Abstract

We investigate the spectral efficiency (SE) limit of pre-filtered modulation in optical fiber communication systems. We show that pre-filtering induced symbol correlation generates a modulation with memory and, thus, a higher SE limit than that of the original memoryless modulation. The SE limits of a series of modulation formats with varying number L of correlated symbols producing 50% of the original bandwidth are evaluated, which approach the SE limit of a modulation format with 50% pre-filtering as L → ∞. We show that the SE limit of a modulation format with L correlated symbols approximates a lower bound for the SE limit of the corresponding pre-filtered format.

© 2010 OSA

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  1. M. Salsi, H. Mardoyan, P. Tran, C. Koebele, E. Dutisseuil, G. Charlet, and S. Bigo, “155x100Gbit/s coherent PDM-QPSK transmission over 7,200km,” Proc. ECOC’09, paper PD2.5 (2009).
  2. A. Sano, H. Masuda, T. Kobayashi, M. Fujiwara, K. Horikoshi, E. Yoshida, Y. Miyamoto, M. Matsui, M. Mizoguchi, H. Yamazaki, Y. Sakamaki, and H. Ishii, “69.1-Tb/s (432x171-Gb/s) C- and extended L-band transmission over 240 km using PDM-16-QAM modulation and digital coherent detection,” Proc. OFC/NFOEC’10, paper PDPB7 (2010).
  3. X. Zhou, J. Yu, M. Huang, Y. Shao, T. Wang, L. Nelson, P. Magill, M. Birk, P. I. Borel, D. W. Peckham, and R. Lingle, Jr., “64-Tb/s (640x107-Gb/s) PDM-36QAM transmission over 320km using both pre- and post-transmission digital equalization,” Proc. OFC/NFOEC’10, paper PDPB9 (2010).
  4. X. Liu, S. Chandrasekhar, B. Chu, P. J. Winzer, A. H. Gnauck, and D. W. Peckham, “Transmission of a 448-Gb/s reduced-guard-interval CO-OFDM signal with a 60-GHz optical bandwidth over 2000 km of ULAF and five 80-GHz-Grid ROADMs,” Proc. OFC/NFOEC’10, paper PDPC2 (2010).
  5. N. Alić, G. C. Papen, R. E. Saperstein, L. B. Milstein, and Y. Fainman, “Signal statistics and maximum likelihood sequence estimation in intensity modulated fiber optic links containing a single optical pre-amplifier,” Opt. Express 13(12), 4568–4579 (2005).
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  6. M. Rubsamen, P. J. Winzer, and R.-J. Essiambre, “MLSE receivers for narrow-band optical filtering,” Proc. OFC/NFOEC‘06, paper OWB6 (2006).
  7. N. Alic, E. Myslivets, and S. Radic, “1.0 bit/s/Hz spectral efficiency in single polarization at 2000km with narrowly filtered intensity modulated signals,” Proc. IEEE Summer Topical Meeting 2008, paper TuD3.4 (2008).
  8. J. X. Cai, Y. Cai, C. R. Davidson, D. Foursa, A. Lucero, O. Sinkin, A. Pilipetskii, G. Mohs, and S. Neal Bergano, “Transmission of 96x100G pre-filtered PDM-RZ-QPSK channels with 300% spectral efficiency over 10,608km and 400% spectral efficiency over 4,368km,” Proc. OFC/NFOEC’10, paper PDPB10 (2010).
  9. Y. Cai, J. X. Cai, C. R. Davidson, D. Foursa, A. Lucero, O. Sinkin, A. Pilipetskii, G. Mohs, and S. Neal Bergano, “High spectral efficiency long-Haul transmission with pre-filtering and maximum a posteriori probability detection,” Proc. ECOC’2010, paper We.7.C.4 (2010).
  10. C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423 and 623–656 (1948).
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    [CrossRef]
  12. Y. Cai, N. Ramanujam, J. M. Morris, T. Adali, G. Lenner, A. B. Puc, and A. Pilipetskii, “Performance limit of forward error correction codes in optical fiber communications,” Proc. OFC/IOOC’01, paper TuF2 (2001).
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    [CrossRef]
  15. I. 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]
  16. Y. Cai, “Performance limits of FEC and modulation formats in optical fiber communications,” Proc. LEOS’06, paper WH1 (2006).
  17. R. 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]
  18. A. D. Ellis, J. Zhao, and D. Cotter, “Approaching the non-linear Shannon limit,” J. Lightwave Technol. 28(4), 423–433 (2010).
    [CrossRef]
  19. D. Arnold, and H. Loeliger, “On the information rate of binary-input channels with memory,” Proc. ICC’2001, pp. 2692–2695 (2001).
  20. T. Cover, and J. Thomas, Elements of Information Theory, John Wiley & Sons, New York (1991).
  21. R. G. Gallager, Information Theory and Reliable Communication, John Wiley & Sons, New York (1968).
  22. J. G. Proakis, Digital Communications, 4th Edition, McGraw-Hill, New York (2001).
  23. N. Alic, M. Karlsson, M. Sköld, O. Milenkovic, P. A. Andrekson, and S. Radic, “Joint statistics and MLSD in filtered incoherent high-speed fiber optic communications,” J. Lightwave Technol. 28(10), 1564–1572 (2010).
    [CrossRef]
  24. Y. Cai, D. Foursa, C. R. Davidson, J.-X. Cai, O. Sinkin, M. Nissov, and A. Pilipetskii, “Experimental demonstration of coherent MAP detection for nonlinearity mitigation in long-haul transmissions,” Proc. OFC/NFOEC’10, paper OTuE1 (2010).
  25. I. B. Djordjevic, L. L. Minkov, and H. G. Batshon, “Mitigation of linear and nonlinear impairments in high-speed optical networks by using LDPC-coded turbo equalization,” IEEE J. Sel. Areas Comm. 26(6), 73–83 (2008).
    [CrossRef]

2010 (3)

2008 (1)

I. B. Djordjevic, L. L. Minkov, and H. G. Batshon, “Mitigation of linear and nonlinear impairments in high-speed optical networks by using LDPC-coded turbo equalization,” IEEE J. Sel. Areas Comm. 26(6), 73–83 (2008).
[CrossRef]

2005 (2)

2004 (1)

J. Kahn and K. Ho, “Spectral efficiency limits and modulation/detection techniques for DWDM systems,” J. Sel. Top. Quantum Electron. 10(2), 259–272 (2004).
[CrossRef]

2003 (1)

2001 (1)

1948 (1)

C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423 and 623–656 (1948).

Alic, N.

Andrekson, P. A.

Ashikhmin, A.

Batshon, H. G.

I. B. Djordjevic, L. L. Minkov, and H. G. Batshon, “Mitigation of linear and nonlinear impairments in high-speed optical networks by using LDPC-coded turbo equalization,” IEEE J. Sel. Areas Comm. 26(6), 73–83 (2008).
[CrossRef]

Cotter, D.

Djordjevic, I.

Djordjevic, I. B.

I. B. Djordjevic, L. L. Minkov, and H. G. Batshon, “Mitigation of linear and nonlinear impairments in high-speed optical networks by using LDPC-coded turbo equalization,” IEEE J. Sel. Areas Comm. 26(6), 73–83 (2008).
[CrossRef]

Ellis, A. D.

Essiambre, R.

Fainman, Y.

Foschini, G. J.

Gabitov, I.

Goebel, B.

Ho, K.

J. Kahn and K. Ho, “Spectral efficiency limits and modulation/detection techniques for DWDM systems,” J. Sel. Top. Quantum Electron. 10(2), 259–272 (2004).
[CrossRef]

Ivkovic, M.

Kahn, J.

J. Kahn and K. Ho, “Spectral efficiency limits and modulation/detection techniques for DWDM systems,” J. Sel. Top. Quantum Electron. 10(2), 259–272 (2004).
[CrossRef]

Karlsson, M.

Kramer, G.

Milenkovic, O.

Milstein, L. B.

Minkov, L. L.

I. B. Djordjevic, L. L. Minkov, and H. G. Batshon, “Mitigation of linear and nonlinear impairments in high-speed optical networks by using LDPC-coded turbo equalization,” IEEE J. Sel. Areas Comm. 26(6), 73–83 (2008).
[CrossRef]

Papen, G. C.

Radic, S.

Saperstein, R. E.

Shannon, C. E.

C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423 and 623–656 (1948).

Sköld, M.

Tang, J.

van Wijngaarden, A. J.

Vasic, B.

Wei, X.

Winzer, P. J.

Zhao, J.

Bell Syst. Tech. J. (1)

C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423 and 623–656 (1948).

IEEE J. Sel. Areas Comm. (1)

I. B. Djordjevic, L. L. Minkov, and H. G. Batshon, “Mitigation of linear and nonlinear impairments in high-speed optical networks by using LDPC-coded turbo equalization,” IEEE J. Sel. Areas Comm. 26(6), 73–83 (2008).
[CrossRef]

J. Lightwave Technol. (6)

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

J. Kahn and K. Ho, “Spectral efficiency limits and modulation/detection techniques for DWDM systems,” J. Sel. Top. Quantum Electron. 10(2), 259–272 (2004).
[CrossRef]

Opt. Express (1)

Other (15)

D. Arnold, and H. Loeliger, “On the information rate of binary-input channels with memory,” Proc. ICC’2001, pp. 2692–2695 (2001).

T. Cover, and J. Thomas, Elements of Information Theory, John Wiley & Sons, New York (1991).

R. G. Gallager, Information Theory and Reliable Communication, John Wiley & Sons, New York (1968).

J. G. Proakis, Digital Communications, 4th Edition, McGraw-Hill, New York (2001).

Y. Cai, “Performance limits of FEC and modulation formats in optical fiber communications,” Proc. LEOS’06, paper WH1 (2006).

Y. Cai, D. Foursa, C. R. Davidson, J.-X. Cai, O. Sinkin, M. Nissov, and A. Pilipetskii, “Experimental demonstration of coherent MAP detection for nonlinearity mitigation in long-haul transmissions,” Proc. OFC/NFOEC’10, paper OTuE1 (2010).

M. Salsi, H. Mardoyan, P. Tran, C. Koebele, E. Dutisseuil, G. Charlet, and S. Bigo, “155x100Gbit/s coherent PDM-QPSK transmission over 7,200km,” Proc. ECOC’09, paper PD2.5 (2009).

A. Sano, H. Masuda, T. Kobayashi, M. Fujiwara, K. Horikoshi, E. Yoshida, Y. Miyamoto, M. Matsui, M. Mizoguchi, H. Yamazaki, Y. Sakamaki, and H. Ishii, “69.1-Tb/s (432x171-Gb/s) C- and extended L-band transmission over 240 km using PDM-16-QAM modulation and digital coherent detection,” Proc. OFC/NFOEC’10, paper PDPB7 (2010).

X. Zhou, J. Yu, M. Huang, Y. Shao, T. Wang, L. Nelson, P. Magill, M. Birk, P. I. Borel, D. W. Peckham, and R. Lingle, Jr., “64-Tb/s (640x107-Gb/s) PDM-36QAM transmission over 320km using both pre- and post-transmission digital equalization,” Proc. OFC/NFOEC’10, paper PDPB9 (2010).

X. Liu, S. Chandrasekhar, B. Chu, P. J. Winzer, A. H. Gnauck, and D. W. Peckham, “Transmission of a 448-Gb/s reduced-guard-interval CO-OFDM signal with a 60-GHz optical bandwidth over 2000 km of ULAF and five 80-GHz-Grid ROADMs,” Proc. OFC/NFOEC’10, paper PDPC2 (2010).

M. Rubsamen, P. J. Winzer, and R.-J. Essiambre, “MLSE receivers for narrow-band optical filtering,” Proc. OFC/NFOEC‘06, paper OWB6 (2006).

N. Alic, E. Myslivets, and S. Radic, “1.0 bit/s/Hz spectral efficiency in single polarization at 2000km with narrowly filtered intensity modulated signals,” Proc. IEEE Summer Topical Meeting 2008, paper TuD3.4 (2008).

J. X. Cai, Y. Cai, C. R. Davidson, D. Foursa, A. Lucero, O. Sinkin, A. Pilipetskii, G. Mohs, and S. Neal Bergano, “Transmission of 96x100G pre-filtered PDM-RZ-QPSK channels with 300% spectral efficiency over 10,608km and 400% spectral efficiency over 4,368km,” Proc. OFC/NFOEC’10, paper PDPB10 (2010).

Y. Cai, J. X. Cai, C. R. Davidson, D. Foursa, A. Lucero, O. Sinkin, A. Pilipetskii, G. Mohs, and S. Neal Bergano, “High spectral efficiency long-Haul transmission with pre-filtering and maximum a posteriori probability detection,” Proc. ECOC’2010, paper We.7.C.4 (2010).

Y. Cai, N. Ramanujam, J. M. Morris, T. Adali, G. Lenner, A. B. Puc, and A. Pilipetskii, “Performance limit of forward error correction codes in optical fiber communications,” Proc. OFC/IOOC’01, paper TuF2 (2001).

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Figures (7)

Fig. 1
Fig. 1

(a) modulated spectrum, (b) in-phase and quadrature components in baseband, and (c) time-domain pulses of a T-second W-bandwidth signal that can approach the SE limit of an AWGN channel.

Fig. 2
Fig. 2

SE limits for unconstrained channel inputs and BPSK modulation [1D limit: from Eq. (3), BPSK: from Eq. (4)].

Fig. 3
Fig. 3

Spectrum and pulses of a binary modulation (a) before and (b) after pre-filtering, where the dashed lines indicate sampling instants.

Fig. 4
Fig. 4

The probability mass function of the signal value evaluated with Eq. (7) for k = 4.

Fig. 5
Fig. 5

Approximations of the SE limit of 50% pre-filtered BPSK by taking a small number (L) of correlated symbols in the SE limit evaluation.

Fig. 6
Fig. 6

Mean and variance of the distance between channel input vectors.

Fig. 7
Fig. 7

Upper and lower bounds on the 50% pre-filtered BPSK and QPSK modulation.

Equations (12)

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C / W = log 2 ( 1 + P N ) ,
sinc ( 2 W t ) = sin ( 2 π W t ) 2 π W t .
C 1D / W = 1 2 log 2 ( 1 + P N ) ,
C B / W = f Y ( y i ) log 2 [ f Y ( y i ) ] d y i log 2 ( 2 π e N ) ,
f Y ( y i ) = 1 2 μ = 1 1 1 2 π N exp [ ( y i μ ) 2 2 N ] .
A 0 = { 1 ,   1 } ,      A k = { A k 1 sinc ( k 1 2 ) ,   A k 1 ,   A k 1 + sinc ( k 1 2 ) } ,
P 0 = { 1 2 ,   1 2 } ,      P k = { 1 4 P k 1 ,   1 2 P k 1 ,   1 4 P k 1 } .
A = { ( x i 1 , x i 2 , ... , x i L ) ,   i = 1 ,   2 ,   ... ,   2 W T } ,
C F / W F = Lim T 2 T W ... f Y ( y 1 , y 2 ,   ... , y L ) log 2 [ f Y ( y 1 , y 2 ,   ... , y L ) ] d y 1 d y 2 ... d y L log 2 ( 2 π e N ) ,
f Y ( y 1 , y 2 ,   ... , y L ) = ( x i 1 , x i 2 , x i L ) A p ( x i 1 , x i 2 , x i L ) j = 1 L 1 2 π N exp [ ( y i x i j ) 2 2 N ] ,
p ( x i 1 , x i 2 , ... , x i L ) = 2 W T ,   i = 1 ,   2 ,   ... ,   2 W T .
D i k = ( x i 1 x k 1 ) 2 + ( x i 2 x k 2 ) 2 + ... + ( x i L x k L ) 2 .

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