Abstract

We introduce a new system configuration to reduce the nonlinear phase noise (NLPN) by splitting the digital back propagation (DBP) between transmitter and receiver, asymmetrically, along with using mid-line optical phase conjugation (OPC). Our analytical results show that the variance of NLPN reduces by a factor of 16 compared to the standard configuration which is the dispersion uncompensated fiber optic link with full DBP at the receiver, i.e., the back propagation for the fiber spans is done entirely at the receiver. Numerical simulations show the same trend as predicted by the analytical model, and show about 2.6 dB and 2 dB improvement in Q-factor, for single channel and 5-channel WDM systems, respectively.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. G. P. Agrawal, Nonlinear Fiber Optics, 5th ed. (Academic, 2012).
  2. S. Kumar and M. J. Deen, Fiber Optic Communications: Fundamentals and Applications (John Wiley & Sons, 2014).
    [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 Telecommunications IV-B, 4th ed., I. P. Kaminow and T. Li, eds. (Academic, 2002).
    [Crossref]
  4. S. Kumar, J. C. Mauro, S. Raghavan, and D. Q. Chowdhury, “Intrachannel nonlinear penalties in dispersion-managed transmission systems,” IEEE J. Sel. Top. Quantum Electron. 8(3), 626–631 (2002).
    [Crossref]
  5. J. P. Gordon and L. F. Mollenauer, “Phase noise in photonic communications systems using linear amplifiers,” Opt. Lett. 15(23), 1351–1353 (1990).
    [Crossref] [PubMed]
  6. R.-J. Essiambre and P. J. Winzer, “Fibre nonlinearities in electronically pre-distorted transmission,” in 2005 European Conference on Optical Communication (IEEE, 2005), pp. 191–192.
  7. E. Ip and J. M. Kahn, “Compensation of dispersion and nonlinear impairments using digital backpropagation,” J. Lightwave Technol. 26(20), 3416–3425 (2008).
    [Crossref]
  8. X. Li, X. Chen, G. Goldfarb, E. Mateo, I. Kim, Fatih 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. A. Mecozzi, “Limits to the long haul coherent transmission set by the Kerr nonlinearity and noise of in-line amplifiers,” J. Lightwave Technol. 12(11), 1993–2000 (1994).
    [Crossref]
  10. K. Ho, “Probability density of nonlinear phase noise,” J. Opt. Soc. Am. B: Opt. Phys. 20(9), 1875–1879 (2003).
    [Crossref]
  11. A. Mecozzi, “Probability density functions of the nonlinear phase noise,” Opt. Lett. 29(7), 673–675 (2004).
    [Crossref] [PubMed]
  12. S. Kumar, “Effect of dispersion on nonlinear phase noise in optical transmission systems,” Opt. Lett. 30(24), 3278–3280 (2005).
    [Crossref]
  13. S. Kumar, “Analysis of nonlinear phase noise in coherent fiber-optic systems based on phase shift keying,” J. Lightwave Technol. 27(21), 4722–4733 (2009).
    [Crossref]
  14. P. Minzioni, V. Pusino, I. Cristiani, L. Marazzi, M. Martinelli, and V. Degiorgio, “Study of the Gordon-Mollenauer effect and of the optical-phase-conjugation compensation method in phase-modulated optical communication systems,” IEEE Photonics J. 2(3), 284–291 (2010).
    [Crossref]
  15. A. Demir, “Nonlinear phase noise in optical fiber communication systems,” J. Lightwave Technol. 25(8), 2002–2032 (2007).
    [Crossref]
  16. N. Ekanayake and H. Herath, “Effect of nonlinear phase noise on the performance of M-ary PSK signals in optical fiber links,” J. Lightwave Technol. 31(3), 447–454 (2013).
    [Crossref]
  17. Z. Xu, P. Y. Kam, and C. Yu, “Adaptive maximum likelihood sequence detection for QPSK coherent optical communication system,” IEEE Photonics Technol. Lett. 26(6), 583–586 (2014).
    [Crossref]
  18. X. Zhu and S. Kumar, “Nonlinear phase noise in coherent optical OFDM transmission systems,” Opt. Express 18(7), 7347–7360 (2010).
    [Crossref] [PubMed]
  19. C. Pan, H. Bülow, W. Idler, L. Schmalen, and F. Kschischang, “Optical nonlinear phase noise compensation for 9 × 32 -Gbaud PolDM-16 QAM transmission using a code-aided expectation-maximization algorithm,” J. Lightwave Technol. 33(17), 3679–3686 (2015).
    [Crossref]
  20. D. Wang, M. Zhang, Z. Li, Y. Cui, J. Liu, Y. Yang, and H. Wang, “Nonlinear decision boundary created by a machine learning-based classifier to mitigate nonlinear phase noise,” in 2015 European Conference on Optical Communication (IEEE, 2015), pp. 1–3.
  21. D. Lavery, D. Ives, G. Liga, A. Alvarado, S. J. Savory, and P. Bayvel, “The benefit of split nonlinearity compensation for single-channel optical fiber communications,” IEEE Photonics Technol. Lett. 28(617), 1803–1806 (2016).
    [Crossref]
  22. D. Semrau, D. Lavery, L. Galdino, R. I. Killey, and P. Bayvel, “The impact of transceiver noise on digital nonlinearity compensation,” J. Lightwave Technol. 36(3), 695–702 (2018).
    [Crossref]
  23. A. Yariv, D. Fekete, and D. M. Pepper, “Compensation for channel dispersion by nonlinear optical phase conjugation,” Opt. Lett. 4(2), 52–54 (1979).
    [Crossref] [PubMed]
  24. S. Watanabe, G. Ishikawa, T. Naito, and T. Chikama, “Generation of optical phase-conjugate waves and compensation for pulse shape distortion in a single-mode fiber,” J. Lightwave Technol. 12(12), 2139–2146(1994).
    [Crossref]
  25. S. Kumar and L. Liu, “Reduction of nonlinear phase noise using optical phase conjugation in quasi-linear optical transmission systems,” Opt. Express 15(5), 2166–2177 (2007).
    [Crossref] [PubMed]
  26. K. P. Ho, “Cross-Phase Modulation-Induced Nonlinear Phase Noise for Quadriphase-Shift-Keying Signals,” in Impact of Nonlinearities on Fiber Optic Communications, S. Kumar, ed. (Springer-Verlag, 2011).
    [Crossref]
  27. P. K. A. Wai and C. R. Menyak, “Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14(2), 148–157 (1996).
    [Crossref]
  28. C. McKinstrie, S. Radic, and C. Xie, “Reduction of soliton phase jitter by in-line phase conjugation,” Opt. Lett. 28(17), 1519–1521 (2003).
    [Crossref] [PubMed]
  29. L. M. Zhang and F. R. Kschischang, “Staircase codes with 6% to 33% overhead,” J. Lightwave Technol. 32(10), 1999–2002 (2014).
    [Crossref]
  30. R. Maher, D. Lavery, D. Millar, A. Alvarado, K. Parsons, R. Killey, and P. Bayvel, “Reach enhancement of 100% for a DP-64QAM super-channel using MC-DBP,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2015), paper Th4D.5.
  31. L. Galdino, D. Semrau, D. Lavery, G. Saavedra, C. B. Czegledi, E. Agrell, R. I. Killey, and P. Bayvel, “On the limits of digital back-propagation in the presence of transceiver noise,” Opt. Express 25(4), 4564–4578 (2017).
    [Crossref] [PubMed]

2018 (1)

D. Semrau, D. Lavery, L. Galdino, R. I. Killey, and P. Bayvel, “The impact of transceiver noise on digital nonlinearity compensation,” J. Lightwave Technol. 36(3), 695–702 (2018).
[Crossref]

2017 (1)

L. Galdino, D. Semrau, D. Lavery, G. Saavedra, C. B. Czegledi, E. Agrell, R. I. Killey, and P. Bayvel, “On the limits of digital back-propagation in the presence of transceiver noise,” Opt. Express 25(4), 4564–4578 (2017).
[Crossref] [PubMed]

2016 (1)

D. Lavery, D. Ives, G. Liga, A. Alvarado, S. J. Savory, and P. Bayvel, “The benefit of split nonlinearity compensation for single-channel optical fiber communications,” IEEE Photonics Technol. Lett. 28(617), 1803–1806 (2016).
[Crossref]

2015 (1)

C. Pan, H. Bülow, W. Idler, L. Schmalen, and F. Kschischang, “Optical nonlinear phase noise compensation for 9 × 32 -Gbaud PolDM-16 QAM transmission using a code-aided expectation-maximization algorithm,” J. Lightwave Technol. 33(17), 3679–3686 (2015).
[Crossref]

2014 (2)

Z. Xu, P. Y. Kam, and C. Yu, “Adaptive maximum likelihood sequence detection for QPSK coherent optical communication system,” IEEE Photonics Technol. Lett. 26(6), 583–586 (2014).
[Crossref]

L. M. Zhang and F. R. Kschischang, “Staircase codes with 6% to 33% overhead,” J. Lightwave Technol. 32(10), 1999–2002 (2014).
[Crossref]

2013 (1)

N. Ekanayake and H. Herath, “Effect of nonlinear phase noise on the performance of M-ary PSK signals in optical fiber links,” J. Lightwave Technol. 31(3), 447–454 (2013).
[Crossref]

2010 (2)

X. Zhu and S. Kumar, “Nonlinear phase noise in coherent optical OFDM transmission systems,” Opt. Express 18(7), 7347–7360 (2010).
[Crossref] [PubMed]

P. Minzioni, V. Pusino, I. Cristiani, L. Marazzi, M. Martinelli, and V. Degiorgio, “Study of the Gordon-Mollenauer effect and of the optical-phase-conjugation compensation method in phase-modulated optical communication systems,” IEEE Photonics J. 2(3), 284–291 (2010).
[Crossref]

2009 (1)

S. Kumar, “Analysis of nonlinear phase noise in coherent fiber-optic systems based on phase shift keying,” J. Lightwave Technol. 27(21), 4722–4733 (2009).
[Crossref]

2008 (2)

E. Ip and J. M. Kahn, “Compensation of dispersion and nonlinear impairments using digital backpropagation,” J. Lightwave Technol. 26(20), 3416–3425 (2008).
[Crossref]

X. Li, X. Chen, G. Goldfarb, E. Mateo, I. Kim, Fatih 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]

2007 (2)

A. Demir, “Nonlinear phase noise in optical fiber communication systems,” J. Lightwave Technol. 25(8), 2002–2032 (2007).
[Crossref]

S. Kumar and L. Liu, “Reduction of nonlinear phase noise using optical phase conjugation in quasi-linear optical transmission systems,” Opt. Express 15(5), 2166–2177 (2007).
[Crossref] [PubMed]

2005 (1)

S. Kumar, “Effect of dispersion on nonlinear phase noise in optical transmission systems,” Opt. Lett. 30(24), 3278–3280 (2005).
[Crossref]

2004 (1)

A. Mecozzi, “Probability density functions of the nonlinear phase noise,” Opt. Lett. 29(7), 673–675 (2004).
[Crossref] [PubMed]

2003 (2)

K. Ho, “Probability density of nonlinear phase noise,” J. Opt. Soc. Am. B: Opt. Phys. 20(9), 1875–1879 (2003).
[Crossref]

C. McKinstrie, S. Radic, and C. Xie, “Reduction of soliton phase jitter by in-line phase conjugation,” Opt. Lett. 28(17), 1519–1521 (2003).
[Crossref] [PubMed]

2002 (1)

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

1996 (1)

P. K. A. Wai and C. R. Menyak, “Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14(2), 148–157 (1996).
[Crossref]

1994 (2)

S. Watanabe, G. Ishikawa, T. Naito, and T. Chikama, “Generation of optical phase-conjugate waves and compensation for pulse shape distortion in a single-mode fiber,” J. Lightwave Technol. 12(12), 2139–2146(1994).
[Crossref]

A. Mecozzi, “Limits to the long haul coherent transmission set by the Kerr nonlinearity and noise of in-line amplifiers,” J. Lightwave Technol. 12(11), 1993–2000 (1994).
[Crossref]

1990 (1)

J. P. Gordon and L. F. Mollenauer, “Phase noise in photonic communications systems using linear amplifiers,” Opt. Lett. 15(23), 1351–1353 (1990).
[Crossref] [PubMed]

1979 (1)

A. Yariv, D. Fekete, and D. M. Pepper, “Compensation for channel dispersion by nonlinear optical phase conjugation,” Opt. Lett. 4(2), 52–54 (1979).
[Crossref] [PubMed]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics, 5th ed. (Academic, 2012).

Agrell, E.

L. Galdino, D. Semrau, D. Lavery, G. Saavedra, C. B. Czegledi, E. Agrell, R. I. Killey, and P. Bayvel, “On the limits of digital back-propagation in the presence of transceiver noise,” Opt. Express 25(4), 4564–4578 (2017).
[Crossref] [PubMed]

Alvarado, A.

D. Lavery, D. Ives, G. Liga, A. Alvarado, S. J. Savory, and P. Bayvel, “The benefit of split nonlinearity compensation for single-channel optical fiber communications,” IEEE Photonics Technol. Lett. 28(617), 1803–1806 (2016).
[Crossref]

R. Maher, D. Lavery, D. Millar, A. Alvarado, K. Parsons, R. Killey, and P. Bayvel, “Reach enhancement of 100% for a DP-64QAM super-channel using MC-DBP,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2015), paper Th4D.5.

Bayvel, P.

D. Semrau, D. Lavery, L. Galdino, R. I. Killey, and P. Bayvel, “The impact of transceiver noise on digital nonlinearity compensation,” J. Lightwave Technol. 36(3), 695–702 (2018).
[Crossref]

L. Galdino, D. Semrau, D. Lavery, G. Saavedra, C. B. Czegledi, E. Agrell, R. I. Killey, and P. Bayvel, “On the limits of digital back-propagation in the presence of transceiver noise,” Opt. Express 25(4), 4564–4578 (2017).
[Crossref] [PubMed]

D. Lavery, D. Ives, G. Liga, A. Alvarado, S. J. Savory, and P. Bayvel, “The benefit of split nonlinearity compensation for single-channel optical fiber communications,” IEEE Photonics Technol. Lett. 28(617), 1803–1806 (2016).
[Crossref]

R. Maher, D. Lavery, D. Millar, A. Alvarado, K. Parsons, R. Killey, and P. Bayvel, “Reach enhancement of 100% for a DP-64QAM super-channel using MC-DBP,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2015), paper Th4D.5.

Bülow, H.

C. Pan, H. Bülow, W. Idler, L. Schmalen, and F. Kschischang, “Optical nonlinear phase noise compensation for 9 × 32 -Gbaud PolDM-16 QAM transmission using a code-aided expectation-maximization algorithm,” J. Lightwave Technol. 33(17), 3679–3686 (2015).
[Crossref]

Chen, X.

X. Li, X. Chen, G. Goldfarb, E. Mateo, I. Kim, Fatih 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]

Chikama, T.

S. Watanabe, G. Ishikawa, T. Naito, and T. Chikama, “Generation of optical phase-conjugate waves and compensation for pulse shape distortion in a single-mode fiber,” J. Lightwave Technol. 12(12), 2139–2146(1994).
[Crossref]

Chowdhury, D. Q.

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

Cristiani, I.

P. Minzioni, V. Pusino, I. Cristiani, L. Marazzi, M. Martinelli, and V. Degiorgio, “Study of the Gordon-Mollenauer effect and of the optical-phase-conjugation compensation method in phase-modulated optical communication systems,” IEEE Photonics J. 2(3), 284–291 (2010).
[Crossref]

Cui, Y.

D. Wang, M. Zhang, Z. Li, Y. Cui, J. Liu, Y. Yang, and H. Wang, “Nonlinear decision boundary created by a machine learning-based classifier to mitigate nonlinear phase noise,” in 2015 European Conference on Optical Communication (IEEE, 2015), pp. 1–3.

Czegledi, C. B.

L. Galdino, D. Semrau, D. Lavery, G. Saavedra, C. B. Czegledi, E. Agrell, R. I. Killey, and P. Bayvel, “On the limits of digital back-propagation in the presence of transceiver noise,” Opt. Express 25(4), 4564–4578 (2017).
[Crossref] [PubMed]

Deen, M. J.

S. Kumar and M. J. Deen, Fiber Optic Communications: Fundamentals and Applications (John Wiley & Sons, 2014).
[Crossref]

Degiorgio, V.

P. Minzioni, V. Pusino, I. Cristiani, L. Marazzi, M. Martinelli, and V. Degiorgio, “Study of the Gordon-Mollenauer effect and of the optical-phase-conjugation compensation method in phase-modulated optical communication systems,” IEEE Photonics J. 2(3), 284–291 (2010).
[Crossref]

Demir, A.

A. Demir, “Nonlinear phase noise in optical fiber communication systems,” J. Lightwave Technol. 25(8), 2002–2032 (2007).
[Crossref]

Ekanayake, N.

N. Ekanayake and H. Herath, “Effect of nonlinear phase noise on the performance of M-ary PSK signals in optical fiber links,” J. Lightwave Technol. 31(3), 447–454 (2013).
[Crossref]

Essiambre, R.-J.

R.-J. Essiambre, G. Raybon, and B. Mikkelsen, “Pseudo-Linear Transmission Of High-Speed TDM Signals: 40 And 160 Gb/s,” in Optical Fiber Telecommunications IV-B, 4th ed., I. P. Kaminow and T. Li, eds. (Academic, 2002).
[Crossref]

R.-J. Essiambre and P. J. Winzer, “Fibre nonlinearities in electronically pre-distorted transmission,” in 2005 European Conference on Optical Communication (IEEE, 2005), pp. 191–192.

Fekete, D.

A. Yariv, D. Fekete, and D. M. Pepper, “Compensation for channel dispersion by nonlinear optical phase conjugation,” Opt. Lett. 4(2), 52–54 (1979).
[Crossref] [PubMed]

Galdino, L.

D. Semrau, D. Lavery, L. Galdino, R. I. Killey, and P. Bayvel, “The impact of transceiver noise on digital nonlinearity compensation,” J. Lightwave Technol. 36(3), 695–702 (2018).
[Crossref]

L. Galdino, D. Semrau, D. Lavery, G. Saavedra, C. B. Czegledi, E. Agrell, R. I. Killey, and P. Bayvel, “On the limits of digital back-propagation in the presence of transceiver noise,” Opt. Express 25(4), 4564–4578 (2017).
[Crossref] [PubMed]

Goldfarb, G.

X. Li, X. Chen, G. Goldfarb, E. Mateo, I. Kim, Fatih 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]

Gordon, J. P.

J. P. Gordon and L. F. Mollenauer, “Phase noise in photonic communications systems using linear amplifiers,” Opt. Lett. 15(23), 1351–1353 (1990).
[Crossref] [PubMed]

Herath, H.

N. Ekanayake and H. Herath, “Effect of nonlinear phase noise on the performance of M-ary PSK signals in optical fiber links,” J. Lightwave Technol. 31(3), 447–454 (2013).
[Crossref]

Ho, K.

K. Ho, “Probability density of nonlinear phase noise,” J. Opt. Soc. Am. B: Opt. Phys. 20(9), 1875–1879 (2003).
[Crossref]

Ho, K. P.

K. P. Ho, “Cross-Phase Modulation-Induced Nonlinear Phase Noise for Quadriphase-Shift-Keying Signals,” in Impact of Nonlinearities on Fiber Optic Communications, S. Kumar, ed. (Springer-Verlag, 2011).
[Crossref]

Idler, W.

C. Pan, H. Bülow, W. Idler, L. Schmalen, and F. Kschischang, “Optical nonlinear phase noise compensation for 9 × 32 -Gbaud PolDM-16 QAM transmission using a code-aided expectation-maximization algorithm,” J. Lightwave Technol. 33(17), 3679–3686 (2015).
[Crossref]

Ip, E.

E. Ip and J. M. Kahn, “Compensation of dispersion and nonlinear impairments using digital backpropagation,” J. Lightwave Technol. 26(20), 3416–3425 (2008).
[Crossref]

Ishikawa, G.

S. Watanabe, G. Ishikawa, T. Naito, and T. Chikama, “Generation of optical phase-conjugate waves and compensation for pulse shape distortion in a single-mode fiber,” J. Lightwave Technol. 12(12), 2139–2146(1994).
[Crossref]

Ives, D.

D. Lavery, D. Ives, G. Liga, A. Alvarado, S. J. Savory, and P. Bayvel, “The benefit of split nonlinearity compensation for single-channel optical fiber communications,” IEEE Photonics Technol. Lett. 28(617), 1803–1806 (2016).
[Crossref]

Kahn, J. M.

E. Ip and J. M. Kahn, “Compensation of dispersion and nonlinear impairments using digital backpropagation,” J. Lightwave Technol. 26(20), 3416–3425 (2008).
[Crossref]

Kam, P. Y.

Z. Xu, P. Y. Kam, and C. Yu, “Adaptive maximum likelihood sequence detection for QPSK coherent optical communication system,” IEEE Photonics Technol. Lett. 26(6), 583–586 (2014).
[Crossref]

Killey, R.

R. Maher, D. Lavery, D. Millar, A. Alvarado, K. Parsons, R. Killey, and P. Bayvel, “Reach enhancement of 100% for a DP-64QAM super-channel using MC-DBP,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2015), paper Th4D.5.

Killey, R. I.

D. Semrau, D. Lavery, L. Galdino, R. I. Killey, and P. Bayvel, “The impact of transceiver noise on digital nonlinearity compensation,” J. Lightwave Technol. 36(3), 695–702 (2018).
[Crossref]

L. Galdino, D. Semrau, D. Lavery, G. Saavedra, C. B. Czegledi, E. Agrell, R. I. Killey, and P. Bayvel, “On the limits of digital back-propagation in the presence of transceiver noise,” Opt. Express 25(4), 4564–4578 (2017).
[Crossref] [PubMed]

Kim, I.

X. Li, X. Chen, G. Goldfarb, E. Mateo, I. Kim, Fatih 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]

Kschischang, F.

C. Pan, H. Bülow, W. Idler, L. Schmalen, and F. Kschischang, “Optical nonlinear phase noise compensation for 9 × 32 -Gbaud PolDM-16 QAM transmission using a code-aided expectation-maximization algorithm,” J. Lightwave Technol. 33(17), 3679–3686 (2015).
[Crossref]

Kschischang, F. R.

L. M. Zhang and F. R. Kschischang, “Staircase codes with 6% to 33% overhead,” J. Lightwave Technol. 32(10), 1999–2002 (2014).
[Crossref]

Kumar, S.

X. Zhu and S. Kumar, “Nonlinear phase noise in coherent optical OFDM transmission systems,” Opt. Express 18(7), 7347–7360 (2010).
[Crossref] [PubMed]

S. Kumar, “Analysis of nonlinear phase noise in coherent fiber-optic systems based on phase shift keying,” J. Lightwave Technol. 27(21), 4722–4733 (2009).
[Crossref]

S. Kumar and L. Liu, “Reduction of nonlinear phase noise using optical phase conjugation in quasi-linear optical transmission systems,” Opt. Express 15(5), 2166–2177 (2007).
[Crossref] [PubMed]

S. Kumar, “Effect of dispersion on nonlinear phase noise in optical transmission systems,” Opt. Lett. 30(24), 3278–3280 (2005).
[Crossref]

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

S. Kumar and M. J. Deen, Fiber Optic Communications: Fundamentals and Applications (John Wiley & Sons, 2014).
[Crossref]

Lavery, D.

D. Semrau, D. Lavery, L. Galdino, R. I. Killey, and P. Bayvel, “The impact of transceiver noise on digital nonlinearity compensation,” J. Lightwave Technol. 36(3), 695–702 (2018).
[Crossref]

L. Galdino, D. Semrau, D. Lavery, G. Saavedra, C. B. Czegledi, E. Agrell, R. I. Killey, and P. Bayvel, “On the limits of digital back-propagation in the presence of transceiver noise,” Opt. Express 25(4), 4564–4578 (2017).
[Crossref] [PubMed]

D. Lavery, D. Ives, G. Liga, A. Alvarado, S. J. Savory, and P. Bayvel, “The benefit of split nonlinearity compensation for single-channel optical fiber communications,” IEEE Photonics Technol. Lett. 28(617), 1803–1806 (2016).
[Crossref]

R. Maher, D. Lavery, D. Millar, A. Alvarado, K. Parsons, R. Killey, and P. Bayvel, “Reach enhancement of 100% for a DP-64QAM super-channel using MC-DBP,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2015), paper Th4D.5.

Li, G.

X. Li, X. Chen, G. Goldfarb, E. Mateo, I. Kim, Fatih 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]

Li, X.

X. Li, X. Chen, G. Goldfarb, E. Mateo, I. Kim, Fatih 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]

Li, Z.

D. Wang, M. Zhang, Z. Li, Y. Cui, J. Liu, Y. Yang, and H. Wang, “Nonlinear decision boundary created by a machine learning-based classifier to mitigate nonlinear phase noise,” in 2015 European Conference on Optical Communication (IEEE, 2015), pp. 1–3.

Liga, G.

D. Lavery, D. Ives, G. Liga, A. Alvarado, S. J. Savory, and P. Bayvel, “The benefit of split nonlinearity compensation for single-channel optical fiber communications,” IEEE Photonics Technol. Lett. 28(617), 1803–1806 (2016).
[Crossref]

Liu, J.

D. Wang, M. Zhang, Z. Li, Y. Cui, J. Liu, Y. Yang, and H. Wang, “Nonlinear decision boundary created by a machine learning-based classifier to mitigate nonlinear phase noise,” in 2015 European Conference on Optical Communication (IEEE, 2015), pp. 1–3.

Liu, L.

S. Kumar and L. Liu, “Reduction of nonlinear phase noise using optical phase conjugation in quasi-linear optical transmission systems,” Opt. Express 15(5), 2166–2177 (2007).
[Crossref] [PubMed]

Maher, R.

R. Maher, D. Lavery, D. Millar, A. Alvarado, K. Parsons, R. Killey, and P. Bayvel, “Reach enhancement of 100% for a DP-64QAM super-channel using MC-DBP,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2015), paper Th4D.5.

Marazzi, L.

P. Minzioni, V. Pusino, I. Cristiani, L. Marazzi, M. Martinelli, and V. Degiorgio, “Study of the Gordon-Mollenauer effect and of the optical-phase-conjugation compensation method in phase-modulated optical communication systems,” IEEE Photonics J. 2(3), 284–291 (2010).
[Crossref]

Martinelli, M.

P. Minzioni, V. Pusino, I. Cristiani, L. Marazzi, M. Martinelli, and V. Degiorgio, “Study of the Gordon-Mollenauer effect and of the optical-phase-conjugation compensation method in phase-modulated optical communication systems,” IEEE Photonics J. 2(3), 284–291 (2010).
[Crossref]

Mateo, E.

X. Li, X. Chen, G. Goldfarb, E. Mateo, I. Kim, Fatih 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]

Mauro, J. C.

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

McKinstrie, C.

C. McKinstrie, S. Radic, and C. Xie, “Reduction of soliton phase jitter by in-line phase conjugation,” Opt. Lett. 28(17), 1519–1521 (2003).
[Crossref] [PubMed]

Mecozzi, A.

A. Mecozzi, “Probability density functions of the nonlinear phase noise,” Opt. Lett. 29(7), 673–675 (2004).
[Crossref] [PubMed]

A. Mecozzi, “Limits to the long haul coherent transmission set by the Kerr nonlinearity and noise of in-line amplifiers,” J. Lightwave Technol. 12(11), 1993–2000 (1994).
[Crossref]

Menyak, C. R.

P. K. A. Wai and C. R. Menyak, “Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14(2), 148–157 (1996).
[Crossref]

Mikkelsen, B.

R.-J. Essiambre, G. Raybon, and B. Mikkelsen, “Pseudo-Linear Transmission Of High-Speed TDM Signals: 40 And 160 Gb/s,” in Optical Fiber Telecommunications IV-B, 4th ed., I. P. Kaminow and T. Li, eds. (Academic, 2002).
[Crossref]

Millar, D.

R. Maher, D. Lavery, D. Millar, A. Alvarado, K. Parsons, R. Killey, and P. Bayvel, “Reach enhancement of 100% for a DP-64QAM super-channel using MC-DBP,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2015), paper Th4D.5.

Minzioni, P.

P. Minzioni, V. Pusino, I. Cristiani, L. Marazzi, M. Martinelli, and V. Degiorgio, “Study of the Gordon-Mollenauer effect and of the optical-phase-conjugation compensation method in phase-modulated optical communication systems,” IEEE Photonics J. 2(3), 284–291 (2010).
[Crossref]

Mollenauer, L. F.

J. P. Gordon and L. F. Mollenauer, “Phase noise in photonic communications systems using linear amplifiers,” Opt. Lett. 15(23), 1351–1353 (1990).
[Crossref] [PubMed]

Naito, T.

S. Watanabe, G. Ishikawa, T. Naito, and T. Chikama, “Generation of optical phase-conjugate waves and compensation for pulse shape distortion in a single-mode fiber,” J. Lightwave Technol. 12(12), 2139–2146(1994).
[Crossref]

Pan, C.

C. Pan, H. Bülow, W. Idler, L. Schmalen, and F. Kschischang, “Optical nonlinear phase noise compensation for 9 × 32 -Gbaud PolDM-16 QAM transmission using a code-aided expectation-maximization algorithm,” J. Lightwave Technol. 33(17), 3679–3686 (2015).
[Crossref]

Parsons, K.

R. Maher, D. Lavery, D. Millar, A. Alvarado, K. Parsons, R. Killey, and P. Bayvel, “Reach enhancement of 100% for a DP-64QAM super-channel using MC-DBP,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2015), paper Th4D.5.

Pepper, D. M.

A. Yariv, D. Fekete, and D. M. Pepper, “Compensation for channel dispersion by nonlinear optical phase conjugation,” Opt. Lett. 4(2), 52–54 (1979).
[Crossref] [PubMed]

Pusino, V.

P. Minzioni, V. Pusino, I. Cristiani, L. Marazzi, M. Martinelli, and V. Degiorgio, “Study of the Gordon-Mollenauer effect and of the optical-phase-conjugation compensation method in phase-modulated optical communication systems,” IEEE Photonics J. 2(3), 284–291 (2010).
[Crossref]

Radic, S.

C. McKinstrie, S. Radic, and C. Xie, “Reduction of soliton phase jitter by in-line phase conjugation,” Opt. Lett. 28(17), 1519–1521 (2003).
[Crossref] [PubMed]

Raghavan, S.

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

Raybon, G.

R.-J. Essiambre, G. Raybon, and B. Mikkelsen, “Pseudo-Linear Transmission Of High-Speed TDM Signals: 40 And 160 Gb/s,” in Optical Fiber Telecommunications IV-B, 4th ed., I. P. Kaminow and T. Li, eds. (Academic, 2002).
[Crossref]

Saavedra, G.

L. Galdino, D. Semrau, D. Lavery, G. Saavedra, C. B. Czegledi, E. Agrell, R. I. Killey, and P. Bayvel, “On the limits of digital back-propagation in the presence of transceiver noise,” Opt. Express 25(4), 4564–4578 (2017).
[Crossref] [PubMed]

Savory, S. J.

D. Lavery, D. Ives, G. Liga, A. Alvarado, S. J. Savory, and P. Bayvel, “The benefit of split nonlinearity compensation for single-channel optical fiber communications,” IEEE Photonics Technol. Lett. 28(617), 1803–1806 (2016).
[Crossref]

Schmalen, L.

C. Pan, H. Bülow, W. Idler, L. Schmalen, and F. Kschischang, “Optical nonlinear phase noise compensation for 9 × 32 -Gbaud PolDM-16 QAM transmission using a code-aided expectation-maximization algorithm,” J. Lightwave Technol. 33(17), 3679–3686 (2015).
[Crossref]

Semrau, D.

D. Semrau, D. Lavery, L. Galdino, R. I. Killey, and P. Bayvel, “The impact of transceiver noise on digital nonlinearity compensation,” J. Lightwave Technol. 36(3), 695–702 (2018).
[Crossref]

L. Galdino, D. Semrau, D. Lavery, G. Saavedra, C. B. Czegledi, E. Agrell, R. I. Killey, and P. Bayvel, “On the limits of digital back-propagation in the presence of transceiver noise,” Opt. Express 25(4), 4564–4578 (2017).
[Crossref] [PubMed]

Wai, P. K. A.

P. K. A. Wai and C. R. Menyak, “Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14(2), 148–157 (1996).
[Crossref]

Wang, D.

D. Wang, M. Zhang, Z. Li, Y. Cui, J. Liu, Y. Yang, and H. Wang, “Nonlinear decision boundary created by a machine learning-based classifier to mitigate nonlinear phase noise,” in 2015 European Conference on Optical Communication (IEEE, 2015), pp. 1–3.

Wang, H.

D. Wang, M. Zhang, Z. Li, Y. Cui, J. Liu, Y. Yang, and H. Wang, “Nonlinear decision boundary created by a machine learning-based classifier to mitigate nonlinear phase noise,” in 2015 European Conference on Optical Communication (IEEE, 2015), pp. 1–3.

Watanabe, S.

S. Watanabe, G. Ishikawa, T. Naito, and T. Chikama, “Generation of optical phase-conjugate waves and compensation for pulse shape distortion in a single-mode fiber,” J. Lightwave Technol. 12(12), 2139–2146(1994).
[Crossref]

Winzer, P. J.

R.-J. Essiambre and P. J. Winzer, “Fibre nonlinearities in electronically pre-distorted transmission,” in 2005 European Conference on Optical Communication (IEEE, 2005), pp. 191–192.

Xie, C.

C. McKinstrie, S. Radic, and C. Xie, “Reduction of soliton phase jitter by in-line phase conjugation,” Opt. Lett. 28(17), 1519–1521 (2003).
[Crossref] [PubMed]

Xu, Z.

Z. Xu, P. Y. Kam, and C. Yu, “Adaptive maximum likelihood sequence detection for QPSK coherent optical communication system,” IEEE Photonics Technol. Lett. 26(6), 583–586 (2014).
[Crossref]

Yaman, Fatih

X. Li, X. Chen, G. Goldfarb, E. Mateo, I. Kim, Fatih 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]

Yang, Y.

D. Wang, M. Zhang, Z. Li, Y. Cui, J. Liu, Y. Yang, and H. Wang, “Nonlinear decision boundary created by a machine learning-based classifier to mitigate nonlinear phase noise,” in 2015 European Conference on Optical Communication (IEEE, 2015), pp. 1–3.

Yariv, A.

A. Yariv, D. Fekete, and D. M. Pepper, “Compensation for channel dispersion by nonlinear optical phase conjugation,” Opt. Lett. 4(2), 52–54 (1979).
[Crossref] [PubMed]

Yu, C.

Z. Xu, P. Y. Kam, and C. Yu, “Adaptive maximum likelihood sequence detection for QPSK coherent optical communication system,” IEEE Photonics Technol. Lett. 26(6), 583–586 (2014).
[Crossref]

Zhang, L. M.

L. M. Zhang and F. R. Kschischang, “Staircase codes with 6% to 33% overhead,” J. Lightwave Technol. 32(10), 1999–2002 (2014).
[Crossref]

Zhang, M.

D. Wang, M. Zhang, Z. Li, Y. Cui, J. Liu, Y. Yang, and H. Wang, “Nonlinear decision boundary created by a machine learning-based classifier to mitigate nonlinear phase noise,” in 2015 European Conference on Optical Communication (IEEE, 2015), pp. 1–3.

Zhu, X.

X. Zhu and S. Kumar, “Nonlinear phase noise in coherent optical OFDM transmission systems,” Opt. Express 18(7), 7347–7360 (2010).
[Crossref] [PubMed]

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

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

IEEE Photonics J. (1)

P. Minzioni, V. Pusino, I. Cristiani, L. Marazzi, M. Martinelli, and V. Degiorgio, “Study of the Gordon-Mollenauer effect and of the optical-phase-conjugation compensation method in phase-modulated optical communication systems,” IEEE Photonics J. 2(3), 284–291 (2010).
[Crossref]

IEEE Photonics Technol. Lett. (2)

Z. Xu, P. Y. Kam, and C. Yu, “Adaptive maximum likelihood sequence detection for QPSK coherent optical communication system,” IEEE Photonics Technol. Lett. 26(6), 583–586 (2014).
[Crossref]

D. Lavery, D. Ives, G. Liga, A. Alvarado, S. J. Savory, and P. Bayvel, “The benefit of split nonlinearity compensation for single-channel optical fiber communications,” IEEE Photonics Technol. Lett. 28(617), 1803–1806 (2016).
[Crossref]

J. Lightwave Technol. (10)

D. Semrau, D. Lavery, L. Galdino, R. I. Killey, and P. Bayvel, “The impact of transceiver noise on digital nonlinearity compensation,” J. Lightwave Technol. 36(3), 695–702 (2018).
[Crossref]

S. Watanabe, G. Ishikawa, T. Naito, and T. Chikama, “Generation of optical phase-conjugate waves and compensation for pulse shape distortion in a single-mode fiber,” J. Lightwave Technol. 12(12), 2139–2146(1994).
[Crossref]

P. K. A. Wai and C. R. Menyak, “Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14(2), 148–157 (1996).
[Crossref]

L. M. Zhang and F. R. Kschischang, “Staircase codes with 6% to 33% overhead,” J. Lightwave Technol. 32(10), 1999–2002 (2014).
[Crossref]

S. Kumar, “Analysis of nonlinear phase noise in coherent fiber-optic systems based on phase shift keying,” J. Lightwave Technol. 27(21), 4722–4733 (2009).
[Crossref]

C. Pan, H. Bülow, W. Idler, L. Schmalen, and F. Kschischang, “Optical nonlinear phase noise compensation for 9 × 32 -Gbaud PolDM-16 QAM transmission using a code-aided expectation-maximization algorithm,” J. Lightwave Technol. 33(17), 3679–3686 (2015).
[Crossref]

A. Demir, “Nonlinear phase noise in optical fiber communication systems,” J. Lightwave Technol. 25(8), 2002–2032 (2007).
[Crossref]

N. Ekanayake and H. Herath, “Effect of nonlinear phase noise on the performance of M-ary PSK signals in optical fiber links,” J. Lightwave Technol. 31(3), 447–454 (2013).
[Crossref]

E. Ip and J. M. Kahn, “Compensation of dispersion and nonlinear impairments using digital backpropagation,” J. Lightwave Technol. 26(20), 3416–3425 (2008).
[Crossref]

A. Mecozzi, “Limits to the long haul coherent transmission set by the Kerr nonlinearity and noise of in-line amplifiers,” J. Lightwave Technol. 12(11), 1993–2000 (1994).
[Crossref]

J. Opt. Soc. Am. B: Opt. Phys. (1)

K. Ho, “Probability density of nonlinear phase noise,” J. Opt. Soc. Am. B: Opt. Phys. 20(9), 1875–1879 (2003).
[Crossref]

Opt. Express (4)

X. Li, X. Chen, G. Goldfarb, E. Mateo, I. Kim, Fatih 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]

X. Zhu and S. Kumar, “Nonlinear phase noise in coherent optical OFDM transmission systems,” Opt. Express 18(7), 7347–7360 (2010).
[Crossref] [PubMed]

L. Galdino, D. Semrau, D. Lavery, G. Saavedra, C. B. Czegledi, E. Agrell, R. I. Killey, and P. Bayvel, “On the limits of digital back-propagation in the presence of transceiver noise,” Opt. Express 25(4), 4564–4578 (2017).
[Crossref] [PubMed]

S. Kumar and L. Liu, “Reduction of nonlinear phase noise using optical phase conjugation in quasi-linear optical transmission systems,” Opt. Express 15(5), 2166–2177 (2007).
[Crossref] [PubMed]

Opt. Lett. (5)

A. Yariv, D. Fekete, and D. M. Pepper, “Compensation for channel dispersion by nonlinear optical phase conjugation,” Opt. Lett. 4(2), 52–54 (1979).
[Crossref] [PubMed]

C. McKinstrie, S. Radic, and C. Xie, “Reduction of soliton phase jitter by in-line phase conjugation,” Opt. Lett. 28(17), 1519–1521 (2003).
[Crossref] [PubMed]

A. Mecozzi, “Probability density functions of the nonlinear phase noise,” Opt. Lett. 29(7), 673–675 (2004).
[Crossref] [PubMed]

S. Kumar, “Effect of dispersion on nonlinear phase noise in optical transmission systems,” Opt. Lett. 30(24), 3278–3280 (2005).
[Crossref]

J. P. Gordon and L. F. Mollenauer, “Phase noise in photonic communications systems using linear amplifiers,” Opt. Lett. 15(23), 1351–1353 (1990).
[Crossref] [PubMed]

Other (7)

R.-J. Essiambre and P. J. Winzer, “Fibre nonlinearities in electronically pre-distorted transmission,” in 2005 European Conference on Optical Communication (IEEE, 2005), pp. 191–192.

G. P. Agrawal, Nonlinear Fiber Optics, 5th ed. (Academic, 2012).

S. Kumar and M. J. Deen, Fiber Optic Communications: Fundamentals and Applications (John Wiley & Sons, 2014).
[Crossref]

R.-J. Essiambre, G. Raybon, and B. Mikkelsen, “Pseudo-Linear Transmission Of High-Speed TDM Signals: 40 And 160 Gb/s,” in Optical Fiber Telecommunications IV-B, 4th ed., I. P. Kaminow and T. Li, eds. (Academic, 2002).
[Crossref]

D. Wang, M. Zhang, Z. Li, Y. Cui, J. Liu, Y. Yang, and H. Wang, “Nonlinear decision boundary created by a machine learning-based classifier to mitigate nonlinear phase noise,” in 2015 European Conference on Optical Communication (IEEE, 2015), pp. 1–3.

R. Maher, D. Lavery, D. Millar, A. Alvarado, K. Parsons, R. Killey, and P. Bayvel, “Reach enhancement of 100% for a DP-64QAM super-channel using MC-DBP,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2015), paper Th4D.5.

K. P. Ho, “Cross-Phase Modulation-Induced Nonlinear Phase Noise for Quadriphase-Shift-Keying Signals,” in Impact of Nonlinearities on Fiber Optic Communications, S. Kumar, ed. (Springer-Verlag, 2011).
[Crossref]

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Figures (11)

Fig. 1
Fig. 1 Close-up representation of a fiber-optic communication link. TF= Transmission fiber.
Fig. 2
Fig. 2 Scheme 1: Full DBP at the receiver. The standard configuration. TF stands for transmission fiber, and VF stands for the virtual fiber.
Fig. 3
Fig. 3 Scheme 2: DBP split between transmitter and receiver. TF = transmission fiber, VF = virtual fiber.
Fig. 4
Fig. 4 Scheme 3: Full DBP at the receiver with OPC set-up. OPC = optical phase conjugation, DPC = digital phase conjugation, TF = transmission fiber.
Fig. 5
Fig. 5 Noise source located at mL has an image at point (Nm)L, from which the distance to the end of the line is mL.
Fig. 6
Fig. 6 Scheme 4: Fiber optic system with a mid-point OPC and asymmetric DBP. OPC = optical phase conjugation, DPC = digital phase conjugation, TF = transmission fiber.
Fig. 7
Fig. 7 Multiplication factor profile. Multiplication factor M(m) is the strength of NLPN originated at the point mL. N = 120.
Fig. 8
Fig. 8 Q-factor performance comparison of the four schemes, for single channel systems.
Fig. 9
Fig. 9 Maximum achievable reach for single channel systems. The forward error correction (FEC) limit is 4.7 × 10−3, (i.e. Q = 8.29 dB) [29].
Fig. 10
Fig. 10 Q-factor performance comparison of the four schemes, for WDM system with 5 channels.
Fig. 11
Fig. 11 Maximum achievable reach for 5 channel WDM systems. The forward error correction (FEC) limit is 4.7 × 10−3, (i.e. Q = 8.29 dB) [29].

Tables (1)

Tables Icon

Table 1 Reach decrease in WDM systems compared to single channel case.

Equations (31)

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

q ( 0 , t ) = A f ( t ) ,
q ( L + , t ) = A f ( t ) e j γ | f ( t ) | 2 L e f f A 2 + n 1 ( t ) ,
n 1 ( t ) = δ A 1 f ( t ) e j θ n 1 ( t ) .
q ( L + , t ) = ( A + δ A 1 ) f ( t ) e j γ | f ( t ) | 2 L e f f A 2 + j θ n 1 ( t ) ,
q ( 2 L , t ) = ( A + δ A 1 ) f ( t ) e α L / 2 + j θ n 1 ( t ) e j γ | f ( t ) | 2 [ A 2 L e f f + ( A + δ A 1 ) 2 L e f f ] .
q ( 2 L , t ) ( A + δ A 1 ) f ( t ) e α L / 2 + j θ n 1 ( t ) e j γ | f ( t ) | 2 L e f f [ 2 A 2 + 2 A δ A 1 ] .
q ( N L + , t ) = ( A + m = 1 N δ A m ) f ( t ) e j θ n ( t ) + j K ( t ) [ N A 2 + 2 A m = 1 N ( N m ) δ A m ] ,
θ n ( t ) = i = 1 N θ n i ( t ) ,
K ( t ) = γ | f ( t ) | 2 L e f f .
q b ( L + , t ) = ( A + m = 1 N δ A m ) f ( t ) e j θ n ( t ) + j K ( t ) [ ( N 1 ) A 2 + 2 A m = 1 N ( N m 1 ) δ A m ] .
q b ( N L + , t ) = ( A + m = 1 N δ A m ) f ( t ) e j θ n ( t ) j 2 A K ( t ) ( m = 1 N m δ A m ) .
δ ϕ N L 1 = 2 A K ( t ) ( m = 1 N M ( m ) δ A m ) .
σ N L 1 2 = δ ϕ N L 1 2 = ( 2 A K ( t ) ) 2 δ A 2 ( m = 1 N m 2 ) ,
σ N L 1 2 ( 2 A K ( t ) ) 2 N 3 .
q b ( N L + , t ) = ( A + m = 1 N δ A m ) f ( t ) e j δ ϕ N L 2 ( t ) + j θ n ( t ) ,
δ ϕ N L 2 ( t ) = 2 A K ( t ) m = 1 N M ( m ) δ A m ,
M ( m ) = N 2 m .
σ N L 2 2 ( 2 A K ( t ) ) 2 ( N 3 4 ) .
q ( N L + , t ) = ( A + m = 1 N δ A m ) f ( t ) e j θ n ( t ) + j 2 A K ( t ) [ m = 1 N 2 m δ A m + m = N 2 + 1 N ( N m ) δ A m ] .
q b ( N L + , t ) = ( A + m = 1 N δ A m ) f ( t ) e j θ n ( t ) + δ ϕ N L 3 ( t ) ,
δ ϕ N L 3 = 2 A K ( t ) [ m = 1 N M ( m ) δ A m ] ,
M ( m ) = { m if 1 m N 2 m N . if N 2 + 1 m N
σ N L 3 2 ( 2 A K ( t ) ) 2 ( N 3 4 ) .
q ( N L + , t ) = ( A + m = 1 N δ A m ) f ( t ) e j θ n ( t ) + j K ( t ) A 2 X N + j 2 A K ( t ) [ m = 1 N 2 m δ A m + m = N 2 + 1 N ( N m ) δ A m ] .
q b ( N L + , t ) = ( A + m = 1 N δ A m ) f ( t ) e j θ n ( t ) + δ ϕ N L 4 ( t ) ,
δ ϕ N L 4 = 2 A K ( t ) [ m = 1 N M ( m ) δ A m ] ,
M ( m ) = { N X m if 1 m N 2 m N ( 1 X ) . if N 2 + 1 m N
σ N L 4 2 ( X ) = ( 2 A K ( t ) ) 2 δ A 2 [ m = 1 N M 2 ( m ) ] ,
δ A 2 = δ A i 2 ,   i = 1 , 2 , , N .
σ N L 4 2 ( 2 A K ( t ) ) 2 ( N 3 16 ) .
Q = 20 l o g 10 ( 2 e r f c i n v [ 2. B E R ] ) ,

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