Abstract

We model the average channel capacity of optical wireless communication systems for cases of weak to strong turbulence channels, using the exponentiation Weibull distribution model. The joint effects of the beam wander and spread, pointing errors, atmospheric attenuation, and the spectral index of non-Kolmogorov turbulence on system performance are included. Our results show that the average capacity decreases steeply as the propagation length L changes from 0 to 200 m and decreases slowly down or tends to a stable value as the propagation length L is greater than 200 m. In the weak turbulence region, by increasing the detection aperture, we can improve the average channel capacity and the atmospheric visibility as an important issue affecting the average channel capacity. In the strong turbulence region, the increase of the radius of the detection aperture cannot reduce the effects of the atmospheric turbulence on the average channel capacity, and the effect of atmospheric visibility on the channel information capacity can be ignored. The effect of the spectral power exponent on the average channel capacity in the strong turbulence region is higher than weak turbulence region. Irrespective of the details determining the turbulent channel, we can say that pointing errors have a significant effect on the average channel capacity of optical wireless communication systems in turbulence channels.

© 2014 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2013 (2)

R. Barrios and F. Dios, “Exponentiated Weibull model for the irradiance probability density function of a laser beam propagating through atmospheric turbulence,” Opt. Laser Technol. 45, 13–20 (2013).
[CrossRef]

R. Barrios and F. Dios, “Exponentiated Weibull fading model for free-space optical links with partially coherent beams under aperture averaging,” Opt. Eng. 52, 046003 (2013).
[CrossRef]

2012 (5)

C. F. Si, Y. X. Zhang, Y. G. Wang, J. Y. Wang, and J. J. Jia, “Average capacity for non-Kolmogorov turbulent slant optical links with beam wander corrected and pointing errors,” Optik 123, 1–5 (2012).
[CrossRef]

R. Barrios and F. Dios, “Exponentiated Weibull distribution family under aperture averaging for Gaussian beam waves,” Opt. Express 20, 13055 (2012).
[CrossRef]

X. Yi, Z. J. Liu, and P. Yue, “Average BER of free-space optical systems in turbulent atmosphere with exponentiated Weibull distribution,” Opt. Lett. 37, 5142–5144 (2012).
[CrossRef]

N. A. Mohammed, A. S. El-Wakeel, and M. H. Aly, “Performance evaluation of FSO link under NRZ-RZ line codes, different weather conditions and receiver types in the presence of pointing errors,” J. Electr. Electron. Eng. 6, 28–35 (2012).

P. Deng, X. Yuan, and D. Huang, “Scintillation of a laser beam propagation through non-Kolmogorov strong turbulence,” Opt. Commun. 285, 880–887 (2012).
[CrossRef]

2011 (2)

2010 (3)

D. T. Wayne, R. L. Phillips, and L. C. Andrews, “Comparing the log-normal and gamma-gamma model to experimental probability density functions of aperture averaging data,” Proc. SPIE 7814, 78140K (2010).
[CrossRef]

B. Epple, “Simplified channel model for simulation of free-space optical communications,” J. Opt. Commun. Netw. 2, 293–304 (2010).
[CrossRef]

C. Liu, Y. Yao, Y. Sun, and X. Zhao, “Average capacity for heterodyne FSO communication systems over gamma-gamma turbulence channels with pointing errors,” Electron. Lett. 46, 12–13 (2010).

2009 (1)

2008 (2)

A. Zilberman, E. Golbraikh, and N. Koperka, “Propagation of electromagnetic waves in Kolmogorov and non-Kolmogorov atmospheric turbulence: Three-layer altitude model,” Appl. Opt. 47, 6385–6391 (2008).
[CrossRef]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47, 026003 (2008).
[CrossRef]

2007 (3)

2003 (1)

2002 (1)

X. Zhu and J. M. Kahn, “Free-space optical communications through atmospheric turbulence channels,” IEEE Trans. Commun. 50, 1293–1300 (2002).
[CrossRef]

2001 (1)

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

2000 (1)

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, “Aperture averaging of optical scintillations: power fluctuations and temporal spectrum,” Waves Random Media 10, 53–70 (2000).
[CrossRef]

1995 (1)

R. R. Beland, “Some aspects of propagation through weak isotropic non-Kolmogorov turbulence,” Proc. SPIE 6, 6–16 (1995).
[CrossRef]

1977 (1)

1967 (1)

Al-Habash, M. A.

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

Aly, M. H.

N. A. Mohammed, A. S. El-Wakeel, and M. H. Aly, “Performance evaluation of FSO link under NRZ-RZ line codes, different weather conditions and receiver types in the presence of pointing errors,” J. Electr. Electron. Eng. 6, 28–35 (2012).

Andrews, L. C.

D. T. Wayne, R. L. Phillips, and L. C. Andrews, “Comparing the log-normal and gamma-gamma model to experimental probability density functions of aperture averaging data,” Proc. SPIE 7814, 78140K (2010).
[CrossRef]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47, 026003 (2008).
[CrossRef]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Scintillation index of optical plane wave propagating through non-Kolmogorov moderate-strong turbulence,” Proc. SPIE 6747, 67470B (2007).
[CrossRef]

F. S. Vetelino, C. Young, and L. C. Andrews, “Fade statistics and aperture averaging for Gaussian beam waves in moderate-to-strong turbulence,” Appl. Opt. 46, 3780–3790 (2007).
[CrossRef]

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

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, “Aperture averaging of optical scintillations: power fluctuations and temporal spectrum,” Waves Random Media 10, 53–70 (2000).
[CrossRef]

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005).

Arnon, S.

Bai, X. Z.

Barrios, R.

R. Barrios and F. Dios, “Exponentiated Weibull model for the irradiance probability density function of a laser beam propagating through atmospheric turbulence,” Opt. Laser Technol. 45, 13–20 (2013).
[CrossRef]

R. Barrios and F. Dios, “Exponentiated Weibull fading model for free-space optical links with partially coherent beams under aperture averaging,” Opt. Eng. 52, 046003 (2013).
[CrossRef]

R. Barrios and F. Dios, “Exponentiated Weibull distribution family under aperture averaging for Gaussian beam waves,” Opt. Express 20, 13055 (2012).
[CrossRef]

Baykal, Y.

Beland, R. R.

R. R. Beland, “Some aspects of propagation through weak isotropic non-Kolmogorov turbulence,” Proc. SPIE 6, 6–16 (1995).
[CrossRef]

Brychkov, Y. A.

A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and Series, Vol. 3: More Special Functions (Gordon and Breach, 1990).

Cao, L.

Cui, L.

Deng, P.

P. Deng, X. Yuan, and D. Huang, “Scintillation of a laser beam propagation through non-Kolmogorov strong turbulence,” Opt. Commun. 285, 880–887 (2012).
[CrossRef]

Dios, F.

R. Barrios and F. Dios, “Exponentiated Weibull fading model for free-space optical links with partially coherent beams under aperture averaging,” Opt. Eng. 52, 046003 (2013).
[CrossRef]

R. Barrios and F. Dios, “Exponentiated Weibull model for the irradiance probability density function of a laser beam propagating through atmospheric turbulence,” Opt. Laser Technol. 45, 13–20 (2013).
[CrossRef]

R. Barrios and F. Dios, “Exponentiated Weibull distribution family under aperture averaging for Gaussian beam waves,” Opt. Express 20, 13055 (2012).
[CrossRef]

El-Wakeel, A. S.

N. A. Mohammed, A. S. El-Wakeel, and M. H. Aly, “Performance evaluation of FSO link under NRZ-RZ line codes, different weather conditions and receiver types in the presence of pointing errors,” J. Electr. Electron. Eng. 6, 28–35 (2012).

Epple, B.

Farid, A.

Ferrero, V.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47, 026003 (2008).
[CrossRef]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Scintillation index of optical plane wave propagating through non-Kolmogorov moderate-strong turbulence,” Proc. SPIE 6747, 67470B (2007).
[CrossRef]

Fried, D. L.

Gerçekcioglu, H.

Golbraikh, E.

Gradshteyn, I.

I. Gradshteyn and I. Ryzhik, Table of Integrals, Series, and Products, A. Jeffrey and D. Zwillinger, eds., 7th ed. (Academic, 2007).

Hopen, C. Y.

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, “Aperture averaging of optical scintillations: power fluctuations and temporal spectrum,” Waves Random Media 10, 53–70 (2000).
[CrossRef]

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005).

Hranilovic, S.

Huang, D.

P. Deng, X. Yuan, and D. Huang, “Scintillation of a laser beam propagation through non-Kolmogorov strong turbulence,” Opt. Commun. 285, 880–887 (2012).
[CrossRef]

Jia, J. J.

C. F. Si, Y. X. Zhang, Y. G. Wang, J. Y. Wang, and J. J. Jia, “Average capacity for non-Kolmogorov turbulent slant optical links with beam wander corrected and pointing errors,” Optik 123, 1–5 (2012).
[CrossRef]

Kahn, J. M.

X. Zhu and J. M. Kahn, “Free-space optical communications through atmospheric turbulence channels,” IEEE Trans. Commun. 50, 1293–1300 (2002).
[CrossRef]

Karagiannidis, G. K.

Koperka, N.

Liu, C.

C. Liu, Y. Yao, Y. Sun, and X. Zhao, “Average capacity for heterodyne FSO communication systems over gamma-gamma turbulence channels with pointing errors,” Electron. Lett. 46, 12–13 (2010).

Liu, Z. J.

Marichev, O. I.

A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and Series, Vol. 3: More Special Functions (Gordon and Breach, 1990).

Mironov, V. L.

Mohammed, N. A.

N. A. Mohammed, A. S. El-Wakeel, and M. H. Aly, “Performance evaluation of FSO link under NRZ-RZ line codes, different weather conditions and receiver types in the presence of pointing errors,” J. Electr. Electron. Eng. 6, 28–35 (2012).

Nosov, V. V.

Phillips, R. L.

D. T. Wayne, R. L. Phillips, and L. C. Andrews, “Comparing the log-normal and gamma-gamma model to experimental probability density functions of aperture averaging data,” Proc. SPIE 7814, 78140K (2010).
[CrossRef]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47, 026003 (2008).
[CrossRef]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Scintillation index of optical plane wave propagating through non-Kolmogorov moderate-strong turbulence,” Proc. SPIE 6747, 67470B (2007).
[CrossRef]

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

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, “Aperture averaging of optical scintillations: power fluctuations and temporal spectrum,” Waves Random Media 10, 53–70 (2000).
[CrossRef]

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005).

Prudnikov, A. P.

A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and Series, Vol. 3: More Special Functions (Gordon and Breach, 1990).

Ryzhik, I.

I. Gradshteyn and I. Ryzhik, Table of Integrals, Series, and Products, A. Jeffrey and D. Zwillinger, eds., 7th ed. (Academic, 2007).

Sandalidis, H. G.

Si, C. F.

C. F. Si, Y. X. Zhang, Y. G. Wang, J. Y. Wang, and J. J. Jia, “Average capacity for non-Kolmogorov turbulent slant optical links with beam wander corrected and pointing errors,” Optik 123, 1–5 (2012).
[CrossRef]

Sun, Y.

C. Liu, Y. Yao, Y. Sun, and X. Zhao, “Average capacity for heterodyne FSO communication systems over gamma-gamma turbulence channels with pointing errors,” Electron. Lett. 46, 12–13 (2010).

Toselli, I.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47, 026003 (2008).
[CrossRef]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Scintillation index of optical plane wave propagating through non-Kolmogorov moderate-strong turbulence,” Proc. SPIE 6747, 67470B (2007).
[CrossRef]

Tsiftsis, T. A.

Vetelino, F. S.

Wang, J. Y.

C. F. Si, Y. X. Zhang, Y. G. Wang, J. Y. Wang, and J. J. Jia, “Average capacity for non-Kolmogorov turbulent slant optical links with beam wander corrected and pointing errors,” Optik 123, 1–5 (2012).
[CrossRef]

Wang, Y. G.

C. F. Si, Y. X. Zhang, Y. G. Wang, J. Y. Wang, and J. J. Jia, “Average capacity for non-Kolmogorov turbulent slant optical links with beam wander corrected and pointing errors,” Optik 123, 1–5 (2012).
[CrossRef]

Wayne, D. T.

D. T. Wayne, R. L. Phillips, and L. C. Andrews, “Comparing the log-normal and gamma-gamma model to experimental probability density functions of aperture averaging data,” Proc. SPIE 7814, 78140K (2010).
[CrossRef]

Xue, B.

Xue, W.

Yao, Y.

C. Liu, Y. Yao, Y. Sun, and X. Zhao, “Average capacity for heterodyne FSO communication systems over gamma-gamma turbulence channels with pointing errors,” Electron. Lett. 46, 12–13 (2010).

Yi, X.

Young, C.

Yuan, X.

P. Deng, X. Yuan, and D. Huang, “Scintillation of a laser beam propagation through non-Kolmogorov strong turbulence,” Opt. Commun. 285, 880–887 (2012).
[CrossRef]

Yue, P.

Zhang, Y. X.

C. F. Si, Y. X. Zhang, Y. G. Wang, J. Y. Wang, and J. J. Jia, “Average capacity for non-Kolmogorov turbulent slant optical links with beam wander corrected and pointing errors,” Optik 123, 1–5 (2012).
[CrossRef]

Zhao, X.

C. Liu, Y. Yao, Y. Sun, and X. Zhao, “Average capacity for heterodyne FSO communication systems over gamma-gamma turbulence channels with pointing errors,” Electron. Lett. 46, 12–13 (2010).

Zheng, S.

Zhou, F. G.

Zhu, X.

X. Zhu and J. M. Kahn, “Free-space optical communications through atmospheric turbulence channels,” IEEE Trans. Commun. 50, 1293–1300 (2002).
[CrossRef]

Zilberman, A.

Appl. Opt. (2)

Electron. Lett. (1)

C. Liu, Y. Yao, Y. Sun, and X. Zhao, “Average capacity for heterodyne FSO communication systems over gamma-gamma turbulence channels with pointing errors,” Electron. Lett. 46, 12–13 (2010).

IEEE Trans. Commun. (1)

X. Zhu and J. M. Kahn, “Free-space optical communications through atmospheric turbulence channels,” IEEE Trans. Commun. 50, 1293–1300 (2002).
[CrossRef]

J. Electr. Electron. Eng. (1)

N. A. Mohammed, A. S. El-Wakeel, and M. H. Aly, “Performance evaluation of FSO link under NRZ-RZ line codes, different weather conditions and receiver types in the presence of pointing errors,” J. Electr. Electron. Eng. 6, 28–35 (2012).

J. Lightwave Technol. (2)

J. Opt. Commun. Netw. (1)

J. Opt. Soc. Am. (2)

Opt. Commun. (1)

P. Deng, X. Yuan, and D. Huang, “Scintillation of a laser beam propagation through non-Kolmogorov strong turbulence,” Opt. Commun. 285, 880–887 (2012).
[CrossRef]

Opt. Eng. (3)

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47, 026003 (2008).
[CrossRef]

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

R. Barrios and F. Dios, “Exponentiated Weibull fading model for free-space optical links with partially coherent beams under aperture averaging,” Opt. Eng. 52, 046003 (2013).
[CrossRef]

Opt. Express (2)

Opt. Laser Technol. (1)

R. Barrios and F. Dios, “Exponentiated Weibull model for the irradiance probability density function of a laser beam propagating through atmospheric turbulence,” Opt. Laser Technol. 45, 13–20 (2013).
[CrossRef]

Opt. Lett. (3)

Optik (1)

C. F. Si, Y. X. Zhang, Y. G. Wang, J. Y. Wang, and J. J. Jia, “Average capacity for non-Kolmogorov turbulent slant optical links with beam wander corrected and pointing errors,” Optik 123, 1–5 (2012).
[CrossRef]

Proc. SPIE (3)

R. R. Beland, “Some aspects of propagation through weak isotropic non-Kolmogorov turbulence,” Proc. SPIE 6, 6–16 (1995).
[CrossRef]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Scintillation index of optical plane wave propagating through non-Kolmogorov moderate-strong turbulence,” Proc. SPIE 6747, 67470B (2007).
[CrossRef]

D. T. Wayne, R. L. Phillips, and L. C. Andrews, “Comparing the log-normal and gamma-gamma model to experimental probability density functions of aperture averaging data,” Proc. SPIE 7814, 78140K (2010).
[CrossRef]

Waves Random Media (1)

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, “Aperture averaging of optical scintillations: power fluctuations and temporal spectrum,” Waves Random Media 10, 53–70 (2000).
[CrossRef]

Other (4)

Wolfram, The Wolfram functions site http://functions.wolfram.com (2001).

I. Gradshteyn and I. Ryzhik, Table of Integrals, Series, and Products, A. Jeffrey and D. Zwillinger, eds., 7th ed. (Academic, 2007).

A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and Series, Vol. 3: More Special Functions (Gordon and Breach, 1990).

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005).

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Figures (5)

Fig. 1.
Fig. 1.

Channel capacity, C/B, of optical communication link as a function of propagation distance L and Cn2.

Fig. 2.
Fig. 2.

Channel capacity, C/B, of optical communication link as a function of propagation distance L and α.

Fig. 3.
Fig. 3.

Channel capacity, C/B, of optical communication link as a function of propagation distance L and D.

Fig. 4.
Fig. 4.

Channel capacity, C/B, of optical communication link as a function of propagation distance L and the atmospheric attenuation hl.

Fig. 5.
Fig. 5.

Channel capacity, C/B, of optical communication link as a function of propagation distance L and the normalized jitter xs/D.

Equations (32)

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

y=hx+n,
C=B0log2(1+h2/N0)fh(h)dh,
hl=exp(βlL),
βl=13V(λ×109550)q(V)[dB/km],
q(V)={1.6forV>50km1.3for6<V<50km0.16V+0.34for1<V<6kmV0.5for0.5<V<1km0forV<0.5km.
fha(ha)=abc(ha/c)b1exp[(ha/c)b]×{1exp[(ha/c)b]}a1,ha>0,
a7.220σI2/3Γ(2.487σI2/60.104)
b1.012(aσI2)13/25+0.142
c=1aΓ(1+1/b)g1(a,b)
g1(a,b)=j=0(1)jΓ(a)j!(j+1)1+1/bΓ(aj),
ρ0={23απ2Γ(1α/2)k2A(α)C˜n2LΓ(α/2)}1/(α2),
A(α)=Γ(α1)sin((α3)π/2)/4π2.
σI2=σ˜R2[(1+2Θ)2+4Λ2](α2)/43(α2)/2sin(απ/4)sin(αφ1/2+φ2)+N(α)σ˜R2Λeα21×{(σpewb)2+(rσpewb)2},
σ˜R2=6.5π2A(α)Γ(1α/2)sin(απ/4)/α×1.23C˜n2k3α/2Lα/2.
wb=w0[1LR0+(1+2w02ρ02)(λLπw02)2]1/2,
φ1=arctan[(1+2Θ)2Λ],φ2=arctan[2Λ(1+2Θ)]
N(α)=4πα21.231αΓ(2α2)A(α)R(α)
R(α)=6.5π2A(α)Γ(1α/2)sin(απ/4)/α.
σpe2{(λL2w0)2(w0ρ0)α2,w0ρ01(λL2w0)2(w0ρ0)α4,w0ρ01.
σI2=exp(0.49σ˜R2[1+fX(α)σ˜R4α2]3α2+0.51σ˜R2(1+fY(α)σ˜R4α2)α22)1+N(α)σ˜R2Λeα21×{(σpewb)2+((rσpe)wb)2},
fX(α)=(1.016B(α)Z(α))(2/(α6))
B(α)=01ξα4(1+2αα1ξ)(α6α2)dξ
Z(α)=64.2α2(A(α)R(α))2(α4)α2Γ(6αα2)×[Γ(1α2)/Γ(α2)]α6α22(α6)(7α)α2
fhp(hp)=γ2A0γ2hpγ21,0hpA0,
fh(h)=dhafh/ha(h/ha)fha(ha),
fh/ha(h/ha)=1hahlfhp(hhahl)=γ2A0γ2hahl(hhahl)γ21,0hA0hahl.
fh(h)=γ2A0hl(hA0hl)γ21hA0hldha(ha)γ2abc(hac)b1×exp[(hac)b]{1exp[(hac)b]}a1.
fh(h)=aγ2cA0hl(hA0hlc)γ21j=0(1)jΓ(a)j!Γ(aj)(1+j)1γ2bΓ[(1γ2b),(1+j)(hA0hlc)b].
C=aBγ2(cA0hl)γ2j=0(1)jΓ(a)j!Γ(aj)(1+j)1γ2b0dhlog2(1+h2N0)(h)γ21Γ[(1γ2b),(1+j)(hcA0hl)b]
On expressingln(1+h2N0)=G2,21,2(h2N|1,11,0),Γ(a,x)=G1,22,0(x|10,a),
C=aBγ2(cA0hl)γ2ln2j=0(1)jΓ(a)j!Γ(aj)(1+j)1γ2b0dh(h)γ21G2,21,2(h2N0|1,11,0)G1,22,0[(1+j)(hcA0hl)b|10,(1γ2/b)].
C=aBγ2K2ln2(N0cA0hl)γ2(2π)[(1γ2b)(1O)+2(1K)]j=0(1)jΓ(a)j!Γ(aj)(1+j)1γ2bGK+2O,2(K+O)2(K+O),O[(T/K)K|Δ(K,1),Δ(O,γ22),Δ(O,1γ22)Δ(K,0),Δ(K,1γ2b),Δ(O,γ22),Δ(O,γ22)],

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