Abstract

We consider the design of compact multisubcarrier constellations for intensity-modulated direct-detected optical systems. The constellations are designed to minimize the average electrical, average optical, and peak optical power for a given minimum distance between constellation points. We formulate the constellation design as a nonconvex optimization problem with second-order cone constraints, (nonconvex) quadratic constraints, and a convex objective function. We show that this problem can be relaxed to a (convex) second-order cone programming (SOCP) problem. We introduce a simple iterative method in which the SOCP relaxation is improved in each iteration. Several numerical simulation examples are provided to illustrate the effectiveness of our method. For the single-subcarrier case, the new constellations are compared with the best known formats in terms of power and spectral efficiency. Our new constellations outperform the corresponding face-centered cubic lattice and quadrature-amplitude-modulation-based constellations, with average electrical and optical power gains in the vicinity of 0.5 dB, for low symbol error rates. The corresponding peak optical power gains are also in the vicinity of 0.5 dB. By studying the mutual information inherent to the new constellations, we show that the potentials are still valid for coded systems. For the two-subcarrier case, we still outperform two-subcarrier schemes based on conventional constellations and optimized single-subcarrier constellations with the same dimensions.

© 2014 Optical Society of America

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2014 (1)

Q. Gao, J. H. Manton, G. Chen, and Y. Hua, “Constellation design for a multicarrier optical wireless communication channel,” IEEE Trans. Commun., vol.  62, no. 1, pp. 214–225, Jan. 2014.
[CrossRef]

2013 (1)

H. Yoon and J. C. Ye, “Motion estimation/compensated compressed sensing using patch-based low rank penalty,” Proc. SPIE, vol.  8858, 88581Y, Sept. 2013.
[CrossRef]

2012 (3)

J. Karout, E. Agrell, K. Szczerba, and M. Karlsson, “Optimizing constellations for single-subcarrier intensity-modulated optical systems,” IEEE Trans. Inf. Theory, vol.  58, no. 7, pp. 4645–4659, July 2012.
[CrossRef]

M. Beko and R. Dinis, “Designing good multi-dimensional constellations,” IEEE Wireless Commun. Lett., vol.  1, no. 3, pp. 221–224, 2012.
[CrossRef]

M. Beko, “Efficient beamforming in cognitive radio multicast transmission,” IEEE Trans. Wireless Commun., vol.  11, no. 11, pp. 4108–4117, 2012.
[CrossRef]

2011 (1)

L. W. Zhang, J. Gu, and X. T. Xiao, “A class of nonlinear Lagrangians for nonconvex second order cone programming,” Comput. Optim. Appl., vol.  49, no. 1, pp. 61–99, May 2011.
[CrossRef]

2010 (3)

2009 (3)

A. Lapidoth, S. M. Moser, and M. A. Wigger, “On the capacity of free-space optical intensity channels,” IEEE Trans. Inf. Theory, vol.  55, no. 10, pp. 4449–4461, Oct. 2009.
[CrossRef]

B. Inan, S. C. J. Lee, S. Randel, I. Neokosmidis, A. M. J. Koonen, and J. W. Walewski, “Impact of LED nonlinearity on discrete multitone modulation,” J. Opt. Commun. Netw., vol.  1, no. 5, pp. 439–451, Oct. 2009.
[CrossRef]

C. Kanzow, I. Ferenczi, and M. Fukushima, “On the local convergence of semismooth Newton methods for linear and nonlinear second-order cone programs without strict complementarity,” SIAM J. Optim., vol.  20, no. 1, pp. 297–320, 2009.
[CrossRef]

2006 (1)

R. Collobert, F. Sinz, J. Weston, and L. Bottou, “Large scale transductive SVMs,” J. Mach. Learn. Res., vol.  7, pp. 1687–1712, 2006.

2004 (1)

S. Hranilovic and F. R. Kschischang, “Capacity bounds for power- and band-limited optical intensity channels corrupted by Gaussian noise,” IEEE Trans. Inf. Theory, vol.  50, no. 5, pp. 784–795, 2004.
[CrossRef]

2003 (2)

S. Hranilovic and F. R. Kschischang, “Optical intensity-modulated direct detection channels: Signal space and lattice codes,” IEEE Trans. Inf. Theory, vol.  49, no. 6, pp. 1385–1399, 2003.
[CrossRef]

T. Ohtsuki, “Multiple-subcarrier modulation in optical wireless communications,” IEEE Commun. Mag., vol.  41, no. 3, pp. 74–79, Mar. 2003.
[CrossRef]

1999 (1)

1997 (1)

J. M. Kahn and J. R. Barry, “Wireless infrared communications,” Proc. IEEE, vol.  85, no. 2, pp. 265–298, 1997.
[CrossRef]

1996 (1)

L. P. Chen and K. Y. Lau, “Regime where zero-bias is the low power solution for digitally modulated laser diodes,” IEEE Photon. Technol. Lett., vol.  8, no. 2, pp. 185–187, 1996.
[CrossRef]

Agrell, E.

J. Karout, E. Agrell, K. Szczerba, and M. Karlsson, “Optimizing constellations for single-subcarrier intensity-modulated optical systems,” IEEE Trans. Inf. Theory, vol.  58, no. 7, pp. 4645–4659, July 2012.
[CrossRef]

J. Karout, E. Agrell, and M. Karlsson, “Power efficient subcarrier modulation for intensity modulated channels,” Opt. Express, vol.  18, no. 17, pp. 17913–17921, Aug. 2010.
[CrossRef]

Barry, J. R.

J. M. Kahn and J. R. Barry, “Wireless infrared communications,” Proc. IEEE, vol.  85, no. 2, pp. 265–298, 1997.
[CrossRef]

J. R. Barry, Wireless Infrared Communications. Norwell, MA: Kluwer Academic, 1994.

Beko, M.

M. Beko and R. Dinis, “Designing good multi-dimensional constellations,” IEEE Wireless Commun. Lett., vol.  1, no. 3, pp. 221–224, 2012.
[CrossRef]

M. Beko, “Efficient beamforming in cognitive radio multicast transmission,” IEEE Trans. Wireless Commun., vol.  11, no. 11, pp. 4108–4117, 2012.
[CrossRef]

Bigot-Astruc, M.

D. Molin, G. Kuyt, M. Bigot-Astruc, and P. Sillard, “Recent advances in MMF technology for data networks,” in Proc. Optical Fiber Communication Conf., Los Angeles, CA, Mar.6–10, 2011, paper OWJ6.

Bottou, L.

R. Collobert, F. Sinz, J. Weston, and L. Bottou, “Large scale transductive SVMs,” J. Mach. Learn. Res., vol.  7, pp. 1687–1712, 2006.

Boyd, S.

S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge, UK: Cambridge University, 2004.

Bradley, P. S.

P. S. Bradley and O. L. Mangasarian, “Feature selection via concave minimization and support vector machines,” in Proc. Int. Conf. on Machine Learning, Madison, WI, July24–27, 1998, pp. 82–90.

Breyer, F.

S. Randel, F. Breyer, and S. C. J. Lee, “High-speed transmission over multimode optical fibers,” in Proc. Optical Fiber Communication Conf., San Diego, CA, Feb.24–28, 2008.

Chen, G.

Q. Gao, J. H. Manton, G. Chen, and Y. Hua, “Constellation design for a multicarrier optical wireless communication channel,” IEEE Trans. Commun., vol.  62, no. 1, pp. 214–225, Jan. 2014.
[CrossRef]

Chen, L. P.

L. P. Chen and K. Y. Lau, “Regime where zero-bias is the low power solution for digitally modulated laser diodes,” IEEE Photon. Technol. Lett., vol.  8, no. 2, pp. 185–187, 1996.
[CrossRef]

Cho, Z. H.

K. S. Kim, Y. D. Son, Z. H. Cho, J. B. Ra, and J. C. Ye, “Globally convergent 3D dynamic PET reconstruction with patch-based non-convex low rank regularization,” in Proc. IEEE Int. Symp. on Biomedical Imaging, San Francisco, CA, Apr.7–11, 2013, pp. 1158–1161.

Collobert, R.

R. Collobert, F. Sinz, J. Weston, and L. Bottou, “Large scale transductive SVMs,” J. Mach. Learn. Res., vol.  7, pp. 1687–1712, 2006.

Conradi, J.

Conway, J. H.

J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, 3rd ed. New York: Springer-Verlag, 1999, p. 21.

Dinis, R.

M. Beko and R. Dinis, “Designing good multi-dimensional constellations,” IEEE Wireless Commun. Lett., vol.  1, no. 3, pp. 221–224, 2012.
[CrossRef]

Essiambre, R.-J.

Farid, A. A.

A. A. Farid and S. Hranilovic, “Capacity bounds for wireless optical intensity channels with Gaussian noise,” IEEE Trans. Inf. Theory, vol.  56, no. 12, pp. 6066–6077, 2010.
[CrossRef]

Ferenczi, I.

C. Kanzow, I. Ferenczi, and M. Fukushima, “On the local convergence of semismooth Newton methods for linear and nonlinear second-order cone programs without strict complementarity,” SIAM J. Optim., vol.  20, no. 1, pp. 297–320, 2009.
[CrossRef]

Foschini, G. J.

Fukushima, M.

C. Kanzow, I. Ferenczi, and M. Fukushima, “On the local convergence of semismooth Newton methods for linear and nonlinear second-order cone programs without strict complementarity,” SIAM J. Optim., vol.  20, no. 1, pp. 297–320, 2009.
[CrossRef]

Gao, Q.

Q. Gao, J. H. Manton, G. Chen, and Y. Hua, “Constellation design for a multicarrier optical wireless communication channel,” IEEE Trans. Commun., vol.  62, no. 1, pp. 214–225, Jan. 2014.
[CrossRef]

Goebel, B.

Gu, J.

L. W. Zhang, J. Gu, and X. T. Xiao, “A class of nonlinear Lagrangians for nonconvex second order cone programming,” Comput. Optim. Appl., vol.  49, no. 1, pp. 61–99, May 2011.
[CrossRef]

Hinedi, S. M.

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

Ho, K.-P.

K.-P. Ho, Phase-Modulated Optical Communication Systems. New York: Springer, 2005.

Hranilovic, S.

A. A. Farid and S. Hranilovic, “Capacity bounds for wireless optical intensity channels with Gaussian noise,” IEEE Trans. Inf. Theory, vol.  56, no. 12, pp. 6066–6077, 2010.
[CrossRef]

S. Hranilovic and F. R. Kschischang, “Capacity bounds for power- and band-limited optical intensity channels corrupted by Gaussian noise,” IEEE Trans. Inf. Theory, vol.  50, no. 5, pp. 784–795, 2004.
[CrossRef]

S. Hranilovic and F. R. Kschischang, “Optical intensity-modulated direct detection channels: Signal space and lattice codes,” IEEE Trans. Inf. Theory, vol.  49, no. 6, pp. 1385–1399, 2003.
[CrossRef]

S. Hranilovic, Wireless Optical Communication Systems. New York: Springer, 2005.

Hua, Y.

Q. Gao, J. H. Manton, G. Chen, and Y. Hua, “Constellation design for a multicarrier optical wireless communication channel,” IEEE Trans. Commun., vol.  62, no. 1, pp. 214–225, Jan. 2014.
[CrossRef]

Inan, B.

Kahn, J. M.

J. M. Kahn and J. R. Barry, “Wireless infrared communications,” Proc. IEEE, vol.  85, no. 2, pp. 265–298, 1997.
[CrossRef]

Kanzow, C.

C. Kanzow, I. Ferenczi, and M. Fukushima, “On the local convergence of semismooth Newton methods for linear and nonlinear second-order cone programs without strict complementarity,” SIAM J. Optim., vol.  20, no. 1, pp. 297–320, 2009.
[CrossRef]

Karlsson, M.

J. Karout, E. Agrell, K. Szczerba, and M. Karlsson, “Optimizing constellations for single-subcarrier intensity-modulated optical systems,” IEEE Trans. Inf. Theory, vol.  58, no. 7, pp. 4645–4659, July 2012.
[CrossRef]

J. Karout, E. Agrell, and M. Karlsson, “Power efficient subcarrier modulation for intensity modulated channels,” Opt. Express, vol.  18, no. 17, pp. 17913–17921, Aug. 2010.
[CrossRef]

Karout, J.

J. Karout, E. Agrell, K. Szczerba, and M. Karlsson, “Optimizing constellations for single-subcarrier intensity-modulated optical systems,” IEEE Trans. Inf. Theory, vol.  58, no. 7, pp. 4645–4659, July 2012.
[CrossRef]

J. Karout, E. Agrell, and M. Karlsson, “Power efficient subcarrier modulation for intensity modulated channels,” Opt. Express, vol.  18, no. 17, pp. 17913–17921, Aug. 2010.
[CrossRef]

Kim, K. S.

K. S. Kim, Y. D. Son, Z. H. Cho, J. B. Ra, and J. C. Ye, “Globally convergent 3D dynamic PET reconstruction with patch-based non-convex low rank regularization,” in Proc. IEEE Int. Symp. on Biomedical Imaging, San Francisco, CA, Apr.7–11, 2013, pp. 1158–1161.

Koonen, A. M. J.

Kramer, G.

Kschischang, F. R.

S. Hranilovic and F. R. Kschischang, “Capacity bounds for power- and band-limited optical intensity channels corrupted by Gaussian noise,” IEEE Trans. Inf. Theory, vol.  50, no. 5, pp. 784–795, 2004.
[CrossRef]

S. Hranilovic and F. R. Kschischang, “Optical intensity-modulated direct detection channels: Signal space and lattice codes,” IEEE Trans. Inf. Theory, vol.  49, no. 6, pp. 1385–1399, 2003.
[CrossRef]

Kuyt, G.

D. Molin, G. Kuyt, M. Bigot-Astruc, and P. Sillard, “Recent advances in MMF technology for data networks,” in Proc. Optical Fiber Communication Conf., Los Angeles, CA, Mar.6–10, 2011, paper OWJ6.

Lanckriet, G. R. G.

B. K. Sriperumbudur and G. R. G. Lanckriet, “On the convergence of the concave-convex procedure,” in Advances in Neural Information Processing Systems 22, Y. Bengio, D. Schuurmans, J. Lafferty, C. K. I. Williams, and A. Culotta, Eds., 2009, pp. 1759–1767.

Lapidoth, A.

A. Lapidoth, S. M. Moser, and M. A. Wigger, “On the capacity of free-space optical intensity channels,” IEEE Trans. Inf. Theory, vol.  55, no. 10, pp. 4449–4461, Oct. 2009.
[CrossRef]

Lau, K. Y.

L. P. Chen and K. Y. Lau, “Regime where zero-bias is the low power solution for digitally modulated laser diodes,” IEEE Photon. Technol. Lett., vol.  8, no. 2, pp. 185–187, 1996.
[CrossRef]

Lee, S. C. J.

B. Inan, S. C. J. Lee, S. Randel, I. Neokosmidis, A. M. J. Koonen, and J. W. Walewski, “Impact of LED nonlinearity on discrete multitone modulation,” J. Opt. Commun. Netw., vol.  1, no. 5, pp. 439–451, Oct. 2009.
[CrossRef]

S. Randel, F. Breyer, and S. C. J. Lee, “High-speed transmission over multimode optical fibers,” in Proc. Optical Fiber Communication Conf., San Diego, CA, Feb.24–28, 2008.

Lindsey, W. C.

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

Mangasarian, O. L.

P. S. Bradley and O. L. Mangasarian, “Feature selection via concave minimization and support vector machines,” in Proc. Int. Conf. on Machine Learning, Madison, WI, July24–27, 1998, pp. 82–90.

Manton, J. H.

Q. Gao, J. H. Manton, G. Chen, and Y. Hua, “Constellation design for a multicarrier optical wireless communication channel,” IEEE Trans. Commun., vol.  62, no. 1, pp. 214–225, Jan. 2014.
[CrossRef]

Molin, D.

D. Molin, G. Kuyt, M. Bigot-Astruc, and P. Sillard, “Recent advances in MMF technology for data networks,” in Proc. Optical Fiber Communication Conf., Los Angeles, CA, Mar.6–10, 2011, paper OWJ6.

Moser, S. M.

A. Lapidoth, S. M. Moser, and M. A. Wigger, “On the capacity of free-space optical intensity channels,” IEEE Trans. Inf. Theory, vol.  55, no. 10, pp. 4449–4461, Oct. 2009.
[CrossRef]

Neokosmidis, I.

Ohtsuki, T.

T. Ohtsuki, “Multiple-subcarrier modulation in optical wireless communications,” IEEE Commun. Mag., vol.  41, no. 3, pp. 74–79, Mar. 2003.
[CrossRef]

Pólik, I.

I. Pólik and T. Terlaky, “Interior point methods for nonlinear optimization,” in Nonlinear Optimization, G. Di Pillo and F. Schoen, Eds., 1st ed. Heidelberg, Germany: Springer, 2010, ch. 4.

Ra, J. B.

K. S. Kim, Y. D. Son, Z. H. Cho, J. B. Ra, and J. C. Ye, “Globally convergent 3D dynamic PET reconstruction with patch-based non-convex low rank regularization,” in Proc. IEEE Int. Symp. on Biomedical Imaging, San Francisco, CA, Apr.7–11, 2013, pp. 1158–1161.

Randel, S.

B. Inan, S. C. J. Lee, S. Randel, I. Neokosmidis, A. M. J. Koonen, and J. W. Walewski, “Impact of LED nonlinearity on discrete multitone modulation,” J. Opt. Commun. Netw., vol.  1, no. 5, pp. 439–451, Oct. 2009.
[CrossRef]

S. Randel, F. Breyer, and S. C. J. Lee, “High-speed transmission over multimode optical fibers,” in Proc. Optical Fiber Communication Conf., San Diego, CA, Feb.24–28, 2008.

Sillard, P.

D. Molin, G. Kuyt, M. Bigot-Astruc, and P. Sillard, “Recent advances in MMF technology for data networks,” in Proc. Optical Fiber Communication Conf., Los Angeles, CA, Mar.6–10, 2011, paper OWJ6.

Simon, M. K.

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

Sinz, F.

R. Collobert, F. Sinz, J. Weston, and L. Bottou, “Large scale transductive SVMs,” J. Mach. Learn. Res., vol.  7, pp. 1687–1712, 2006.

Sloane, N. J. A.

J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, 3rd ed. New York: Springer-Verlag, 1999, p. 21.

Son, Y. D.

K. S. Kim, Y. D. Son, Z. H. Cho, J. B. Ra, and J. C. Ye, “Globally convergent 3D dynamic PET reconstruction with patch-based non-convex low rank regularization,” in Proc. IEEE Int. Symp. on Biomedical Imaging, San Francisco, CA, Apr.7–11, 2013, pp. 1158–1161.

Sriperumbudur, B. K.

B. K. Sriperumbudur and G. R. G. Lanckriet, “On the convergence of the concave-convex procedure,” in Advances in Neural Information Processing Systems 22, Y. Bengio, D. Schuurmans, J. Lafferty, C. K. I. Williams, and A. Culotta, Eds., 2009, pp. 1759–1767.

Szczerba, K.

J. Karout, E. Agrell, K. Szczerba, and M. Karlsson, “Optimizing constellations for single-subcarrier intensity-modulated optical systems,” IEEE Trans. Inf. Theory, vol.  58, no. 7, pp. 4645–4659, July 2012.
[CrossRef]

Terlaky, T.

I. Pólik and T. Terlaky, “Interior point methods for nonlinear optimization,” in Nonlinear Optimization, G. Di Pillo and F. Schoen, Eds., 1st ed. Heidelberg, Germany: Springer, 2010, ch. 4.

Vandenberghe, L.

S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge, UK: Cambridge University, 2004.

Walewski, J. W.

Walklin, S.

Weston, J.

R. Collobert, F. Sinz, J. Weston, and L. Bottou, “Large scale transductive SVMs,” J. Mach. Learn. Res., vol.  7, pp. 1687–1712, 2006.

Wigger, M. A.

A. Lapidoth, S. M. Moser, and M. A. Wigger, “On the capacity of free-space optical intensity channels,” IEEE Trans. Inf. Theory, vol.  55, no. 10, pp. 4449–4461, Oct. 2009.
[CrossRef]

Winzer, P. J.

Xiao, X. T.

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Figures (9)

Fig. 1.
Fig. 1.

CCPP¯e, M=128 (left); CCPP¯o, M=128 (middle); CCPP^o, M=128 (right).

Fig. 2.
Fig. 2.

A3P¯e, M=128 (left); A3P¯o, M=128 (middle); A3P^o, M=128 (right).

Fig. 3.
Fig. 3.

128̆-QAM.

Fig. 4.
Fig. 4.

Performance comparison between the star-shaped QAM constellations, FCC lattice (A3) constellations, and new constellations optimized for the average electrical power and uncoded transmission.

Fig. 5.
Fig. 5.

Performance comparison between the star-shaped QAM constellations, FCC lattice (A3) constellations, and new constellations optimized for the average optical power and uncoded transmission.

Fig. 6.
Fig. 6.

Performance comparison between the star-shaped QAM constellations, FCC lattice (A3) constellations, and new constellations optimized for the peak optical power and uncoded transmission.

Fig. 7.
Fig. 7.

Spectral efficiency of the star-shaped QAM constellations, FCC lattice (A3) constellations, and new constellations versus Eb/N0 for coded transmission.

Fig. 8.
Fig. 8.

Spectral efficiency of the star-shaped QAM constellations, FCC lattice (A3) constellations, and new constellations versus γP¯o for coded transmission.

Fig. 9.
Fig. 9.

Spectral efficiency of the star-shaped QAM constellations, FCC lattice (A3) constellations, and new constellations versus γP^o for coded transmission.

Tables (8)

Tables Icon

TABLE I Average Electrical Power of Our Best Constellations, as Well as the Constellations of [8], Star-Shaped QAM Constellations, and FCC Lattice (A3) Constellations, Optimized for the Average Electrical Power

Tables Icon

TABLE II Average Optical Power of Our Best Constellations, as Well as the Constellations of [8], Star-Shaped QAM Constellations, and FCC Lattice (A3) Constellations, Optimized for the Average Optical Power

Tables Icon

TABLE III Peak Optical Power of Our Best Constellations, as Well as the Constellations of [8], Star-Shaped QAM Constellations, and FCC Lattice (A3) Constellations, Optimized for the Peak Optical Power

Tables Icon

TABLE IV ANNN of Our Best Constellations, as Well as the Constellations of [8], Star-Shaped QAM Constellations, and FCC Lattice (A3) Constellations, Optimized for the Average Electrical Power

Tables Icon

TABLE V ANNN of Our Best Constellations, as Well as the Constellations of [8], Star-Shaped QAM Constellations, and FCC Lattice (A3) Constellations, Optimized for the Average Optical Power

Tables Icon

TABLE VI ANNN of Our Best Constellations, as Well as the Constellations of [8], Star-Shaped QAM Constellations, and FCC Lattice (A3) Constellations, Optimized for the Peak Optical Power

Tables Icon

TABLE VII Average Electrical, Average Optical, and Peak Optical Power of Our Best Constellations for T=2

Tables Icon

TABLE VIII Average Electrical, Average Optical, and Peak Optical Power of Star-Shaped QAM Constellations for T=2

Equations (16)

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

M={(s1,,sM):sisj2D2,ij,andsiϒ},
ϒ={wR2T+1:w1j=1T2(w2j2+w2j+12)},
f(C)=maxi=1,,M{si,1+j=1T2(si,2j2+si,2j+12)}
minimizef(C),
sisj2D2,1i<jM,
si,1j=1T2(si,2j2+si,2j+12),i=1,,M.
minimizef(C)sR3M,
2s(0)TEijss(0)TEijs(0)1,1i<jM,
f1TEis2((f2TEis)2+(f3TEis)2),i=1,,M,
γP¯o(dB)5log10EbN0+10log10E{si,1}E{si2}
γP^o(dB)5log10EbN0+10log10maxi{si,1+2(si,22+si,32)}E{si2}
SERANNN·Q(2·ACG·EbN0),
η=RbW(bits/s/Hz),
CCPP¯e={(0,0,0),(0.8165,0.5752,0.0499),(0.8165,0.3308,0.4732),(1.6024,1.0979,0.2804),(0.8165,0.2444,0.5231),(1.6451,0.0156,0.0333),(1.3608,0.9586,0.0832),(1.6293,0.6062,0.9797),(1.8086,0.9893,0.8104),(2.3291,0.7132,0.0025),(1.4628,0.3797,0.9622),(2.4002,1.6777,0.2569),(2.3514,0.0225,0.6745),(1.5527,0.0658,1.0959),(2.2108,1.3812,0.7323),(3.0053,0.7449,0.7336),(2.1723,1.0705,1.1017),(2.4621,0.0044,0.6861),(1.5397,0.8459,0.6853),(2.4901,0.7394,0.0087),(2.4171,1.6744,0.3431),(2.1915,0.6348,1.4137),(2.2977,1.4961,0.6336),(2.4358,0.7394,1.5556),(2.6249,1.4191,1.1964),(2.3127,0.5354,1.5452),(3.1342,0.0229,0.0538),(3.0445,0.1445,1.5927),(2.3573,0.2532,1.6475),(3.1095,0.7555,0.6264),(3.0606,0.7139,0.8122),(3.0041,0.8261,0.8621)},
CCPP¯0={(0,0,0,),(1.3608,0.9619,0.0272),(0.8165,0.2744,0.5080),(0.8165,0.3027,0.4916),(0.8165,0.5771,0.0163),(1.3608,0.5045,0.8194),(1.6327,0.5796,0.9985),(1.6330,0,0),(1.3608,0.4574,0.8466),(1.6293,1.1475,0.1021),(2.0928,1.3620,0.5787),(2.0380,0.0234,1.4409),(1.6293,0.7145,0.9038),(2.4336,1.7198,0.0576),(2.1267,1.1208,1.0027),(2.0716,0.1963,1.4517),(2.4283,0.5603,0.2314),(2.5895,0.9042,1.5922),(2.4383,0.4956,0.3254),(2.1187,1.4387,0.4180),(2.4336,0.8945,1.4700),(2.4491,0.8847,1.4888),(2.5408,0.3377,0.6567),(2.7523,0.8692,1.7412),(2.6377,0.3957,0.7325),(2.8409,0.1232,2.0050),(2.8440,0.1560,2.0049),(2.0380,1.1923,0.8094),(2.7695,1.9533,0.1401),(2.8403,1.7086,1.0556),(2.8720,1.7049,1.1034),(2.8335,1.8124,0.8542)},
CCPP^0={(0,0,0),(0.8165,0.0426,0.5758),(0.8704,0.4750,0.3914),(0.8513,0.5208,0.3018),(1.5793,0.3022,1.0751),(1.5482,0.6533,0.8785),(1.6208,0.6847,0.9190),(1.6636,1.0839,0.4572),(1.3106,0.8160,0.4392),(1.7039,1.2034,0.0581),(1.7613,0.3442,1.1970),(2.2684,1.4246,0.7371),(2.2684,1.3470,0.8709),(2.2684,1.0055,1.2497),(2.2684,0.4462,1.5407),(2.2684,0.2517,1.5644),(1.7684,0.2114,0.0506),(2.2684,1.5968,0.1520),(2.8330,1.2034,0.0581),(2.7755,0.3442,1.1970),(2.2684,1.1964,1.0684),(2.7684,0.2114,0.0506),(2.2684,0.6014,0.2482),(2.9886,0.6533,0.8785),(2.9161,0.6847,0.9190),(3.6856,0.5208,0.3018),(2.9575,0.3022,1.0751),(2.8732,1.0839,0.4572),(3.2263,0.8160,0.4392),(3.7203,0.0426,0.5758),(3.6665,0.4750,0.3914),(4.5368,0,0)},