Abstract

In this paper, we show numerically and experimentally that expectation maximization (EM) algorithm is a powerful tool in combating system impairments such as fibre nonlinearities, inphase and quadrature (I/Q) modulator imperfections and laser linewidth. The EM algorithm is an iterative algorithm that can be used to compensate for the impairments which have an imprint on a signal constellation, i.e. rotation and distortion of the constellation points. The EM is especially effective for combating non-linear phase noise (NLPN). It is because NLPN severely distorts the signal constellation and this can be tracked by the EM. The gain in the nonlinear system tolerance for the system under consideration is shown to be dependent on the transmission scenario. We show experimentally that for a dispersion managed polarization multiplexed 16-QAM system at 14 Gbaud a gain in the nonlinear system tolerance of up to 3 dB can be obtained. For, a dispersion unmanaged system this gain reduces to 0.5 dB.

© 2012 OSA

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References

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    [CrossRef]
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    [CrossRef]
  5. 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, 2570–2576 (2011).
    [CrossRef]
  6. N. Stojanovic, Y. Huang, F. N. Hauske, Y. Fang, M. Chen, C. Xie, and Q. Xiong, “Mlse-based nonlinearity mitigation for wdm 112 gbit/s pdm-qpsk transmissions with digital coherent receiver,” in Proc. of OFC, paper OTu3C.5, Los Angeles, California, USA, (2011).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  17. F. Vacondio, O. Rival, C. Simonneau, E. Grellier, L. Lorcy, J.-C. Antona, S. Bigo, and A. Bononi, “On nonlinear distortions of highly dispersive optical coherent systems,” Opt. Express20, 1022–1032 (2012).
    [CrossRef] [PubMed]
  18. A. Bononi, N. Rossi, and P. Serena, “Transmission limitations due to fiber nonlinearity,” in Proc. of OFC, paper OWO7, Los Angeles, California, USA, (2011).
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    [CrossRef] [PubMed]

2012 (2)

2011 (3)

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, 2570–2576 (2011).
[CrossRef]

D. Rafique, J. Zhao, and A. D. Ellis, “Compensation of nonlinear fibre impairments in coherent systems employing spectrally efficient modulation formats,” IEICE Trans. on Commun.E94-B, 1815–1822 (2011).
[CrossRef]

D. Zibar, J. C. R. F. de Olivera, V. B. Ribeiro, A. Paradisi, J. C. Diniz, K. J. Larsen, and I. T. Monroy, “Experimental investigation and digital compensation of dgd for 112 gb/s pdm-qpsk clock recovery,” Opt. Express19, 429–437 (2011).
[CrossRef]

2010 (5)

2008 (1)

2007 (1)

1977 (1)

A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximum likelihood from incomplete data via the em algorithm,” J. Roy. Stat Soc. Series B.39, 1–38 (1977).

Antona, J.-C.

Bai, N.

E. Ip, N. Bai, and T. Wang, “Complexity versus performance tradeoff for fiber nonlinearity compensation using frequency-shaped, multi-subband backpropagation,” in Proc. of OFC, paper OThF4, Los Angeles, California, USA, (2011).

Bayvel, P.

Behrens, C.

Bigo, S.

Bishop, C. M.

C. M. Bishop, Pattern Recognition and Machine Learning (Springer, 2006).

Bononi, A.

F. Vacondio, O. Rival, C. Simonneau, E. Grellier, L. Lorcy, J.-C. Antona, S. Bigo, and A. Bononi, “On nonlinear distortions of highly dispersive optical coherent systems,” Opt. Express20, 1022–1032 (2012).
[CrossRef] [PubMed]

A. Bononi, N. Rossi, and P. Serena, “Transmission limitations due to fiber nonlinearity,” in Proc. of OFC, paper OWO7, Los Angeles, California, USA, (2011).

Borkowski, R.

D. Zibar, O. Winther, N. Franceshi, R. Borkowski, A. Caballero, A. Valeria, N. M. Schmidt, G. G. Neil, B. Mao, Y. Ye, J. K. Larsen, and T. I. Monroy, “Nonlinear impairment compensation using expectation maximization for pdm 16-qam systems,” in Proc. of ECOC, paper Th1D2, Amsterdam, The Netherlands, (2012).

Bosco, G.

Buhl, L.

Caballero, A.

N. G. Gonzalez, D. Zibar, A. Caballero, and I. T. Monroy, “Experimental 2.5-gb/s qpsk wdm phasemodulated radio-over-fiber link with digital demodulation by a k-means algorithm,” IEEE Photon. Technol. Lett.22, 335–337 (2010).
[CrossRef]

D. Zibar, O. Winther, N. Franceshi, R. Borkowski, A. Caballero, A. Valeria, N. M. Schmidt, G. G. Neil, B. Mao, Y. Ye, J. K. Larsen, and T. I. Monroy, “Nonlinear impairment compensation using expectation maximization for pdm 16-qam systems,” in Proc. of ECOC, paper Th1D2, Amsterdam, The Netherlands, (2012).

Carena, A.

Chen, M.

N. Stojanovic, Y. Huang, F. N. Hauske, Y. Fang, M. Chen, C. Xie, and Q. Xiong, “Mlse-based nonlinearity mitigation for wdm 112 gbit/s pdm-qpsk transmissions with digital coherent receiver,” in Proc. of OFC, paper OTu3C.5, Los Angeles, California, USA, (2011).

Curri, V.

de Olivera, J. C. R. F.

D. Zibar, J. C. R. F. de Olivera, V. B. Ribeiro, A. Paradisi, J. C. Diniz, K. J. Larsen, and I. T. Monroy, “Experimental investigation and digital compensation of dgd for 112 gb/s pdm-qpsk clock recovery,” Opt. Express19, 429–437 (2011).
[CrossRef]

Dempster, A. P.

A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximum likelihood from incomplete data via the em algorithm,” J. Roy. Stat Soc. Series B.39, 1–38 (1977).

Diniz, J. C.

D. Zibar, J. C. R. F. de Olivera, V. B. Ribeiro, A. Paradisi, J. C. Diniz, K. J. Larsen, and I. T. Monroy, “Experimental investigation and digital compensation of dgd for 112 gb/s pdm-qpsk clock recovery,” Opt. Express19, 429–437 (2011).
[CrossRef]

Doerr, C.

Dou, L.

Ellis, A. D.

D. Rafique, J. Zhao, and A. D. Ellis, “Compensation of nonlinear fibre impairments in coherent systems employing spectrally efficient modulation formats,” IEICE Trans. on Commun.E94-B, 1815–1822 (2011).
[CrossRef]

Essiambre, R.-J.

Fang, Y.

N. Stojanovic, Y. Huang, F. N. Hauske, Y. Fang, M. Chen, C. Xie, and Q. Xiong, “Mlse-based nonlinearity mitigation for wdm 112 gbit/s pdm-qpsk transmissions with digital coherent receiver,” in Proc. of OFC, paper OTu3C.5, Los Angeles, California, USA, (2011).

Fechtel, S.

H. Meyr, M. Moeneclaey, and S. Fechtel, Digital Communication Receivers / Synchronization, Channel Estimation, and Signal Processing (Wiley, 1998).

Forghieri, F.

Foschini, G.

Franceshi, N.

D. Zibar, O. Winther, N. Franceshi, R. Borkowski, A. Caballero, A. Valeria, N. M. Schmidt, G. G. Neil, B. Mao, Y. Ye, J. K. Larsen, and T. I. Monroy, “Nonlinear impairment compensation using expectation maximization for pdm 16-qam systems,” in Proc. of ECOC, paper Th1D2, Amsterdam, The Netherlands, (2012).

Gnauck, A.

Goebel, B.

Gonzalez, N. G.

N. G. Gonzalez, D. Zibar, A. Caballero, and I. T. Monroy, “Experimental 2.5-gb/s qpsk wdm phasemodulated radio-over-fiber link with digital demodulation by a k-means algorithm,” IEEE Photon. Technol. Lett.22, 335–337 (2010).
[CrossRef]

Grellier, E.

Hauske, F. N.

N. Stojanovic, Y. Huang, F. N. Hauske, Y. Fang, M. Chen, C. Xie, and Q. Xiong, “Mlse-based nonlinearity mitigation for wdm 112 gbit/s pdm-qpsk transmissions with digital coherent receiver,” in Proc. of OFC, paper OTu3C.5, Los Angeles, California, USA, (2011).

Hoshida, T.

Huang, Y.

N. Stojanovic, Y. Huang, F. N. Hauske, Y. Fang, M. Chen, C. Xie, and Q. Xiong, “Mlse-based nonlinearity mitigation for wdm 112 gbit/s pdm-qpsk transmissions with digital coherent receiver,” in Proc. of OFC, paper OTu3C.5, Los Angeles, California, USA, (2011).

Ip, E.

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

E. Ip, N. Bai, and T. Wang, “Complexity versus performance tradeoff for fiber nonlinearity compensation using frequency-shaped, multi-subband backpropagation,” in Proc. of OFC, paper OThF4, Los Angeles, California, USA, (2011).

Kahn, J.

Killey, R. I.

Kramer, G.

Kurzweil, J.

J. Kurzweil, An Introduction to Digital Communications (John Wiley, 2000).

Laird, N. M.

A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximum likelihood from incomplete data via the em algorithm,” J. Roy. Stat Soc. Series B.39, 1–38 (1977).

Larsen, J. K.

D. Zibar, O. Winther, N. Franceshi, R. Borkowski, A. Caballero, A. Valeria, N. M. Schmidt, G. G. Neil, B. Mao, Y. Ye, J. K. Larsen, and T. I. Monroy, “Nonlinear impairment compensation using expectation maximization for pdm 16-qam systems,” in Proc. of ECOC, paper Th1D2, Amsterdam, The Netherlands, (2012).

Larsen, K. J.

D. Zibar, J. C. R. F. de Olivera, V. B. Ribeiro, A. Paradisi, J. C. Diniz, K. J. Larsen, and I. T. Monroy, “Experimental investigation and digital compensation of dgd for 112 gb/s pdm-qpsk clock recovery,” Opt. Express19, 429–437 (2011).
[CrossRef]

Lau, A.

Lavery, D.

Li, L.

Lorcy, L.

Magarini, M.

Makovejs, S.

Mao, B.

D. Zibar, O. Winther, N. Franceshi, R. Borkowski, A. Caballero, A. Valeria, N. M. Schmidt, G. G. Neil, B. Mao, Y. Ye, J. K. Larsen, and T. I. Monroy, “Nonlinear impairment compensation using expectation maximization for pdm 16-qam systems,” in Proc. of ECOC, paper Th1D2, Amsterdam, The Netherlands, (2012).

Meyr, H.

H. Meyr, M. Moeneclaey, and S. Fechtel, Digital Communication Receivers / Synchronization, Channel Estimation, and Signal Processing (Wiley, 1998).

Millar, D. S.

Moeneclaey, M.

H. Meyr, M. Moeneclaey, and S. Fechtel, Digital Communication Receivers / Synchronization, Channel Estimation, and Signal Processing (Wiley, 1998).

Monroy, I. T.

D. Zibar, J. C. R. F. de Olivera, V. B. Ribeiro, A. Paradisi, J. C. Diniz, K. J. Larsen, and I. T. Monroy, “Experimental investigation and digital compensation of dgd for 112 gb/s pdm-qpsk clock recovery,” Opt. Express19, 429–437 (2011).
[CrossRef]

N. G. Gonzalez, D. Zibar, A. Caballero, and I. T. Monroy, “Experimental 2.5-gb/s qpsk wdm phasemodulated radio-over-fiber link with digital demodulation by a k-means algorithm,” IEEE Photon. Technol. Lett.22, 335–337 (2010).
[CrossRef]

Monroy, T. I.

D. Zibar, O. Winther, N. Franceshi, R. Borkowski, A. Caballero, A. Valeria, N. M. Schmidt, G. G. Neil, B. Mao, Y. Ye, J. K. Larsen, and T. I. Monroy, “Nonlinear impairment compensation using expectation maximization for pdm 16-qam systems,” in Proc. of ECOC, paper Th1D2, Amsterdam, The Netherlands, (2012).

Neil, G. G.

D. Zibar, O. Winther, N. Franceshi, R. Borkowski, A. Caballero, A. Valeria, N. M. Schmidt, G. G. Neil, B. Mao, Y. Ye, J. K. Larsen, and T. I. Monroy, “Nonlinear impairment compensation using expectation maximization for pdm 16-qam systems,” in Proc. of ECOC, paper Th1D2, Amsterdam, The Netherlands, (2012).

Paradisi, A.

D. Zibar, J. C. R. F. de Olivera, V. B. Ribeiro, A. Paradisi, J. C. Diniz, K. J. Larsen, and I. T. Monroy, “Experimental investigation and digital compensation of dgd for 112 gb/s pdm-qpsk clock recovery,” Opt. Express19, 429–437 (2011).
[CrossRef]

Poggiolini, P.

Rafique, D.

D. Rafique, J. Zhao, and A. D. Ellis, “Compensation of nonlinear fibre impairments in coherent systems employing spectrally efficient modulation formats,” IEICE Trans. on Commun.E94-B, 1815–1822 (2011).
[CrossRef]

Rasmussen, J. C.

Ribeiro, V. B.

D. Zibar, J. C. R. F. de Olivera, V. B. Ribeiro, A. Paradisi, J. C. Diniz, K. J. Larsen, and I. T. Monroy, “Experimental investigation and digital compensation of dgd for 112 gb/s pdm-qpsk clock recovery,” Opt. Express19, 429–437 (2011).
[CrossRef]

Rival, O.

Rossi, N.

A. Bononi, N. Rossi, and P. Serena, “Transmission limitations due to fiber nonlinearity,” in Proc. of OFC, paper OWO7, Los Angeles, California, USA, (2011).

Rubin, D. B.

A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximum likelihood from incomplete data via the em algorithm,” J. Roy. Stat Soc. Series B.39, 1–38 (1977).

Savory, S. J.

Schmidt, N. M.

D. Zibar, O. Winther, N. Franceshi, R. Borkowski, A. Caballero, A. Valeria, N. M. Schmidt, G. G. Neil, B. Mao, Y. Ye, J. K. Larsen, and T. I. Monroy, “Nonlinear impairment compensation using expectation maximization for pdm 16-qam systems,” in Proc. of ECOC, paper Th1D2, Amsterdam, The Netherlands, (2012).

Serena, P.

A. Bononi, N. Rossi, and P. Serena, “Transmission limitations due to fiber nonlinearity,” in Proc. of OFC, paper OWO7, Los Angeles, California, USA, (2011).

Simonneau, C.

Stojanovic, N.

N. Stojanovic, Y. Huang, F. N. Hauske, Y. Fang, M. Chen, C. Xie, and Q. Xiong, “Mlse-based nonlinearity mitigation for wdm 112 gbit/s pdm-qpsk transmissions with digital coherent receiver,” in Proc. of OFC, paper OTu3C.5, Los Angeles, California, USA, (2011).

Tao, Z.

Vacondio, F.

Valeria, A.

D. Zibar, O. Winther, N. Franceshi, R. Borkowski, A. Caballero, A. Valeria, N. M. Schmidt, G. G. Neil, B. Mao, Y. Ye, J. K. Larsen, and T. I. Monroy, “Nonlinear impairment compensation using expectation maximization for pdm 16-qam systems,” in Proc. of ECOC, paper Th1D2, Amsterdam, The Netherlands, (2012).

Wang, T.

E. Ip, N. Bai, and T. Wang, “Complexity versus performance tradeoff for fiber nonlinearity compensation using frequency-shaped, multi-subband backpropagation,” in Proc. of OFC, paper OThF4, Los Angeles, California, USA, (2011).

Winther, O.

D. Zibar, O. Winther, N. Franceshi, R. Borkowski, A. Caballero, A. Valeria, N. M. Schmidt, G. G. Neil, B. Mao, Y. Ye, J. K. Larsen, and T. I. Monroy, “Nonlinear impairment compensation using expectation maximization for pdm 16-qam systems,” in Proc. of ECOC, paper Th1D2, Amsterdam, The Netherlands, (2012).

Winzer, P.

Xie, C.

N. Stojanovic, Y. Huang, F. N. Hauske, Y. Fang, M. Chen, C. Xie, and Q. Xiong, “Mlse-based nonlinearity mitigation for wdm 112 gbit/s pdm-qpsk transmissions with digital coherent receiver,” in Proc. of OFC, paper OTu3C.5, Los Angeles, California, USA, (2011).

Xiong, Q.

N. Stojanovic, Y. Huang, F. N. Hauske, Y. Fang, M. Chen, C. Xie, and Q. Xiong, “Mlse-based nonlinearity mitigation for wdm 112 gbit/s pdm-qpsk transmissions with digital coherent receiver,” in Proc. of OFC, paper OTu3C.5, Los Angeles, California, USA, (2011).

Yan, W.

Ye, Y.

D. Zibar, O. Winther, N. Franceshi, R. Borkowski, A. Caballero, A. Valeria, N. M. Schmidt, G. G. Neil, B. Mao, Y. Ye, J. K. Larsen, and T. I. Monroy, “Nonlinear impairment compensation using expectation maximization for pdm 16-qam systems,” in Proc. of ECOC, paper Th1D2, Amsterdam, The Netherlands, (2012).

Zhao, J.

D. Rafique, J. Zhao, and A. D. Ellis, “Compensation of nonlinear fibre impairments in coherent systems employing spectrally efficient modulation formats,” IEICE Trans. on Commun.E94-B, 1815–1822 (2011).
[CrossRef]

Zibar, D.

D. Zibar, J. C. R. F. de Olivera, V. B. Ribeiro, A. Paradisi, J. C. Diniz, K. J. Larsen, and I. T. Monroy, “Experimental investigation and digital compensation of dgd for 112 gb/s pdm-qpsk clock recovery,” Opt. Express19, 429–437 (2011).
[CrossRef]

N. G. Gonzalez, D. Zibar, A. Caballero, and I. T. Monroy, “Experimental 2.5-gb/s qpsk wdm phasemodulated radio-over-fiber link with digital demodulation by a k-means algorithm,” IEEE Photon. Technol. Lett.22, 335–337 (2010).
[CrossRef]

D. Zibar, O. Winther, N. Franceshi, R. Borkowski, A. Caballero, A. Valeria, N. M. Schmidt, G. G. Neil, B. Mao, Y. Ye, J. K. Larsen, and T. I. Monroy, “Nonlinear impairment compensation using expectation maximization for pdm 16-qam systems,” in Proc. of ECOC, paper Th1D2, Amsterdam, The Netherlands, (2012).

IEEE J Sel. Top. Quantum Electron (1)

S. J. Savory, “Digital coherent optical receivers: Algorithms and subsystems,” IEEE J Sel. Top. Quantum Electron16, 1164–1179 (2010).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

N. G. Gonzalez, D. Zibar, A. Caballero, and I. T. Monroy, “Experimental 2.5-gb/s qpsk wdm phasemodulated radio-over-fiber link with digital demodulation by a k-means algorithm,” IEEE Photon. Technol. Lett.22, 335–337 (2010).
[CrossRef]

IEICE Trans. on Commun. (1)

D. Rafique, J. Zhao, and A. D. Ellis, “Compensation of nonlinear fibre impairments in coherent systems employing spectrally efficient modulation formats,” IEICE Trans. on Commun.E94-B, 1815–1822 (2011).
[CrossRef]

J. Lightwave Technol. (6)

J. Roy. Stat Soc. Series B. (1)

A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximum likelihood from incomplete data via the em algorithm,” J. Roy. Stat Soc. Series B.39, 1–38 (1977).

Opt. Express (3)

Other (7)

H. Meyr, M. Moeneclaey, and S. Fechtel, Digital Communication Receivers / Synchronization, Channel Estimation, and Signal Processing (Wiley, 1998).

J. Kurzweil, An Introduction to Digital Communications (John Wiley, 2000).

C. M. Bishop, Pattern Recognition and Machine Learning (Springer, 2006).

N. Stojanovic, Y. Huang, F. N. Hauske, Y. Fang, M. Chen, C. Xie, and Q. Xiong, “Mlse-based nonlinearity mitigation for wdm 112 gbit/s pdm-qpsk transmissions with digital coherent receiver,” in Proc. of OFC, paper OTu3C.5, Los Angeles, California, USA, (2011).

A. Bononi, N. Rossi, and P. Serena, “Transmission limitations due to fiber nonlinearity,” in Proc. of OFC, paper OWO7, Los Angeles, California, USA, (2011).

E. Ip, N. Bai, and T. Wang, “Complexity versus performance tradeoff for fiber nonlinearity compensation using frequency-shaped, multi-subband backpropagation,” in Proc. of OFC, paper OThF4, Los Angeles, California, USA, (2011).

D. Zibar, O. Winther, N. Franceshi, R. Borkowski, A. Caballero, A. Valeria, N. M. Schmidt, G. G. Neil, B. Mao, Y. Ye, J. K. Larsen, and T. I. Monroy, “Nonlinear impairment compensation using expectation maximization for pdm 16-qam systems,” in Proc. of ECOC, paper Th1D2, Amsterdam, The Netherlands, (2012).

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Figures (10)

Fig. 1
Fig. 1

Schematic diagram of the set-up used for simulations and experiment. PD: photodiode, PBS: polarization beam splitter, A/D: analog-to-digital converter, LO: local oscillator

Fig. 2
Fig. 2

Impact of different impairments on signal constellation for a 16-QAM signal. (a) Constellation of a signal dominated by additive noise. (b) Constellation of a signal dominated by phase noise. (c) Constellation of a signal dominated by non-linear phase noise.

Fig. 3
Fig. 3

Flow-chart illustrating the steps performed by the EM algorithm and subsequent signal demodulation. Niter denotes the number of specified iteration.

Fig. 4
Fig. 4

BER as a function of combined laser linewidth for the back-to-back-case. The modulator modulation depth, m = Vpp/Vπ = 2.12, I/Q imbalance: 5% and OSNR is 25 dB.

Fig. 5
Fig. 5

The total number of spans is 12 and the combined laser linewidth is 200 kHz. (a) BER as a function of span input power for NLPN dominated transmission link, (dispersion numerically set to zero). (b) BER as a function of span input power for dispersion managed link. Transmission link consists of SSMF and DCF.

Fig. 6
Fig. 6

The total number of spans is 12. (a) BER as a function of span input power for dispersion unmanaged link. The combined laser linewidth is 200 kHz. (b) BER as a function of combined laser linewidth for dispersion unmanaged link.

Fig. 7
Fig. 7

Recovered constellation diagram impaired by nonlinear phase noise. Only a single transmission span is considered

Fig. 8
Fig. 8

(a) Constellation diagram of the demodulated signal after 800 km of transmission through dispersion managed link. (b) BER as a function of span input power for dispersion managed link after 800 km of transmission.

Fig. 9
Fig. 9

(a) Constellation diagram of the demodulated signal after 800 km of transmission through dispersion unmanaged link. (b) BER as a function of span input power for dispersion unmanaged link after 800 km of transmission.

Fig. 10
Fig. 10

(a) BER as a function of span input power for dispersion unmanaged link after 240 km of transmission. (b) BER as a function of span input power for dispersion unmanaged link after 400 km of transmission.

Equations (15)

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

p ( x ) = k = 1 M π k N ( x | μ k , Σ k ) ,
N ( x | μ k , Σ k ) = 1 2 π | Σ k | 1 / 2 e 1 2 ( x μ k ) T Σ k 1 ( x μ k ) ,
Σ k = [ var ( x 1 ) cov ( x 1 , x 2 ) cov ( x 1 , x 2 ) var ( x 2 ) ] [ σ 1 , 1 2 σ 1 , 2 2 σ 2 , 1 2 σ 2 , 2 2 ] .
Σ k = Σ = [ σ 2 0 0 σ 2 ] .
k ^ = argmax k p ( k | x )
p ( k | x ) = π k N ( x | μ k , Σ k ) l = 1 M π l N ( x | μ k , Σ k ) .
k ^ = argmax k { 1 2 x T Σ k 1 x + w k T x + w k 0 }
k ^ = argmax k { w k T x + w k 0 } .
Ξ ^ = argmax Ξ p ( X | Ξ ) ,
p ( X | Ξ ) = n = 1 N p ( x n | Ξ ) = n = 1 N k = 1 M π k N ( x n | μ k , Σ k ) .
E-step : γ n k p ( k | x n ) = π k N ( x n | μ k , Σ k ) l = 1 M π l N ( x n | μ l , Σ l ) for n = 1 , , N and k = 1 , , M
M-step : N k = n = 1 N γ n k
π k = N k N
μ k = 1 N k n = 1 N γ n k x n
Σ k = 1 N k n = 1 N γ n k ( x n μ k ) ( x n μ k ) T for k = 1 , , M ,

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