Abstract

By exploiting the electromagnetic wave’s four-dimensional signal space, we find that for the additive white Gaussian noise channel, the modulation format with best sensitivity to be an 8-level format with 1.76 dB asymptotic gain over BPSK, for uncoded optical transmission with coherent detection. Low-complexity modulators are presented for the format, as well as an interpretation in terms of quantum-limited sensitivity.

© 2009 Optical Society of America

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References

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  1. S. Tsukamoto, D. Ly-Gagnon, K. Katoh, and K. Kikuchi, "Coherent demodulation of 40-Gbit/s polarizationmultiplexed QPSK signals with 16-GHz spacing after 200-km transmission," in Proceedings of Optical Fiber Communication and National Fiber Optic Engineers Conference, OFC/NFOEC, Vol. 6, p. PDP29 (2005).
  2. G. Charlet, M. Salsi, J. Renaudier, O. Pardo, H. Mardoyan, and S. Bigo, "Performance comparison of singlypolarised and polarisation-multiplexed coherent transmission at 10 Gbauds under linear impairments," Electron. Lett. 43(20), 1109-1111 (2007).
    [CrossRef]
  3. H. Sun, K. Wu, and K. Roberts, "Real-time measurements of a 40 Gb/s coherent system," Opt. Express 16(2), 873-879 (2008).
    [CrossRef]
  4. E. Ip, A. Lau, D. Barros, and J. Kahn, "Coherent detection in optical fiber systems," Opt. Express 16(2), 753-791 (2008). Erratum, Vol. 16, no. 26, p. 21943, (2008).
    [CrossRef]
  5. J. R. Pierce, "Optical Channels: Practical Limits with Photon Counting," IEEE Trans. Commun. 26(12), 1819- 1821 (1978).
    [CrossRef]
  6. S. Benedetto and E. Biglieri, Principles of Digital Transmission: With Wireless Applications (Kluwer Academic Publishers, 1999).
  7. S. Betti, F. Curti, G. De Marchis, and E. Iannone, "A novel multilevel coherent optical system: 4-quadrature signaling," J. Lightwave Technol. 9(4), 514-523 (1991).
    [CrossRef]
  8. R. L. Graham and N. J. A. Sloane, "Penny-packing and two-dimensional codes," Discrete and Computational Geometry 5(1), 1-11 (1990).
  9. N. J. A. Sloane, R. H. Hardin, T. S. Duff, and J. H. Conway, "Minimal-Energy Clusters of Hard Spheres," Discrete and Computational Geometry 14(3), 237-259 (1995).
  10. N. J. A. Sloane, R. H. Hardin, T. S. Duff, and J. H. Conway, "Minimal-Energy Clusters, Library of 3-d clusters, Library of 4-d clusters," (1997). URLhttp://www.research.att.com/∼njas/cluster/.
  11. J. Kahn and K.-P. Ho, "Spectral efficiency limits and modulation/detection techniques for DWDM systems," IEEE J. Sel. Top. Quantum Electron. 10(2), 259-272 (2004).
  12. M. K. Simon, S. Hinedi, and W. C. Lindsey, Digital Communication Techniques: Signal Design and Detection (Prentice-Hall, 1995).
  13. L. Xiao and X. Dong, "New analytical expressions for orthogonal, biorthogonal, and transorthogonal signaling in Nakagami fading channels with diversity reception," IEEE Trans.Wireless Commun. 4(4), 1418-1424 (2005).
    [CrossRef]
  14. D. Saha and T. Birdsall, "Quadrature-quadrature phase-shift keying," IEEE Trans. Commun. 37(5), 437-448 (1989).
    [CrossRef]

2008 (2)

E. Ip, A. Lau, D. Barros, and J. Kahn, "Coherent detection in optical fiber systems," Opt. Express 16(2), 753-791 (2008). Erratum, Vol. 16, no. 26, p. 21943, (2008).
[CrossRef]

H. Sun, K. Wu, and K. Roberts, "Real-time measurements of a 40 Gb/s coherent system," Opt. Express 16(2), 873-879 (2008).
[CrossRef]

2007 (1)

G. Charlet, M. Salsi, J. Renaudier, O. Pardo, H. Mardoyan, and S. Bigo, "Performance comparison of singlypolarised and polarisation-multiplexed coherent transmission at 10 Gbauds under linear impairments," Electron. Lett. 43(20), 1109-1111 (2007).
[CrossRef]

2005 (1)

L. Xiao and X. Dong, "New analytical expressions for orthogonal, biorthogonal, and transorthogonal signaling in Nakagami fading channels with diversity reception," IEEE Trans.Wireless Commun. 4(4), 1418-1424 (2005).
[CrossRef]

2004 (1)

J. Kahn and K.-P. Ho, "Spectral efficiency limits and modulation/detection techniques for DWDM systems," IEEE J. Sel. Top. Quantum Electron. 10(2), 259-272 (2004).

1995 (1)

N. J. A. Sloane, R. H. Hardin, T. S. Duff, and J. H. Conway, "Minimal-Energy Clusters of Hard Spheres," Discrete and Computational Geometry 14(3), 237-259 (1995).

1991 (1)

S. Betti, F. Curti, G. De Marchis, and E. Iannone, "A novel multilevel coherent optical system: 4-quadrature signaling," J. Lightwave Technol. 9(4), 514-523 (1991).
[CrossRef]

1990 (1)

R. L. Graham and N. J. A. Sloane, "Penny-packing and two-dimensional codes," Discrete and Computational Geometry 5(1), 1-11 (1990).

1989 (1)

D. Saha and T. Birdsall, "Quadrature-quadrature phase-shift keying," IEEE Trans. Commun. 37(5), 437-448 (1989).
[CrossRef]

1978 (1)

J. R. Pierce, "Optical Channels: Practical Limits with Photon Counting," IEEE Trans. Commun. 26(12), 1819- 1821 (1978).
[CrossRef]

Barros, D.

E. Ip, A. Lau, D. Barros, and J. Kahn, "Coherent detection in optical fiber systems," Opt. Express 16(2), 753-791 (2008). Erratum, Vol. 16, no. 26, p. 21943, (2008).
[CrossRef]

Betti, S.

S. Betti, F. Curti, G. De Marchis, and E. Iannone, "A novel multilevel coherent optical system: 4-quadrature signaling," J. Lightwave Technol. 9(4), 514-523 (1991).
[CrossRef]

Bigo, S.

G. Charlet, M. Salsi, J. Renaudier, O. Pardo, H. Mardoyan, and S. Bigo, "Performance comparison of singlypolarised and polarisation-multiplexed coherent transmission at 10 Gbauds under linear impairments," Electron. Lett. 43(20), 1109-1111 (2007).
[CrossRef]

Birdsall, T.

D. Saha and T. Birdsall, "Quadrature-quadrature phase-shift keying," IEEE Trans. Commun. 37(5), 437-448 (1989).
[CrossRef]

Charlet, G.

G. Charlet, M. Salsi, J. Renaudier, O. Pardo, H. Mardoyan, and S. Bigo, "Performance comparison of singlypolarised and polarisation-multiplexed coherent transmission at 10 Gbauds under linear impairments," Electron. Lett. 43(20), 1109-1111 (2007).
[CrossRef]

Conway, J. H.

N. J. A. Sloane, R. H. Hardin, T. S. Duff, and J. H. Conway, "Minimal-Energy Clusters of Hard Spheres," Discrete and Computational Geometry 14(3), 237-259 (1995).

Curti, F.

S. Betti, F. Curti, G. De Marchis, and E. Iannone, "A novel multilevel coherent optical system: 4-quadrature signaling," J. Lightwave Technol. 9(4), 514-523 (1991).
[CrossRef]

De Marchis, G.

S. Betti, F. Curti, G. De Marchis, and E. Iannone, "A novel multilevel coherent optical system: 4-quadrature signaling," J. Lightwave Technol. 9(4), 514-523 (1991).
[CrossRef]

Dong, X.

L. Xiao and X. Dong, "New analytical expressions for orthogonal, biorthogonal, and transorthogonal signaling in Nakagami fading channels with diversity reception," IEEE Trans.Wireless Commun. 4(4), 1418-1424 (2005).
[CrossRef]

Duff, T. S.

N. J. A. Sloane, R. H. Hardin, T. S. Duff, and J. H. Conway, "Minimal-Energy Clusters of Hard Spheres," Discrete and Computational Geometry 14(3), 237-259 (1995).

Graham, R. L.

R. L. Graham and N. J. A. Sloane, "Penny-packing and two-dimensional codes," Discrete and Computational Geometry 5(1), 1-11 (1990).

Hardin, R. H.

N. J. A. Sloane, R. H. Hardin, T. S. Duff, and J. H. Conway, "Minimal-Energy Clusters of Hard Spheres," Discrete and Computational Geometry 14(3), 237-259 (1995).

Ho, K.-P.

J. Kahn and K.-P. Ho, "Spectral efficiency limits and modulation/detection techniques for DWDM systems," IEEE J. Sel. Top. Quantum Electron. 10(2), 259-272 (2004).

Iannone, E.

S. Betti, F. Curti, G. De Marchis, and E. Iannone, "A novel multilevel coherent optical system: 4-quadrature signaling," J. Lightwave Technol. 9(4), 514-523 (1991).
[CrossRef]

Ip, E.

E. Ip, A. Lau, D. Barros, and J. Kahn, "Coherent detection in optical fiber systems," Opt. Express 16(2), 753-791 (2008). Erratum, Vol. 16, no. 26, p. 21943, (2008).
[CrossRef]

Kahn, J.

E. Ip, A. Lau, D. Barros, and J. Kahn, "Coherent detection in optical fiber systems," Opt. Express 16(2), 753-791 (2008). Erratum, Vol. 16, no. 26, p. 21943, (2008).
[CrossRef]

J. Kahn and K.-P. Ho, "Spectral efficiency limits and modulation/detection techniques for DWDM systems," IEEE J. Sel. Top. Quantum Electron. 10(2), 259-272 (2004).

Lau, A.

E. Ip, A. Lau, D. Barros, and J. Kahn, "Coherent detection in optical fiber systems," Opt. Express 16(2), 753-791 (2008). Erratum, Vol. 16, no. 26, p. 21943, (2008).
[CrossRef]

Mardoyan, H.

G. Charlet, M. Salsi, J. Renaudier, O. Pardo, H. Mardoyan, and S. Bigo, "Performance comparison of singlypolarised and polarisation-multiplexed coherent transmission at 10 Gbauds under linear impairments," Electron. Lett. 43(20), 1109-1111 (2007).
[CrossRef]

Pardo, O.

G. Charlet, M. Salsi, J. Renaudier, O. Pardo, H. Mardoyan, and S. Bigo, "Performance comparison of singlypolarised and polarisation-multiplexed coherent transmission at 10 Gbauds under linear impairments," Electron. Lett. 43(20), 1109-1111 (2007).
[CrossRef]

Pierce, J. R.

J. R. Pierce, "Optical Channels: Practical Limits with Photon Counting," IEEE Trans. Commun. 26(12), 1819- 1821 (1978).
[CrossRef]

Renaudier, J.

G. Charlet, M. Salsi, J. Renaudier, O. Pardo, H. Mardoyan, and S. Bigo, "Performance comparison of singlypolarised and polarisation-multiplexed coherent transmission at 10 Gbauds under linear impairments," Electron. Lett. 43(20), 1109-1111 (2007).
[CrossRef]

Roberts, K.

Saha, D.

D. Saha and T. Birdsall, "Quadrature-quadrature phase-shift keying," IEEE Trans. Commun. 37(5), 437-448 (1989).
[CrossRef]

Salsi, M.

G. Charlet, M. Salsi, J. Renaudier, O. Pardo, H. Mardoyan, and S. Bigo, "Performance comparison of singlypolarised and polarisation-multiplexed coherent transmission at 10 Gbauds under linear impairments," Electron. Lett. 43(20), 1109-1111 (2007).
[CrossRef]

Sloane, N. J. A.

N. J. A. Sloane, R. H. Hardin, T. S. Duff, and J. H. Conway, "Minimal-Energy Clusters of Hard Spheres," Discrete and Computational Geometry 14(3), 237-259 (1995).

R. L. Graham and N. J. A. Sloane, "Penny-packing and two-dimensional codes," Discrete and Computational Geometry 5(1), 1-11 (1990).

Sun, H.

Wu, K.

Xiao, L.

L. Xiao and X. Dong, "New analytical expressions for orthogonal, biorthogonal, and transorthogonal signaling in Nakagami fading channels with diversity reception," IEEE Trans.Wireless Commun. 4(4), 1418-1424 (2005).
[CrossRef]

Discrete and Computational Geometry (2)

R. L. Graham and N. J. A. Sloane, "Penny-packing and two-dimensional codes," Discrete and Computational Geometry 5(1), 1-11 (1990).

N. J. A. Sloane, R. H. Hardin, T. S. Duff, and J. H. Conway, "Minimal-Energy Clusters of Hard Spheres," Discrete and Computational Geometry 14(3), 237-259 (1995).

Electron. Lett. (1)

G. Charlet, M. Salsi, J. Renaudier, O. Pardo, H. Mardoyan, and S. Bigo, "Performance comparison of singlypolarised and polarisation-multiplexed coherent transmission at 10 Gbauds under linear impairments," Electron. Lett. 43(20), 1109-1111 (2007).
[CrossRef]

Erratum (1)

E. Ip, A. Lau, D. Barros, and J. Kahn, "Coherent detection in optical fiber systems," Opt. Express 16(2), 753-791 (2008). Erratum, Vol. 16, no. 26, p. 21943, (2008).
[CrossRef]

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

J. Kahn and K.-P. Ho, "Spectral efficiency limits and modulation/detection techniques for DWDM systems," IEEE J. Sel. Top. Quantum Electron. 10(2), 259-272 (2004).

IEEE Trans. Commun. (2)

J. R. Pierce, "Optical Channels: Practical Limits with Photon Counting," IEEE Trans. Commun. 26(12), 1819- 1821 (1978).
[CrossRef]

D. Saha and T. Birdsall, "Quadrature-quadrature phase-shift keying," IEEE Trans. Commun. 37(5), 437-448 (1989).
[CrossRef]

IEEE Trans.Wireless Commun. (1)

L. Xiao and X. Dong, "New analytical expressions for orthogonal, biorthogonal, and transorthogonal signaling in Nakagami fading channels with diversity reception," IEEE Trans.Wireless Commun. 4(4), 1418-1424 (2005).
[CrossRef]

J. Lightwave Technol. (1)

S. Betti, F. Curti, G. De Marchis, and E. Iannone, "A novel multilevel coherent optical system: 4-quadrature signaling," J. Lightwave Technol. 9(4), 514-523 (1991).
[CrossRef]

Opt. Express (1)

Other (4)

S. Tsukamoto, D. Ly-Gagnon, K. Katoh, and K. Kikuchi, "Coherent demodulation of 40-Gbit/s polarizationmultiplexed QPSK signals with 16-GHz spacing after 200-km transmission," in Proceedings of Optical Fiber Communication and National Fiber Optic Engineers Conference, OFC/NFOEC, Vol. 6, p. PDP29 (2005).

M. K. Simon, S. Hinedi, and W. C. Lindsey, Digital Communication Techniques: Signal Design and Detection (Prentice-Hall, 1995).

S. Benedetto and E. Biglieri, Principles of Digital Transmission: With Wireless Applications (Kluwer Academic Publishers, 1999).

N. J. A. Sloane, R. H. Hardin, T. S. Duff, and J. H. Conway, "Minimal-Energy Clusters, Library of 3-d clusters, Library of 4-d clusters," (1997). URLhttp://www.research.att.com/∼njas/cluster/.

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Figures (3)

Fig. 1.
Fig. 1.

Spectral efficiency vs. sensitivity penalty 1/γ for optimum M-ary constellations in N=2,3, and 4 dimensions. Coordinates (N,M) refer to the optimum M-ary constellation in dimension N. The points from M=2 to 32 are joined by lines as a guide to the eye. Also shown are M-PSK (for M=3 to 8) and 16-QAM for comparison.

Fig. 2.
Fig. 2.

(a) A transmitter configuration for PS-QPSK based on a standard QPSK transmitter and a polarization modulator (PolM), where Bi denotes driving bits to the PS-QPSK channel. (b) Alternative transmitter for PS-QPSK using 8 out of the 16 symbols of a DP-QPSK transmitter, where bi denotes DP-QPSK driving bits.

Fig. 3.
Fig. 3.

BER vs. Eb /N 0 for PS-QPSK and BPSK over an AWGN channel.

Equations (3)

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SERPSQPSK =11π 0(1erfcx)3e(xEsN0)2 d x
= 12 erfc EsN0 +1π 0(33erfcx+erfc2x) . erfc (x)e(xEsx02) d x
EbN0 =nbNAnsp

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