Abstract

This paper elaborates on the end-to-end capacity of dual-hop free-space optical (FSO) communication systems employing amplify-and-forward (AF) relaying, assuming channel state information is only known at the receiving terminals. The relay is assumed to either possess perfect channel state information or have a fixed gain. The performance of the considered system is affected by the combined effects of atmospheric turbulence-induced fading, pointing errors (i.e., misalignment fading), and path loss. Atmospheric turbulence conditions are modeled using the gamma–gamma distribution. For the system under consideration, accurate analytical approximations as well as upper bounds to the ergodic capacity are derived. In addition, bound approximations in the high signal-to-noise ratio regime are deduced that provide valuable insights into the impact of model parameters on the capacity of AF FSO dual-hop relaying systems. Numerically evaluated and computer simulation results are further provided to demonstrate the validity of the proposed mathematical analysis.

© 2013 Optical Society of America

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2013 (1)

2012 (6)

A. Stassinakis, H. Nistazakis, and G. Tombras, “Comparative performance study of one or multiple receivers schemes for FSO links over gamma–gamma turbulence channels,” J. Mod. Opt., vol.  59, pp. 1023–1031, 2012.
[CrossRef]

M. Safari, M. M. Rad, and M. Uysal, “Multi-hop relaying over the atmospheric Poisson channel: Outage analysis and optimization,” IEEE Trans. Commun., vol.  60, no. 3, pp. 817–829, Mar. 2012.
[CrossRef]

M. A. Kashani, M. M. Rad, M. Safari, and M. Uysal, “All-optical amplify-and-forward relaying system for atmospheric channels,” IEEE Commun. Lett., vol.  16, no. 10, pp. 1684–1687, Oct. 2012.
[CrossRef]

E. Bayaki, D. S. Michalopoulos, and R. Schober, “EDFA-based all-optical relaying in free-space optical systems,” IEEE Trans. Commun., vol.  60, no. 12, pp. 3797–3807, Dec. 2012.
[CrossRef]

K. Peppas, F. Lazarakis, A. Alexandridis, and K. Dangakis, “Simple, accurate formula for the average bit error probability of multiple-input multiple-output free-space optical links over negative exponential turbulence channels,” Opt. Lett., vol.  37, pp. 3243–3245, Aug. 2012.
[CrossRef]

K. Peppas, A. Stassinakis, G. Topalis, H. Nistazakis, and G. Tombras, “Average capacity of optical wireless communication systems over I-K atmospheric turbulence channels,” J. Opt. Commun. Netw., vol.  4, pp. 1026–1032, 2012.
[CrossRef]

2011 (3)

2010 (5)

H. Henniger and O. Wilfert, “An introduction to free-space optical communications,” Radioengineering, vol.  19, no. 2, pp. 203–212, 2010.

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., vol.  46, no. 12, pp. 851–853, June 2010.
[CrossRef]

S. Jin, M. R. McKay, C. Zhong, and K.-K. Wong, “Ergodic capacity analysis of amplify-and-forward MIMO dual-hop systems,” IEEE Trans. Inf. Theory, vol.  56, no. 5, pp. 2204–2224, May 2010.
[CrossRef]

C. Liu, Y. Yao, Y. Sun, and X. Zhao, “Analysis of average capacity for free-space optical links with pointing errors over gamma–gamma turbulence channels,” Chin. Opt. Lett., vol.  8, no. 6, pp. 537–540, June 2010.
[CrossRef]

C. Datsikas, K. Peppas, N. Sagias, and G. Tombras, “Serial free-space optical relaying communications over gamma–gamma atmospheric turbulence channels,” J. Opt. Commun. Netw., vol.  2, pp. 576–586, Aug. 2010.
[CrossRef]

2009 (7)

D. K. Borah and D. G. Voelz, “Pointing error effects on free-space optical communication links in the presence of atmospheric turbulence,” J. Lightwave Technol., vol.  27, no. 18, pp. 3965–3973, Sept. 2009.
[CrossRef]

H. G. Sandalidis, T. A. Tsiftsis, and G. K. Karagiannidis, “Optical wireless communications with heterodyne detection over turbulence channels with pointing errors,” J. Lightwave Technol., vol.  27, no. 20, pp. 4440–4445, Oct. 2009.
[CrossRef]

M. Kamiri and N. Nasiri-Kerari, “BER analysis of cooperative systems in free-space optical networks,” J. Lightwave Technol., vol.  27, no. 24, pp. 5639–5647, Dec. 2009.
[CrossRef]

G. Farhadi and N. C. Beaulieu, “On the ergodic capacity of multi-hop wireless relaying systems,” IEEE Trans. Wireless Commun., vol.  8, no. 5, pp. 2286–2291, May 2009.
[CrossRef]

Y. Han, S. H. Ting, C. K. Ho, and W. H. Chin, “Performance bounds for two-way amplify-and-forward relaying,” IEEE Trans. Wireless Commun., vol.  8, no. 1, pp. 432–439, Jan. 2009.
[CrossRef]

E. Bayaki, R. Schober, and R. Mallik, “Performance analysis of MIMO free-space optical systems in gamma–gamma fading,” IEEE Trans. Commun., vol.  57, no. 11, pp. 3415–3424, Nov. 2009.
[CrossRef]

A. Garcia-Zambrana, C. Castillo-Vazquez, B. Castillo-Vazquez, and A. Hiniesta-Gomez, “Selection transmit diversity for FSO links over strong atmospheric turbulence channels,” IEEE Photon. Technol. Lett., vol.  21, no. 14, pp. 1017–1019, 2009.
[CrossRef]

2008 (2)

M. Safari and M. Uysal, “Do we really need space-time block coding for free-space optical communication with direct detection?” IEEE Trans. Wireless Commun., vol.  7, no. 11, pp. 4445–4448, Nov. 2008.
[CrossRef]

M. Safari and M. Uysal, “Relay-assisted free-space optical communication,” IEEE Trans. Wireless Commun., vol.  7, no. 12, pp. 5441–5449, Dec. 2008.
[CrossRef]

2007 (3)

S. M. Navidpour, M. Uysal, and M. Kavehrad, “Performance of free-space optical transmission with spatial diversity,” IEEE Trans. Wireless Commun., vol.  6, no. 8, pp. 2813–2819, Aug. 2007.
[CrossRef]

A. Farid and S. Hranilovic, “Outage capacity optimization for free-space optical links with pointing errors,” J. Lightwave Technol., vol.  25, no. 7, pp. 1702–1710, 2007.
[CrossRef]

M. D. Yacoub, “The α−μ distribution: A physical fading model for the Stacy distribution,” IEEE Trans. Veh. Technol., vol.  56, no. 1, pp. 27–34, Jan. 2007.

2005 (2)

J. Perez, J. Ibanez, L. Vielva, and I. Santamaria, “Closed-form approximation for the outage capacity of orthogonal STBC,” IEEE Commun. Lett., vol.  9, no. 11, pp. 961–963, Nov. 2005.
[CrossRef]

A. K. Majumdar, “Free-space laser communication performance in the atmospheric channel,” J. Opt. Fiber Commun. Rep., vol.  2, pp. 345–396, 2005.
[CrossRef]

2004 (2)

D. Kedar and S. Arnon, “Urban optical wireless communication networks: The main challenges and possible solutions,” IEEE Commun. Mag., vol.  42, no. 5, pp. S2–S7, May 2004.
[CrossRef]

J. N. Laneman, D. N. C. Tse, and G. W. Wornell, “Cooperative diversity in wireless networks: Efficient protocols and outage behavior,” IEEE Trans. Inf. Theory, vol.  50, no. 12, pp. 3062–3080, Dec. 2004.
[CrossRef]

2003 (3)

2002 (1)

X. Zhu and J. M. Kahn, “Free-space optical communications through atmospheric turbulence channels,” IEEE Trans. Commun., vol.  50, no. 8, pp. 1293–1300, 2002.
[CrossRef]

2001 (3)

I. I. Kim, B. McArthur, and E. Korevaar, “Comparison of laser beam propagation at 785 nm and 1550 nm in fog and haze for optical wireless communications,” Proc. SPIE, vol.  4214, pp. 26–37, Feb. 2001.
[CrossRef]

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance PDF of a laser beam propagating through turbulent media,” Opt. Eng., vol.  40, no. 8, pp. 1554–1562, 2001.
[CrossRef]

S. Shamai and S. Verdú, “The impact of frequency-flat fading on the spectral efficiency of CDMA,” IEEE Trans. Inf. Theory, vol.  47, no. 4, pp. 1302–1327, May 2001.
[CrossRef]

1985 (1)

J. D. Barry and G. S. Mecherle, “Beam pointing error as a significant design parameter for satellite-borne, free-space optical communication systems,” Opt. Eng., vol.  24, no. 6, pp. 241049–241054, Dec. 1985.
[CrossRef]

1981 (1)

N. Cressie, A. S. Davis, J. L. Folks, and G. E. Policello, “The moment-generating function and negative integer moments,” Am. Stat., vol.  35, pp. 148–150, Aug. 1981.

1977 (1)

B. D. Carter and M. D. Springer, “The distribution of products, quotients and powers of independent H-function variates,” SIAM J. Appl. Math., vol.  33, no. 4, pp. 542–558, 1977.
[CrossRef]

Abou-Rjeily, C.

Abramovitz, M.

M. Abramovitz and I. Stegun, Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables, New York: Dover, 1964.

Akella, J.

J. Akella, M. Yuksel, and S. Kalyanaraman, “Error analysis of multihop free-space optical communication,” in Proc. IEEE Int. Conf. on Communications (ICC), Seoul, South Korea, May 2005.

Alexandridis, A.

Al-Habash, M. A.

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance PDF of a laser beam propagating through turbulent media,” Opt. Eng., vol.  40, no. 8, pp. 1554–1562, 2001.
[CrossRef]

Alouini, M. S.

M. O. Hasna and M. S. Alouini, “End-to-end performance of transmission systems with relays over Rayleigh fading channels,” IEEE Trans. Wireless Commun., vol.  2, no. 6, pp. 1126–1131, Nov. 2003.
[CrossRef]

Alouini, M.-S.

F. Yilmaz and M.-S. Alouini, “Product of the powers of generalized Nakagami-m variates and performance of cascaded fading channels,” in IEEE Global Telecommunications Conf., 2009, pp. 1–8.

Andrews, L.

L. Andrews and R. L. Philips, Laser Beam Propagation Through Random Media. SPIE, 2005.

L. Andrews, R. L. Philips, and C. Y. Hopen, Laser Beam Scintillation With Applications. SPIE, 2001.

Andrews, L. C.

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance PDF of a laser beam propagating through turbulent media,” Opt. Eng., vol.  40, no. 8, pp. 1554–1562, 2001.
[CrossRef]

Arnon, S.

D. Kedar and S. Arnon, “Urban optical wireless communication networks: The main challenges and possible solutions,” IEEE Commun. Mag., vol.  42, no. 5, pp. S2–S7, May 2004.
[CrossRef]

S. Arnon, “Effects of atmospheric turbulence and building sway on optical wireless communication systems,” Opt. Lett., vol.  28, no. 2, pp. 129–131, Jan. 2003.
[CrossRef]

D. Kedar and S. Arnon, “Optical wireless communication through fog in the presence of pointing errors,” Appl. Opt., vol.  42, no. 24, pp. 4946–4954, Aug. 2003.
[CrossRef]

S. Arnon, “Optical wireless communications,” in Encyclopedia of Optical Engineering (EOE), R. G. Driggers, Ed. New York: Marcel Dekker, 2003, pp. 1866–1886, Invited.

Barry, J. D.

J. D. Barry and G. S. Mecherle, “Beam pointing error as a significant design parameter for satellite-borne, free-space optical communication systems,” Opt. Eng., vol.  24, no. 6, pp. 241049–241054, Dec. 1985.
[CrossRef]

Bayaki, E.

E. Bayaki, D. S. Michalopoulos, and R. Schober, “EDFA-based all-optical relaying in free-space optical systems,” IEEE Trans. Commun., vol.  60, no. 12, pp. 3797–3807, Dec. 2012.
[CrossRef]

E. Bayaki, R. Schober, and R. Mallik, “Performance analysis of MIMO free-space optical systems in gamma–gamma fading,” IEEE Trans. Commun., vol.  57, no. 11, pp. 3415–3424, Nov. 2009.
[CrossRef]

Beaulieu, N. C.

G. Farhadi and N. C. Beaulieu, “On the ergodic capacity of multi-hop wireless relaying systems,” IEEE Trans. Wireless Commun., vol.  8, no. 5, pp. 2286–2291, May 2009.
[CrossRef]

Borah, D. K.

Brychkov, Y. A.

A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and Series Volume 3: More Special Functions, 1st ed. Gordon and Breach Science Publishers, 1986.

Carter, B. D.

B. D. Carter and M. D. Springer, “The distribution of products, quotients and powers of independent H-function variates,” SIAM J. Appl. Math., vol.  33, no. 4, pp. 542–558, 1977.
[CrossRef]

Castillo-Vazquez, B.

A. Garcia-Zambrana, C. Castillo-Vazquez, B. Castillo-Vazquez, and A. Hiniesta-Gomez, “Selection transmit diversity for FSO links over strong atmospheric turbulence channels,” IEEE Photon. Technol. Lett., vol.  21, no. 14, pp. 1017–1019, 2009.
[CrossRef]

Castillo-Vazquez, C.

A. Garcia-Zambrana, C. Castillo-Vazquez, B. Castillo-Vazquez, and A. Hiniesta-Gomez, “Selection transmit diversity for FSO links over strong atmospheric turbulence channels,” IEEE Photon. Technol. Lett., vol.  21, no. 14, pp. 1017–1019, 2009.
[CrossRef]

Chatzidiamantis, N. D.

Chin, W. H.

Y. Han, S. H. Ting, C. K. Ho, and W. H. Chin, “Performance bounds for two-way amplify-and-forward relaying,” IEEE Trans. Wireless Commun., vol.  8, no. 1, pp. 432–439, Jan. 2009.
[CrossRef]

Cressie, N.

N. Cressie, A. S. Davis, J. L. Folks, and G. E. Policello, “The moment-generating function and negative integer moments,” Am. Stat., vol.  35, pp. 148–150, Aug. 1981.

Dangakis, K.

Datsikas, C.

Davis, A. S.

N. Cressie, A. S. Davis, J. L. Folks, and G. E. Policello, “The moment-generating function and negative integer moments,” Am. Stat., vol.  35, pp. 148–150, Aug. 1981.

Farhadi, G.

G. Farhadi and N. C. Beaulieu, “On the ergodic capacity of multi-hop wireless relaying systems,” IEEE Trans. Wireless Commun., vol.  8, no. 5, pp. 2286–2291, May 2009.
[CrossRef]

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Folks, J. L.

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C. Liu, Y. Yao, Y. Sun, and X. Zhao, “Analysis of average capacity for free-space optical links with pointing errors over gamma–gamma turbulence channels,” Chin. Opt. Lett., vol.  8, no. 6, pp. 537–540, June 2010.
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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., vol.  46, no. 12, pp. 851–853, June 2010.
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C. Liu, Y. Yao, Y. Sun, and X. Zhao, “Analysis of average capacity for free-space optical links with pointing errors over gamma–gamma turbulence channels,” Chin. Opt. Lett., vol.  8, no. 6, pp. 537–540, June 2010.
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Zhong, C.

S. Jin, M. R. McKay, C. Zhong, and K.-K. Wong, “Ergodic capacity analysis of amplify-and-forward MIMO dual-hop systems,” IEEE Trans. Inf. Theory, vol.  56, no. 5, pp. 2204–2224, May 2010.
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N. Cressie, A. S. Davis, J. L. Folks, and G. E. Policello, “The moment-generating function and negative integer moments,” Am. Stat., vol.  35, pp. 148–150, Aug. 1981.

Appl. Opt. (1)

Chin. Opt. Lett. (1)

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., vol.  46, no. 12, pp. 851–853, June 2010.
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IEEE Commun. Lett. (2)

J. Perez, J. Ibanez, L. Vielva, and I. Santamaria, “Closed-form approximation for the outage capacity of orthogonal STBC,” IEEE Commun. Lett., vol.  9, no. 11, pp. 961–963, Nov. 2005.
[CrossRef]

M. A. Kashani, M. M. Rad, M. Safari, and M. Uysal, “All-optical amplify-and-forward relaying system for atmospheric channels,” IEEE Commun. Lett., vol.  16, no. 10, pp. 1684–1687, Oct. 2012.
[CrossRef]

IEEE Commun. Mag. (1)

D. Kedar and S. Arnon, “Urban optical wireless communication networks: The main challenges and possible solutions,” IEEE Commun. Mag., vol.  42, no. 5, pp. S2–S7, May 2004.
[CrossRef]

IEEE Photon. Technol. Lett. (1)

A. Garcia-Zambrana, C. Castillo-Vazquez, B. Castillo-Vazquez, and A. Hiniesta-Gomez, “Selection transmit diversity for FSO links over strong atmospheric turbulence channels,” IEEE Photon. Technol. Lett., vol.  21, no. 14, pp. 1017–1019, 2009.
[CrossRef]

IEEE Trans. Commun. (5)

E. Bayaki, R. Schober, and R. Mallik, “Performance analysis of MIMO free-space optical systems in gamma–gamma fading,” IEEE Trans. Commun., vol.  57, no. 11, pp. 3415–3424, Nov. 2009.
[CrossRef]

M. Safari, M. M. Rad, and M. Uysal, “Multi-hop relaying over the atmospheric Poisson channel: Outage analysis and optimization,” IEEE Trans. Commun., vol.  60, no. 3, pp. 817–829, Mar. 2012.
[CrossRef]

H. G. Sandalidis, “Coded free-space optical links over strong turbulence and misalignment fading channels,” IEEE Trans. Commun., vol.  59, no. 3, pp. 669–674, Mar. 2011.
[CrossRef]

E. Bayaki, D. S. Michalopoulos, and R. Schober, “EDFA-based all-optical relaying in free-space optical systems,” IEEE Trans. Commun., vol.  60, no. 12, pp. 3797–3807, Dec. 2012.
[CrossRef]

X. Zhu and J. M. Kahn, “Free-space optical communications through atmospheric turbulence channels,” IEEE Trans. Commun., vol.  50, no. 8, pp. 1293–1300, 2002.
[CrossRef]

IEEE Trans. Inf. Theory (3)

S. Jin, M. R. McKay, C. Zhong, and K.-K. Wong, “Ergodic capacity analysis of amplify-and-forward MIMO dual-hop systems,” IEEE Trans. Inf. Theory, vol.  56, no. 5, pp. 2204–2224, May 2010.
[CrossRef]

S. Shamai and S. Verdú, “The impact of frequency-flat fading on the spectral efficiency of CDMA,” IEEE Trans. Inf. Theory, vol.  47, no. 4, pp. 1302–1327, May 2001.
[CrossRef]

J. N. Laneman, D. N. C. Tse, and G. W. Wornell, “Cooperative diversity in wireless networks: Efficient protocols and outage behavior,” IEEE Trans. Inf. Theory, vol.  50, no. 12, pp. 3062–3080, Dec. 2004.
[CrossRef]

IEEE Trans. Veh. Technol. (1)

M. D. Yacoub, “The α−μ distribution: A physical fading model for the Stacy distribution,” IEEE Trans. Veh. Technol., vol.  56, no. 1, pp. 27–34, Jan. 2007.

IEEE Trans. Wireless Commun. (6)

M. Safari and M. Uysal, “Relay-assisted free-space optical communication,” IEEE Trans. Wireless Commun., vol.  7, no. 12, pp. 5441–5449, Dec. 2008.
[CrossRef]

S. M. Navidpour, M. Uysal, and M. Kavehrad, “Performance of free-space optical transmission with spatial diversity,” IEEE Trans. Wireless Commun., vol.  6, no. 8, pp. 2813–2819, Aug. 2007.
[CrossRef]

M. Safari and M. Uysal, “Do we really need space-time block coding for free-space optical communication with direct detection?” IEEE Trans. Wireless Commun., vol.  7, no. 11, pp. 4445–4448, Nov. 2008.
[CrossRef]

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Figures (7)

Fig. 1.
Fig. 1.

Dual-hop optical wireless communication system.

Fig. 2.
Fig. 2.

Average channel capacity of a dual-hop CSI-assisted optical wireless system as a function of the transmitted optical power, P t , assuming clear weather conditions, z 1 = 2000 m , z 2 = 3000 m , P t , 1 = 0.4 P t , P t , 2 = 0.6 P t , W z / r = 10 , and various values of σ s / r .

Fig. 3.
Fig. 3.

Average channel capacity of a dual-hop CSI-assisted optical wireless system as a function of the transmitted optical power, P t , assuming light fog conditions, z 1 = 800 m , z 2 = 900 m , P t , 1 = 0.4 P t , P t , 2 = 0.6 P t , W z / r = 10 , and various values of σ s / r .

Fig. 4.
Fig. 4.

Average channel capacity of a dual-hop CSI-assisted optical wireless system as a function of the transmitted optical power, P t , assuming light fog conditions, z 1 = 800 m , z 2 = 900 m , P t , 1 = 0.4 P t , P t , 2 = 0.6 P t , σ s / r = 2 , and various values of W z / r .

Fig. 5.
Fig. 5.

Average channel capacity of a dual-hop semi-blind optical wireless system as a function of the transmitted optical power, P t , assuming clear weather conditions, z 1 = 2000 m , z 2 = 3000 m , P t , 1 = 0.4 P t , P t , 2 = 0.6 P t , and various values of W z / r and σ s / r .

Fig. 6.
Fig. 6.

5%-outage capacity of a dual-hop semi-blind optical wireless system as a function of the transmitted optical power, P t , assuming clear weather conditions, z 1 = 2000 m , z 2 = 3000 m , P t , 1 = 0.4 P t , P t , 2 = 0.6 P t , and various values of W z / r and σ s / r .

Fig. 7.
Fig. 7.

Average channel capacity of a dual-hop CSI-assisted optical wireless system as a function of ρ . Simulation parameters are as follows. Clear weather conditions: z 1 = 1000 m , z 2 = 2000 m . Light fog conditions: z 1 = 800 m , z 2 = 900 m . Scenario A: W z , 1 / r = 16 , σ s , 1 / r = 4 , W z , 2 / r = 4 , σ s , 2 / r = 2 . Scenario B: W z , 1 / r = 4 , σ s , 1 / r = 2 , W z , 2 / r = 16 , σ s , 2 / r = 4 . No pointing errors: z 1 = 2000 m , z 2 = 3000 m .

Equations (49)

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y k = R k h k x k 1 + n k ,
h k = h k , h k , s h k , p ,
ω = ω 1 ω 2 C 2 + C 1 ω 1 + ω 2 ,
ω k = 2 P t , k 2 R k 2 h k 2 σ n , k 2 .
h k , ( z ) = exp ( S z ) ,
S = 3.91 V ( λ 550 ) q ,
f h k , s ( x ) = 2 ( α k β k ) ( α k + β k ) / 2 Γ ( α k ) Γ ( β k ) x α k + β k 2 1 K α k β k [ 2 α k β k x ] ,
α k = { exp [ 0.49 Σ k 2 ( 1 + 0.18 d k 2 + 0.56 Σ k 12 / 5 ) 7 / 6 ] 1 } 1
β k = { exp [ 0.51 Σ k 2 ( 1 + 0.69 Σ k 12 / 5 ) 5 / 6 ( 1 + 0.9 d k 2 + 0.62 d k 2 Σ k 12 / 5 ) 5 / 6 ] 1 } 1 .
h k , p ( a ; z ) A 0 , k exp ( 2 a 2 W z , k , eq 2 ) ,
f a ( a ) = a σ s , k 2 exp ( a 2 2 σ s , k 2 ) , a > 0 ,
f h k , p ( x ) = ξ 2 A 0 , k ξ 2 x ξ 2 1 , 0 x A 0 , k ,
f h k ( x ) = α k β k ξ k 2 A 0 , k h k , Γ ( α k ) Γ ( β k ) × G 1 , 3 3 , 0 [ α k β k A 0 , k h k , x | ξ k 2 ξ k 2 1 , α k 1 , β k 1 ] .
f ω k ( x ) = ξ k 2 2 Γ ( α k ) Γ ( β k ) x G 1 , 3 3 , 0 [ α k β k σ n , k 2 x 2 P t , k R k A 0 , k h k , | ξ k 2 + 1 ξ k 2 , α k , β k ] .
C erg E C = 1 2 0 log 2 ( 1 + ω ) f ω ( ω ) d ω .
f ω ( x ) α μ μ x α μ 1 ω ^ α μ Γ ( μ ) exp ( μ x α ω ^ α ) .
Γ 2 ( μ + 1 / α ) Γ ( μ ) Γ ( μ + 2 / α ) Γ 2 ( μ + 1 / α ) = E 2 ω E ω 2 E 2 ω ,
Γ 2 ( μ + 2 / α ) Γ ( μ ) Γ ( μ + 4 / α ) Γ 2 ( μ + 2 / α ) = E 2 ω 2 E ω 4 E 2 ω 2 .
E ω n = 2 k = 1 2 [ α k + β k ] 6 + 5 n π 2 Γ ( n ) k = 1 2 [ ξ k 2 Γ ( α k ) Γ ( β k ) ] ( P t , 2 R 2 A 0 , 2 h l , 2 α 2 β 2 σ n , 2 ) 2 n × G 7 , 7 6 , 6 [ ( α 2 β 2 σ n , 2 P t , 1 R 1 A 0 1 h l , 1 α 1 β 1 σ n , 1 P t , 2 R 2 A 0 2 h l , 2 ) 2 | 1 , 1 β 1 2 , 1 α 1 2 , 1 ξ 1 2 2 , 1 β 1 2 , 1 α 1 2 , 1 + n + ξ 2 2 2 β 2 + 1 2 + n , α 2 + 1 2 + n , ξ 2 2 2 + n , β 2 2 + n , α 2 2 + n , n , ξ 1 2 2 ] .
E ω n = 2 α 2 + β 2 3 + n π Γ ( n ) ( ξ 1 2 + 2 n ) k = 1 2 [ ξ k 2 Γ ( α k ) Γ ( β k ) ] ( P t , 1 R 1 A 0 , 1 h l , 1 α 1 β 1 σ n , 1 ) 2 n Γ ( 2 n + α 1 ) Γ ( 2 n + β 1 ) × G 2 , 6 6 , 1 [ C α 2 2 β 2 2 σ n , 2 2 32 P t , 2 2 R 2 2 A 0 2 2 h l , 2 2 | 1 n , 1 + ξ 2 2 2 β 2 + 1 2 , α 2 + 1 2 , ξ 2 2 2 , β 2 2 , α 2 2 , 0 ] .
ω ^ = μ 1 α Γ ( μ ) E ω Γ ( μ + 1 α ) .
C erg μ μ 2 Γ ( μ ) ln 2 0 x μ 1 exp ( μ x ) ln ( 1 + ω ^ x 1 α ) d x .
C erg 1 2 Γ ( μ ) ln 2 H 3 , 2 1 , 3 [ ω ^ μ 1 α | ( 1 , 1 ) , ( 1 , 1 ) , ( 1 μ , 1 α ) ( 1 , 1 ) , ( 0 , 1 ) ] .
C q = μ C + 2 σ C erfc 1 ( 2 q 50 ) ,
E C 2 μ μ ( ln 2 ) 2 4 Γ ( μ ) 0 x μ 1 exp ( μ x ) ln 2 ( 1 + ω ^ x 1 α ) d x .
C erg 1 2 log 2 ( 1 + E ω ) ,
ω ω eq = 1 2 ω 1 ω 2 = k = 1 2 [ P t , k R k σ n , k h k ] .
C erg 1 2 ln 2 E ln ( 1 + ω eq ) = 1 2 ln 2 0 ln ( 1 + x ) f ω eq ( x ) d x .
C erg 1 2 ln 2 k = 1 2 [ ξ k 2 Γ ( α k ) Γ ( β k ) ] G 4 , 8 8 , 1 [ k = 1 2 [ α k β k σ n , k A 0 , k h k , P t , k R k ] | 0 , 1 , ξ 1 2 + 1 , ξ 2 2 + 1 0 , 0 , ξ 1 2 , ξ 2 2 , α 1 , α 2 , β 1 , β 2 ] .
C erg 2 log 2 ( P t ) + log 2 [ 2 ρ ( 1 ρ ) R 1 R 2 σ n , 1 σ n , 2 C ] + 1 ln 2 k = 1 2 J k 1 2 ln 2 C direct ,
J k = ξ k 2 Γ ( α k ) Γ ( β k ) G 3 , 4 4 , 0 [ α k β k A 0 k h l , k | 1 , 1 , 1 + ξ k 2 ξ k 2 , α k , β k , 0 ] ξ k 2 Γ ( α k ) Γ ( β k ) G 3 , 4 3 , 2 [ α k β k A 0 k h l , k | 1 , 1 , 1 + ξ k 2 ξ k 2 , α k , β k , 0 ] γ
C direct = 2 α k + β k 3 ξ k 2 π Γ ( α k ) Γ ( β k ) × G 3 , 7 7 , 1 [ α k 2 β k 2 σ n , k 2 C 32 P t , k 2 R k 2 A 0 , k 2 h k , 2 | 0 , 1 , 1 + ξ k 2 2 0 , 0 , α k + 1 2 , β k + 1 2 , ξ k 2 2 , α k 2 , β k 2 ] .
C erg = S ( log 2 P t L ) + o ( 1 ) ,
S = lim P t C erg log 2 P t ,
L = lim P t ( log 2 P t C erg S ) .
C erg log 2 ( P t ) + log 2 [ 2 ρ ( 1 ρ ) R 1 R 2 ] + 1 ln 2 k = 1 2 J k 1 2 ln 2 E log 2 [ 2 ρ 2 R 1 2 h 1 2 σ n , 2 2 + 2 ( 1 ρ ) 2 R 2 2 h 2 2 σ n , 1 2 ] .
G 2 = 2 α 1 + β 1 3 ξ 1 2 π Γ ( α 1 ) Γ ( β 1 ) × G 2 , 6 6 , 1 [ α 1 2 β 1 2 σ n , 1 2 32 P t , 1 2 R 1 2 A 0 1 2 h l , 1 2 | 1 , 1 + ξ 1 2 2 β 1 + 1 2 , α 1 + 1 2 , ξ 1 2 2 , β 1 2 , α 1 2 , 1 ] .
E { Y n } = 1 Γ ( n ) 0 s n 1 M Y ( s ) d s .
E ω n = 1 Γ ( n ) 0 s n 1 M 1 / ω 1 ( s ) M 1 / ω 2 ( s ) d s .
M 1 / ω k ( s ) = ξ k 2 2 Γ ( α k ) Γ ( β k ) 0 x 1 exp ( s x ) × G 1 , 3 3 , 0 [ α k β k σ n , k 2 x 2 P t , k R k A 0 , k h k , | ξ k 2 + 1 ξ k 2 , α k , β k ] d x .
M 1 / ω k ( s ) = 2 α k + β k 3 ξ k 2 π Γ ( α k ) Γ ( β k ) × G 1 , 6 6 , 0 [ α k 2 β k 2 σ n , k 2 32 P t , k 2 R k 2 A 0 , k 2 h k , 2 s | ξ k 2 2 + 1 0 , α k + 1 2 , β k + 1 2 , ξ k 2 2 , α k 2 , β k 2 ] .
E ω n = 0 0 ( ω 1 ω 2 C + ω 2 ) n f ω 1 ( ω 1 ) f ω 2 ( ω 2 ) d ω 1 d ω 2 .
E ω n = k = 1 2 [ ξ k 2 2 Γ ( α k ) Γ ( β k ) ] × 0 G 1 , 3 3 , 0 ω 1 n 1 [ α 1 β 1 σ n , 1 2 ω 1 2 P t , 1 R 1 A 0 , 1 h l , 1 | ξ 1 2 + 1 ξ 1 2 , α 1 , β 1 ] d ω 1 × 0 ω 2 n 1 ( C + ω 2 ) n G 1 , 3 3 , 0 [ α 2 β 2 σ n , 2 2 ω 2 2 P t , 2 R 2 A 0 , 2 h l , 2 | ξ 2 2 + 1 ξ 2 2 , α 2 , β 2 ] d ω 2 .
I 1 = 2 n ξ 1 2 Γ ( 2 n + β 1 ) Γ ( 2 n + α 1 ) Γ ( α 1 ) Γ ( β 1 ) ( ξ 1 2 + 2 n ) ( P t , 1 R 1 A 0 , 1 h l , 1 α 1 β 1 σ 1 ) 2 n .
I 2 = 2 α 2 + β 2 3 ξ 2 2 π Γ ( α 2 ) Γ ( β 2 ) Γ ( n ) × G 2 , 6 6 , 1 [ C α 2 2 β 2 2 σ n , 2 2 32 P t , 2 2 R 2 2 A 0 2 2 h l , 2 2 | 1 n , 1 + ξ 2 2 2 β 2 + 1 2 , α 2 + 1 2 , ξ 2 2 2 , β 2 2 , α 2 2 , 0 ] .
f ω eq ( x ) = k = 1 2 [ α k β k ξ k 2 σ n , k P t , k R k A 0 , k h k , Γ ( α k ) Γ ( β k ) ] × G 2 , 6 6 , 0 [ k = 1 2 [ α k β k σ n , k A 0 , k h k , P t , k R k ] x | { ξ k 2 } k = 1 2 { ξ k 2 1 , α k 1 , β k 1 } k = 1 2 ] .
C erg 1 2 ln 2 E ln ( ω 1 ω 2 C + ω 2 ) = 1 2 ln 2 [ E ln ω 1 + E ln ω 2 E ln ( C + ω 2 ) ] .
C erg 2 log 2 ( P t ) + log 2 [ 2 ρ ( 1 ρ ) R 1 R 2 σ n , 1 σ n , 2 C ] + 1 ln 2 k = 1 2 E ln h k 1 2 ln 2 E ln ( 1 + h 2 C ) .
ln z = G 2 , 3 1 , 2 [ x | 1 , 1 1 , 0 , 0 ] G 1 , 2 2 , 0 [ x | 1 0 , 0 ] γ .