Abstract

We propose a method to estimate the lower bound of achievable information rates (AIRs) of high speed orthogonal frequency-division multiplexing (OFDM) in spatial division multiplexing (SDM) optical long-haul transmission systems. The estimation of AIR is based on the forward recursion of multidimensional super-symbol efficient sliding-window Bahl–Cocke–Jelinek–Raviv (BCJR) algorithm. We consider most of the degradations of fiber links including nonlinear effects in few-mode fiber (FMF). This method does not consider the SDM as a simple multiplexer of independent data streams, but provides a super-symbol version for AIR calculation over spatial channels. This super-symbol version of AIR calculation algorithm, in principle, can be used for arbitrary multiple-input-multiple-output (MIMO)-SDM system with channel memory consideration. We illustrate this method by performing Monte Carlo simulations in a complete FMF model. Both channel model and algorithm for calculation of the AIRs are described in details. We also compare the AIRs results for QPSK/16QAM in both single mode fiber (SMF)- and FMF-based optical OFDM transmission.

© 2015 Optical Society of America

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References

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  1. I. B. Djordjevic, M. Cvijetic, and C. Lin, “Multidimensional signaling and coding enabling multi-Tb/s optical transport and networking: multidimensional aspects of coded modulation,” IEEE Signal Proc. 31(2), 104–117 (2014).
    [Crossref]
  2. R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R. Essiambre, P. J. Winzer, D. W. Peckham, A. H. McCurdy, and R. Lingle, “Mode-division multiplexing over 96 km of few-mode fiber using coherent 6 ×6 MIMO processing,” J. Lightwave Technol. 30(4), 521–531 (2012).
    [Crossref]
  3. I. B. Djordjevic, “Spatial-domain-based hybrid multidimensional coded-modulation schemes enabling multi-Tb/s optical transport,” J. Lightwave Technol. 30(14), 2315–2328 (2012).
    [Crossref]
  4. R. J. Essiambre, G. J. Foschini, G. Kramer, and P. J. Winzer, “Capacity limits of information transport in fiber-optic networks,” Phys. Rev. Lett. 101(16), 163901 (2008).
    [Crossref] [PubMed]
  5. X. Chen, A. Li, G. Gao, A. A. Amin, and W. Shieh, “Characterization of fiber nonlinearity and analysis of its impact on link capacity limit of two-mode fibers,” IEEE Photon. J. 4(2), 455–460 (2012).
    [Crossref]
  6. A. Li, X. Chen, D. Che, Y. Wang, and W. Shieh, “Capacity Limit of Few-Mode Fibers for Space-Division Multiplexed Coherent Optical OFDM Superchannel,” in Asia Communications and Photonics Conference (ACP’2014) AW3F.1.
    [Crossref]
  7. P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fibre communications,” Nature 411(6841), 1027–1030 (2001).
    [Crossref] [PubMed]
  8. P. J. Winzer and G. J. Foschini, “MIMO capacities and outage probabilities in spatially multiplexed optical transport systems,” Opt. Express 19(17), 16680–16696 (2011).
    [Crossref] [PubMed]
  9. M. Secondini, E. Forestieri, and G. Prati, “Achievable information rate in nonlinear WDM fiber-optic systems with arbitrary modulation formats and dispersion maps,” J. Lightwave Technol. 31(23), 3839–3852 (2013).
    [Crossref]
  10. P. Poggiolini, G. Bosco, A. Carena, V. Curri, Y. Jiang, and F. Forghieri, “The GN-model of fiber non-linear propagation and its applications,” J. Lightwave Technol. 32(4), 694–721 (2014).
    [Crossref]
  11. E. Agrell, A. Alvarado, G. Durisi, and M. Karlsson, “Capacity of a nonlinear optical channel with finite memory,” J. Lightwave Technol. 32(16), 2862–2876 (2014).
    [Crossref]
  12. E. Meron, M. Feder, and M. Shtaif, “On the achievable communication rates of generalized soliton transmission systems,” [Online]. arXiv:1207.0297 (Available: http://arxiv.org/abs/1207.0297 )
  13. R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. 28(4), 662–701 (2010).
    [Crossref]
  14. D. M. Arnold, H.-A. Loeliger, P. O. Vontobel, A. Kavcic, and W. Zeng, “Simulation-based computation of information rates for channels with memory,” IEEE Trans. Inf. Theory 52(8), 3498–3508 (2006).
    [Crossref]
  15. C. Lin, I. B. Djordjevic, M. Cvijetic, and D. Zou, “Mode-Multiplexed Multi-Tb/s Superchannel Transmission with Advanced Multidimensional Signaling in the Presence of Fiber Nonlinearities,” IEEE Trans. Commun. 62(7), 2507–2514 (2014).
    [Crossref]
  16. S. Mumtaz, R.-J. Essiambre, and G. P. Agrawal, “Nonlinear propagation in multimode and multicore fibers: generalization of the Manakov equations,” J. Lightwave Technol. 31(3), 398–406 (2013).
    [Crossref]
  17. C. Lin, I. B. Djordjevic, D. Zou, M. Arabaci, and M. Cvijetic, “Non-binary LDPC coded mode-multiplexed coherent optical OFDM 1.28 Tbit/s 16-QAM signal transmission over 2000-km of few-mode fibers with mode dependent loss,” IEEE Photon. J. 4(5), 1922–1929 (2012).
    [Crossref]
  18. A. Mecozzi, C. Antonelli, and M. Shtaif, “Nonlinear propagation in multi-mode fibers in the strong coupling regime,” Opt. Express 20(11), 11673–11678 (2012).
    [Crossref] [PubMed]
  19. S. O. Arik, D. Askarov, and J. M. Kahn, “Effect of mode coupling on signal processing complexity in mode-division multiplexing,” J. Lightwave Technol. 31(3), 423–431 (2013).
    [Crossref]
  20. M. Cvijetic, I. B. Djordjevic, Advanced Optical Communication Systems and Networks, (Boston-London: Artech House, 2013).
  21. K.-P. Ho and J. M. Kahn, “Statistics of group delays in multimode fiber with strong mode coupling,” J. Lightwave Technol. 29(21), 3119–3128 (2011).
    [Crossref]
  22. I. B. Djordjevic, B. Vasic, M. Ivkovic, and I. Gabitov, “Achievable information rates for high-speed long-haul optical transmission,” J. Lightwave Technol. 23(11), 3755–3763 (2005).
    [Crossref]
  23. C. Lin, S. Chandrasekhar, and P. J. Winzer, “Experimental Study of the Limits of Digital Nonlinearity Compensation in DWDM Systems,” Optical Fiber Communications Conference (OFC’ 15), Th4D.4.
    [Crossref]

2014 (4)

I. B. Djordjevic, M. Cvijetic, and C. Lin, “Multidimensional signaling and coding enabling multi-Tb/s optical transport and networking: multidimensional aspects of coded modulation,” IEEE Signal Proc. 31(2), 104–117 (2014).
[Crossref]

C. Lin, I. B. Djordjevic, M. Cvijetic, and D. Zou, “Mode-Multiplexed Multi-Tb/s Superchannel Transmission with Advanced Multidimensional Signaling in the Presence of Fiber Nonlinearities,” IEEE Trans. Commun. 62(7), 2507–2514 (2014).
[Crossref]

P. Poggiolini, G. Bosco, A. Carena, V. Curri, Y. Jiang, and F. Forghieri, “The GN-model of fiber non-linear propagation and its applications,” J. Lightwave Technol. 32(4), 694–721 (2014).
[Crossref]

E. Agrell, A. Alvarado, G. Durisi, and M. Karlsson, “Capacity of a nonlinear optical channel with finite memory,” J. Lightwave Technol. 32(16), 2862–2876 (2014).
[Crossref]

2013 (3)

2012 (5)

2011 (2)

2010 (1)

2008 (1)

R. J. Essiambre, G. J. Foschini, G. Kramer, and P. J. Winzer, “Capacity limits of information transport in fiber-optic networks,” Phys. Rev. Lett. 101(16), 163901 (2008).
[Crossref] [PubMed]

2006 (1)

D. M. Arnold, H.-A. Loeliger, P. O. Vontobel, A. Kavcic, and W. Zeng, “Simulation-based computation of information rates for channels with memory,” IEEE Trans. Inf. Theory 52(8), 3498–3508 (2006).
[Crossref]

2005 (1)

2001 (1)

P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fibre communications,” Nature 411(6841), 1027–1030 (2001).
[Crossref] [PubMed]

Agrawal, G. P.

Agrell, E.

Alvarado, A.

Amin, A. A.

X. Chen, A. Li, G. Gao, A. A. Amin, and W. Shieh, “Characterization of fiber nonlinearity and analysis of its impact on link capacity limit of two-mode fibers,” IEEE Photon. J. 4(2), 455–460 (2012).
[Crossref]

Antonelli, C.

Arabaci, M.

C. Lin, I. B. Djordjevic, D. Zou, M. Arabaci, and M. Cvijetic, “Non-binary LDPC coded mode-multiplexed coherent optical OFDM 1.28 Tbit/s 16-QAM signal transmission over 2000-km of few-mode fibers with mode dependent loss,” IEEE Photon. J. 4(5), 1922–1929 (2012).
[Crossref]

Arik, S. O.

Arnold, D. M.

D. M. Arnold, H.-A. Loeliger, P. O. Vontobel, A. Kavcic, and W. Zeng, “Simulation-based computation of information rates for channels with memory,” IEEE Trans. Inf. Theory 52(8), 3498–3508 (2006).
[Crossref]

Askarov, D.

Bolle, C.

Bosco, G.

Burrows, E. C.

Carena, A.

Chen, X.

X. Chen, A. Li, G. Gao, A. A. Amin, and W. Shieh, “Characterization of fiber nonlinearity and analysis of its impact on link capacity limit of two-mode fibers,” IEEE Photon. J. 4(2), 455–460 (2012).
[Crossref]

Curri, V.

Cvijetic, M.

I. B. Djordjevic, M. Cvijetic, and C. Lin, “Multidimensional signaling and coding enabling multi-Tb/s optical transport and networking: multidimensional aspects of coded modulation,” IEEE Signal Proc. 31(2), 104–117 (2014).
[Crossref]

C. Lin, I. B. Djordjevic, M. Cvijetic, and D. Zou, “Mode-Multiplexed Multi-Tb/s Superchannel Transmission with Advanced Multidimensional Signaling in the Presence of Fiber Nonlinearities,” IEEE Trans. Commun. 62(7), 2507–2514 (2014).
[Crossref]

C. Lin, I. B. Djordjevic, D. Zou, M. Arabaci, and M. Cvijetic, “Non-binary LDPC coded mode-multiplexed coherent optical OFDM 1.28 Tbit/s 16-QAM signal transmission over 2000-km of few-mode fibers with mode dependent loss,” IEEE Photon. J. 4(5), 1922–1929 (2012).
[Crossref]

Djordjevic, I. B.

C. Lin, I. B. Djordjevic, M. Cvijetic, and D. Zou, “Mode-Multiplexed Multi-Tb/s Superchannel Transmission with Advanced Multidimensional Signaling in the Presence of Fiber Nonlinearities,” IEEE Trans. Commun. 62(7), 2507–2514 (2014).
[Crossref]

I. B. Djordjevic, M. Cvijetic, and C. Lin, “Multidimensional signaling and coding enabling multi-Tb/s optical transport and networking: multidimensional aspects of coded modulation,” IEEE Signal Proc. 31(2), 104–117 (2014).
[Crossref]

C. Lin, I. B. Djordjevic, D. Zou, M. Arabaci, and M. Cvijetic, “Non-binary LDPC coded mode-multiplexed coherent optical OFDM 1.28 Tbit/s 16-QAM signal transmission over 2000-km of few-mode fibers with mode dependent loss,” IEEE Photon. J. 4(5), 1922–1929 (2012).
[Crossref]

I. B. Djordjevic, “Spatial-domain-based hybrid multidimensional coded-modulation schemes enabling multi-Tb/s optical transport,” J. Lightwave Technol. 30(14), 2315–2328 (2012).
[Crossref]

I. B. Djordjevic, B. Vasic, M. Ivkovic, and I. Gabitov, “Achievable information rates for high-speed long-haul optical transmission,” J. Lightwave Technol. 23(11), 3755–3763 (2005).
[Crossref]

Durisi, G.

Esmaeelpour, M.

Essiambre, R.

Essiambre, R. J.

R. J. Essiambre, G. J. Foschini, G. Kramer, and P. J. Winzer, “Capacity limits of information transport in fiber-optic networks,” Phys. Rev. Lett. 101(16), 163901 (2008).
[Crossref] [PubMed]

Essiambre, R.-J.

Forestieri, E.

Forghieri, F.

Foschini, G. J.

Gabitov, I.

Gao, G.

X. Chen, A. Li, G. Gao, A. A. Amin, and W. Shieh, “Characterization of fiber nonlinearity and analysis of its impact on link capacity limit of two-mode fibers,” IEEE Photon. J. 4(2), 455–460 (2012).
[Crossref]

Gnauck, A. H.

Goebel, B.

Ho, K.-P.

Ivkovic, M.

Jiang, Y.

Kahn, J. M.

Karlsson, M.

Kavcic, A.

D. M. Arnold, H.-A. Loeliger, P. O. Vontobel, A. Kavcic, and W. Zeng, “Simulation-based computation of information rates for channels with memory,” IEEE Trans. Inf. Theory 52(8), 3498–3508 (2006).
[Crossref]

Kramer, G.

R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. 28(4), 662–701 (2010).
[Crossref]

R. J. Essiambre, G. J. Foschini, G. Kramer, and P. J. Winzer, “Capacity limits of information transport in fiber-optic networks,” Phys. Rev. Lett. 101(16), 163901 (2008).
[Crossref] [PubMed]

Li, A.

X. Chen, A. Li, G. Gao, A. A. Amin, and W. Shieh, “Characterization of fiber nonlinearity and analysis of its impact on link capacity limit of two-mode fibers,” IEEE Photon. J. 4(2), 455–460 (2012).
[Crossref]

Lin, C.

I. B. Djordjevic, M. Cvijetic, and C. Lin, “Multidimensional signaling and coding enabling multi-Tb/s optical transport and networking: multidimensional aspects of coded modulation,” IEEE Signal Proc. 31(2), 104–117 (2014).
[Crossref]

C. Lin, I. B. Djordjevic, M. Cvijetic, and D. Zou, “Mode-Multiplexed Multi-Tb/s Superchannel Transmission with Advanced Multidimensional Signaling in the Presence of Fiber Nonlinearities,” IEEE Trans. Commun. 62(7), 2507–2514 (2014).
[Crossref]

C. Lin, I. B. Djordjevic, D. Zou, M. Arabaci, and M. Cvijetic, “Non-binary LDPC coded mode-multiplexed coherent optical OFDM 1.28 Tbit/s 16-QAM signal transmission over 2000-km of few-mode fibers with mode dependent loss,” IEEE Photon. J. 4(5), 1922–1929 (2012).
[Crossref]

Lingle, R.

Loeliger, H.-A.

D. M. Arnold, H.-A. Loeliger, P. O. Vontobel, A. Kavcic, and W. Zeng, “Simulation-based computation of information rates for channels with memory,” IEEE Trans. Inf. Theory 52(8), 3498–3508 (2006).
[Crossref]

McCurdy, A. H.

Mecozzi, A.

Mitra, P. P.

P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fibre communications,” Nature 411(6841), 1027–1030 (2001).
[Crossref] [PubMed]

Mumtaz, S.

Peckham, D. W.

Poggiolini, P.

Prati, G.

Randel, S.

Ryf, R.

Secondini, M.

Shieh, W.

X. Chen, A. Li, G. Gao, A. A. Amin, and W. Shieh, “Characterization of fiber nonlinearity and analysis of its impact on link capacity limit of two-mode fibers,” IEEE Photon. J. 4(2), 455–460 (2012).
[Crossref]

Shtaif, M.

Sierra, A.

Stark, J. B.

P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fibre communications,” Nature 411(6841), 1027–1030 (2001).
[Crossref] [PubMed]

Vasic, B.

Vontobel, P. O.

D. M. Arnold, H.-A. Loeliger, P. O. Vontobel, A. Kavcic, and W. Zeng, “Simulation-based computation of information rates for channels with memory,” IEEE Trans. Inf. Theory 52(8), 3498–3508 (2006).
[Crossref]

Winzer, P. J.

Zeng, W.

D. M. Arnold, H.-A. Loeliger, P. O. Vontobel, A. Kavcic, and W. Zeng, “Simulation-based computation of information rates for channels with memory,” IEEE Trans. Inf. Theory 52(8), 3498–3508 (2006).
[Crossref]

Zou, D.

C. Lin, I. B. Djordjevic, M. Cvijetic, and D. Zou, “Mode-Multiplexed Multi-Tb/s Superchannel Transmission with Advanced Multidimensional Signaling in the Presence of Fiber Nonlinearities,” IEEE Trans. Commun. 62(7), 2507–2514 (2014).
[Crossref]

C. Lin, I. B. Djordjevic, D. Zou, M. Arabaci, and M. Cvijetic, “Non-binary LDPC coded mode-multiplexed coherent optical OFDM 1.28 Tbit/s 16-QAM signal transmission over 2000-km of few-mode fibers with mode dependent loss,” IEEE Photon. J. 4(5), 1922–1929 (2012).
[Crossref]

IEEE Photon. J. (2)

X. Chen, A. Li, G. Gao, A. A. Amin, and W. Shieh, “Characterization of fiber nonlinearity and analysis of its impact on link capacity limit of two-mode fibers,” IEEE Photon. J. 4(2), 455–460 (2012).
[Crossref]

C. Lin, I. B. Djordjevic, D. Zou, M. Arabaci, and M. Cvijetic, “Non-binary LDPC coded mode-multiplexed coherent optical OFDM 1.28 Tbit/s 16-QAM signal transmission over 2000-km of few-mode fibers with mode dependent loss,” IEEE Photon. J. 4(5), 1922–1929 (2012).
[Crossref]

IEEE Signal Proc. (1)

I. B. Djordjevic, M. Cvijetic, and C. Lin, “Multidimensional signaling and coding enabling multi-Tb/s optical transport and networking: multidimensional aspects of coded modulation,” IEEE Signal Proc. 31(2), 104–117 (2014).
[Crossref]

IEEE Trans. Commun. (1)

C. Lin, I. B. Djordjevic, M. Cvijetic, and D. Zou, “Mode-Multiplexed Multi-Tb/s Superchannel Transmission with Advanced Multidimensional Signaling in the Presence of Fiber Nonlinearities,” IEEE Trans. Commun. 62(7), 2507–2514 (2014).
[Crossref]

IEEE Trans. Inf. Theory (1)

D. M. Arnold, H.-A. Loeliger, P. O. Vontobel, A. Kavcic, and W. Zeng, “Simulation-based computation of information rates for channels with memory,” IEEE Trans. Inf. Theory 52(8), 3498–3508 (2006).
[Crossref]

J. Lightwave Technol. (10)

R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. 28(4), 662–701 (2010).
[Crossref]

S. Mumtaz, R.-J. Essiambre, and G. P. Agrawal, “Nonlinear propagation in multimode and multicore fibers: generalization of the Manakov equations,” J. Lightwave Technol. 31(3), 398–406 (2013).
[Crossref]

M. Secondini, E. Forestieri, and G. Prati, “Achievable information rate in nonlinear WDM fiber-optic systems with arbitrary modulation formats and dispersion maps,” J. Lightwave Technol. 31(23), 3839–3852 (2013).
[Crossref]

P. Poggiolini, G. Bosco, A. Carena, V. Curri, Y. Jiang, and F. Forghieri, “The GN-model of fiber non-linear propagation and its applications,” J. Lightwave Technol. 32(4), 694–721 (2014).
[Crossref]

E. Agrell, A. Alvarado, G. Durisi, and M. Karlsson, “Capacity of a nonlinear optical channel with finite memory,” J. Lightwave Technol. 32(16), 2862–2876 (2014).
[Crossref]

R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R. Essiambre, P. J. Winzer, D. W. Peckham, A. H. McCurdy, and R. Lingle, “Mode-division multiplexing over 96 km of few-mode fiber using coherent 6 ×6 MIMO processing,” J. Lightwave Technol. 30(4), 521–531 (2012).
[Crossref]

I. B. Djordjevic, “Spatial-domain-based hybrid multidimensional coded-modulation schemes enabling multi-Tb/s optical transport,” J. Lightwave Technol. 30(14), 2315–2328 (2012).
[Crossref]

S. O. Arik, D. Askarov, and J. M. Kahn, “Effect of mode coupling on signal processing complexity in mode-division multiplexing,” J. Lightwave Technol. 31(3), 423–431 (2013).
[Crossref]

K.-P. Ho and J. M. Kahn, “Statistics of group delays in multimode fiber with strong mode coupling,” J. Lightwave Technol. 29(21), 3119–3128 (2011).
[Crossref]

I. B. Djordjevic, B. Vasic, M. Ivkovic, and I. Gabitov, “Achievable information rates for high-speed long-haul optical transmission,” J. Lightwave Technol. 23(11), 3755–3763 (2005).
[Crossref]

Nature (1)

P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fibre communications,” Nature 411(6841), 1027–1030 (2001).
[Crossref] [PubMed]

Opt. Express (2)

Phys. Rev. Lett. (1)

R. J. Essiambre, G. J. Foschini, G. Kramer, and P. J. Winzer, “Capacity limits of information transport in fiber-optic networks,” Phys. Rev. Lett. 101(16), 163901 (2008).
[Crossref] [PubMed]

Other (4)

A. Li, X. Chen, D. Che, Y. Wang, and W. Shieh, “Capacity Limit of Few-Mode Fibers for Space-Division Multiplexed Coherent Optical OFDM Superchannel,” in Asia Communications and Photonics Conference (ACP’2014) AW3F.1.
[Crossref]

E. Meron, M. Feder, and M. Shtaif, “On the achievable communication rates of generalized soliton transmission systems,” [Online]. arXiv:1207.0297 (Available: http://arxiv.org/abs/1207.0297 )

M. Cvijetic, I. B. Djordjevic, Advanced Optical Communication Systems and Networks, (Boston-London: Artech House, 2013).

C. Lin, S. Chandrasekhar, and P. J. Winzer, “Experimental Study of the Limits of Digital Nonlinearity Compensation in DWDM Systems,” Optical Fiber Communications Conference (OFC’ 15), Th4D.4.
[Crossref]

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Figures (5)

Fig. 1
Fig. 1 OFDM system: (a) overall system configuration, (b) modulation and demodulation; (c) and compensation scheme.
Fig. 2
Fig. 2 Data block arrangement for AIR calculation in multidimensional systems.
Fig. 3
Fig. 3 Finite state machine description for AIR calculation.
Fig. 4
Fig. 4 Received constellations: (a), (c) and (e) correspond to QPSK over SMF for number of spans equal to 50, 100, and 150, respectively. On the other hand, (b), (d) and (f) correspond to 16QAM over FMF (with two spatial and two polarization modes) for number of spans equal to 1, 50, and 100, respectively.
Fig. 5
Fig. 5 AIR results for CO-OFDM transmission over: (a) SMF and (b) FMF.

Tables (1)

Tables Icon

Table 1 Fiber parameters and AIR calculation OFDM system parameters

Equations (32)

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

E = [ E 1 E 2 ] T = [ E 1x E 1y E 2x E 2y ] T ,
d E dz =( D ^ + N ^ ) E ,
D ^ = 1 2 α β 1 dy dx j β 2 2 t 2 + 1 3! β 3 3 t 3 ,
β 1 =UΑ U ,
Α=( τ 1 + τ PMD 1 2 σ 1 0 2×2 0 2×2 τ N + τ PMD N 2 σ 1 ),
τ PMD i = D PMD 1 2 2 L correlation ,
N ^ i =jγ[ f self 8 9 | E i | 2 I + ki N f cross 4 3 | E k | 2 I 1 3 ( E i σ 3 E i ) σ 3 ]i=1,2,,N,
γ= 2 πn 2 f ref cA eff ,
x =( x 1 x 2 ,..., x n ),
x i =( x i,1 x i,2 x i,N )= {0,1,...,M1} N ,
y i =( y i,1 y i,2 y i,N ) N ,
I(Y;X)=H(Y)H(Y|X),
H(Y)=E[ log 2 Pr(Y)]= lim n 1 n log 2 Pr( y[1,n] ),
log 2 Pr( y[1,n] )= i=1 n log 2 Pr( y i | y[1,i1] ) .
H(Y)= lim n 1 n i=1 n log 2 Pr( y i | y[1,i1] ) .
H(Y|X)=E[ log 2 Pr(Y|X)]= lim n 1 n log 2 Pr( y[1,n] | x[1,n] ) = lim n 1 n i=1 n log 2 Pr( y i | y[1,i1] , x[1,n] ) .
I(Y;X)= lim n 1 n log 2 Pr( y i | y[1,i1] , x[1,n] ) lim n 1 n i=1 n log 2 Pr( y[1,n] ) = lim n 1 n i=1 n d I i ,
d I i = log 2 Pr( y i | y[1,i1] , x[1,n] ) log 2 Pr( y i | y[1,i1] ).
s i =[ x ia ,..., x i1 , x i , x i+1 ,..., x i+b ],
s i ={0,1,...,D},
m=a+b..
D= M dim = M (m+1)N .
Pr( y[ia,i+b] | s i ).
f(z)=sgn(z)( 1 μ )( (1+μ) | z | 1)| z |V.
N edges = M N .
Pr(x)= 1 M N .
α i (s)=log[ Pr( s i =s, y[1,i] ) ];
γ i (s',s)=log[ Pr( y i |s)Pr( x i ) ].
γ i (s',s)=log[ Pr( s i =s, y i , s i1 =s') ];
α i (s)= max * [ α i1 (s')+ γ i (s',s) ],
max * (c,d)=ln( e c + e d )=max(c,d)+ln(1+ e | yx | ).
{ α 0 (s)={s,s0} α 0 (s=0)=0 α 1 (s)={s} .

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