Abstract

Adaptive modulation and coding can provide robust and spectrally efficient transmission over terrestrial free-space optical channels. Three adaptive modulation schemes are considered in this paper: (i) variable-rate variable-power adaptation, (ii) channel inversion, and (iii) truncated channel inversion schemes. It is shown that a simple channel inversion scheme performs comparable to a variable-rate variable-power adaptation scheme in the weak turbulence regime but faces significant performance degradation in the strong turbulence regime. We further study adaptive coding based on large-girth quasi-cyclic low-density parity-check- (LDPC-) coded modulation. It is shown by simulation that deep fades of the order of 30dB and above in the regime of strong turbulence can be tolerated with the proposed scheme. It is demonstrated that communication in the saturation regime is possible with the proposed adaptive LDPC-coded modulation. We also determine the spectral efficiencies for the proposed adaptive modulation and adaptive coding schemes.

© 2010 Optical Society of America

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References

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  1. L. C. Andrews, R. L. Philips, Laser Beam Propagation Through Random Media, SPIE Press, 2005.
    [Crossref]
  2. H. Willebrand, B. S. Ghuman, Free-Space Optics: Enabling Optical Connectivity in Today’s Networks, Sams Publishing, 2002.
  3. I. B. Djordjevic, B. Vasic, M. A. Neifeld, “LDPC coded OFDM over the atmospheric turbulence channel,” Opt. Express, vol. 15, no. 10, pp. 6332–6346, May 2007.
  4. I. B. Djordjevic, S. Denic, J. Anguita, B. Vasic, M. A. Neifeld, “LDPC-coded MIMO optical communication over the atmospheric turbulence channel,” J. Lightwave Technol., vol. 26, no. 5, pp. 478–487, Mar. 2008.
    [Crossref]
  5. M. Luby, “LT codes,” Proc. IEEE Symp. on Foundations of Computer Science (FOCS02), 2002, p. 43.
  6. A. Shokrollahi, “Raptor codes,” IEEE Trans. Inf. Theory, vol. 52, pp. 2551–2567, June 2006.
    [Crossref]
  7. J. A. Anguita, M. A. Neifeld, B. Hildner, B. Vasic, “Rateless coding on experimental temporally correlated FSO channels,” J. Lightwave Technol., vol. 28, no. 7, pp. 990–1002, Apr. 2010.
    [Crossref]
  8. D. J. C. MacKay, “Fountain codes,” IEE Proc.-Commun., vol. 152, pp. 1062–1068, Dec. 2005.
    [Crossref]
  9. S. Denic, I. B. Djordjevic, J. Anguita, B. Vasic, M. A. Neifeld, “Information theoretic limits for free-space optical channels with and without memory,” J. Lightwave Technol., vol. 26, no. 19, pp. 3376–3384, Oct. 2008.
    [Crossref]
  10. T. M. Cover, J. A. Thomas, Elements of Information Theory, New York: Wiley, 1991.
    [Crossref]
  11. A. Goldsmith, Wireless Communications, Cambridge: Cambridge U. Press, 2005.
    [Crossref]
  12. A. Goldsmith, S.-G. Chua, “Adaptive coded modulation for fading channels,” IEEE Trans. Commun., vol. 46, pp. 595–601, May 1998.
    [Crossref]
  13. J. Liao, S. Deng, K. Connor, Z. R. Huang, “Antenna integration with laser diodes and photodetectors for a miniaturized dual-mode wireless transceiver,” in Proc. 58th Electronic Component Technology Conf. (ECTC), Orlando, FL, May 2008, pp. 1864–1868.
  14. M. A. Al-Habash, L. C. Andrews, R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. (Bellingham), vol. 40, pp. 1554–1562, 2001.
    [Crossref]
  15. J. A. Anguita, I. B. Djordjevic, M. A. Neifeld, B. V. Vasic, “Shannon capacities and error-correction codes for optical atmospheric turbulent channels,” BioInterphases, vol. 4, no. 9, pp. 586–601, Sept. 2005.
  16. J. Li, M. Uysal, “Optical wireless communications: system model, capacity and coding,” in Proc. 2003 58th IEEE Vehicular Technology Conf., Oct. 2003, vol. 1, pp. 168–172.
  17. S. Hranilovic, 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, May 2004.
    [Crossref]
  18. J. G. Proakis, Digital Communications, Boston, MA: McGraw-Hill, 2001.
  19. J. G. Proakis, Digital Communications, Boston: McGraw-Hill, 2001.
  20. G. Ungerboeck, “Channel coding with multilevel/phase signals,” IEEE Trans. Inf. Theory, vol. 28, pp. 55–67, Jan. 1982.
    [Crossref]
  21. G. D. Forney, “Coset codes—Part I: introduction and geometrical classification,” IEEE Trans. Inf. Theory, vol. 34, pp. 1123–1151, Sept. 1988.
    [Crossref]
  22. M. P. C. Fossorier, “Quasi-cyclic low-density parity-check codes from circulant permutation matrices,” IEEE Trans. Inf. Theory, vol. 50, pp. 1788–1794, Aug. 2004.
    [Crossref]
  23. I. B. Djordjevic, L. Xu, T. Wang, M. Cvijetic, “Large girth low-density parity-check codes for long-haul high-speed optical communications,” in Proc. OFC/NFOEC 2008, San Diego, CA, Feb. 2008, paper JWA53.
  24. X.-Y. Hu, E. Eleftheriou, D.-M. Arnold, A. Dholakia, “Efficient implementations of the sum-product algorithm for decoding of LDPC codes,” in Proc. IEEE GLOBECOM, vol. 2, Nov. 2001, pp. 1036–1036E.
  25. A. A. Farid, S. Hranilovic, “Upper and lower bounds on the capacity of wireless optical intensity channels,” in Proc. ISIT 2007, Nice, France, June 2007, pp. 2416–2420.

2010 (1)

2008 (2)

2007 (1)

I. B. Djordjevic, B. Vasic, M. A. Neifeld, “LDPC coded OFDM over the atmospheric turbulence channel,” Opt. Express, vol. 15, no. 10, pp. 6332–6346, May 2007.

2006 (1)

A. Shokrollahi, “Raptor codes,” IEEE Trans. Inf. Theory, vol. 52, pp. 2551–2567, June 2006.
[Crossref]

2005 (2)

J. A. Anguita, I. B. Djordjevic, M. A. Neifeld, B. V. Vasic, “Shannon capacities and error-correction codes for optical atmospheric turbulent channels,” BioInterphases, vol. 4, no. 9, pp. 586–601, Sept. 2005.

D. J. C. MacKay, “Fountain codes,” IEE Proc.-Commun., vol. 152, pp. 1062–1068, Dec. 2005.
[Crossref]

2004 (2)

M. P. C. Fossorier, “Quasi-cyclic low-density parity-check codes from circulant permutation matrices,” IEEE Trans. Inf. Theory, vol. 50, pp. 1788–1794, Aug. 2004.
[Crossref]

S. Hranilovic, 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, May 2004.
[Crossref]

2001 (1)

M. A. Al-Habash, L. C. Andrews, R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. (Bellingham), vol. 40, pp. 1554–1562, 2001.
[Crossref]

1998 (1)

A. Goldsmith, S.-G. Chua, “Adaptive coded modulation for fading channels,” IEEE Trans. Commun., vol. 46, pp. 595–601, May 1998.
[Crossref]

1988 (1)

G. D. Forney, “Coset codes—Part I: introduction and geometrical classification,” IEEE Trans. Inf. Theory, vol. 34, pp. 1123–1151, Sept. 1988.
[Crossref]

1982 (1)

G. Ungerboeck, “Channel coding with multilevel/phase signals,” IEEE Trans. Inf. Theory, vol. 28, pp. 55–67, Jan. 1982.
[Crossref]

Al-Habash, M. A.

M. A. Al-Habash, L. C. Andrews, R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. (Bellingham), vol. 40, pp. 1554–1562, 2001.
[Crossref]

Andrews, L. C.

M. A. Al-Habash, L. C. Andrews, R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. (Bellingham), vol. 40, pp. 1554–1562, 2001.
[Crossref]

L. C. Andrews, R. L. Philips, Laser Beam Propagation Through Random Media, SPIE Press, 2005.
[Crossref]

Anguita, J.

Anguita, J. A.

J. A. Anguita, M. A. Neifeld, B. Hildner, B. Vasic, “Rateless coding on experimental temporally correlated FSO channels,” J. Lightwave Technol., vol. 28, no. 7, pp. 990–1002, Apr. 2010.
[Crossref]

J. A. Anguita, I. B. Djordjevic, M. A. Neifeld, B. V. Vasic, “Shannon capacities and error-correction codes for optical atmospheric turbulent channels,” BioInterphases, vol. 4, no. 9, pp. 586–601, Sept. 2005.

Arnold, D.-M.

X.-Y. Hu, E. Eleftheriou, D.-M. Arnold, A. Dholakia, “Efficient implementations of the sum-product algorithm for decoding of LDPC codes,” in Proc. IEEE GLOBECOM, vol. 2, Nov. 2001, pp. 1036–1036E.

Chua, S.-G.

A. Goldsmith, S.-G. Chua, “Adaptive coded modulation for fading channels,” IEEE Trans. Commun., vol. 46, pp. 595–601, May 1998.
[Crossref]

Connor, K.

J. Liao, S. Deng, K. Connor, Z. R. Huang, “Antenna integration with laser diodes and photodetectors for a miniaturized dual-mode wireless transceiver,” in Proc. 58th Electronic Component Technology Conf. (ECTC), Orlando, FL, May 2008, pp. 1864–1868.

Cover, T. M.

T. M. Cover, J. A. Thomas, Elements of Information Theory, New York: Wiley, 1991.
[Crossref]

Cvijetic, M.

I. B. Djordjevic, L. Xu, T. Wang, M. Cvijetic, “Large girth low-density parity-check codes for long-haul high-speed optical communications,” in Proc. OFC/NFOEC 2008, San Diego, CA, Feb. 2008, paper JWA53.

Deng, S.

J. Liao, S. Deng, K. Connor, Z. R. Huang, “Antenna integration with laser diodes and photodetectors for a miniaturized dual-mode wireless transceiver,” in Proc. 58th Electronic Component Technology Conf. (ECTC), Orlando, FL, May 2008, pp. 1864–1868.

Denic, S.

Dholakia, A.

X.-Y. Hu, E. Eleftheriou, D.-M. Arnold, A. Dholakia, “Efficient implementations of the sum-product algorithm for decoding of LDPC codes,” in Proc. IEEE GLOBECOM, vol. 2, Nov. 2001, pp. 1036–1036E.

Djordjevic, I. B.

I. B. Djordjevic, S. Denic, J. Anguita, B. Vasic, M. A. Neifeld, “LDPC-coded MIMO optical communication over the atmospheric turbulence channel,” J. Lightwave Technol., vol. 26, no. 5, pp. 478–487, Mar. 2008.
[Crossref]

S. Denic, I. B. Djordjevic, J. Anguita, B. Vasic, M. A. Neifeld, “Information theoretic limits for free-space optical channels with and without memory,” J. Lightwave Technol., vol. 26, no. 19, pp. 3376–3384, Oct. 2008.
[Crossref]

I. B. Djordjevic, B. Vasic, M. A. Neifeld, “LDPC coded OFDM over the atmospheric turbulence channel,” Opt. Express, vol. 15, no. 10, pp. 6332–6346, May 2007.

J. A. Anguita, I. B. Djordjevic, M. A. Neifeld, B. V. Vasic, “Shannon capacities and error-correction codes for optical atmospheric turbulent channels,” BioInterphases, vol. 4, no. 9, pp. 586–601, Sept. 2005.

I. B. Djordjevic, L. Xu, T. Wang, M. Cvijetic, “Large girth low-density parity-check codes for long-haul high-speed optical communications,” in Proc. OFC/NFOEC 2008, San Diego, CA, Feb. 2008, paper JWA53.

Eleftheriou, E.

X.-Y. Hu, E. Eleftheriou, D.-M. Arnold, A. Dholakia, “Efficient implementations of the sum-product algorithm for decoding of LDPC codes,” in Proc. IEEE GLOBECOM, vol. 2, Nov. 2001, pp. 1036–1036E.

Farid, A. A.

A. A. Farid, S. Hranilovic, “Upper and lower bounds on the capacity of wireless optical intensity channels,” in Proc. ISIT 2007, Nice, France, June 2007, pp. 2416–2420.

Forney, G. D.

G. D. Forney, “Coset codes—Part I: introduction and geometrical classification,” IEEE Trans. Inf. Theory, vol. 34, pp. 1123–1151, Sept. 1988.
[Crossref]

Fossorier, M. P. C.

M. P. C. Fossorier, “Quasi-cyclic low-density parity-check codes from circulant permutation matrices,” IEEE Trans. Inf. Theory, vol. 50, pp. 1788–1794, Aug. 2004.
[Crossref]

Ghuman, B. S.

H. Willebrand, B. S. Ghuman, Free-Space Optics: Enabling Optical Connectivity in Today’s Networks, Sams Publishing, 2002.

Goldsmith, A.

A. Goldsmith, S.-G. Chua, “Adaptive coded modulation for fading channels,” IEEE Trans. Commun., vol. 46, pp. 595–601, May 1998.
[Crossref]

A. Goldsmith, Wireless Communications, Cambridge: Cambridge U. Press, 2005.
[Crossref]

Hildner, B.

Hranilovic, S.

S. Hranilovic, 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, May 2004.
[Crossref]

A. A. Farid, S. Hranilovic, “Upper and lower bounds on the capacity of wireless optical intensity channels,” in Proc. ISIT 2007, Nice, France, June 2007, pp. 2416–2420.

Hu, X.-Y.

X.-Y. Hu, E. Eleftheriou, D.-M. Arnold, A. Dholakia, “Efficient implementations of the sum-product algorithm for decoding of LDPC codes,” in Proc. IEEE GLOBECOM, vol. 2, Nov. 2001, pp. 1036–1036E.

Huang, Z. R.

J. Liao, S. Deng, K. Connor, Z. R. Huang, “Antenna integration with laser diodes and photodetectors for a miniaturized dual-mode wireless transceiver,” in Proc. 58th Electronic Component Technology Conf. (ECTC), Orlando, FL, May 2008, pp. 1864–1868.

Kschischang, F. R.

S. Hranilovic, 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, May 2004.
[Crossref]

Li, J.

J. Li, M. Uysal, “Optical wireless communications: system model, capacity and coding,” in Proc. 2003 58th IEEE Vehicular Technology Conf., Oct. 2003, vol. 1, pp. 168–172.

Liao, J.

J. Liao, S. Deng, K. Connor, Z. R. Huang, “Antenna integration with laser diodes and photodetectors for a miniaturized dual-mode wireless transceiver,” in Proc. 58th Electronic Component Technology Conf. (ECTC), Orlando, FL, May 2008, pp. 1864–1868.

Luby, M.

M. Luby, “LT codes,” Proc. IEEE Symp. on Foundations of Computer Science (FOCS02), 2002, p. 43.

MacKay, D. J. C.

D. J. C. MacKay, “Fountain codes,” IEE Proc.-Commun., vol. 152, pp. 1062–1068, Dec. 2005.
[Crossref]

Neifeld, M. A.

Philips, R. L.

L. C. Andrews, R. L. Philips, Laser Beam Propagation Through Random Media, SPIE Press, 2005.
[Crossref]

Phillips, R. L.

M. A. Al-Habash, L. C. Andrews, R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. (Bellingham), vol. 40, pp. 1554–1562, 2001.
[Crossref]

Proakis, J. G.

J. G. Proakis, Digital Communications, Boston, MA: McGraw-Hill, 2001.

J. G. Proakis, Digital Communications, Boston: McGraw-Hill, 2001.

Shokrollahi, A.

A. Shokrollahi, “Raptor codes,” IEEE Trans. Inf. Theory, vol. 52, pp. 2551–2567, June 2006.
[Crossref]

Thomas, J. A.

T. M. Cover, J. A. Thomas, Elements of Information Theory, New York: Wiley, 1991.
[Crossref]

Ungerboeck, G.

G. Ungerboeck, “Channel coding with multilevel/phase signals,” IEEE Trans. Inf. Theory, vol. 28, pp. 55–67, Jan. 1982.
[Crossref]

Uysal, M.

J. Li, M. Uysal, “Optical wireless communications: system model, capacity and coding,” in Proc. 2003 58th IEEE Vehicular Technology Conf., Oct. 2003, vol. 1, pp. 168–172.

Vasic, B.

Vasic, B. V.

J. A. Anguita, I. B. Djordjevic, M. A. Neifeld, B. V. Vasic, “Shannon capacities and error-correction codes for optical atmospheric turbulent channels,” BioInterphases, vol. 4, no. 9, pp. 586–601, Sept. 2005.

Wang, T.

I. B. Djordjevic, L. Xu, T. Wang, M. Cvijetic, “Large girth low-density parity-check codes for long-haul high-speed optical communications,” in Proc. OFC/NFOEC 2008, San Diego, CA, Feb. 2008, paper JWA53.

Willebrand, H.

H. Willebrand, B. S. Ghuman, Free-Space Optics: Enabling Optical Connectivity in Today’s Networks, Sams Publishing, 2002.

Xu, L.

I. B. Djordjevic, L. Xu, T. Wang, M. Cvijetic, “Large girth low-density parity-check codes for long-haul high-speed optical communications,” in Proc. OFC/NFOEC 2008, San Diego, CA, Feb. 2008, paper JWA53.

BioInterphases (1)

J. A. Anguita, I. B. Djordjevic, M. A. Neifeld, B. V. Vasic, “Shannon capacities and error-correction codes for optical atmospheric turbulent channels,” BioInterphases, vol. 4, no. 9, pp. 586–601, Sept. 2005.

IEE Proc.-Commun. (1)

D. J. C. MacKay, “Fountain codes,” IEE Proc.-Commun., vol. 152, pp. 1062–1068, Dec. 2005.
[Crossref]

IEEE Trans. Commun. (1)

A. Goldsmith, S.-G. Chua, “Adaptive coded modulation for fading channels,” IEEE Trans. Commun., vol. 46, pp. 595–601, May 1998.
[Crossref]

IEEE Trans. Inf. Theory (5)

A. Shokrollahi, “Raptor codes,” IEEE Trans. Inf. Theory, vol. 52, pp. 2551–2567, June 2006.
[Crossref]

G. Ungerboeck, “Channel coding with multilevel/phase signals,” IEEE Trans. Inf. Theory, vol. 28, pp. 55–67, Jan. 1982.
[Crossref]

G. D. Forney, “Coset codes—Part I: introduction and geometrical classification,” IEEE Trans. Inf. Theory, vol. 34, pp. 1123–1151, Sept. 1988.
[Crossref]

M. P. C. Fossorier, “Quasi-cyclic low-density parity-check codes from circulant permutation matrices,” IEEE Trans. Inf. Theory, vol. 50, pp. 1788–1794, Aug. 2004.
[Crossref]

S. Hranilovic, 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, May 2004.
[Crossref]

J. Lightwave Technol. (3)

Opt. Eng. (Bellingham) (1)

M. A. Al-Habash, L. C. Andrews, R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. (Bellingham), vol. 40, pp. 1554–1562, 2001.
[Crossref]

Opt. Express (1)

I. B. Djordjevic, B. Vasic, M. A. Neifeld, “LDPC coded OFDM over the atmospheric turbulence channel,” Opt. Express, vol. 15, no. 10, pp. 6332–6346, May 2007.

Other (12)

J. Li, M. Uysal, “Optical wireless communications: system model, capacity and coding,” in Proc. 2003 58th IEEE Vehicular Technology Conf., Oct. 2003, vol. 1, pp. 168–172.

J. Liao, S. Deng, K. Connor, Z. R. Huang, “Antenna integration with laser diodes and photodetectors for a miniaturized dual-mode wireless transceiver,” in Proc. 58th Electronic Component Technology Conf. (ECTC), Orlando, FL, May 2008, pp. 1864–1868.

M. Luby, “LT codes,” Proc. IEEE Symp. on Foundations of Computer Science (FOCS02), 2002, p. 43.

L. C. Andrews, R. L. Philips, Laser Beam Propagation Through Random Media, SPIE Press, 2005.
[Crossref]

H. Willebrand, B. S. Ghuman, Free-Space Optics: Enabling Optical Connectivity in Today’s Networks, Sams Publishing, 2002.

T. M. Cover, J. A. Thomas, Elements of Information Theory, New York: Wiley, 1991.
[Crossref]

A. Goldsmith, Wireless Communications, Cambridge: Cambridge U. Press, 2005.
[Crossref]

J. G. Proakis, Digital Communications, Boston, MA: McGraw-Hill, 2001.

J. G. Proakis, Digital Communications, Boston: McGraw-Hill, 2001.

I. B. Djordjevic, L. Xu, T. Wang, M. Cvijetic, “Large girth low-density parity-check codes for long-haul high-speed optical communications,” in Proc. OFC/NFOEC 2008, San Diego, CA, Feb. 2008, paper JWA53.

X.-Y. Hu, E. Eleftheriou, D.-M. Arnold, A. Dholakia, “Efficient implementations of the sum-product algorithm for decoding of LDPC codes,” in Proc. IEEE GLOBECOM, vol. 2, Nov. 2001, pp. 1036–1036E.

A. A. Farid, S. Hranilovic, “Upper and lower bounds on the capacity of wireless optical intensity channels,” in Proc. ISIT 2007, Nice, France, June 2007, pp. 2416–2420.

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Figures (6)

Fig. 1
Fig. 1

Block diagram of the adaptive modulation and coding terrestrial FSO system with RF feedback. LLRs, log-likelihood ratios; APP, a posteriori probability.

Fig. 2
Fig. 2

Bit error probability versus SNR per bit for MPAM on the AWGN channel.

Fig. 3
Fig. 3

Spectral efficiencies against symbol SNR for different target bit probabilities of error: (a) in the weak turbulence regime and (b) in the strong turbulence regime.

Fig. 4
Fig. 4

Nonadaptive uncoded MPAM bit error probabilities against bit SNR.

Fig. 5
Fig. 5

Nonadaptive LDPC-coded bit error probabilities versus SNR per bit for MPAM on the AWGN channel, under the weak turbulence regime and under the strong turbulence regime.

Fig. 6
Fig. 6

Spectral efficiencies against symbol SNR for adaptive LDPC-coded MPAM.

Equations (23)

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

y t = R i t x t + n t ,
p ( i t ) = 2 ( α β ) ( α + β ) 2 Γ ( α ) Γ ( β ) i t ( α + β ) 2 1 K α β ( 2 α β i t ) , i t > 0 ,
α = 1 exp [ 0.49 σ R 2 ( 1 + 1.11 σ R 12 5 ) 7 6 ] 1 ,
β = 1 exp [ 0.51 σ R 2 ( 1 + 0.69 σ R 12 5 ) 5 6 ] 1 ,
σ R 2 = 1.23 C n 2 k 7 6 L 11 6 ,
α M = 1 + M β M β σ I 2 1 , β M = M β ,
P b M 1 M log 2 M erfc ( 3 Γ 0 2 ( M 1 ) ( 2 M 1 ) ) ,
erfc ( z ) = 2 π z exp ( u 2 ) d u .
P b 0.2 exp [ 1.85 Γ 0 2 2.19 log 2 M 1 ] .
m = log 2 M = 1 2.19 log 2 ( 1 + K Γ 0 ) , K = 1.85 ln ( 5 P b ) .
m ( i t ) = 1 2.19 log 2 [ 1 + K Γ ( i t ) P ( i t ) P ] , Γ ( i t ) = i t 2 Γ 0 .
L [ P ( i t ) ] = 0 m ( i t ) p ( i t ) d i t + λ [ 0 P ( i t ) p ( i t ) d i t P ] ,
K Γ 0 P ( i t ) P = { 1 i tsh 2 1 i t 2 , i t i tsh 0 , i t < i tsh } .
i tsh ( 1 i tsh 2 1 i t 2 ) p ( i t ) d i t = K Γ 0 .
R B = 1 2.19 i tsh log 2 ( i t 2 i tsh 2 ) p ( i t ) d i t [ bits s Hz ] .
P ( i t ) P = 1 i t 2 E [ 1 i t 2 ] , E [ 1 i t 2 ] = 0 1 i t 2 p ( i t ) d i t .
R B = 1 2.19 log 2 ( 1 + K Γ 0 1 E [ 1 i t 2 ] ) .
P ( i t ) P = { 1 i t 2 E i tsh [ 1 i t 2 ] , i i tsh 0 , i < i tsh } ,
E i tsh [ 1 i t 2 ] = i tsh 1 i t 2 p ( i t ) d i t .
R B = 1 2.19 max i tsh { log 2 ( 1 + K Γ 0 1 E i tsh [ 1 i t 2 ] ) P ( i t i tsh ) } ,
P ( i t i tsh ) = i tsh p ( i t ) d ( i t ) .
P b 0.2 exp [ 1.85 G c Γ 0 2 2.19 log 2 M 1 ] .
H = [ I I I I I P S [ 1 ] P S [ 2 ] P S [ c 1 ] I P 2 S [ 1 ] P 2 S [ 2 ] P 2 S [ c 1 ] I P ( r 1 ) S [ 1 ] P ( r 1 ) S [ 2 ] P ( r 1 ) S [ c 1 ] ] ,