Abstract

We analytically obtain the exact fundamental limit of 4-level amplitude shifted keying formats (4ASK) with direct detection optically preamplified receivers. The optimum multilevel spacing and the corresponding decision thresholds, which depend both on the signal to noise ratio and optical bandwidth, are obtained numerically considering the Chi-squared distribution of each level. The quantum limit under the optimum level spacing is 127.5 photons/bit, which is about 0.2 dB smaller than the results by the Gaussian approximation. Over a broad range of the signal to noise ratio and the optical bandwidth, we have found that not only the bit error rate but also the optimum level spacing are well predicted by Gaussian method, although the three decision thresholds are all underestimated.

© 2007 Optical Society of America

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  1. S. Walklin and J. Conradi, "Multilevel signaling for increasing the reach of 10Gbps lightwave systems," J. Lightwave Technol. 17, 2235 (1999).
    [CrossRef]
  2. S. K. Ibrahim, S. Bhandare, and R. Noe, "Narrowband 20-Gbps quaternary intensity modulation generated by duobinary 10-Gbps modulaltion in 2 quadratures," ECOC 2005 Paper Th 2.6.5.
  3. S. K. Ibrahim, S. Bhandare, and R. Noe, "20-Gbps quaternary intensity modulation based on duobinary modulation with unequal amplitude in two polarizations," IEEE Photon. Technol. Lett. 14, 1482 (2006).
    [CrossRef]
  4. S. K. Ibrahim, S. Bhandare, and R. Noe, "Narrowband 2x10 Gbit/s Quaternary Intensity Modulation Based on Duobinary Modulation in Two Polarizations with Unequal Amplitudes," OFC 2006 Paper OThI2.
  5. J. Zhao, L. Huo, C. K. Chan, L. K. Chen, and C. Lin, "Analytical investigation of optimization, performance bound, and chromatic dispersion tolerances of 4-amplitude-shifted keying format," OFC 2006 Paper JThB15.
  6. J. Rebola and A. V. T. Cartaxo, "On the quaternary level spacing signaling optimization for increasing the transmission distance in optical communication systems," in Proc. of ConfTele2001 3rd Conference on Telecommunications, Figueira da Foz 23, 514 (2001).
  7. N. Avlonitis, E. M. Yeatman, M. Jones, and A. Hadjifotiou, "Multilevel amplitude shift keying in dispersion uncompensated optical systems," IEE Proc. Optoelectron. 153, 101 (2006).
    [CrossRef]
  8. P. J. Winzer, "Optically preamplifier receiver with low quantum limit," Electron. Lett. 37, 582 (2001).
    [CrossRef]
  9. L. Ribeiro, J. Da Rocha, and J. Pinto, "Performance evaluation of EDFA preamplified receivers taking into account intersymbol interference," J. Lightwave Technol. 13, 225 (1995).
    [CrossRef]
  10. P. A. Humblet, and M. Azizoglu, "On the bit error rate of lightwave systems with optical amplifier," J. Lightwave Technol. 9, 1576 (1991).
    [CrossRef]
  11. D. Marcuse, "Derivation of analytical expressions for the bit-error probability in lightwave systems with optical amplifiers," J. Lightwave Technol. 8, 1816 (1990).
    [CrossRef]
  12. J. P. Gordon and L. F. Mollenauer, "Effects of fiber nonlinearities and amplifier spacing on ultra-long distance transmission," J. Lightwave Technol. 9, 170 (1991).
    [CrossRef]
  13. L. W. Couch, Digital and Analog Communication Systems (Prentice Hall, Upper Saddle River, N.J., 2001).
  14. C. W. Helstrom, "Computing the generalized Marcum Q-function," IEEE Transactions on Information Theory 38, 1422 (1992).
    [CrossRef]
  15. G. Bosco, A. Carena, V. Curri, R. Gaudino, and P. Poggiolini, "Quantum limit of direct-dection receivers using duobinary transmission," IEEE Photon. Technol. Lett. 15, 102 (2003).
    [CrossRef]
  16. J. M. Kahn and K. P. Ho, "Spectral efficiency limits and modulation/detection techniques for DWDM systems," IEEE J. Sel. Top. Quantum Electron. 10, 259 (2004).
    [CrossRef]
  17. G. Bosco and P. Poggiolini, "The Impact of receiver imperfections on the performance of optical direct-detection DPSK," J. Lightwave Technol. 23, 842 (2005).
    [CrossRef]
  18. K. Sekine, N. Kikuchi, and S. Sasaki, "Modulation parameter tolerance for 8- and 16-APSK (amplitude- and phase-shift keying) signals," OFC 2006 Paper JThB13.

2006

S. K. Ibrahim, S. Bhandare, and R. Noe, "20-Gbps quaternary intensity modulation based on duobinary modulation with unequal amplitude in two polarizations," IEEE Photon. Technol. Lett. 14, 1482 (2006).
[CrossRef]

N. Avlonitis, E. M. Yeatman, M. Jones, and A. Hadjifotiou, "Multilevel amplitude shift keying in dispersion uncompensated optical systems," IEE Proc. Optoelectron. 153, 101 (2006).
[CrossRef]

2005

2004

J. M. Kahn and K. P. Ho, "Spectral efficiency limits and modulation/detection techniques for DWDM systems," IEEE J. Sel. Top. Quantum Electron. 10, 259 (2004).
[CrossRef]

2003

G. Bosco, A. Carena, V. Curri, R. Gaudino, and P. Poggiolini, "Quantum limit of direct-dection receivers using duobinary transmission," IEEE Photon. Technol. Lett. 15, 102 (2003).
[CrossRef]

2001

P. J. Winzer, "Optically preamplifier receiver with low quantum limit," Electron. Lett. 37, 582 (2001).
[CrossRef]

1999

1995

L. Ribeiro, J. Da Rocha, and J. Pinto, "Performance evaluation of EDFA preamplified receivers taking into account intersymbol interference," J. Lightwave Technol. 13, 225 (1995).
[CrossRef]

1992

C. W. Helstrom, "Computing the generalized Marcum Q-function," IEEE Transactions on Information Theory 38, 1422 (1992).
[CrossRef]

1991

J. P. Gordon and L. F. Mollenauer, "Effects of fiber nonlinearities and amplifier spacing on ultra-long distance transmission," J. Lightwave Technol. 9, 170 (1991).
[CrossRef]

P. A. Humblet, and M. Azizoglu, "On the bit error rate of lightwave systems with optical amplifier," J. Lightwave Technol. 9, 1576 (1991).
[CrossRef]

1990

D. Marcuse, "Derivation of analytical expressions for the bit-error probability in lightwave systems with optical amplifiers," J. Lightwave Technol. 8, 1816 (1990).
[CrossRef]

Avlonitis, N.

N. Avlonitis, E. M. Yeatman, M. Jones, and A. Hadjifotiou, "Multilevel amplitude shift keying in dispersion uncompensated optical systems," IEE Proc. Optoelectron. 153, 101 (2006).
[CrossRef]

Azizoglu, M.

P. A. Humblet, and M. Azizoglu, "On the bit error rate of lightwave systems with optical amplifier," J. Lightwave Technol. 9, 1576 (1991).
[CrossRef]

Bhandare, S.

S. K. Ibrahim, S. Bhandare, and R. Noe, "20-Gbps quaternary intensity modulation based on duobinary modulation with unequal amplitude in two polarizations," IEEE Photon. Technol. Lett. 14, 1482 (2006).
[CrossRef]

Bosco, G.

G. Bosco and P. Poggiolini, "The Impact of receiver imperfections on the performance of optical direct-detection DPSK," J. Lightwave Technol. 23, 842 (2005).
[CrossRef]

G. Bosco, A. Carena, V. Curri, R. Gaudino, and P. Poggiolini, "Quantum limit of direct-dection receivers using duobinary transmission," IEEE Photon. Technol. Lett. 15, 102 (2003).
[CrossRef]

Carena, A.

G. Bosco, A. Carena, V. Curri, R. Gaudino, and P. Poggiolini, "Quantum limit of direct-dection receivers using duobinary transmission," IEEE Photon. Technol. Lett. 15, 102 (2003).
[CrossRef]

Conradi, J.

Curri, V.

G. Bosco, A. Carena, V. Curri, R. Gaudino, and P. Poggiolini, "Quantum limit of direct-dection receivers using duobinary transmission," IEEE Photon. Technol. Lett. 15, 102 (2003).
[CrossRef]

Da Rocha, J.

L. Ribeiro, J. Da Rocha, and J. Pinto, "Performance evaluation of EDFA preamplified receivers taking into account intersymbol interference," J. Lightwave Technol. 13, 225 (1995).
[CrossRef]

Gaudino, R.

G. Bosco, A. Carena, V. Curri, R. Gaudino, and P. Poggiolini, "Quantum limit of direct-dection receivers using duobinary transmission," IEEE Photon. Technol. Lett. 15, 102 (2003).
[CrossRef]

Gordon, J. P.

J. P. Gordon and L. F. Mollenauer, "Effects of fiber nonlinearities and amplifier spacing on ultra-long distance transmission," J. Lightwave Technol. 9, 170 (1991).
[CrossRef]

Hadjifotiou, A.

N. Avlonitis, E. M. Yeatman, M. Jones, and A. Hadjifotiou, "Multilevel amplitude shift keying in dispersion uncompensated optical systems," IEE Proc. Optoelectron. 153, 101 (2006).
[CrossRef]

Helstrom, C. W.

C. W. Helstrom, "Computing the generalized Marcum Q-function," IEEE Transactions on Information Theory 38, 1422 (1992).
[CrossRef]

Ho, K. P.

J. M. Kahn and K. P. Ho, "Spectral efficiency limits and modulation/detection techniques for DWDM systems," IEEE J. Sel. Top. Quantum Electron. 10, 259 (2004).
[CrossRef]

Humblet, P. A.

P. A. Humblet, and M. Azizoglu, "On the bit error rate of lightwave systems with optical amplifier," J. Lightwave Technol. 9, 1576 (1991).
[CrossRef]

Ibrahim, S. K.

S. K. Ibrahim, S. Bhandare, and R. Noe, "20-Gbps quaternary intensity modulation based on duobinary modulation with unequal amplitude in two polarizations," IEEE Photon. Technol. Lett. 14, 1482 (2006).
[CrossRef]

Jones, M.

N. Avlonitis, E. M. Yeatman, M. Jones, and A. Hadjifotiou, "Multilevel amplitude shift keying in dispersion uncompensated optical systems," IEE Proc. Optoelectron. 153, 101 (2006).
[CrossRef]

Kahn, J. M.

J. M. Kahn and K. P. Ho, "Spectral efficiency limits and modulation/detection techniques for DWDM systems," IEEE J. Sel. Top. Quantum Electron. 10, 259 (2004).
[CrossRef]

Marcuse, D.

D. Marcuse, "Derivation of analytical expressions for the bit-error probability in lightwave systems with optical amplifiers," J. Lightwave Technol. 8, 1816 (1990).
[CrossRef]

Mollenauer, L. F.

J. P. Gordon and L. F. Mollenauer, "Effects of fiber nonlinearities and amplifier spacing on ultra-long distance transmission," J. Lightwave Technol. 9, 170 (1991).
[CrossRef]

Noe, R.

S. K. Ibrahim, S. Bhandare, and R. Noe, "20-Gbps quaternary intensity modulation based on duobinary modulation with unequal amplitude in two polarizations," IEEE Photon. Technol. Lett. 14, 1482 (2006).
[CrossRef]

Pinto, J.

L. Ribeiro, J. Da Rocha, and J. Pinto, "Performance evaluation of EDFA preamplified receivers taking into account intersymbol interference," J. Lightwave Technol. 13, 225 (1995).
[CrossRef]

Poggiolini, P.

G. Bosco and P. Poggiolini, "The Impact of receiver imperfections on the performance of optical direct-detection DPSK," J. Lightwave Technol. 23, 842 (2005).
[CrossRef]

G. Bosco, A. Carena, V. Curri, R. Gaudino, and P. Poggiolini, "Quantum limit of direct-dection receivers using duobinary transmission," IEEE Photon. Technol. Lett. 15, 102 (2003).
[CrossRef]

Ribeiro, L.

L. Ribeiro, J. Da Rocha, and J. Pinto, "Performance evaluation of EDFA preamplified receivers taking into account intersymbol interference," J. Lightwave Technol. 13, 225 (1995).
[CrossRef]

Walklin, S.

Winzer, P. J.

P. J. Winzer, "Optically preamplifier receiver with low quantum limit," Electron. Lett. 37, 582 (2001).
[CrossRef]

Yeatman, E. M.

N. Avlonitis, E. M. Yeatman, M. Jones, and A. Hadjifotiou, "Multilevel amplitude shift keying in dispersion uncompensated optical systems," IEE Proc. Optoelectron. 153, 101 (2006).
[CrossRef]

Electron. Lett.

P. J. Winzer, "Optically preamplifier receiver with low quantum limit," Electron. Lett. 37, 582 (2001).
[CrossRef]

IEE Proc. Optoelectron.

N. Avlonitis, E. M. Yeatman, M. Jones, and A. Hadjifotiou, "Multilevel amplitude shift keying in dispersion uncompensated optical systems," IEE Proc. Optoelectron. 153, 101 (2006).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

J. M. Kahn and K. P. Ho, "Spectral efficiency limits and modulation/detection techniques for DWDM systems," IEEE J. Sel. Top. Quantum Electron. 10, 259 (2004).
[CrossRef]

IEEE Photon. Technol. Lett.

G. Bosco, A. Carena, V. Curri, R. Gaudino, and P. Poggiolini, "Quantum limit of direct-dection receivers using duobinary transmission," IEEE Photon. Technol. Lett. 15, 102 (2003).
[CrossRef]

S. K. Ibrahim, S. Bhandare, and R. Noe, "20-Gbps quaternary intensity modulation based on duobinary modulation with unequal amplitude in two polarizations," IEEE Photon. Technol. Lett. 14, 1482 (2006).
[CrossRef]

IEEE Transactions on Information Theory

C. W. Helstrom, "Computing the generalized Marcum Q-function," IEEE Transactions on Information Theory 38, 1422 (1992).
[CrossRef]

J. Lightwave Technol.

S. Walklin and J. Conradi, "Multilevel signaling for increasing the reach of 10Gbps lightwave systems," J. Lightwave Technol. 17, 2235 (1999).
[CrossRef]

G. Bosco and P. Poggiolini, "The Impact of receiver imperfections on the performance of optical direct-detection DPSK," J. Lightwave Technol. 23, 842 (2005).
[CrossRef]

L. Ribeiro, J. Da Rocha, and J. Pinto, "Performance evaluation of EDFA preamplified receivers taking into account intersymbol interference," J. Lightwave Technol. 13, 225 (1995).
[CrossRef]

P. A. Humblet, and M. Azizoglu, "On the bit error rate of lightwave systems with optical amplifier," J. Lightwave Technol. 9, 1576 (1991).
[CrossRef]

D. Marcuse, "Derivation of analytical expressions for the bit-error probability in lightwave systems with optical amplifiers," J. Lightwave Technol. 8, 1816 (1990).
[CrossRef]

J. P. Gordon and L. F. Mollenauer, "Effects of fiber nonlinearities and amplifier spacing on ultra-long distance transmission," J. Lightwave Technol. 9, 170 (1991).
[CrossRef]

Other

L. W. Couch, Digital and Analog Communication Systems (Prentice Hall, Upper Saddle River, N.J., 2001).

S. K. Ibrahim, S. Bhandare, and R. Noe, "Narrowband 2x10 Gbit/s Quaternary Intensity Modulation Based on Duobinary Modulation in Two Polarizations with Unequal Amplitudes," OFC 2006 Paper OThI2.

J. Zhao, L. Huo, C. K. Chan, L. K. Chen, and C. Lin, "Analytical investigation of optimization, performance bound, and chromatic dispersion tolerances of 4-amplitude-shifted keying format," OFC 2006 Paper JThB15.

J. Rebola and A. V. T. Cartaxo, "On the quaternary level spacing signaling optimization for increasing the transmission distance in optical communication systems," in Proc. of ConfTele2001 3rd Conference on Telecommunications, Figueira da Foz 23, 514 (2001).

K. Sekine, N. Kikuchi, and S. Sasaki, "Modulation parameter tolerance for 8- and 16-APSK (amplitude- and phase-shift keying) signals," OFC 2006 Paper JThB13.

S. K. Ibrahim, S. Bhandare, and R. Noe, "Narrowband 20-Gbps quaternary intensity modulation generated by duobinary 10-Gbps modulaltion in 2 quadratures," ECOC 2005 Paper Th 2.6.5.

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Figures (4)

Fig. 1.
Fig. 1.

The bit error rate for 4ASK as a function of photons/bit with various M.

Fig. 2.
Fig. 2.

The optimum normalized multilevel spacing and thresholds as a function of photons/ bit with (a) M = 1, (b) M=10, (c) M=30, (d) M=50. (e) The optimum normalized multilevel spacing and thresholds at a bit error rate of 10-9 as a function of M.

Fig. 3.
Fig. 3.

Required photons/bit to achieve a bit error rate of 10-9 for both the binary and the quaternary formats.

Fig. 4.
Fig. 4.

The bit error rate versus the photons/bit for the RZ-4ASK format for the back to back and CD = 510ps/nm with the exact and Gaussian approximation methods.

Equations (17)

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

s ( t ) = { k a k p ( t kT ) }
f S y ( χ ) = ( χ S y ) ( M 1 ) 2 exp ( χ S y ) I M 1 ( 2 χ S y )
f S 0 ( χ ) = χ M 1 e χ ( M 1 ) !
P ( e S 0 ) = e γ 1 n = 0 M 1 1 n ! γ 1 n
P ( e S i ) = 1 Q M ( 2 S i , 2 γ i ) + Q M ( 2 S i , 2 γ i + 1 ) for i = 1,2
P ( e S 3 ) = 1 Q M ( 2 S 3 , 2 γ 3 ) )
P e = 1 2 { 1 4 P ( e S 0 ) + 1 4 P ( e S 1 ) + 1 4 P ( e S 2 ) + 1 4 P ( e S 3 ) }
P e S 1 = P e S 2 = P e γ 1 = P e γ 2 = P e γ 3 = 0
Q M a b b = b M a M 1 e 1 2 ( a 2 + b 2 ) I M 1 ( ab )
Q M a b a = b M a M 1 e 1 2 ( a 2 + b 2 ) I M ( ab )
Q M a b = Q a b + e 1 2 ( a 2 + b 2 ) n = 1 M 1 ( a b ) n I n ( ab ) and I n ( ab ) = I n ( ab ) .
p e γ 1 = 0 = > e a 1 2 2 I M 1 ( a 1 b 1 ) * ( M 1 ) ! ( a 1 b 1 2 ) ( 1 M ) = 1
P e S 1 = 0 = > ( b 1 b 2 ) M e 1 2 ( b 1 2 b 2 2 ) I M ( a 1 b 1 ) I M ( a 1 b 2 ) = 1
P e γ 2 = 0 = > ( a 2 a 1 ) M 1 e 1 2 ( a 1 2 a 2 2 ) I M 1 ( a 1 b 2 ) I M 1 ( a 2 b 2 ) = 1
P e S 2 = 0 = > ( b 2 b 3 ) M e 1 2 ( b 2 2 b 3 2 ) I M ( a 2 b 2 ) I M ( a 2 b 3 ) = 1
P e γ 3 = 0 = > ( a 3 a 2 ) M 1 e 1 2 ( a 2 2 a 3 2 ) I M 1 ( a 2 b 3 ) I M 1 ( a 3 b 3 ) = 1
Pe = 1 8 i = 1 3 erfc ( Q i 2 ) = 3 8 erfc ( Q 2 )

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