Abstract

A mutual information inspired nonbinary coded modulation design with non-uniform shaping is proposed. Instead of traditional power of two signal constellation sizes, we design 5-QAM, 7-QAM and 9-QAM constellations, which can be used in adaptive optical networks. The non-uniform shaping and LDPC code rate are jointly considered in the design, which results in a better performance scheme for the same SNR values. The matched nonbinary (NB) LDPC code is used for this scheme, which further improves the coding gain and the overall performance. We analyze both coding performance and system SNR performance. We show that the proposed NB LDPC-coded 9-QAM has more than 2dB gain in symbol SNR compared to traditional LDPC-coded star-8-QAM. On the other hand, the proposed NB LDPC-coded 5-QAM and 7-QAM have even better performance than LDPC-coded QPSK.

© 2016 Optical Society of America

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References

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  1. I. B. Djordjevic and T. Wang, “Multiple component codes based generalized LDPC codes for high-speed optical transport,” Opt. Express 22(14), 16694–16705 (2014).
    [Crossref] [PubMed]
  2. D. Zou and I. B. Djordjevic, “FPGA implementation of concatenated non-binary QC-LDPC codes for high-speed optical transport,” Opt. Express 23(11), 14501–14509 (2015).
    [Crossref] [PubMed]
  3. C. Lin, I. B. Djordjevic, M. Cvijetic, and D. Zou, “Mode-Multiplexed Multi-Tb/s Superchannel Transmission with Advanced Multidimensional Signaling in the Presence of Fiber Nonlinearities,” IEEE Trans. Commun. 62(7), 2507–2514 (2014).
    [Crossref]
  4. 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.
  5. L. Beygi, E. Agrell, J. M. Kahn, and M. Karlsson, “Rate-adaptive coded modulation for fiber-optic communications,” J. Lightwave Technol. 32(2), 333–343 (2014).
    [Crossref]
  6. M. P. Yankov, D. Zibar, K. J. Larsen, L. P. Christensen, and S. Forchhammer, “Constellation shaping for fiber-optic channels with QAM and high spectral efficiency,” IEEE Photonics Technol. Lett. 26(23), 2407–2410 (2014).
    [Crossref]
  7. T. Fehenberger, D. Lavery, R. Maher, A. Alvarado, P. Bayvel, and N. Hanik, “Sensitivity gains by mismatched probabilistic shaping for optical communication systems,” IEEE Photonics Technol. Lett. 28(7), 786–789 (2016).
    [Crossref]
  8. 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]
  9. A. R. Calderbank and L. H. Ozarow, “Nonequiprobable signaling on the Gaussian channel. information theory,” IEEE Trans. 36(4), 726–740 (1990).
  10. C. Lin, I. B. Djordjevic, and D. Zou, “Achievable information rates calculation for optical OFDM transmission over few-mode fiber long-haul transmission systems,” Opt. Express 23(13), 16846–16856 (2015).
    [Crossref] [PubMed]
  11. I. B. Djordjevic, “On advanced FEC and coded modulation for ultra-high-speed optical transmission,” IEEE Comm. Surv. and Tutor. PP(99), 1–31 (2016).
    [Crossref]
  12. F. R. Kschischang and S. Pasupathy, “Optimal nonuniform signaling for Gaussian channels,” IEEE Trans. Inf. Theory 39(3), 913–929 (1993).
    [Crossref]
  13. M. Abramowitz and I. A. Stegun, “Handbook of mathematical functions.” Appl. Math. Series 55, 62 (1966).
  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. A. Alvarado, A. Erik, L. Domanic, M. Robert, and B. Polina, “Replacing the soft FEC limit paradigm in the design of optical communication systems,” arXiv preprint arXiv:1503–05477 (2015).

2016 (2)

T. Fehenberger, D. Lavery, R. Maher, A. Alvarado, P. Bayvel, and N. Hanik, “Sensitivity gains by mismatched probabilistic shaping for optical communication systems,” IEEE Photonics Technol. Lett. 28(7), 786–789 (2016).
[Crossref]

I. B. Djordjevic, “On advanced FEC and coded modulation for ultra-high-speed optical transmission,” IEEE Comm. Surv. and Tutor. PP(99), 1–31 (2016).
[Crossref]

2015 (2)

2014 (4)

C. Lin, I. B. Djordjevic, M. Cvijetic, and D. Zou, “Mode-Multiplexed Multi-Tb/s Superchannel Transmission with Advanced Multidimensional Signaling in the Presence of Fiber Nonlinearities,” IEEE Trans. Commun. 62(7), 2507–2514 (2014).
[Crossref]

L. Beygi, E. Agrell, J. M. Kahn, and M. Karlsson, “Rate-adaptive coded modulation for fiber-optic communications,” J. Lightwave Technol. 32(2), 333–343 (2014).
[Crossref]

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

I. B. Djordjevic and T. Wang, “Multiple component codes based generalized LDPC codes for high-speed optical transport,” Opt. Express 22(14), 16694–16705 (2014).
[Crossref] [PubMed]

1993 (2)

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]

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

1990 (1)

A. R. Calderbank and L. H. Ozarow, “Nonequiprobable signaling on the Gaussian channel. information theory,” IEEE Trans. 36(4), 726–740 (1990).

Agrell, E.

Alvarado, A.

T. Fehenberger, D. Lavery, R. Maher, A. Alvarado, P. Bayvel, and N. Hanik, “Sensitivity gains by mismatched probabilistic shaping for optical communication systems,” IEEE Photonics Technol. Lett. 28(7), 786–789 (2016).
[Crossref]

Bayvel, P.

T. Fehenberger, D. Lavery, R. Maher, A. Alvarado, P. Bayvel, and N. Hanik, “Sensitivity gains by mismatched probabilistic shaping for optical communication systems,” IEEE Photonics Technol. Lett. 28(7), 786–789 (2016).
[Crossref]

Beygi, L.

Calderbank, A. R.

A. R. Calderbank and L. H. Ozarow, “Nonequiprobable signaling on the Gaussian channel. information theory,” IEEE Trans. 36(4), 726–740 (1990).

Christensen, L. P.

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

Cvijetic, M.

C. Lin, I. B. Djordjevic, M. Cvijetic, and D. Zou, “Mode-Multiplexed Multi-Tb/s Superchannel Transmission with Advanced Multidimensional Signaling in the Presence of Fiber Nonlinearities,” IEEE Trans. Commun. 62(7), 2507–2514 (2014).
[Crossref]

Djordjevic, I. B.

Fehenberger, T.

T. Fehenberger, D. Lavery, R. Maher, A. Alvarado, P. Bayvel, and N. Hanik, “Sensitivity gains by mismatched probabilistic shaping for optical communication systems,” IEEE Photonics Technol. Lett. 28(7), 786–789 (2016).
[Crossref]

Forchhammer, S.

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

Hanik, N.

T. Fehenberger, D. Lavery, R. Maher, A. Alvarado, P. Bayvel, and N. Hanik, “Sensitivity gains by mismatched probabilistic shaping for optical communication systems,” IEEE Photonics Technol. Lett. 28(7), 786–789 (2016).
[Crossref]

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]

Kahn, J. M.

Karlsson, M.

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. R.

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

Larsen, K. J.

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

Lavery, D.

T. Fehenberger, D. Lavery, R. Maher, A. Alvarado, P. Bayvel, and N. Hanik, “Sensitivity gains by mismatched probabilistic shaping for optical communication systems,” IEEE Photonics Technol. Lett. 28(7), 786–789 (2016).
[Crossref]

Lin, C.

C. Lin, I. B. Djordjevic, and D. Zou, “Achievable information rates calculation for optical OFDM transmission over few-mode fiber long-haul transmission systems,” Opt. Express 23(13), 16846–16856 (2015).
[Crossref] [PubMed]

C. Lin, I. B. Djordjevic, M. Cvijetic, and D. Zou, “Mode-Multiplexed Multi-Tb/s Superchannel Transmission with Advanced Multidimensional Signaling in the Presence of Fiber Nonlinearities,” IEEE Trans. Commun. 62(7), 2507–2514 (2014).
[Crossref]

Maher, R.

T. Fehenberger, D. Lavery, R. Maher, A. Alvarado, P. Bayvel, and N. Hanik, “Sensitivity gains by mismatched probabilistic shaping for optical communication systems,” IEEE Photonics Technol. Lett. 28(7), 786–789 (2016).
[Crossref]

Ozarow, L. H.

A. R. Calderbank and L. H. Ozarow, “Nonequiprobable signaling on the Gaussian channel. information theory,” IEEE Trans. 36(4), 726–740 (1990).

Pasupathy, S.

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

Wang, T.

Yankov, M. P.

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

Zibar, D.

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

Zou, D.

IEEE Comm. Surv. and Tutor. (1)

I. B. Djordjevic, “On advanced FEC and coded modulation for ultra-high-speed optical transmission,” IEEE Comm. Surv. and Tutor. PP(99), 1–31 (2016).
[Crossref]

IEEE Photonics Technol. Lett. (2)

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

T. Fehenberger, D. Lavery, R. Maher, A. Alvarado, P. Bayvel, and N. Hanik, “Sensitivity gains by mismatched probabilistic shaping for optical communication systems,” IEEE Photonics Technol. Lett. 28(7), 786–789 (2016).
[Crossref]

IEEE Trans. (1)

A. R. Calderbank and L. H. Ozarow, “Nonequiprobable signaling on the Gaussian channel. information theory,” IEEE Trans. 36(4), 726–740 (1990).

IEEE Trans. Commun. (1)

C. Lin, I. B. Djordjevic, M. Cvijetic, and D. Zou, “Mode-Multiplexed Multi-Tb/s Superchannel Transmission with Advanced Multidimensional Signaling in the Presence of Fiber Nonlinearities,” IEEE Trans. Commun. 62(7), 2507–2514 (2014).
[Crossref]

IEEE Trans. Inf. Theory (2)

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]

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

J. Lightwave Technol. (1)

Opt. Express (3)

Other (4)

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.

M. Abramowitz and I. A. Stegun, “Handbook of mathematical functions.” Appl. Math. Series 55, 62 (1966).

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.

A. Alvarado, A. Erik, L. Domanic, M. Robert, and B. Polina, “Replacing the soft FEC limit paradigm in the design of optical communication systems,” arXiv preprint arXiv:1503–05477 (2015).

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Figures (10)

Fig. 1
Fig. 1 Proposed generalized nonuniform nonbinary coded modulation scheme.
Fig. 2
Fig. 2 Non-uniform Huffman code for (a) 5-QAM (b) 7-QAM and (c) 9-QAM.
Fig. 3
Fig. 3 (a) The 9-QAM constellation design with changing magnitude and (b) gap to Shannon limit with two variables optimization.
Fig. 4
Fig. 4 Third layer magnitude optimization of 9-QAM.
Fig. 5
Fig. 5 Nonuniform signal constellations obtained by proposed optimization method: (a) 5-QAM, (b) 7-QAM, and (c) 9-QAM received constellations at SNR = 6dB.
Fig. 6
Fig. 6 Achievable information rate vs. SNR for various proposed constellations.
Fig. 7
Fig. 7 Tanner graph representation of GF(32) LDPC code.
Fig. 8
Fig. 8 BCJR based trellis check node processor. V2C: variable-to-check node message.
Fig. 9
Fig. 9 Pre-and-post FEC byte error rate performance.
Fig. 10
Fig. 10 Post-FEC BER vs. SNR performance.

Equations (13)

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

q=[ q 1 , q 1 ,..., q M ]
I( X,Y )=H(X)H( X|Y ) = x=1 M p x log 2 ( p x ) + x=1 M p x + p( y|x ) log 2 p( y|x ) p x x=1 M p( y|x ) p x dy
p=Rq+(1R)m
α i ( s )=log[ Pr( s i =s, y [ 1,i ] ) ]
β i ( s )=log[ Pr( s i =s, y [ i,n ] ) ]
γ i ( s ,s )=log[ Pr( y i |s )Pr( x i ) ]
γ i ( s ,s )=log[ Pr( s i =s, y i , s i1 = s ) ]
α i ( s )= max [ α i1 ( s )+ γ i ( s ,s ) ]
β i ( s )= max [ β i+1 ( s )+ γ i ( s ,s ) ]
max ( c,d )=ln( e c + e d )=max( c,d )+ln( 1+ e | yx | )
{ α 0 ( s )= { sS,s0 } α 0 ( s=0 )=0 α 1 ( s )= { sS }
{ β 0 ( s )= { sS,s0 } β n+1 ( s=0 )=0 β 1 ( s )= { sS }
V2 C i = max ( γ i , α i , β i+1 ).

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