Abstract

We propose a maximum a posteriori probability (MAP) turbo equalizer based on the sliding-window multilevel Bahl–Cocke–Jelinek–Raviv algorithm. This scheme is suitable for simultaneous nonlinear and linear impairment mitigation in multilevel coded-modulation schemes with coherent detection. The proposed scheme employs large-girth quasi-cyclic LDPC codes as channel codes. We demonstrate the efficiency of this method in dealing with fiber nonlinearities by performing Monte Carlo simulations. In addition, we provide the experimental results that demonstrate the efficiency of this method in dealing with polarization mode dispersion. We also study the ultimate channel capacity limits, assuming an independent identically distributed source.

© 2009 Optical Society of America

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    [CrossRef]
  4. R.-J. Essiambre, G. J. Foschini, G. Kramer, P. J. Winzer, “Capacity limits of information transport in fiber-optic networks,” Phys. Rev. Lett., vol. 101, paper 163901, Oct. 2008.
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  21. J. Hou, P. H. Siegel, L. B. Milstein, H. D. Pfitser, “Capacity approaching bandwidth-efficient coded modulation schemes based on low-density parity-check codes,” IEEE Trans. Inf. Theory, vol. 49, no. 9, pp. 2141–2155, Sept. 2003.
    [CrossRef]
  22. G. Caire, G. Taricco, E. Biglieri, “Bit-interleaved coded modulation,” IEEE Trans. Inf. Theory, vol. 44, pp. 927–946, May 1998.
    [CrossRef]
  23. H. Xiao-Yu, E. Eleftheriou, D.-M. Arnold, A. Dholakia, “Efficient implementations of the sum-product algorithm for decoding of LDPC codes,” in Proc. IEEE GLOBECOM, vol. 2, Nov. 2001, pp. 1036–1036E.
  24. G. Colavolpe, G. Ferrari, R. Raheli, “Reduced-sate BCJR-type algorithms,” IEEE J. Sel. Areas Commun., vol. 19, pp. 848–858, May 2001.
    [CrossRef]
  25. I. B. Djordjevic, B. Vasic, “Nonlinear BCJR equalizer for suppression of intrachannel nonlinearities in 40 Gb∕s optical communication systems,” Opt. Express, vol. 14, pp. 4625–4635, 2006.
    [CrossRef] [PubMed]
  26. T. M. Cover, J. A. Thomas, Elements of Information Theory. New York: Wiley, 1991.
    [CrossRef]
  27. F. M. Reza, An Introduction to Information Theory. New York: McGraw-Hill, 1961.
  28. W. T. Webb, R. Steele, “Variable rate QAM for mobile radio,” IEEE Trans. Commun., vol. 43, pp. 2223–2230, July 1995.
    [CrossRef]
  29. J. G. Proakis, Digital Communications. Boston: McGraw Hill, 2001.

2008

S. J. Savory, “Digital filters for coherent optical receivers,” Opt. Express, vol. 16, pp. 804–817, Jan. 2008.
[CrossRef] [PubMed]

I. B. Djordjevic, L. L. Minkov, H. G. Batshon, “Mitigation of linear and nonlinear impairments in high-speed optical networks by using LDPC-coded turbo equalization,” IEEE J. Sel. Areas Commun., vol. 26, no. 6, pp. 73–83, Aug. 2008.
[CrossRef]

R.-J. Essiambre, G. J. Foschini, G. Kramer, P. J. Winzer, “Capacity limits of information transport in fiber-optic networks,” Phys. Rev. Lett., vol. 101, paper 163901, Oct. 2008.
[CrossRef] [PubMed]

I. B. Djordjevic, L. Xu, T. Wang, “Simultaneous chromatic dispersion and PMD compensation by using coded-OFDM and girth-10 LCPC codes,” Opt. Express, vol. 16, no. 14, pp. 10269–10278, July 2008.
[CrossRef] [PubMed]

2007

2006

2005

2004

M. P. C. Fossorier, “Quasi-cyclic low-density parity-check codes from circulant permutation matricies,” JETP, vol. 50, pp. 1788–1794, Aug. 2004.

J. M. Kahn, K.-P. Ho, “Spectral efficiency limits and modulation/detection techniques for DWDM systems,” IEEE J. Sel. Top. Quantum Electron., vol. 10, no. 2, pp. 259–272, Mar./Apr. 2004.
[CrossRef]

2003

K. S. Turitsyn, S. A. Derevyanko, I. V. Yurkevich, S. K. Turitsyn, “Information capacity of optical fiber channels with zero average dispersion,” Phys. Rev. Lett., vol. 91, no. 20, paper 203901, Nov. 2003.
[CrossRef] [PubMed]

J. Hou, P. H. Siegel, L. B. Milstein, H. D. Pfitser, “Capacity approaching bandwidth-efficient coded modulation schemes based on low-density parity-check codes,” IEEE Trans. Inf. Theory, vol. 49, no. 9, pp. 2141–2155, Sept. 2003.
[CrossRef]

2002

2001

P. P. Mitra, J. B. Stark, “Nonlinear limits to the information capacity of optical fiber communications,” Nature, vol. 411, no. 6841, pp. 1027–1030, June 2001.
[CrossRef] [PubMed]

S. ten Brink, “Convergence behavior of iteratively decoded parallel concatenated codes,” IEEE Trans. Commun., vol. 40, pp. 1727–1737, Oct. 2001.
[CrossRef]

G. Colavolpe, G. Ferrari, R. Raheli, “Reduced-sate BCJR-type algorithms,” IEEE J. Sel. Areas Commun., vol. 19, pp. 848–858, May 2001.
[CrossRef]

1998

G. Caire, G. Taricco, E. Biglieri, “Bit-interleaved coded modulation,” IEEE Trans. Inf. Theory, vol. 44, pp. 927–946, May 1998.
[CrossRef]

1995

W. T. Webb, R. Steele, “Variable rate QAM for mobile radio,” IEEE Trans. Commun., vol. 43, pp. 2223–2230, July 1995.
[CrossRef]

1974

L. R. Bahl, J. Cocke, F. Jelinek, J. Raviv, “Optimal decoding of linear codes for minimizing symbol error rate,” IEEE Trans. Inf. Theory, vol. IT-20, pp. 284–287, Mar. 1974.
[CrossRef]

Alic, N.

N. Alic, G. C. Papen, R. E. Saperstein, R. Jiang, C. Marki, Y. Fainman, S. Radic, “Experimental demonstration of 10 Gb∕s NRZ extended dispersion-limited reach over 600 km-SMF link without optical dispersion compensation,” in Optical Fiber Communication Conf. and Expo. and the Nat. Fiber Optic Engineers Conf., 5–10 March 2006, paper OWB7.

Arnold, D.-M.

H. Xiao-Yu, E. Eleftheriou, D.-M. Arnold, A. Dholakia, “Efficient implementations of the sum-product algorithm for decoding of LDPC codes,” in Proc. IEEE GLOBECOM, vol. 2, Nov. 2001, pp. 1036–1036E.

Bahl, L. R.

L. R. Bahl, J. Cocke, F. Jelinek, J. Raviv, “Optimal decoding of linear codes for minimizing symbol error rate,” IEEE Trans. Inf. Theory, vol. IT-20, pp. 284–287, Mar. 1974.
[CrossRef]

Batshon, H. G.

I. B. Djordjevic, L. L. Minkov, H. G. Batshon, “Mitigation of linear and nonlinear impairments in high-speed optical networks by using LDPC-coded turbo equalization,” IEEE J. Sel. Areas Commun., vol. 26, no. 6, pp. 73–83, Aug. 2008.
[CrossRef]

Biglieri, E.

G. Caire, G. Taricco, E. Biglieri, “Bit-interleaved coded modulation,” IEEE Trans. Inf. Theory, vol. 44, pp. 927–946, May 1998.
[CrossRef]

Caire, G.

G. Caire, G. Taricco, E. Biglieri, “Bit-interleaved coded modulation,” IEEE Trans. Inf. Theory, vol. 44, pp. 927–946, May 1998.
[CrossRef]

Cocke, J.

L. R. Bahl, J. Cocke, F. Jelinek, J. Raviv, “Optimal decoding of linear codes for minimizing symbol error rate,” IEEE Trans. Inf. Theory, vol. IT-20, pp. 284–287, Mar. 1974.
[CrossRef]

Colavolpe, G.

G. Colavolpe, G. Ferrari, R. Raheli, “Reduced-sate BCJR-type algorithms,” IEEE J. Sel. Areas Commun., vol. 19, pp. 848–858, May 2001.
[CrossRef]

Costello, D. J.

S. Lin, D. J. Costello, Error Control Coding: Fundamentals and Applications. Upper Sadle River: Pearson Prentice-Hall, 2004.

Cover, T. M.

T. M. Cover, J. A. Thomas, Elements of Information Theory. New York: Wiley, 1991.
[CrossRef]

Derevyanko, S. A.

K. S. Turitsyn, S. A. Derevyanko, I. V. Yurkevich, S. K. Turitsyn, “Information capacity of optical fiber channels with zero average dispersion,” Phys. Rev. Lett., vol. 91, no. 20, paper 203901, Nov. 2003.
[CrossRef] [PubMed]

Dholakia, A.

H. Xiao-Yu, E. Eleftheriou, D.-M. Arnold, A. Dholakia, “Efficient implementations of the sum-product algorithm for decoding of LDPC codes,” in Proc. IEEE GLOBECOM, vol. 2, Nov. 2001, pp. 1036–1036E.

Djordjevic, I.

M. Ivkovic, I. Djordjevic, P. Rajkovic, B. Vasic, “Pulse energy probability density functions for long-haul optical fiber transmission systems by using instantons and Edgeworth expansion,” IEEE Photon. Technol. Lett., vol. 19, no. 20, pp. 1604–1606, Oct. 2007.
[CrossRef]

Djordjevic, I. B.

Eleftheriou, E.

H. Xiao-Yu, E. Eleftheriou, D.-M. Arnold, A. Dholakia, “Efficient implementations of the sum-product algorithm for decoding of LDPC codes,” in Proc. IEEE GLOBECOM, vol. 2, Nov. 2001, pp. 1036–1036E.

Essiambre, R.-J.

R.-J. Essiambre, G. J. Foschini, G. Kramer, P. J. Winzer, “Capacity limits of information transport in fiber-optic networks,” Phys. Rev. Lett., vol. 101, paper 163901, Oct. 2008.
[CrossRef] [PubMed]

Fainman, Y.

N. Alic, G. C. Papen, R. E. Saperstein, R. Jiang, C. Marki, Y. Fainman, S. Radic, “Experimental demonstration of 10 Gb∕s NRZ extended dispersion-limited reach over 600 km-SMF link without optical dispersion compensation,” in Optical Fiber Communication Conf. and Expo. and the Nat. Fiber Optic Engineers Conf., 5–10 March 2006, paper OWB7.

Ferrari, G.

G. Colavolpe, G. Ferrari, R. Raheli, “Reduced-sate BCJR-type algorithms,” IEEE J. Sel. Areas Commun., vol. 19, pp. 848–858, May 2001.
[CrossRef]

Foschini, G. J.

R.-J. Essiambre, G. J. Foschini, G. Kramer, P. J. Winzer, “Capacity limits of information transport in fiber-optic networks,” Phys. Rev. Lett., vol. 101, paper 163901, Oct. 2008.
[CrossRef] [PubMed]

Fossorier, M. P. C.

M. P. C. Fossorier, “Quasi-cyclic low-density parity-check codes from circulant permutation matricies,” JETP, vol. 50, pp. 1788–1794, Aug. 2004.

Gabitov, I.

Ho, K.-P.

J. M. Kahn, K.-P. Ho, “Spectral efficiency limits and modulation/detection techniques for DWDM systems,” IEEE J. Sel. Top. Quantum Electron., vol. 10, no. 2, pp. 259–272, Mar./Apr. 2004.
[CrossRef]

Hou, J.

J. Hou, P. H. Siegel, L. B. Milstein, H. D. Pfitser, “Capacity approaching bandwidth-efficient coded modulation schemes based on low-density parity-check codes,” IEEE Trans. Inf. Theory, vol. 49, no. 9, pp. 2141–2155, Sept. 2003.
[CrossRef]

Ip, E.

E. Ip, J. M. Kahn, “Nonlinear impairment compensation using backpropagation,” in Optical Fibre, New Developments. Vienna, Austria: In-Tech, to be published.

Ivkovic, M.

Jelinek, F.

L. R. Bahl, J. Cocke, F. Jelinek, J. Raviv, “Optimal decoding of linear codes for minimizing symbol error rate,” IEEE Trans. Inf. Theory, vol. IT-20, pp. 284–287, Mar. 1974.
[CrossRef]

Jiang, R.

N. Alic, G. C. Papen, R. E. Saperstein, R. Jiang, C. Marki, Y. Fainman, S. Radic, “Experimental demonstration of 10 Gb∕s NRZ extended dispersion-limited reach over 600 km-SMF link without optical dispersion compensation,” in Optical Fiber Communication Conf. and Expo. and the Nat. Fiber Optic Engineers Conf., 5–10 March 2006, paper OWB7.

Kahn, J. M.

J. M. Kahn, K.-P. Ho, “Spectral efficiency limits and modulation/detection techniques for DWDM systems,” IEEE J. Sel. Top. Quantum Electron., vol. 10, no. 2, pp. 259–272, Mar./Apr. 2004.
[CrossRef]

E. Ip, J. M. Kahn, “Nonlinear impairment compensation using backpropagation,” in Optical Fibre, New Developments. Vienna, Austria: In-Tech, to be published.

Kramer, G.

R.-J. Essiambre, G. J. Foschini, G. Kramer, P. J. Winzer, “Capacity limits of information transport in fiber-optic networks,” Phys. Rev. Lett., vol. 101, paper 163901, Oct. 2008.
[CrossRef] [PubMed]

Lin, S.

S. Lin, D. J. Costello, Error Control Coding: Fundamentals and Applications. Upper Sadle River: Pearson Prentice-Hall, 2004.

Ma, Y.

Marki, C.

N. Alic, G. C. Papen, R. E. Saperstein, R. Jiang, C. Marki, Y. Fainman, S. Radic, “Experimental demonstration of 10 Gb∕s NRZ extended dispersion-limited reach over 600 km-SMF link without optical dispersion compensation,” in Optical Fiber Communication Conf. and Expo. and the Nat. Fiber Optic Engineers Conf., 5–10 March 2006, paper OWB7.

Milstein, L. B.

J. Hou, P. H. Siegel, L. B. Milstein, H. D. Pfitser, “Capacity approaching bandwidth-efficient coded modulation schemes based on low-density parity-check codes,” IEEE Trans. Inf. Theory, vol. 49, no. 9, pp. 2141–2155, Sept. 2003.
[CrossRef]

Minkov, L. L.

I. B. Djordjevic, L. L. Minkov, H. G. Batshon, “Mitigation of linear and nonlinear impairments in high-speed optical networks by using LDPC-coded turbo equalization,” IEEE J. Sel. Areas Commun., vol. 26, no. 6, pp. 73–83, Aug. 2008.
[CrossRef]

Mitra, P. P.

P. P. Mitra, J. B. Stark, “Nonlinear limits to the information capacity of optical fiber communications,” Nature, vol. 411, no. 6841, pp. 1027–1030, June 2001.
[CrossRef] [PubMed]

Papen, G. C.

N. Alic, G. C. Papen, R. E. Saperstein, R. Jiang, C. Marki, Y. Fainman, S. Radic, “Experimental demonstration of 10 Gb∕s NRZ extended dispersion-limited reach over 600 km-SMF link without optical dispersion compensation,” in Optical Fiber Communication Conf. and Expo. and the Nat. Fiber Optic Engineers Conf., 5–10 March 2006, paper OWB7.

Pfitser, H. D.

J. Hou, P. H. Siegel, L. B. Milstein, H. D. Pfitser, “Capacity approaching bandwidth-efficient coded modulation schemes based on low-density parity-check codes,” IEEE Trans. Inf. Theory, vol. 49, no. 9, pp. 2141–2155, Sept. 2003.
[CrossRef]

Proakis, J. G.

J. G. Proakis, Digital Communications. Boston: McGraw Hill, 2001.

Radic, S.

N. Alic, G. C. Papen, R. E. Saperstein, R. Jiang, C. Marki, Y. Fainman, S. Radic, “Experimental demonstration of 10 Gb∕s NRZ extended dispersion-limited reach over 600 km-SMF link without optical dispersion compensation,” in Optical Fiber Communication Conf. and Expo. and the Nat. Fiber Optic Engineers Conf., 5–10 March 2006, paper OWB7.

Raheli, R.

G. Colavolpe, G. Ferrari, R. Raheli, “Reduced-sate BCJR-type algorithms,” IEEE J. Sel. Areas Commun., vol. 19, pp. 848–858, May 2001.
[CrossRef]

Rajkovic, P.

M. Ivkovic, I. Djordjevic, P. Rajkovic, B. Vasic, “Pulse energy probability density functions for long-haul optical fiber transmission systems by using instantons and Edgeworth expansion,” IEEE Photon. Technol. Lett., vol. 19, no. 20, pp. 1604–1606, Oct. 2007.
[CrossRef]

Raviv, J.

L. R. Bahl, J. Cocke, F. Jelinek, J. Raviv, “Optimal decoding of linear codes for minimizing symbol error rate,” IEEE Trans. Inf. Theory, vol. IT-20, pp. 284–287, Mar. 1974.
[CrossRef]

Reza, F. M.

F. M. Reza, An Introduction to Information Theory. New York: McGraw-Hill, 1961.

Ryan, W. E.

W. E. Ryan, “Concatenated convolutional codes and iterative decoding,” in Wiley Encyclopedia of Telecommunications, J. G. Proakis, ed. New York: Wiley, 2002.

Saperstein, R. E.

N. Alic, G. C. Papen, R. E. Saperstein, R. Jiang, C. Marki, Y. Fainman, S. Radic, “Experimental demonstration of 10 Gb∕s NRZ extended dispersion-limited reach over 600 km-SMF link without optical dispersion compensation,” in Optical Fiber Communication Conf. and Expo. and the Nat. Fiber Optic Engineers Conf., 5–10 March 2006, paper OWB7.

Savory, S. J.

Shieh, W.

Siegel, P. H.

J. Hou, P. H. Siegel, L. B. Milstein, H. D. Pfitser, “Capacity approaching bandwidth-efficient coded modulation schemes based on low-density parity-check codes,” IEEE Trans. Inf. Theory, vol. 49, no. 9, pp. 2141–2155, Sept. 2003.
[CrossRef]

Stark, J. B.

P. P. Mitra, J. B. Stark, “Nonlinear limits to the information capacity of optical fiber communications,” Nature, vol. 411, no. 6841, pp. 1027–1030, June 2001.
[CrossRef] [PubMed]

Steele, R.

W. T. Webb, R. Steele, “Variable rate QAM for mobile radio,” IEEE Trans. Commun., vol. 43, pp. 2223–2230, July 1995.
[CrossRef]

Tang, J.

Tang, Y.

Taricco, G.

G. Caire, G. Taricco, E. Biglieri, “Bit-interleaved coded modulation,” IEEE Trans. Inf. Theory, vol. 44, pp. 927–946, May 1998.
[CrossRef]

ten Brink, S.

S. ten Brink, “Convergence behavior of iteratively decoded parallel concatenated codes,” IEEE Trans. Commun., vol. 40, pp. 1727–1737, Oct. 2001.
[CrossRef]

Thomas, J. A.

T. M. Cover, J. A. Thomas, Elements of Information Theory. New York: Wiley, 1991.
[CrossRef]

Turitsyn, K. S.

K. S. Turitsyn, S. A. Derevyanko, I. V. Yurkevich, S. K. Turitsyn, “Information capacity of optical fiber channels with zero average dispersion,” Phys. Rev. Lett., vol. 91, no. 20, paper 203901, Nov. 2003.
[CrossRef] [PubMed]

Turitsyn, S. K.

K. S. Turitsyn, S. A. Derevyanko, I. V. Yurkevich, S. K. Turitsyn, “Information capacity of optical fiber channels with zero average dispersion,” Phys. Rev. Lett., vol. 91, no. 20, paper 203901, Nov. 2003.
[CrossRef] [PubMed]

Vasic, B.

Wang, T.

Webb, W. T.

W. T. Webb, R. Steele, “Variable rate QAM for mobile radio,” IEEE Trans. Commun., vol. 43, pp. 2223–2230, July 1995.
[CrossRef]

Winzer, P. J.

R.-J. Essiambre, G. J. Foschini, G. Kramer, P. J. Winzer, “Capacity limits of information transport in fiber-optic networks,” Phys. Rev. Lett., vol. 101, paper 163901, Oct. 2008.
[CrossRef] [PubMed]

Xiao-Yu, H.

H. Xiao-Yu, E. Eleftheriou, D.-M. Arnold, A. Dholakia, “Efficient implementations of the sum-product algorithm for decoding of LDPC codes,” in Proc. IEEE GLOBECOM, vol. 2, Nov. 2001, pp. 1036–1036E.

Xu, L.

Yi, X.

Yurkevich, I. V.

K. S. Turitsyn, S. A. Derevyanko, I. V. Yurkevich, S. K. Turitsyn, “Information capacity of optical fiber channels with zero average dispersion,” Phys. Rev. Lett., vol. 91, no. 20, paper 203901, Nov. 2003.
[CrossRef] [PubMed]

IEEE J. Sel. Areas Commun.

I. B. Djordjevic, L. L. Minkov, H. G. Batshon, “Mitigation of linear and nonlinear impairments in high-speed optical networks by using LDPC-coded turbo equalization,” IEEE J. Sel. Areas Commun., vol. 26, no. 6, pp. 73–83, Aug. 2008.
[CrossRef]

G. Colavolpe, G. Ferrari, R. Raheli, “Reduced-sate BCJR-type algorithms,” IEEE J. Sel. Areas Commun., vol. 19, pp. 848–858, May 2001.
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

J. M. Kahn, K.-P. Ho, “Spectral efficiency limits and modulation/detection techniques for DWDM systems,” IEEE J. Sel. Top. Quantum Electron., vol. 10, no. 2, pp. 259–272, Mar./Apr. 2004.
[CrossRef]

IEEE Photon. Technol. Lett.

M. Ivkovic, I. Djordjevic, P. Rajkovic, B. Vasic, “Pulse energy probability density functions for long-haul optical fiber transmission systems by using instantons and Edgeworth expansion,” IEEE Photon. Technol. Lett., vol. 19, no. 20, pp. 1604–1606, Oct. 2007.
[CrossRef]

IEEE Trans. Commun.

W. T. Webb, R. Steele, “Variable rate QAM for mobile radio,” IEEE Trans. Commun., vol. 43, pp. 2223–2230, July 1995.
[CrossRef]

S. ten Brink, “Convergence behavior of iteratively decoded parallel concatenated codes,” IEEE Trans. Commun., vol. 40, pp. 1727–1737, Oct. 2001.
[CrossRef]

IEEE Trans. Inf. Theory

L. R. Bahl, J. Cocke, F. Jelinek, J. Raviv, “Optimal decoding of linear codes for minimizing symbol error rate,” IEEE Trans. Inf. Theory, vol. IT-20, pp. 284–287, Mar. 1974.
[CrossRef]

J. Hou, P. H. Siegel, L. B. Milstein, H. D. Pfitser, “Capacity approaching bandwidth-efficient coded modulation schemes based on low-density parity-check codes,” IEEE Trans. Inf. Theory, vol. 49, no. 9, pp. 2141–2155, Sept. 2003.
[CrossRef]

G. Caire, G. Taricco, E. Biglieri, “Bit-interleaved coded modulation,” IEEE Trans. Inf. Theory, vol. 44, pp. 927–946, May 1998.
[CrossRef]

J. Lightwave Technol.

JETP

M. P. C. Fossorier, “Quasi-cyclic low-density parity-check codes from circulant permutation matricies,” JETP, vol. 50, pp. 1788–1794, Aug. 2004.

Nature

P. P. Mitra, J. B. Stark, “Nonlinear limits to the information capacity of optical fiber communications,” Nature, vol. 411, no. 6841, pp. 1027–1030, June 2001.
[CrossRef] [PubMed]

Opt. Express

Phys. Rev. Lett.

R.-J. Essiambre, G. J. Foschini, G. Kramer, P. J. Winzer, “Capacity limits of information transport in fiber-optic networks,” Phys. Rev. Lett., vol. 101, paper 163901, Oct. 2008.
[CrossRef] [PubMed]

K. S. Turitsyn, S. A. Derevyanko, I. V. Yurkevich, S. K. Turitsyn, “Information capacity of optical fiber channels with zero average dispersion,” Phys. Rev. Lett., vol. 91, no. 20, paper 203901, Nov. 2003.
[CrossRef] [PubMed]

Other

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Figures (9)

Fig. 1
Fig. 1

A portion of the trellis for the four-level BCJR equalizer with memory 2 m + 1 = 3 .

Fig. 2
Fig. 2

Proposed coded-modulation and turbo equalization schemes: (a) transmitter configuration and (b) receiver configuration. PM, phase modulator; DFB, distributed-feedback laser.

Fig. 3
Fig. 3

Forward/backward recursion steps for M = 4 -level BCJR equalizer: (a) the forward recursion step, (b) the backward recursion step.

Fig. 4
Fig. 4

Dispersion map under study.

Fig. 5
Fig. 5

BER performance of turbo equalizer based on 4-level BCJR equalizer for QPSK with Gray mapping. The dispersion map used for backpropagation is different from that shown in Fig 4. It is based on only single-mode fiber with EDFAs deployed every 100 km .

Fig. 6
Fig. 6

IID information capacities for linear channel model and different signal constellation sizes. (64-star QAM contains 8 rings with 8 points each, 256-star QAM contains 16 rings with 16 points, and 1024-star QAM contains 16 rings with 64 points.) The SNR is defined as E s N 0 , where E s is the symbol energy and N 0 is the power spectral density.

Fig. 7
Fig. 7

IID information capacity for QPSK and 8PSK with a symbol rate of 50 GS s against the transmission distance.

Fig. 8
Fig. 8

Experimental setup for the polarization multiplexed BPSK study. CW Laser, continuous-wave laser; PM, phase modulator; ASE, amplified spontaneous emission noise source; 3 dB , 3 dB coupler.

Fig. 9
Fig. 9

BER performance of the multilevel turbo for PMD cpmpensation.

Tables (1)

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Table 1 Fiber Parameters

Equations (15)

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α j ( s ) = max * s [ α j 1 ( s ) + γ j ( s , s ) ] ,
β j 1 ( s ) = max * s [ β j ( s ) + γ j ( s , s ) ] ,
γ j ( s , s ) = log [ p ( y j | x [ j m , j + m ] ) P ( x j ) ] .
α 0 ( s ) = { 0 , s = s 0 , s s 0 } and β n ( s ) = { 0 , s = s 0 , s s 0 } ,
Λ ( x j = δ ) = max * ( s , s ) : x j = δ [ α j 1 ( s ) + γ j ( s , s ) + β j ( s ) ] max * ( s , s ) : x j = δ 0 [ α j 1 ( s ) + γ j ( s , s ) + β j ( s ) ] ,
L ( c ̂ k ) = log x j : c k = 0 exp [ Λ ( x j ) ] x j : c k = 1 exp [ Λ ( x j ) ] ,
L LDPC , e [ c k ( t ) ] = L LDPC ( c k ( t ) ) L LDPC ( c k ( t 1 ) ) ,
L BCJR , a ( x j ) = log [ P ( x j ) ] = k = 0 l 1 ( 1 c k ) L D , e ( c k ) .
H = [ I I I I I P S [ 1 ] P S [ 2 ] P S [ c 1 ] I P 2 S [ 1 ] P 2 S [ 2 ] P 2 S [ c 1 ] I P ( r 1 ) S [ 1 ] P ( r 1 ) S [ 2 ] P ( r 1 ) S [ c 1 ] ] ,
I ( Y ; X ) = H ( Y ) H ( Y | X ) ,
E [ log 2 P ( Y ) ] = lim n ( 1 n ) log 2 P ( y [ 1 , n ] ) ,
I ( Y ; X ) = lim n 1 n [ i = 1 n log 2 P ( y i | y [ 1 , i 1 ] , x [ 1 , n ] ) i = 1 n log 2 P ( y i | y [ 1 , i 1 ] ) ] .
α j ( s ) = max * s [ α j 1 ( s ) + γ j ( s , s ) log 2 M ] ,
γ j ( s , s ) = log [ p ( y j | x [ j m , j + m ] ) ] .
log 2 P ( y i | y [ 1 , i 1 ] ) = max * s α i ( s ) ,