1273 (1998).
[Crossref]

1994 (1)

D. Marcuse, A. R. Chraplyvy, and R. W. Tkach, “Dependence of cross-phase modulation on channel number in fiber WDM systems,” J. Lightwave Technol. 12(5), 885–890 (1994).
[Crossref]

1950 (1)

S. F. Boys, “Electronic wave functions. I. A general method of calculation for the stationary states of any molecular system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 200(1063), 542–554 (1950).
[Crossref]

Allen, C.

R. Hui, Y. Wang, K. Demarest, and C. Allen, “Frequency response of cross-phase modulation in multispan WDM optical fiber systems,” IEEE Photon. Technol. Lett. 10(9), 1271–1273 (1998).
[Crossref]

Boys, S. F.

S. F. Boys, “Electronic wave functions. I. A general method of calculation for the stationary states of any molecular system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 200(1063), 542–554 (1950).
[Crossref]

Cartaxo, A.

Cartledge, J. C.

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), 1–3.
[Crossref]

Chandrasekhar, S.

X. Liu, S. Chandrasekhar, and P. J. Winzer, “Digital signal processing techniques enabling multi-Tb/s superchannel transmission: an overview of recent advances in DSP-enabled superchannels,” IEEE Signal Process. Mag. 31(2), 16–24 (2014).
[Crossref]

Chen, X.

Chowdhury, D.

S. Kumar, J. Mauro, S. Raghavan, and D. Chowdhury, “Intrachannel nonlinear penalties in dispersion-managed transmission systems,” IEEE J. Sel. Top. Quantum Electron. 8(3), 626–631 (2002).
[Crossref]

Chraplyvy, A. R.

D. Marcuse, A. R. Chraplyvy, and R. W. Tkach, “Dependence of cross-phase modulation on channel number in fiber WDM systems,” J. Lightwave Technol. 12(5), 885–890 (1994).
[Crossref]

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.

Demarest, K.

R. Hui, Y. Wang, K. Demarest, and C. Allen, “Frequency response of cross-phase modulation in multispan WDM optical fiber systems,” IEEE Photon. Technol. Lett. 10(9), 1271–1273 (1998).
[Crossref]

Dou, L.

Essiambre, R.

Feder, M.

Gao, Y.

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), 1–3.
[Crossref]

Goldfarb, G.

Hoffmann, S.

Hoshida, T.

Hui, R.

R. Hui, Y. Wang, K. Demarest, and C. Allen, “Frequency response of cross-phase modulation in multispan WDM optical fiber systems,” IEEE Photon. Technol. Lett. 10(9), 1271–1273 (1998).
[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), 1–3.
[Crossref]

Kim, I.

Kumar, S.

Li, G.

Li, L.

Li, X.

Liang, X.

Liu, X.

X. Liu, S. Chandrasekhar, and P. J. Winzer, “Digital signal processing techniques enabling multi-Tb/s superchannel transmission: an overview of recent advances in DSP-enabled superchannels,” IEEE Signal Process. Mag. 31(2), 16–24 (2014).
[Crossref]

Marcuse, D.

D. Marcuse, A. R. Chraplyvy, and R. W. Tkach, “Dependence of cross-phase modulation on channel number in fiber WDM systems,” J. Lightwave Technol. 12(5), 885–890 (1994).
[Crossref]

Mateo, E.

Mateo, E. F.

Mauro, J.

S. Kumar, J. Mauro, S. Raghavan, and D. Chowdhury, “Intrachannel nonlinear penalties in dispersion-managed transmission systems,” IEEE J. Sel. Top. Quantum Electron. 8(3), 626–631 (2002).
[Crossref]

Mecozzi, A.

Noé, R.

Pfau, T.

Raghavan, S.

S. Kumar, J. Mauro, S. Raghavan, and D. Chowdhury, “Intrachannel nonlinear penalties in dispersion-managed transmission systems,” IEEE J. Sel. Top. Quantum Electron. 8(3), 626–631 (2002).
[Crossref]

Rasmussen, J. C.

Shao, J.

S. Kumar and J. Shao, “Optical back propagation with optimal step size for fiber optic transmission systems,” IEEE Photon. Technol. Lett. 25(5), 523–526 (2013).
[Crossref]

J. Shao and S. Kumar, “Optical backpropagation for fiber-optic communications using optical phase conjugation at the receiver,” Opt. Lett. 37(15), 3012–3014 (2012).
[Crossref] [PubMed]

Shtaif, M.

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]

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]

Tao, Z.

Tkach, R. W.

D. Marcuse, A. R. Chraplyvy, and R. W. Tkach, “Dependence of cross-phase modulation on channel number in fiber WDM systems,” J. Lightwave Technol. 12(5), 885–890 (1994).
[Crossref]

Wang, Y.

R. Hui, Y. Wang, K. Demarest, and C. Allen, “Frequency response of cross-phase modulation in multispan WDM optical fiber systems,” IEEE Photon. Technol. Lett. 10(9), 1271–1273 (1998).
[Crossref]

Winzer, P. J.

X. Liu, S. Chandrasekhar, and P. J. Winzer, “Digital signal processing techniques enabling multi-Tb/s superchannel transmission: an overview of recent advances in DSP-enabled superchannels,” IEEE Signal Process. Mag. 31(2), 16–24 (2014).
[Crossref]

Yam, S. S.-H.

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), 1–3.
[Crossref]

Yaman, F.

Yan, W.

Yang, D.

Zeng, T.

Zhou, X.

Zhu, L.

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

S. Kumar, J. Mauro, S. Raghavan, and D. Chowdhury, “Intrachannel nonlinear penalties in dispersion-managed transmission systems,” IEEE J. Sel. Top. Quantum Electron. 8(3), 626–631 (2002).
[Crossref]

IEEE Photon. Technol. Lett. (3)

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]

R. Hui, Y. Wang, K. Demarest, and C. Allen, “Frequency response of cross-phase modulation in multispan WDM optical fiber systems,” IEEE Photon. Technol. Lett. 10(9), 1271–1273 (1998).
[Crossref]

S. Kumar and J. Shao, “Optical back propagation with optimal step size for fiber optic transmission systems,” IEEE Photon. Technol. Lett. 25(5), 523–526 (2013).
[Crossref]

IEEE Signal Process. Mag. (1)

X. Liu, S. Chandrasekhar, and P. J. Winzer, “Digital signal processing techniques enabling multi-Tb/s superchannel transmission: an overview of recent advances in DSP-enabled superchannels,” IEEE Signal Process. Mag. 31(2), 16–24 (2014).
[Crossref]

J. Lightwave Technol. (7)

Opt. Express (6)

Opt. Lett. (1)

Proc. R. Soc. Lond. A Math. Phys. Sci. (1)

S. F. Boys, “Electronic wave functions. I. A general method of calculation for the stationary states of any molecular system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 200(1063), 542–554 (1950).
[Crossref]

Other (6)

R.-J. Essiambre, G. Raybon, and B. Mikkelsen, “Pseudo-linear transmission of high-speed TDM signals: 40 and 160 Gb/s,” in Optical Fiber Telecommunication IVB, I. P. Kaminow and T. Li, ed. (Academic Press, 2002).

T. Oyama, H. Nakashima, S. Oda, T. Yamauchi, Z. Tao, T. Hoshida, and J. C. Rasmussen, “Robust and efficient receiver-side compensation method for intra-channel nonlinear effects,” in Optical Fiber Communication Conference, (Optical Society of America, 2014), paper Tu3A.3.
[Crossref]

Y. Fan, L. Dou, Z. Tao, T. Hoshida, and J. C. Rasmussen, “A high performance nonlinear compensation algorithm with reduced complexity based on XPM model,” in Optical Fiber Communication Conference, (Optical Society of America, 2014), paper Th2A.8.
[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), 1–3.
[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.
[Crossref]

X. Liu and S. Chandrasekhar, “Superchannel for next-generation optical networks,” in Optical Fiber Communication Conference,(Optical Society of America, 2014), paper W1H.5.
[Crossref]

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1
Fig. 1

Schematic of a dispersion-managed fiber-optic WDM system using perturbation-based nonlinearity compensation. Tx: transmitter; MUX: multiplexer; DCF: dispersion compensating fiber; G1, G2: amplifier gains; DMUX: demultiplexer; LO: local oscillator; ADC: analog-to-digital convertor; DC: dispersion compensator; CPR: carrier phase recovery; Ymn: coefficient matrix stored in a lookup table; DSP: digital signal processing.

Fig. 2
Fig. 2

Constellations of recovered signals in a 2-channel WDM system: (a) linear compensation only; (b) nonlinear compensation of both intra-channel and XPM distortions using the perturbation technique. (average power per channel Pave = −6 dBm)

Fig. 3
Fig. 3

Q versus average launch power per channel in a 2-channel WDM system.

Fig. 4
Fig. 4

Diagram of a hybrid nonlinearity compensation scheme using DBP for intra-channel impairments compensation and the perturbation technique for XPM compensation. LPF: low pass filter.

Fig. 5
Fig. 5

Constellations of recovered signals in a 2-channel WDM system: (a) linear compensation only; (b) nonlinearity compensation using intra-channel DBP only (step size = 40 km); (c) nonlinearity compensation using intra-channel DBP and the perturbation technique for XPM; (d) after hard-decision correction. (average power per channel Pave = −3 dBm)

Fig. 6
Fig. 6

Diagram of a nonlinearity compensation scheme using hard-decision correction.

Fig. 7
Fig. 7

Q-factor versus normalized threshold distance.

Fig. 8
Fig. 8

Q-factor versus average launch power per channel in a 2-channel WDM system: (a) intra-channel DBP step size = 2 km; (b) intra-channel DBP step size = 40 km.

Fig. 9
Fig. 9

Q-factor versus average launch power per channel in a 5-channel WDM system: (a) intra-channel DBP step size = 2 km; (b) intra-channel DBP step size = 40 km.

Tables (1)

Tables Icon

Table1 Fitting parameters optimized by LSM

Equations (33)

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

j u z β 2 (z) 2 2 u T 2 +γ(z) | u | 2 u=0,
u(z,T)=exp[ w(z)/2 ]q(z,T),
w(z)= 0 z α(s)ds ,γ(z)= γ 0 exp[ w(z) ],
u= u 1 + u 2 ,
j u k z β 2 (z) 2 2 u k T 2 =γ(z)[ | u k | 2 +2 | u l | 2 ] u k ,k=1,2andl=3k.
u 1 (0,T)= P a 0 g(0,T),
u 2 (0,T)= P n= N sym N sym b n g(0,Tn T s )exp(jΩT),
g(0,T)=exp( T 2 2 T 0 2 ),
u k = u k (0) + γ 0 u k (1) + γ 0 2 u k (2) +...,k=1,2,
j u k (1) z β 2 (z) 2 2 u k (1) T 2 = e w(z) [ | u k (0) | 2 +2 | u l (0) | 2 ] u k (0) ,k=1,2andl=3k.
Δ u 1 XPM (T)= γ 0 u 1 (1),XPM =2j γ 0 P 3/2 a 0 m= N sym N sym n= N sym N sym b m b n X mn ( L tot ,T),
X mn ( L tot ,T)= 0 L tot η (s) δ( L tot ,s)R(s) exp[ (D+jT) 2 δ( L tot ,s) ]ds,
η(z)= T 0 3 exp[ w(z) ] T 1 (z) | T 1 (z) | 2 , η (z)=η(z)exp( k=1 3 C k 2 R k + C 2 R )
δ(z,s)=1/Rj2[ S(z)S(s) ],S(z)= 0 z β 2 (s)ds ,
C 1 (z)=m T s +S(z)Ω, C 2 (z)=n T s +S(z)Ω, C 3 (z)=0,
R 1 = R 3 = 1 2 T 1 2 , R 2 = 1 2 ( T 1 ) 2 , T 1 = T 0 2 jS(z) ,
R= R 1 + R 2 + R 3 ,C= C 1 R 1 + C 2 R 2 + C 3 R 3 ,D=jC/R.
Δ u 1 intra (T)= γ 0 u 1 (1),intra =j γ 0 P 3/2 a 0 m= N sym N sym n= N sym N sym a m a n X mn ( L tot ,T),
h (T)= k=1 K ξ k exp[ ( T μ k T s ) 2 2 ( θ k T s ) 2 ] ,
Δ u 1 intra,NG (T)= γ 0 u 1 (1),intra,NG =j γ 0 P 3/2 a 0 m= N sym N sym n= N sym N sym a m a n Y mn ( L tot ,T),
Δ u 1 XPM,NG (T)= γ 0 u 1 (1),XPM,NG =2j γ 0 P 3/2 a 0 m= N sym N sym n= N sym N sym b m b n Y mn ( L tot ,T),
Y mn = 0 L tot k 1 =1 K k 2 =1 K k 3 =1 K η (s) δ( L tot ,s)R(s) exp[ (D+jT) 2 δ( L tot ,s) ]ds,
T 1,k = ( θ k T s ) 2 jS(z) ,η(z)= e w(z) ξ k 1 ξ k 2 ξ k 3 θ k 1 θ k 2 θ k 3 T s 3 T 1, k 1 T 1, k 2 T 1, k 3 ,
C 1 (z)= τ m, k 1 , C 2 (z)= τ n, k 2 , C 3 (z)= μ k 3 T s ,
R 1 (z)= 1 2 T 1, k 1 2 , R 2 (z)= 1 2 ( T 1, k 2 ) 2 , R 3 (z)= 1 2 T 1, k 3 2 ,
τ n,k =(n+ μ k ) T s +S(z)Ω.
Δ u 1 XPM,NG (T)=2j γ 0 P 3/2 l=M M a l m= N sym N sym n= N sym N sym b m b n Y mn (l) ( L tot ,T),
Δ u 1 intra,NG (T)=j γ 0 P 3/2 l=M M a l m= N sym N sym n= N sym N sym a m a n Y mn (l) ( L tot ,T).
Δ u XPM = k=1 N nb Δ u 1,k XPM,NG ,
u out = u in +Δ u intra +Δ u XPM +Δ u H ,
u comp = u out Δ u intra Δ u XPM .
x(t)=sinc( t T s ) cos(aπt/ T s ) 1 (2at/ T s ) 2 .
| si g out si g constel |>r

Metrics