Abstract

The Poisson photon-counting model is accurate for optical channels with low received intensity, such as long-range intersatellite optical wireless links. This work considers the computation of the channel capacity and the design of capacity-approaching, nonuniform signaling for discrete-time Poisson channels in the presence of dark current and underaverage and peak amplitude constraints. Although the capacity of this channel is unknown, numerical computation of the channel capacity is implemented using a particle method. A nonuniform mapper is coupled to a low-density parity check code and a joint demapper–decoder is designed based on the sum-product algorithm. Simulations indicate near-capacity performance of the proposed coding system and significant gains over information rates using traditional uniform signaling. A key observation of this work is that significant gains in rate can be achieved for the same average power consumption by using optical transceivers with nonuniform signaling and a modest increase in peak power.

© 2013 Optical Society of America

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References

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  1. R. J. McEliece, E. R. Rodemich, and A. L. Rubin, “The practical limits of photon communication,” Deep Space Network Progress Report 42–55, Jet Propulsion Laboratory, Pasadena, CA, 1979, pp. 63–67.
  2. R. J. McEliece, “Practical codes for photon communication,” IEEE Trans. Inf. Theory, vol.  IT-27, no. 4, pp. 393–398, July 1981.
    [CrossRef]
  3. R. M. Gagliardi and S. Karp, Optical Communications. New York, NY: John Wiley & Sons, 1995.
  4. R. Fields, C. Lunde, R. Wong, J. Wicker, J. Jordan, B. Hansen, G. Muehlnikel, W. Scheel, U. Sterr, R. Kahle, and R. Meyer, “NFIRE-to-TerraSAR-X laser communication results: satellite pointing, disturbances, and other attributes consistent with successful performance,” Proc. SPIE, vol.  7330, 73300Q, 2009.
    [CrossRef]
  5. S. Shamai, “Capacity of a pulse amplitude modulated direct detection photon channel,” Proc. Inst. Elec. Eng., vol.  137, no. 6, pp. 424–430, Dec. 1990.
    [CrossRef]
  6. A. Lapidoth and S. M. Moser, “On the capacity of the discrete-time Poisson channel,” IEEE Trans. Inf. Theory, vol.  55, no. 1, pp. 303–322, Jan. 2009.
    [CrossRef]
  7. A. Lapidoth, J. H. Shapiro, V. Venkatesan, and L. Wang, “The discrete-time Poisson channel at low input powers,” IEEE Trans. Inf. Theory, vol.  57, no. 6, pp. 3260–3272, June 2011.
    [CrossRef]
  8. A. Martinez, “Spectral efficiency of optical direct detection,” J. Opt. Soc. Am. B, vol.  24, no. 4, pp. 739–749, Apr. 2007.
    [CrossRef]
  9. R. Gallager, Information Theory and Reliable Communication. New York, NY: John Wiley & Sons, 1968.
  10. J. Jiang and K. R. Narayanan, “Multilevel coding for channels with non-uniform inputs and rateless transmission over the BSC,” in Proc. IEEE Int. Symp. Information Theory, Seattle, WA, 2006, pp. 518–522.
  11. A. A. Farid and S. Hranilovic, “Channel capacity and non-uniform signalling for free-space optical intensity channels,” IEEE J. Sel. Areas Commun., vol.  27, no. 9, 1553–1563, Dec. 2009.
    [CrossRef]
  12. J. Dauwels, “Computation of the capacity of continuous memoryless channels and the rate distortion function of memoryless continuous sources,” Institute of Electronics, Information and Communication Engineers, Japan, Tech. Rep. IT2006 7-12, 2006.
  13. H. Hemmati, Deep Space Optical Communications. New York, NY: John Wiley & Sons, 2006.
  14. S. B. Alexander, Optical Communications Receiver Design. Bellingham, WA: SPIE Optical Engineering Press, 1997.
  15. R. Blahut, “Computation of channel capacity and rate-distortion functions,” IEEE Trans. Inf. Theory, vol.  18, pp. 460–473, 1972.
    [CrossRef]
  16. C. C. Chen and C. S. Gardner, “Impact of random pointing and tracking errors on the design of coherent and incoherent optical intersatellite communication links,” IEEE Trans. Commun., vol.  37, pp. 252–260, Mar. 1989.
    [CrossRef]
  17. F. R. Kschischang, B. J. Frey, and H.-A. Loeliger, “Factor graphs and the sum-product algorithm,” IEEE Trans. Inf. Theory, vol.  47, no. 2, pp. 498–519, Feb. 2001.
    [CrossRef]
  18. T. Richardson, A. Shokrollahi, and R. Urbanke, “Design of capacity-approaching irregular low-density parity-check codes,” IEEE Trans. Inf. Theory, vol.  47, pp. 619–637, Feb. 2001.
    [CrossRef]
  19. EPFL Information Processing Group, “LDPC degree distribution optimizer for Gaussian channel” [Online]. Available: http://lthcwww.epfl.ch/research/ldpcopt .
  20. Signal Processing Microelectronics, University of Newcastle, “LOPT—online optimisation of LDPC and RA degree distributions” [Online]. Available: http://sonic.newcastle.edu.au/ldpc/lopt/index.php .

2011 (1)

A. Lapidoth, J. H. Shapiro, V. Venkatesan, and L. Wang, “The discrete-time Poisson channel at low input powers,” IEEE Trans. Inf. Theory, vol.  57, no. 6, pp. 3260–3272, June 2011.
[CrossRef]

2009 (3)

A. A. Farid and S. Hranilovic, “Channel capacity and non-uniform signalling for free-space optical intensity channels,” IEEE J. Sel. Areas Commun., vol.  27, no. 9, 1553–1563, Dec. 2009.
[CrossRef]

R. Fields, C. Lunde, R. Wong, J. Wicker, J. Jordan, B. Hansen, G. Muehlnikel, W. Scheel, U. Sterr, R. Kahle, and R. Meyer, “NFIRE-to-TerraSAR-X laser communication results: satellite pointing, disturbances, and other attributes consistent with successful performance,” Proc. SPIE, vol.  7330, 73300Q, 2009.
[CrossRef]

A. Lapidoth and S. M. Moser, “On the capacity of the discrete-time Poisson channel,” IEEE Trans. Inf. Theory, vol.  55, no. 1, pp. 303–322, Jan. 2009.
[CrossRef]

2007 (1)

2001 (2)

F. R. Kschischang, B. J. Frey, and H.-A. Loeliger, “Factor graphs and the sum-product algorithm,” IEEE Trans. Inf. Theory, vol.  47, no. 2, pp. 498–519, Feb. 2001.
[CrossRef]

T. Richardson, A. Shokrollahi, and R. Urbanke, “Design of capacity-approaching irregular low-density parity-check codes,” IEEE Trans. Inf. Theory, vol.  47, pp. 619–637, Feb. 2001.
[CrossRef]

1990 (1)

S. Shamai, “Capacity of a pulse amplitude modulated direct detection photon channel,” Proc. Inst. Elec. Eng., vol.  137, no. 6, pp. 424–430, Dec. 1990.
[CrossRef]

1989 (1)

C. C. Chen and C. S. Gardner, “Impact of random pointing and tracking errors on the design of coherent and incoherent optical intersatellite communication links,” IEEE Trans. Commun., vol.  37, pp. 252–260, Mar. 1989.
[CrossRef]

1981 (1)

R. J. McEliece, “Practical codes for photon communication,” IEEE Trans. Inf. Theory, vol.  IT-27, no. 4, pp. 393–398, July 1981.
[CrossRef]

1972 (1)

R. Blahut, “Computation of channel capacity and rate-distortion functions,” IEEE Trans. Inf. Theory, vol.  18, pp. 460–473, 1972.
[CrossRef]

Alexander, S. B.

S. B. Alexander, Optical Communications Receiver Design. Bellingham, WA: SPIE Optical Engineering Press, 1997.

Blahut, R.

R. Blahut, “Computation of channel capacity and rate-distortion functions,” IEEE Trans. Inf. Theory, vol.  18, pp. 460–473, 1972.
[CrossRef]

Chen, C. C.

C. C. Chen and C. S. Gardner, “Impact of random pointing and tracking errors on the design of coherent and incoherent optical intersatellite communication links,” IEEE Trans. Commun., vol.  37, pp. 252–260, Mar. 1989.
[CrossRef]

Dauwels, J.

J. Dauwels, “Computation of the capacity of continuous memoryless channels and the rate distortion function of memoryless continuous sources,” Institute of Electronics, Information and Communication Engineers, Japan, Tech. Rep. IT2006 7-12, 2006.

Farid, A. A.

A. A. Farid and S. Hranilovic, “Channel capacity and non-uniform signalling for free-space optical intensity channels,” IEEE J. Sel. Areas Commun., vol.  27, no. 9, 1553–1563, Dec. 2009.
[CrossRef]

Fields, R.

R. Fields, C. Lunde, R. Wong, J. Wicker, J. Jordan, B. Hansen, G. Muehlnikel, W. Scheel, U. Sterr, R. Kahle, and R. Meyer, “NFIRE-to-TerraSAR-X laser communication results: satellite pointing, disturbances, and other attributes consistent with successful performance,” Proc. SPIE, vol.  7330, 73300Q, 2009.
[CrossRef]

Frey, B. J.

F. R. Kschischang, B. J. Frey, and H.-A. Loeliger, “Factor graphs and the sum-product algorithm,” IEEE Trans. Inf. Theory, vol.  47, no. 2, pp. 498–519, Feb. 2001.
[CrossRef]

Gagliardi, R. M.

R. M. Gagliardi and S. Karp, Optical Communications. New York, NY: John Wiley & Sons, 1995.

Gallager, R.

R. Gallager, Information Theory and Reliable Communication. New York, NY: John Wiley & Sons, 1968.

Gardner, C. S.

C. C. Chen and C. S. Gardner, “Impact of random pointing and tracking errors on the design of coherent and incoherent optical intersatellite communication links,” IEEE Trans. Commun., vol.  37, pp. 252–260, Mar. 1989.
[CrossRef]

Hansen, B.

R. Fields, C. Lunde, R. Wong, J. Wicker, J. Jordan, B. Hansen, G. Muehlnikel, W. Scheel, U. Sterr, R. Kahle, and R. Meyer, “NFIRE-to-TerraSAR-X laser communication results: satellite pointing, disturbances, and other attributes consistent with successful performance,” Proc. SPIE, vol.  7330, 73300Q, 2009.
[CrossRef]

Hemmati, H.

H. Hemmati, Deep Space Optical Communications. New York, NY: John Wiley & Sons, 2006.

Hranilovic, S.

A. A. Farid and S. Hranilovic, “Channel capacity and non-uniform signalling for free-space optical intensity channels,” IEEE J. Sel. Areas Commun., vol.  27, no. 9, 1553–1563, Dec. 2009.
[CrossRef]

Jiang, J.

J. Jiang and K. R. Narayanan, “Multilevel coding for channels with non-uniform inputs and rateless transmission over the BSC,” in Proc. IEEE Int. Symp. Information Theory, Seattle, WA, 2006, pp. 518–522.

Jordan, J.

R. Fields, C. Lunde, R. Wong, J. Wicker, J. Jordan, B. Hansen, G. Muehlnikel, W. Scheel, U. Sterr, R. Kahle, and R. Meyer, “NFIRE-to-TerraSAR-X laser communication results: satellite pointing, disturbances, and other attributes consistent with successful performance,” Proc. SPIE, vol.  7330, 73300Q, 2009.
[CrossRef]

Kahle, R.

R. Fields, C. Lunde, R. Wong, J. Wicker, J. Jordan, B. Hansen, G. Muehlnikel, W. Scheel, U. Sterr, R. Kahle, and R. Meyer, “NFIRE-to-TerraSAR-X laser communication results: satellite pointing, disturbances, and other attributes consistent with successful performance,” Proc. SPIE, vol.  7330, 73300Q, 2009.
[CrossRef]

Karp, S.

R. M. Gagliardi and S. Karp, Optical Communications. New York, NY: John Wiley & Sons, 1995.

Kschischang, F. R.

F. R. Kschischang, B. J. Frey, and H.-A. Loeliger, “Factor graphs and the sum-product algorithm,” IEEE Trans. Inf. Theory, vol.  47, no. 2, pp. 498–519, Feb. 2001.
[CrossRef]

Lapidoth, A.

A. Lapidoth, J. H. Shapiro, V. Venkatesan, and L. Wang, “The discrete-time Poisson channel at low input powers,” IEEE Trans. Inf. Theory, vol.  57, no. 6, pp. 3260–3272, June 2011.
[CrossRef]

A. Lapidoth and S. M. Moser, “On the capacity of the discrete-time Poisson channel,” IEEE Trans. Inf. Theory, vol.  55, no. 1, pp. 303–322, Jan. 2009.
[CrossRef]

Loeliger, H.-A.

F. R. Kschischang, B. J. Frey, and H.-A. Loeliger, “Factor graphs and the sum-product algorithm,” IEEE Trans. Inf. Theory, vol.  47, no. 2, pp. 498–519, Feb. 2001.
[CrossRef]

Lunde, C.

R. Fields, C. Lunde, R. Wong, J. Wicker, J. Jordan, B. Hansen, G. Muehlnikel, W. Scheel, U. Sterr, R. Kahle, and R. Meyer, “NFIRE-to-TerraSAR-X laser communication results: satellite pointing, disturbances, and other attributes consistent with successful performance,” Proc. SPIE, vol.  7330, 73300Q, 2009.
[CrossRef]

Martinez, A.

McEliece, R. J.

R. J. McEliece, “Practical codes for photon communication,” IEEE Trans. Inf. Theory, vol.  IT-27, no. 4, pp. 393–398, July 1981.
[CrossRef]

R. J. McEliece, E. R. Rodemich, and A. L. Rubin, “The practical limits of photon communication,” Deep Space Network Progress Report 42–55, Jet Propulsion Laboratory, Pasadena, CA, 1979, pp. 63–67.

Meyer, R.

R. Fields, C. Lunde, R. Wong, J. Wicker, J. Jordan, B. Hansen, G. Muehlnikel, W. Scheel, U. Sterr, R. Kahle, and R. Meyer, “NFIRE-to-TerraSAR-X laser communication results: satellite pointing, disturbances, and other attributes consistent with successful performance,” Proc. SPIE, vol.  7330, 73300Q, 2009.
[CrossRef]

Moser, S. M.

A. Lapidoth and S. M. Moser, “On the capacity of the discrete-time Poisson channel,” IEEE Trans. Inf. Theory, vol.  55, no. 1, pp. 303–322, Jan. 2009.
[CrossRef]

Muehlnikel, G.

R. Fields, C. Lunde, R. Wong, J. Wicker, J. Jordan, B. Hansen, G. Muehlnikel, W. Scheel, U. Sterr, R. Kahle, and R. Meyer, “NFIRE-to-TerraSAR-X laser communication results: satellite pointing, disturbances, and other attributes consistent with successful performance,” Proc. SPIE, vol.  7330, 73300Q, 2009.
[CrossRef]

Narayanan, K. R.

J. Jiang and K. R. Narayanan, “Multilevel coding for channels with non-uniform inputs and rateless transmission over the BSC,” in Proc. IEEE Int. Symp. Information Theory, Seattle, WA, 2006, pp. 518–522.

Richardson, T.

T. Richardson, A. Shokrollahi, and R. Urbanke, “Design of capacity-approaching irregular low-density parity-check codes,” IEEE Trans. Inf. Theory, vol.  47, pp. 619–637, Feb. 2001.
[CrossRef]

Rodemich, E. R.

R. J. McEliece, E. R. Rodemich, and A. L. Rubin, “The practical limits of photon communication,” Deep Space Network Progress Report 42–55, Jet Propulsion Laboratory, Pasadena, CA, 1979, pp. 63–67.

Rubin, A. L.

R. J. McEliece, E. R. Rodemich, and A. L. Rubin, “The practical limits of photon communication,” Deep Space Network Progress Report 42–55, Jet Propulsion Laboratory, Pasadena, CA, 1979, pp. 63–67.

Scheel, W.

R. Fields, C. Lunde, R. Wong, J. Wicker, J. Jordan, B. Hansen, G. Muehlnikel, W. Scheel, U. Sterr, R. Kahle, and R. Meyer, “NFIRE-to-TerraSAR-X laser communication results: satellite pointing, disturbances, and other attributes consistent with successful performance,” Proc. SPIE, vol.  7330, 73300Q, 2009.
[CrossRef]

Shamai, S.

S. Shamai, “Capacity of a pulse amplitude modulated direct detection photon channel,” Proc. Inst. Elec. Eng., vol.  137, no. 6, pp. 424–430, Dec. 1990.
[CrossRef]

Shapiro, J. H.

A. Lapidoth, J. H. Shapiro, V. Venkatesan, and L. Wang, “The discrete-time Poisson channel at low input powers,” IEEE Trans. Inf. Theory, vol.  57, no. 6, pp. 3260–3272, June 2011.
[CrossRef]

Shokrollahi, A.

T. Richardson, A. Shokrollahi, and R. Urbanke, “Design of capacity-approaching irregular low-density parity-check codes,” IEEE Trans. Inf. Theory, vol.  47, pp. 619–637, Feb. 2001.
[CrossRef]

Sterr, U.

R. Fields, C. Lunde, R. Wong, J. Wicker, J. Jordan, B. Hansen, G. Muehlnikel, W. Scheel, U. Sterr, R. Kahle, and R. Meyer, “NFIRE-to-TerraSAR-X laser communication results: satellite pointing, disturbances, and other attributes consistent with successful performance,” Proc. SPIE, vol.  7330, 73300Q, 2009.
[CrossRef]

Urbanke, R.

T. Richardson, A. Shokrollahi, and R. Urbanke, “Design of capacity-approaching irregular low-density parity-check codes,” IEEE Trans. Inf. Theory, vol.  47, pp. 619–637, Feb. 2001.
[CrossRef]

Venkatesan, V.

A. Lapidoth, J. H. Shapiro, V. Venkatesan, and L. Wang, “The discrete-time Poisson channel at low input powers,” IEEE Trans. Inf. Theory, vol.  57, no. 6, pp. 3260–3272, June 2011.
[CrossRef]

Wang, L.

A. Lapidoth, J. H. Shapiro, V. Venkatesan, and L. Wang, “The discrete-time Poisson channel at low input powers,” IEEE Trans. Inf. Theory, vol.  57, no. 6, pp. 3260–3272, June 2011.
[CrossRef]

Wicker, J.

R. Fields, C. Lunde, R. Wong, J. Wicker, J. Jordan, B. Hansen, G. Muehlnikel, W. Scheel, U. Sterr, R. Kahle, and R. Meyer, “NFIRE-to-TerraSAR-X laser communication results: satellite pointing, disturbances, and other attributes consistent with successful performance,” Proc. SPIE, vol.  7330, 73300Q, 2009.
[CrossRef]

Wong, R.

R. Fields, C. Lunde, R. Wong, J. Wicker, J. Jordan, B. Hansen, G. Muehlnikel, W. Scheel, U. Sterr, R. Kahle, and R. Meyer, “NFIRE-to-TerraSAR-X laser communication results: satellite pointing, disturbances, and other attributes consistent with successful performance,” Proc. SPIE, vol.  7330, 73300Q, 2009.
[CrossRef]

IEEE J. Sel. Areas Commun. (1)

A. A. Farid and S. Hranilovic, “Channel capacity and non-uniform signalling for free-space optical intensity channels,” IEEE J. Sel. Areas Commun., vol.  27, no. 9, 1553–1563, Dec. 2009.
[CrossRef]

IEEE Trans. Commun. (1)

C. C. Chen and C. S. Gardner, “Impact of random pointing and tracking errors on the design of coherent and incoherent optical intersatellite communication links,” IEEE Trans. Commun., vol.  37, pp. 252–260, Mar. 1989.
[CrossRef]

IEEE Trans. Inf. Theory (6)

F. R. Kschischang, B. J. Frey, and H.-A. Loeliger, “Factor graphs and the sum-product algorithm,” IEEE Trans. Inf. Theory, vol.  47, no. 2, pp. 498–519, Feb. 2001.
[CrossRef]

T. Richardson, A. Shokrollahi, and R. Urbanke, “Design of capacity-approaching irregular low-density parity-check codes,” IEEE Trans. Inf. Theory, vol.  47, pp. 619–637, Feb. 2001.
[CrossRef]

R. J. McEliece, “Practical codes for photon communication,” IEEE Trans. Inf. Theory, vol.  IT-27, no. 4, pp. 393–398, July 1981.
[CrossRef]

A. Lapidoth and S. M. Moser, “On the capacity of the discrete-time Poisson channel,” IEEE Trans. Inf. Theory, vol.  55, no. 1, pp. 303–322, Jan. 2009.
[CrossRef]

A. Lapidoth, J. H. Shapiro, V. Venkatesan, and L. Wang, “The discrete-time Poisson channel at low input powers,” IEEE Trans. Inf. Theory, vol.  57, no. 6, pp. 3260–3272, June 2011.
[CrossRef]

R. Blahut, “Computation of channel capacity and rate-distortion functions,” IEEE Trans. Inf. Theory, vol.  18, pp. 460–473, 1972.
[CrossRef]

J. Opt. Soc. Am. B (1)

Proc. Inst. Elec. Eng. (1)

S. Shamai, “Capacity of a pulse amplitude modulated direct detection photon channel,” Proc. Inst. Elec. Eng., vol.  137, no. 6, pp. 424–430, Dec. 1990.
[CrossRef]

Proc. SPIE (1)

R. Fields, C. Lunde, R. Wong, J. Wicker, J. Jordan, B. Hansen, G. Muehlnikel, W. Scheel, U. Sterr, R. Kahle, and R. Meyer, “NFIRE-to-TerraSAR-X laser communication results: satellite pointing, disturbances, and other attributes consistent with successful performance,” Proc. SPIE, vol.  7330, 73300Q, 2009.
[CrossRef]

Other (9)

R. J. McEliece, E. R. Rodemich, and A. L. Rubin, “The practical limits of photon communication,” Deep Space Network Progress Report 42–55, Jet Propulsion Laboratory, Pasadena, CA, 1979, pp. 63–67.

R. M. Gagliardi and S. Karp, Optical Communications. New York, NY: John Wiley & Sons, 1995.

R. Gallager, Information Theory and Reliable Communication. New York, NY: John Wiley & Sons, 1968.

J. Jiang and K. R. Narayanan, “Multilevel coding for channels with non-uniform inputs and rateless transmission over the BSC,” in Proc. IEEE Int. Symp. Information Theory, Seattle, WA, 2006, pp. 518–522.

EPFL Information Processing Group, “LDPC degree distribution optimizer for Gaussian channel” [Online]. Available: http://lthcwww.epfl.ch/research/ldpcopt .

Signal Processing Microelectronics, University of Newcastle, “LOPT—online optimisation of LDPC and RA degree distributions” [Online]. Available: http://sonic.newcastle.edu.au/ldpc/lopt/index.php .

J. Dauwels, “Computation of the capacity of continuous memoryless channels and the rate distortion function of memoryless continuous sources,” Institute of Electronics, Information and Communication Engineers, Japan, Tech. Rep. IT2006 7-12, 2006.

H. Hemmati, Deep Space Optical Communications. New York, NY: John Wiley & Sons, 2006.

S. B. Alexander, Optical Communications Receiver Design. Bellingham, WA: SPIE Optical Engineering Press, 1997.

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Figures (11)

Fig. 1.
Fig. 1.

Bounds L(n), U(n) on channel capacity (on top of each other) and closed form bounds from [6] with A/ε=4 and λ=3.

Fig. 2.
Fig. 2.

Capacity achieving input distributions for A/ε=4 and λ=3.

Fig. 3.
Fig. 3.

Bounds L(n), U(n) on channel capacity (on top of each other) and closed form bounds from [6] with A/ε=2 and λ=3.

Fig. 4.
Fig. 4.

Channel capacity for LEO intersatellite link versus A/ε with ε=7.5027 and λ=3.6649. For comparison, the gain in rate versus uniform binary signaling, C0, is also presented.

Fig. 5.
Fig. 5.

Channel capacity and mutual information for constellations from the constrained particle method for ε=4dB and A/ε=4.

Fig. 6.
Fig. 6.

The optimum and the proposed input distribution for different λ when ε=4dB and A/ε=4. For (a) and (b) N=2 and for (c) and (d) N=3.

Fig. 7.
Fig. 7.

System model for the developed encoding method and mapping scheme.

Fig. 8.
Fig. 8.

Developed factor graph for joint demapping and decoding.

Fig. 9.
Fig. 9.

BER versus 1/λ for the nonuniform signaling using finite-length LDPC codes for ε=1.21dB, A/ε=4. The value of 1/λ corresponding to optimal uniform (out of range) and nonuniform signaling at 0.24bits/channel use is presented for comparison.

Fig. 10.
Fig. 10.

BER versus 1/λ for the nonuniform signaling using finite-length LDPC codes with R=1.136bits/channel use as ε=7.5027, A/ε=2.62.

Fig. 11.
Fig. 11.

BER versus 1/λ for the nonuniform signaling using finite-length LDPC codes as ε=4dB, A/ε=4. The 1/λ corresponding to optimal uniform signaling and nonuniform signaling with R=0.75bits/channel use is presented for comparison.

Tables (1)

Tables Icon

TABLE I Terminal Characteristics

Equations (25)

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

PY|X(y|x)=(x+λ)yy!e(x+λ),xR+,yZ+,
0XAandE(X)ε.
CmaxpX(·)ϵPI(X;Y)=maxpX(·)ϵPxpX(x)[yPY|X(y/x)logPY|X(y|x)PY(y)]dx,
P{pX(x):0ApX(x)dx=1,pX(x)0,EpX(·){X}ε}.
pX(x)p^X(x)=i=1Mpiδ(xxi),
p(k)argmaxpI({(x(k1),p)}),(W-step),
x(k)argmaxxI({(x,p(k))}).(X-step).
CL(n)=I({(xi(n),pi(n))}),
CU(n)=maxxX[D(PY|X(y|x)P^(y)(n))s(n)x]+s(n)i=1Mpi(n)xi(n),
ε=PTηTηRηλwhc(λw4πz)2GTLT(πdrλw)2T=7.5027,
GT=(πdtλw)2,
LT=exp(GTθT2).
λ=W(λw)π(dr2)2(Δλ)π4Ω2Tλwhc=3.6649,
(W1(i)W2(i))fX:X={A,W1(i)=W2(i)=1,0,otherwise.
PY|W(y|w1=1)=w2PY|X(y,w2|w1=1)=12PY|X(y|A)+12PY|X(y|0),PY|W(y|w1=0)=PY|X(y|0),
μfw1(i)(w1(i)=1)=μxif(xi=A)μw2(i)f(w2(i)=1)+μxif(xi=0)μw2(i)f(w2(i)=0),μfw1(i)(w1(i)=0)=μxif(xi=0)μw2(i)f(w2(i)=1)+μxif(xi=0)μw2(i)f(w2(i)=0).
mxif=lnμxif(xi=0)μxif(xi=1)=lnP(xi=0|yi)P(xi=1|yi)=ln3P(yi|xi=0)P(yi|xi=A).
λ(x)=0.5513x+0.2031x2+0.0917x4+0.0045x6+0.017x7+0.0995x8+0.033x9,ρ(x)=x2.
pX*(x)=0.4444δ(x)+0.2001δ(x7.1835)+0.0883δ(x9.2315)+0.2672δ(x19.6503).
p^X*(x)=0.5δ(x)+0.25δ(x8.76)+0.25δ(x19.6505),
(W1(i)W2(i))fX:X={A0,W1(i)=0A1,W1(i)=1,W2(i)=0A2,W1(i)=1,W2(i)=1,
λ(x)=0.181804x+0.197579x2+0.011671x3+0.098834x4+0.063856x5+0.239152x24+0.207105x25,ρ(x)=0.839350x10+0.160650x11
p(0)=58,p(A2)=28,p(A)=18,
λ(x)=0.602x+0.238x2+0.0309x3+0.021x4+0.0491x5+0.0141x6+0.0208x7+0.023x8+0.001x9,ρ(x)=0.0001x+0.1017x2+0.8982x3
W=[W1W2W3]fX:X={A0,W1=0,A0,W1=1,W2=1,W3=0,A1,W1=1,W2=0,W3=0,A1,W1=1,W2=0,W3=1,A2W1=W2=W3=1.