Abstract

In this paper, we first describe a 9-symbol non-uniform signaling scheme based on Huffman code, in which different symbols are transmitted with different probabilities. By using the Huffman procedure, prefix code is designed to approach the optimal performance. Then, we introduce an algorithm to determine the optimal signal constellation sets for our proposed non-uniform scheme with the criterion of maximizing constellation figure of merit (CFM). The proposed nonuniform polarization multiplexed signaling 9-QAM scheme has the same spectral efficiency as the conventional 8-QAM. Additionally, we propose a specially designed GF(32) nonbinary quasi-cyclic LDPC code for the coded modulation system based on the 9-QAM non-uniform scheme. Further, we study the efficiency of our proposed non-uniform 9-QAM, combined with nonbinary LDPC coding, and demonstrate by Monte Carlo simulation that the proposed GF(23) nonbinary LDPC coded 9-QAM scheme outperforms nonbinary LDPC coded uniform 8-QAM by at least 0.8dB.

© 2016 Optical Society of America

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References

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  1. P. Winzer, “Beyond 100G Ethernet,” IEEE Commun. Mag. 48(7), 26–30 (2010).
    [Crossref]
  2. M. Seimetz, High-Order Modulation for Optical Fiber Transmission (Springer, 2009).
  3. T. Liu and I. B. Djordjevic, “On the optimum signal constellation design for high-speed optical transport networks,” Opt. Express 20(18), 20396–20406 (2012).
    [Crossref] [PubMed]
  4. N. J. A. Sloane, R. H. Hardin, T. D. S. Duff, and J. H. Conway, “Minimal-energy cluster of hard sphere,” Discrete Comput. Geom. 14(3), 237–259 (1995).
    [Crossref]
  5. C. Hager, A. Graell i Amat, A. Alvarado, and E. Agrell, “Design of APSK constellations for coherent optical channels with nonlinear phase noise,” IEEE Trans. Commun. 61(8), 3362–3373 (2013).
    [Crossref]
  6. J. E. Porath and T. Aulin, “Design of multidimensional signal constellations,” IEEE Proc. Commun. 150(5), 317–323 (2003).
    [Crossref]
  7. M. Karlsson and E. Agrell, “Four-dimensional optimized constellations for coherent optical transmission systems,” in European Conference Optical Communication 2010 (Optical Society of America, 2010), paper. We.8.C.3.
    [Crossref]
  8. G. Takahara, F. Alajaji, N. C. Beaulieu, and H. Kuai, “Constellation Mappings for Two-Dimensional Signaling of Nonuniform Sources,” IEEE Trans. Commun. 51(3), 400–408 (2003).
    [Crossref]
  9. F. Kschischang and S. Pasupathy, “Optimal non-uniform signaling for gaussian channels,” IEEE Trans. Inf. Theory 39(3), 913–929 (1993).
    [Crossref]
  10. F. Buchali, G. Bocherer, W. Idler, L. Schmalen, P. Schulte, and F. Steiner, “Experimental demonstration of capacity increase and rate-adaptation by probabilistically shaped 64-QAM,” in European Conference Optical Communication 2015 (Optical Society of America, 2015), paper PDP.3.4.
    [Crossref]
  11. M. Yanko, D. Zibar, K. J. Larsen, L. P. B. Christensen, and S. Forchammer, “Constellation shaping for fiber-optic channels with QAM and high spectral efficiency,” IEEE Photon. Technol. Lett. 26(23), 2407–2410 (2014).
    [Crossref]
  12. T. Liu and I. B. Djordjevic, “LDPC-coded BICM-ID based nonuniform signaling for ultra-high-speed optical transport,” in Optical Fiber Communication 2016 (Optical Society of America, 2016), paper M3A.3.
  13. A. K. Khandani and P. Kabal, “Shaping multidimensional signal spaces. Part I: optimal shaping shell mapping,” IEEE Trans. Inf. Theory 39(6), 1799–1808 (1993).
    [Crossref]
  14. M. Arabaci, I. B. Djordjevic, R. Saunders, and R. M. Marcoccia, “Polarization-multiplexed rate-adaptive non-binary-quasi-cyclic-LDPC-coded multilevel modulation with coherent detection for optical transport networks” in International Conference on Transparent Optical Networks (2009), paper. Tu.B2.2.
  15. I. B. Djordjevic, L. L. Minkov, L. Xu, and T. Wang, “Suppression of fiber nonlinearities and PMD in coded-modulation schemes with coherent detection by using turbo equalization,” J. Opt. Commun. 1(6), 555–564 (2009).
    [Crossref]
  16. T. H. Lotz, X. Liu, S. Chandrasekhar, P. J. Winzer, H. Haunstein, S. Randel, S. Corteselli, B. Zhu, and D. W. Peckham, “Coded PDM-OFDM transmission with shaped 256-iterative-polar-modulation achieving 11.15-b/s/Hz intrachannel spectral efficiency and 800-km reach,” J. Lightwave Technol. 31(4), 538–545 (2013).
    [Crossref]

2014 (1)

M. Yanko, D. Zibar, K. J. Larsen, L. P. B. Christensen, and S. Forchammer, “Constellation shaping for fiber-optic channels with QAM and high spectral efficiency,” IEEE Photon. Technol. Lett. 26(23), 2407–2410 (2014).
[Crossref]

2013 (2)

2012 (1)

2010 (1)

P. Winzer, “Beyond 100G Ethernet,” IEEE Commun. Mag. 48(7), 26–30 (2010).
[Crossref]

2009 (1)

I. B. Djordjevic, L. L. Minkov, L. Xu, and T. Wang, “Suppression of fiber nonlinearities and PMD in coded-modulation schemes with coherent detection by using turbo equalization,” J. Opt. Commun. 1(6), 555–564 (2009).
[Crossref]

2003 (1)

G. Takahara, F. Alajaji, N. C. Beaulieu, and H. Kuai, “Constellation Mappings for Two-Dimensional Signaling of Nonuniform Sources,” IEEE Trans. Commun. 51(3), 400–408 (2003).
[Crossref]

1995 (1)

N. J. A. Sloane, R. H. Hardin, T. D. S. Duff, and J. H. Conway, “Minimal-energy cluster of hard sphere,” Discrete Comput. Geom. 14(3), 237–259 (1995).
[Crossref]

1993 (2)

F. Kschischang and S. Pasupathy, “Optimal non-uniform signaling for gaussian channels,” IEEE Trans. Inf. Theory 39(3), 913–929 (1993).
[Crossref]

A. K. Khandani and P. Kabal, “Shaping multidimensional signal spaces. Part I: optimal shaping shell mapping,” IEEE Trans. Inf. Theory 39(6), 1799–1808 (1993).
[Crossref]

Agrell, E.

C. Hager, A. Graell i Amat, A. Alvarado, and E. Agrell, “Design of APSK constellations for coherent optical channels with nonlinear phase noise,” IEEE Trans. Commun. 61(8), 3362–3373 (2013).
[Crossref]

Alajaji, F.

G. Takahara, F. Alajaji, N. C. Beaulieu, and H. Kuai, “Constellation Mappings for Two-Dimensional Signaling of Nonuniform Sources,” IEEE Trans. Commun. 51(3), 400–408 (2003).
[Crossref]

Alvarado, A.

C. Hager, A. Graell i Amat, A. Alvarado, and E. Agrell, “Design of APSK constellations for coherent optical channels with nonlinear phase noise,” IEEE Trans. Commun. 61(8), 3362–3373 (2013).
[Crossref]

Beaulieu, N. C.

G. Takahara, F. Alajaji, N. C. Beaulieu, and H. Kuai, “Constellation Mappings for Two-Dimensional Signaling of Nonuniform Sources,” IEEE Trans. Commun. 51(3), 400–408 (2003).
[Crossref]

Chandrasekhar, S.

Christensen, L. P. B.

M. Yanko, D. Zibar, K. J. Larsen, L. P. B. Christensen, and S. Forchammer, “Constellation shaping for fiber-optic channels with QAM and high spectral efficiency,” IEEE Photon. Technol. Lett. 26(23), 2407–2410 (2014).
[Crossref]

Conway, J. H.

N. J. A. Sloane, R. H. Hardin, T. D. S. Duff, and J. H. Conway, “Minimal-energy cluster of hard sphere,” Discrete Comput. Geom. 14(3), 237–259 (1995).
[Crossref]

Corteselli, S.

Djordjevic, I. B.

T. Liu and I. B. Djordjevic, “On the optimum signal constellation design for high-speed optical transport networks,” Opt. Express 20(18), 20396–20406 (2012).
[Crossref] [PubMed]

I. B. Djordjevic, L. L. Minkov, L. Xu, and T. Wang, “Suppression of fiber nonlinearities and PMD in coded-modulation schemes with coherent detection by using turbo equalization,” J. Opt. Commun. 1(6), 555–564 (2009).
[Crossref]

Duff, T. D. S.

N. J. A. Sloane, R. H. Hardin, T. D. S. Duff, and J. H. Conway, “Minimal-energy cluster of hard sphere,” Discrete Comput. Geom. 14(3), 237–259 (1995).
[Crossref]

Forchammer, S.

M. Yanko, D. Zibar, K. J. Larsen, L. P. B. Christensen, and S. Forchammer, “Constellation shaping for fiber-optic channels with QAM and high spectral efficiency,” IEEE Photon. Technol. Lett. 26(23), 2407–2410 (2014).
[Crossref]

Graell i Amat, A.

C. Hager, A. Graell i Amat, A. Alvarado, and E. Agrell, “Design of APSK constellations for coherent optical channels with nonlinear phase noise,” IEEE Trans. Commun. 61(8), 3362–3373 (2013).
[Crossref]

Hager, C.

C. Hager, A. Graell i Amat, A. Alvarado, and E. Agrell, “Design of APSK constellations for coherent optical channels with nonlinear phase noise,” IEEE Trans. Commun. 61(8), 3362–3373 (2013).
[Crossref]

Hardin, R. H.

N. J. A. Sloane, R. H. Hardin, T. D. S. Duff, and J. H. Conway, “Minimal-energy cluster of hard sphere,” Discrete Comput. Geom. 14(3), 237–259 (1995).
[Crossref]

Haunstein, H.

Kabal, P.

A. K. Khandani and P. Kabal, “Shaping multidimensional signal spaces. Part I: optimal shaping shell mapping,” IEEE Trans. Inf. Theory 39(6), 1799–1808 (1993).
[Crossref]

Khandani, A. K.

A. K. Khandani and P. Kabal, “Shaping multidimensional signal spaces. Part I: optimal shaping shell mapping,” IEEE Trans. Inf. Theory 39(6), 1799–1808 (1993).
[Crossref]

Kschischang, F.

F. Kschischang and S. Pasupathy, “Optimal non-uniform signaling for gaussian channels,” IEEE Trans. Inf. Theory 39(3), 913–929 (1993).
[Crossref]

Kuai, H.

G. Takahara, F. Alajaji, N. C. Beaulieu, and H. Kuai, “Constellation Mappings for Two-Dimensional Signaling of Nonuniform Sources,” IEEE Trans. Commun. 51(3), 400–408 (2003).
[Crossref]

Larsen, K. J.

M. Yanko, D. Zibar, K. J. Larsen, L. P. B. Christensen, and S. Forchammer, “Constellation shaping for fiber-optic channels with QAM and high spectral efficiency,” IEEE Photon. Technol. Lett. 26(23), 2407–2410 (2014).
[Crossref]

Liu, T.

Liu, X.

Lotz, T. H.

Minkov, L. L.

I. B. Djordjevic, L. L. Minkov, L. Xu, and T. Wang, “Suppression of fiber nonlinearities and PMD in coded-modulation schemes with coherent detection by using turbo equalization,” J. Opt. Commun. 1(6), 555–564 (2009).
[Crossref]

Pasupathy, S.

F. Kschischang and S. Pasupathy, “Optimal non-uniform signaling for gaussian channels,” IEEE Trans. Inf. Theory 39(3), 913–929 (1993).
[Crossref]

Peckham, D. W.

Randel, S.

Sloane, N. J. A.

N. J. A. Sloane, R. H. Hardin, T. D. S. Duff, and J. H. Conway, “Minimal-energy cluster of hard sphere,” Discrete Comput. Geom. 14(3), 237–259 (1995).
[Crossref]

Takahara, G.

G. Takahara, F. Alajaji, N. C. Beaulieu, and H. Kuai, “Constellation Mappings for Two-Dimensional Signaling of Nonuniform Sources,” IEEE Trans. Commun. 51(3), 400–408 (2003).
[Crossref]

Wang, T.

I. B. Djordjevic, L. L. Minkov, L. Xu, and T. Wang, “Suppression of fiber nonlinearities and PMD in coded-modulation schemes with coherent detection by using turbo equalization,” J. Opt. Commun. 1(6), 555–564 (2009).
[Crossref]

Winzer, P.

P. Winzer, “Beyond 100G Ethernet,” IEEE Commun. Mag. 48(7), 26–30 (2010).
[Crossref]

Winzer, P. J.

Xu, L.

I. B. Djordjevic, L. L. Minkov, L. Xu, and T. Wang, “Suppression of fiber nonlinearities and PMD in coded-modulation schemes with coherent detection by using turbo equalization,” J. Opt. Commun. 1(6), 555–564 (2009).
[Crossref]

Yanko, M.

M. Yanko, D. Zibar, K. J. Larsen, L. P. B. Christensen, and S. Forchammer, “Constellation shaping for fiber-optic channels with QAM and high spectral efficiency,” IEEE Photon. Technol. Lett. 26(23), 2407–2410 (2014).
[Crossref]

Zhu, B.

Zibar, D.

M. Yanko, D. Zibar, K. J. Larsen, L. P. B. Christensen, and S. Forchammer, “Constellation shaping for fiber-optic channels with QAM and high spectral efficiency,” IEEE Photon. Technol. Lett. 26(23), 2407–2410 (2014).
[Crossref]

Discrete Comput. Geom. (1)

N. J. A. Sloane, R. H. Hardin, T. D. S. Duff, and J. H. Conway, “Minimal-energy cluster of hard sphere,” Discrete Comput. Geom. 14(3), 237–259 (1995).
[Crossref]

IEEE Commun. Mag. (1)

P. Winzer, “Beyond 100G Ethernet,” IEEE Commun. Mag. 48(7), 26–30 (2010).
[Crossref]

IEEE Photon. Technol. Lett. (1)

M. Yanko, D. Zibar, K. J. Larsen, L. P. B. Christensen, and S. Forchammer, “Constellation shaping for fiber-optic channels with QAM and high spectral efficiency,” IEEE Photon. Technol. Lett. 26(23), 2407–2410 (2014).
[Crossref]

IEEE Trans. Commun. (2)

G. Takahara, F. Alajaji, N. C. Beaulieu, and H. Kuai, “Constellation Mappings for Two-Dimensional Signaling of Nonuniform Sources,” IEEE Trans. Commun. 51(3), 400–408 (2003).
[Crossref]

C. Hager, A. Graell i Amat, A. Alvarado, and E. Agrell, “Design of APSK constellations for coherent optical channels with nonlinear phase noise,” IEEE Trans. Commun. 61(8), 3362–3373 (2013).
[Crossref]

IEEE Trans. Inf. Theory (2)

F. Kschischang and S. Pasupathy, “Optimal non-uniform signaling for gaussian channels,” IEEE Trans. Inf. Theory 39(3), 913–929 (1993).
[Crossref]

A. K. Khandani and P. Kabal, “Shaping multidimensional signal spaces. Part I: optimal shaping shell mapping,” IEEE Trans. Inf. Theory 39(6), 1799–1808 (1993).
[Crossref]

J. Lightwave Technol. (1)

J. Opt. Commun. (1)

I. B. Djordjevic, L. L. Minkov, L. Xu, and T. Wang, “Suppression of fiber nonlinearities and PMD in coded-modulation schemes with coherent detection by using turbo equalization,” J. Opt. Commun. 1(6), 555–564 (2009).
[Crossref]

Opt. Express (1)

Other (6)

M. Arabaci, I. B. Djordjevic, R. Saunders, and R. M. Marcoccia, “Polarization-multiplexed rate-adaptive non-binary-quasi-cyclic-LDPC-coded multilevel modulation with coherent detection for optical transport networks” in International Conference on Transparent Optical Networks (2009), paper. Tu.B2.2.

T. Liu and I. B. Djordjevic, “LDPC-coded BICM-ID based nonuniform signaling for ultra-high-speed optical transport,” in Optical Fiber Communication 2016 (Optical Society of America, 2016), paper M3A.3.

F. Buchali, G. Bocherer, W. Idler, L. Schmalen, P. Schulte, and F. Steiner, “Experimental demonstration of capacity increase and rate-adaptation by probabilistically shaped 64-QAM,” in European Conference Optical Communication 2015 (Optical Society of America, 2015), paper PDP.3.4.
[Crossref]

M. Seimetz, High-Order Modulation for Optical Fiber Transmission (Springer, 2009).

J. E. Porath and T. Aulin, “Design of multidimensional signal constellations,” IEEE Proc. Commun. 150(5), 317–323 (2003).
[Crossref]

M. Karlsson and E. Agrell, “Four-dimensional optimized constellations for coherent optical transmission systems,” in European Conference Optical Communication 2010 (Optical Society of America, 2010), paper. We.8.C.3.
[Crossref]

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Figures (10)

Fig. 1
Fig. 1 Non-uniform signaling enabled by Huffman code.
Fig. 2
Fig. 2 Optimal constellation design algorithm for non-uniform signaling scheme.
Fig. 3
Fig. 3 The optimization procedure for the constellation design algorithm.
Fig. 4
Fig. 4 The mapping rule for star 8-QAM.
Fig. 5
Fig. 5 Tanner graph of the proposed nonbinary QC-LDPC.
Fig. 6
Fig. 6 Trellis graph of the BCJR algorithm-based check-node processor.
Fig. 7
Fig. 7 Non-uniform 9-QAM coded modulation scheme based on nonbinary LDPC coding.
Fig. 8
Fig. 8 Interleaver structure of 9-QAM.
Fig. 9
Fig. 9 SER performance vs. SNR.
Fig. 10
Fig. 10 BER performance vs. SNR.

Equations (2)

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S= i=1 4 i P i N i ,
CFM d min 2 E ,

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