Abstract

This paper focuses on the ergodic capacity analysis in the context of cooperative free-space optical (FSO) systems when the line of sight is available. Novel asymptotic closed-form expressions for the ergodic capacity corresponding to two different decode-and-forward (DF) strategies are obtained for a cooperative FSO communication system. Here, the atmospheric turbulence is modeled by a gamma-gamma distribution of parameters α and β which allows to study a wide range of turbulence conditions (moderate-to-strong) as well as the effect of the misalignment with zero boresight. It is demonstrated that cooperative communications are able to achieve not only a better performance in terms of the error rate performance as well as outage probability than direct transmission, but also in terms of the channel capacity in the context of FSO systems without much increase in hardware. In this way, a 3-way FSO communication setup is considered, in which the cooperative protocol can be applied to achieve a greater ergodic capacity compared to a direct transmission. It can be concluded that a greater and robust capacity strongly dependent on the relay location is achieved compared to a direct transmission without cooperative communication when line of sight is available. Here, the line of sight is taken into account in order to achieve a significant robustness under different turbulence conditions and more severe pointing errors regardless of the relay location. Simulation results are further demonstrated to confirm the accuracy and usefulness of the derived results.

© 2015 Optical Society of America

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References

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2015 (5)

R. Boluda-Ruiz, A. García-Zambrana, B. Castillo-Vázquez, and C. Castillo-Vázquez, “Impact of relay placement on diversity order in adaptive selective DF relay-assisted FSO communications,” Opt. Express 23(3), 2600–2617 (2015).
[Crossref] [PubMed]

C. del Castillo-Vazquez, R. Boluda-Ruiz, B. del Castillo-Vazquez, and A. Garcia-Zambrana, “Outage performance of DF relay-assisted FSO communications using time-diversity,” IEEE Photon. Technol. Lett. 27(11), 1149–1152 (2015).

M. Aggarwal, P. Garg, and P. Puri, “Ergodic capacity of SIM based DF relayed optical wireless communication systems,” IEEE Photon. Technol. Lett. 27(10), 1104–1107 (2015).
[Crossref]

M. Aggarwal, P. Garg, and P. Puri, “Exact capacity of amplify-and-forward relayed optical wireless communication systems,” IEEE Photon. Technol. Lett. 27(8), 903–906 (2015).
[Crossref]

E. Zedini, I. S. Ansari, and M.-S. Alouini, “Performance analysis of mixed Nakagami-and Gamma–Gamma Dual-Hop FSO Transmission Systems,” IEEE Photonics J. 7(1), 1–20 (2015).

2014 (3)

L. Yang, X. Gao, and M.-S. Alouini, “Performance analysis of free-space optical communication systems with multiuser diversity over stmospheric turbulence channels,” IEEE Photonics J. 6(2), 7901217 (2014).
[Crossref]

M. A. Khalighi and M. Uysal, “Survey on free space optical communication: A communication theory perspective,” IEEE Communications Surveys Tutorials 16(4), 2231–2258 (2014).
[Crossref]

R. Boluda-Ruiz, A. Garcia-Zambrana, C. Castillo-Vazquez, and B. Castillo-Vazquez, “Adaptive selective relaying in cooperative free-space optical systems over atmospheric turbulence and misalignment fading channels,” Opt. Express 22(13), 16629–16644 (2014).
[Crossref] [PubMed]

2013 (7)

2012 (2)

K. P. Peppas, “A new formula for the average bit error probability of dual-hop amplify-and-forward relaying systems over generalized shadowed fading channels,” IEEE Wireless Commun. Lett. 1(2), 85–88 (2012).
[Crossref]

A. Garcia-Zambrana, C. Castillo-Vazquez, B. Castillo-Vazquez, and R. Boluda-Ruiz, “Bit detect and forward relaying for FSO links using equal gain combining over gamma-gamma atmospheric turbulence channels with pointing errors,” Opt. Express 20(15), 16394–16409 (2012).
[Crossref]

2011 (1)

I. S. Ansari, S. Al-Ahmadi, F. Yilmaz, M.-S. Alouini, and H. Yanikomeroglu, “A new formula for the BER of binary modulations with dual-branch selection over generalized-K,” IEEE Trans. Commun. 59(10), 2654–2658 (2011).
[Crossref]

2010 (4)

N. Wang and J. Cheng, “Moment-based estimation for the shape parameters of the gamma-gamma atmospheric turbulence model,” Opt. Express 18(12), 12824–12831 (2010).
[Crossref] [PubMed]

A. García-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Space-time trellis coding with transmit laser selection for FSO links over strong atmospheric turbulence channels,” Opt. Express 18(6), 5356–5366 (2010).
[Crossref] [PubMed]

M. Karimi and M. Nasiri-Kenari, “Outage analysis of relay-assisted free-space optical communications,” IET Communications 4(12), 1423–1432 (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,” Chinese Opt. Lett. 8(6), 537–540 (2010).
[Crossref]

2009 (4)

2008 (3)

H. Sandalidis and T. Tsiftsis, “Outage probability and ergodic capacity of free-space optical links over strong turbulence,” Electron. Lett. 44(1), 46–47 (2008).
[Crossref]

S. Z. Denic, I. Djordjevic, J. Anguita, B. Vasic, and M. A. Neifeld, “Information theoretic limits for free-space optical channels with and without memory,” J. Lightwave Technol. 26(19), 3376–3384 (2008).
[Crossref]

M. Safari and M. Uysal, “Relay-assisted free-space optical communication,” IEEE Trans. Wireless Commun. 7(12), 5441–5449 (2008).
[Crossref]

2007 (1)

2006 (1)

M. Uysal, J. Li, and M. Yu, “Error rate performance analysis of coded free-space optical links over gamma-gamma atmospheric turbulence channels,” IEEE Trans. Wireless Commun. 5(6), 1229–1233 (2006).
[Crossref]

2005 (1)

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, 8 (2001).
[Crossref]

1998 (1)

E. Biglieri, J. Proakis, and S. Shamai, “Fading channels: information-theoretic and communications aspects,” IEEE Trans. Inf. Theory 44(6), 2619–2692 (1998).
[Crossref]

Abou-Rjeily, C.

C. Abou-Rjeily, “Achievable Diversity Orders of Decode-and-Forward Cooperative Protocols over Gamma-Gamma Fading FSO Links,” IEEE Trans. Commun. 61(9), 3919–3930 (2013).
[Crossref]

C. Abou-Rjeily, “Performance Analysis of Selective Relaying in Cooperative Free-Space Optical Systems,” J. Lightwave Technol. 31(18), 2965–2973 (2013).
[Crossref]

Aggarwal, M.

M. Aggarwal, P. Garg, and P. Puri, “Ergodic capacity of SIM based DF relayed optical wireless communication systems,” IEEE Photon. Technol. Lett. 27(10), 1104–1107 (2015).
[Crossref]

M. Aggarwal, P. Garg, and P. Puri, “Exact capacity of amplify-and-forward relayed optical wireless communication systems,” IEEE Photon. Technol. Lett. 27(8), 903–906 (2015).
[Crossref]

Al-Ahmadi, S.

I. S. Ansari, S. Al-Ahmadi, F. Yilmaz, M.-S. Alouini, and H. Yanikomeroglu, “A new formula for the BER of binary modulations with dual-branch selection over generalized-K,” IEEE Trans. Commun. 59(10), 2654–2658 (2011).
[Crossref]

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, 8 (2001).
[Crossref]

Alouini, M.

F. Benkhelifa, Z. Rezki, and M. Alouini, “Low SNR Capacity of FSO Links over Gamma-Gamma Atmospheric Turbulence Channels,” IEEE Commun. Lett. 17(6), 1264–1267 (2013).
[Crossref]

I. Ansari, F. Yilmaz, and M. Alouini, “A unified performance of free-space optical links over Gamma-Gamma turbulence channels with pointing errors,” submitted to IEEE Trans. Communications, technical report available at http://hdl.handle.net/10754/305353 (2015).

Alouini, M.-S.

E. Zedini, I. S. Ansari, and M.-S. Alouini, “Performance analysis of mixed Nakagami-and Gamma–Gamma Dual-Hop FSO Transmission Systems,” IEEE Photonics J. 7(1), 1–20 (2015).

L. Yang, X. Gao, and M.-S. Alouini, “Performance analysis of free-space optical communication systems with multiuser diversity over stmospheric turbulence channels,” IEEE Photonics J. 6(2), 7901217 (2014).
[Crossref]

I. S. Ansari, S. Al-Ahmadi, F. Yilmaz, M.-S. Alouini, and H. Yanikomeroglu, “A new formula for the BER of binary modulations with dual-branch selection over generalized-K,” IEEE Trans. Commun. 59(10), 2654–2658 (2011).
[Crossref]

F. Yilmaz and M.-S. Alouini, “Novel asymptotic results on the high-order statistics of the channel capacity over generalized fading channels,” in Signal Processing Advances in Wireless Communications (SPAWC), 2012 IEEE 13th International Workshop on, pp. 389–393 (IEEE, 2012).
[Crossref]

M. K. Simon and M.-S. Alouini, Digital Communications Over Fading Channels, 2nd ed. (Wiley-IEEE Press, New Jersey, 2005).

Andrews, L.

L. Andrews, R. Phillips, and C. Hopen, Laser Beam Scintillation with Applications, vol. 99 (SPIE Press, 2001).
[Crossref]

Andrews, L. C.

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, 8 (2001).
[Crossref]

Anees, S.

S. Anees and M. R. Bhatnagar, “On the capacity of decode-and-forward dual-hop free space optical communication systems,” in Wireless Communications and Networking Conference (WCNC), 2014 IEEE, pp. 18–23 (IEEE, 2014).
[Crossref]

Anguita, J.

Ansari, I.

I. Ansari, F. Yilmaz, and M. Alouini, “A unified performance of free-space optical links over Gamma-Gamma turbulence channels with pointing errors,” submitted to IEEE Trans. Communications, technical report available at http://hdl.handle.net/10754/305353 (2015).

Ansari, I. S.

E. Zedini, I. S. Ansari, and M.-S. Alouini, “Performance analysis of mixed Nakagami-and Gamma–Gamma Dual-Hop FSO Transmission Systems,” IEEE Photonics J. 7(1), 1–20 (2015).

I. S. Ansari, S. Al-Ahmadi, F. Yilmaz, M.-S. Alouini, and H. Yanikomeroglu, “A new formula for the BER of binary modulations with dual-branch selection over generalized-K,” IEEE Trans. Commun. 59(10), 2654–2658 (2011).
[Crossref]

Benkhelifa, F.

F. Benkhelifa, Z. Rezki, and M. Alouini, “Low SNR Capacity of FSO Links over Gamma-Gamma Atmospheric Turbulence Channels,” IEEE Commun. Lett. 17(6), 1264–1267 (2013).
[Crossref]

Bhatnagar, M. R.

M. R. Bhatnagar, “Average BER analysis of relay selection based decode-and-forward cooperative communication over Gamma-Gamma fading FSO links,” in Communications (ICC), 2013 IEEE International Conference on, pp. 3142–3147 (IEEE, 2013).
[Crossref]

S. Anees and M. R. Bhatnagar, “On the capacity of decode-and-forward dual-hop free space optical communication systems,” in Wireless Communications and Networking Conference (WCNC), 2014 IEEE, pp. 18–23 (IEEE, 2014).
[Crossref]

Biglieri, E.

E. Biglieri, J. Proakis, and S. Shamai, “Fading channels: information-theoretic and communications aspects,” IEEE Trans. Inf. Theory 44(6), 2619–2692 (1998).
[Crossref]

Boluda-Ruiz, R.

Brychkov, Y. A.

A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and series Volume 3: More Special Functions, vol. 3 (Gordon and Breach Science Publishers, 1999).

Castillo-Vazquez, B.

Castillo-Vazquez, C.

Castillo-Vázquez, B.

Castillo-Vázquez, C.

Castillo-Vázquez, M.

Chatzidiamantis, N. D.

Chen, M.

J.-Y. Wang, J.-B. Wang, M. Chen, Q.-S. Hu, N. Huang, R. Guan, and L. Jia, “Free-space optical communications using all-optical relays over weak turbulence channels with pointing errors,” in Wireless Communications & Signal Processing (WCSP), 2013 International Conference on, pp. 1–6 (IEEE, 2013).

Cheng, J.

M. Z. Hassan, M. J. Hossain, and J. Cheng, “Ergodic capacity comparison of optical wireless communications using adaptive transmissions,” Opt. Express 21(17), 20,346–20,362 (2013).
[Crossref]

N. Wang and J. Cheng, “Moment-based estimation for the shape parameters of the gamma-gamma atmospheric turbulence model,” Opt. Express 18(12), 12824–12831 (2010).
[Crossref] [PubMed]

Dai, L.

J. Zhang, L. Dai, Y. Han, Y. Zhang, and Z. Wang, “On the Ergodic Capacity of MIMO Free-Space Optical Systems over Turbulence Channels,” IEEE J. Sel. Areas Commun. (to be published) (2015).
[Crossref]

del Castillo-Vazquez, B.

C. del Castillo-Vazquez, R. Boluda-Ruiz, B. del Castillo-Vazquez, and A. Garcia-Zambrana, “Outage performance of DF relay-assisted FSO communications using time-diversity,” IEEE Photon. Technol. Lett. 27(11), 1149–1152 (2015).

del Castillo-Vazquez, C.

C. del Castillo-Vazquez, R. Boluda-Ruiz, B. del Castillo-Vazquez, and A. Garcia-Zambrana, “Outage performance of DF relay-assisted FSO communications using time-diversity,” IEEE Photon. Technol. Lett. 27(11), 1149–1152 (2015).

Denic, S. Z.

Djordjevic, I.

Dordevic, G.

M. Petkovic and G. Dordevic, “Effects of pointing errors on average capacity of FSO links over gamma-gamma turbulence channel,” in Telecommunication in Modern Satellite, Cable and Broadcasting Services (TELSIKS), 2013 11th International Conference on, vol. 02, pp. 481–484 (2013).
[Crossref]

Fafalios, M. E.

Farid, A. A.

Galambos, J.

J. Galambos and I. Simonelli, Products of Random Variables: Applications to Problems of Physics and to Arithmetical Functions (CRC Press, 2004).

Gao, X.

L. Yang, X. Gao, and M.-S. Alouini, “Performance analysis of free-space optical communication systems with multiuser diversity over stmospheric turbulence channels,” IEEE Photonics J. 6(2), 7901217 (2014).
[Crossref]

Garcia-Zambrana, A.

García-Zambrana, A.

Garg, P.

M. Aggarwal, P. Garg, and P. Puri, “Ergodic capacity of SIM based DF relayed optical wireless communication systems,” IEEE Photon. Technol. Lett. 27(10), 1104–1107 (2015).
[Crossref]

M. Aggarwal, P. Garg, and P. Puri, “Exact capacity of amplify-and-forward relayed optical wireless communication systems,” IEEE Photon. Technol. Lett. 27(8), 903–906 (2015).
[Crossref]

Garrido-Balsells, J. M.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 7th ed. (Academic Press Inc., 2007).

Guan, R.

J.-Y. Wang, J.-B. Wang, M. Chen, Q.-S. Hu, N. Huang, R. Guan, and L. Jia, “Free-space optical communications using all-optical relays over weak turbulence channels with pointing errors,” in Wireless Communications & Signal Processing (WCSP), 2013 International Conference on, pp. 1–6 (IEEE, 2013).

Guillen i Fabregas, A.

N. Letzepis and A. Guillen i Fabregas, “Outage probability of the Gaussian MIMO free-space optical channel with PPM,” IEEE Trans. Commun. 57(12), 3682–3690 (2009).
[Crossref]

Han, Y.

J. Zhang, L. Dai, Y. Han, Y. Zhang, and Z. Wang, “On the Ergodic Capacity of MIMO Free-Space Optical Systems over Turbulence Channels,” IEEE J. Sel. Areas Commun. (to be published) (2015).
[Crossref]

Hassan, M. Z.

M. Z. Hassan, M. J. Hossain, and J. Cheng, “Ergodic capacity comparison of optical wireless communications using adaptive transmissions,” Opt. Express 21(17), 20,346–20,362 (2013).
[Crossref]

Hopen, C.

L. Andrews, R. Phillips, and C. Hopen, Laser Beam Scintillation with Applications, vol. 99 (SPIE Press, 2001).
[Crossref]

Hossain, M. J.

M. Z. Hassan, M. J. Hossain, and J. Cheng, “Ergodic capacity comparison of optical wireless communications using adaptive transmissions,” Opt. Express 21(17), 20,346–20,362 (2013).
[Crossref]

Hranilovic, S.

Hu, Q.-S.

J.-Y. Wang, J.-B. Wang, M. Chen, Q.-S. Hu, N. Huang, R. Guan, and L. Jia, “Free-space optical communications using all-optical relays over weak turbulence channels with pointing errors,” in Wireless Communications & Signal Processing (WCSP), 2013 International Conference on, pp. 1–6 (IEEE, 2013).

Huang, N.

J.-Y. Wang, J.-B. Wang, M. Chen, Q.-S. Hu, N. Huang, R. Guan, and L. Jia, “Free-space optical communications using all-optical relays over weak turbulence channels with pointing errors,” in Wireless Communications & Signal Processing (WCSP), 2013 International Conference on, pp. 1–6 (IEEE, 2013).

Jia, L.

J.-Y. Wang, J.-B. Wang, M. Chen, Q.-S. Hu, N. Huang, R. Guan, and L. Jia, “Free-space optical communications using all-optical relays over weak turbulence channels with pointing errors,” in Wireless Communications & Signal Processing (WCSP), 2013 International Conference on, pp. 1–6 (IEEE, 2013).

Jurado-Navas, A.

Karagianni, E. A.

Karagiannidis, G. K.

Karimi, M.

M. Karimi and M. Nasiri-Kenari, “Outage analysis of relay-assisted free-space optical communications,” IET Communications 4(12), 1423–1432 (2010).
[Crossref]

M. Karimi and M. Nasiri-Kenari, “BER analysis of cooperative systems in free-space optical networks,” J. Light-wave Technol. 27(24), 5639–5647 (2009).
[Crossref]

Khalighi, M. A.

M. A. Khalighi and M. Uysal, “Survey on free space optical communication: A communication theory perspective,” IEEE Communications Surveys Tutorials 16(4), 2231–2258 (2014).
[Crossref]

Kim, I. I.

I. I. Kim, B. McArthur, and E. J. Korevaar, “Comparison of laser beam propagation at 785 nm and 1550 nm in fog and haze for optical wireless communications,” in Information Technologies 2000, pp. 26–37 (International Society for Optics and Photonics, 2001).

Korevaar, E. J.

I. I. Kim, B. McArthur, and E. J. Korevaar, “Comparison of laser beam propagation at 785 nm and 1550 nm in fog and haze for optical wireless communications,” in Information Technologies 2000, pp. 26–37 (International Society for Optics and Photonics, 2001).

Kriezis, E. E.

Letzepis, N.

N. Letzepis and A. Guillen i Fabregas, “Outage probability of the Gaussian MIMO free-space optical channel with PPM,” IEEE Trans. Commun. 57(12), 3682–3690 (2009).
[Crossref]

Li, J.

M. Uysal, J. Li, and M. Yu, “Error rate performance analysis of coded free-space optical links over gamma-gamma atmospheric turbulence channels,” IEEE Trans. Wireless Commun. 5(6), 1229–1233 (2006).
[Crossref]

Liu, C.

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,” Chinese Opt. Lett. 8(6), 537–540 (2010).
[Crossref]

Majumdar, A. K.

A. K. Majumdar and J. C. Ricklin, Free-Space Laser Communications: Principles and Advances, vol. 2 (Springer Science & Business Media, 2010).

Marichev, O. I.

A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and series Volume 3: More Special Functions, vol. 3 (Gordon and Breach Science Publishers, 1999).

McArthur, B.

I. I. Kim, B. McArthur, and E. J. Korevaar, “Comparison of laser beam propagation at 785 nm and 1550 nm in fog and haze for optical wireless communications,” in Information Technologies 2000, pp. 26–37 (International Society for Optics and Photonics, 2001).

Michalopoulos, D. S.

Nasiri-Kenari, M.

M. Karimi and M. Nasiri-Kenari, “Outage analysis of relay-assisted free-space optical communications,” IET Communications 4(12), 1423–1432 (2010).
[Crossref]

M. Karimi and M. Nasiri-Kenari, “BER analysis of cooperative systems in free-space optical networks,” J. Light-wave Technol. 27(24), 5639–5647 (2009).
[Crossref]

Neifeld, M.

Neifeld, M. A.

Nistazakis, H. E.

Paris, J. F.

Peppas, K. P.

K. P. Peppas, A. N. Stassinakis, H. E. Nistazakis, and G. S. Tombras, “Capacity analysis of dual amplify-and-forward relayed free-space optical communication systems over turbulence channels with pointing errors,” J. Opt. Commun. Netw. 5(9), 1032–1042 (2013).
[Crossref]

K. P. Peppas, “A new formula for the average bit error probability of dual-hop amplify-and-forward relaying systems over generalized shadowed fading channels,” IEEE Wireless Commun. Lett. 1(2), 85–88 (2012).
[Crossref]

Petkovic, M.

M. Petkovic and G. Dordevic, “Effects of pointing errors on average capacity of FSO links over gamma-gamma turbulence channel,” in Telecommunication in Modern Satellite, Cable and Broadcasting Services (TELSIKS), 2013 11th International Conference on, vol. 02, pp. 481–484 (2013).
[Crossref]

Phillips, R.

L. Andrews, R. Phillips, and C. Hopen, Laser Beam Scintillation with Applications, vol. 99 (SPIE Press, 2001).
[Crossref]

Phillips, R. L.

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, 8 (2001).
[Crossref]

Proakis, J.

E. Biglieri, J. Proakis, and S. Shamai, “Fading channels: information-theoretic and communications aspects,” IEEE Trans. Inf. Theory 44(6), 2619–2692 (1998).
[Crossref]

Prudnikov, A. P.

A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and series Volume 3: More Special Functions, vol. 3 (Gordon and Breach Science Publishers, 1999).

Puerta-Notario, A.

Puri, P.

M. Aggarwal, P. Garg, and P. Puri, “Exact capacity of amplify-and-forward relayed optical wireless communication systems,” IEEE Photon. Technol. Lett. 27(8), 903–906 (2015).
[Crossref]

M. Aggarwal, P. Garg, and P. Puri, “Ergodic capacity of SIM based DF relayed optical wireless communication systems,” IEEE Photon. Technol. Lett. 27(10), 1104–1107 (2015).
[Crossref]

Rezki, Z.

F. Benkhelifa, Z. Rezki, and M. Alouini, “Low SNR Capacity of FSO Links over Gamma-Gamma Atmospheric Turbulence Channels,” IEEE Commun. Lett. 17(6), 1264–1267 (2013).
[Crossref]

Ricklin, J. C.

A. K. Majumdar and J. C. Ricklin, Free-Space Laser Communications: Principles and Advances, vol. 2 (Springer Science & Business Media, 2010).

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 7th ed. (Academic Press Inc., 2007).

Safari, M.

M. Safari and M. Uysal, “Relay-assisted free-space optical communication,” IEEE Trans. Wireless Commun. 7(12), 5441–5449 (2008).
[Crossref]

Sandalidis, H.

H. Sandalidis and T. Tsiftsis, “Outage probability and ergodic capacity of free-space optical links over strong turbulence,” Electron. Lett. 44(1), 46–47 (2008).
[Crossref]

Sandalidis, H. G.

Schober, R.

Shamai, S.

E. Biglieri, J. Proakis, and S. Shamai, “Fading channels: information-theoretic and communications aspects,” IEEE Trans. Inf. Theory 44(6), 2619–2692 (1998).
[Crossref]

Simon, M. K.

M. K. Simon and M.-S. Alouini, Digital Communications Over Fading Channels, 2nd ed. (Wiley-IEEE Press, New Jersey, 2005).

Simonelli, I.

J. Galambos and I. Simonelli, Products of Random Variables: Applications to Problems of Physics and to Arithmetical Functions (CRC Press, 2004).

Stassinakis, A. N.

Sun, Y.

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,” Chinese Opt. Lett. 8(6), 537–540 (2010).
[Crossref]

Tombras, G. S.

Tsiftsis, T.

H. Sandalidis and T. Tsiftsis, “Outage probability and ergodic capacity of free-space optical links over strong turbulence,” Electron. Lett. 44(1), 46–47 (2008).
[Crossref]

Tsiftsis, T. A.

Tsigopoulos, A. D.

Uysal, M.

M. A. Khalighi and M. Uysal, “Survey on free space optical communication: A communication theory perspective,” IEEE Communications Surveys Tutorials 16(4), 2231–2258 (2014).
[Crossref]

M. Safari and M. Uysal, “Relay-assisted free-space optical communication,” IEEE Trans. Wireless Commun. 7(12), 5441–5449 (2008).
[Crossref]

M. Uysal, J. Li, and M. Yu, “Error rate performance analysis of coded free-space optical links over gamma-gamma atmospheric turbulence channels,” IEEE Trans. Wireless Commun. 5(6), 1229–1233 (2006).
[Crossref]

Vasic, B.

Wang, J.-B.

J.-Y. Wang, J.-B. Wang, M. Chen, Q.-S. Hu, N. Huang, R. Guan, and L. Jia, “Free-space optical communications using all-optical relays over weak turbulence channels with pointing errors,” in Wireless Communications & Signal Processing (WCSP), 2013 International Conference on, pp. 1–6 (IEEE, 2013).

Wang, J.-Y.

J.-Y. Wang, J.-B. Wang, M. Chen, Q.-S. Hu, N. Huang, R. Guan, and L. Jia, “Free-space optical communications using all-optical relays over weak turbulence channels with pointing errors,” in Wireless Communications & Signal Processing (WCSP), 2013 International Conference on, pp. 1–6 (IEEE, 2013).

Wang, N.

Wang, Z.

J. Zhang, L. Dai, Y. Han, Y. Zhang, and Z. Wang, “On the Ergodic Capacity of MIMO Free-Space Optical Systems over Turbulence Channels,” IEEE J. Sel. Areas Commun. (to be published) (2015).
[Crossref]

Yang, L.

L. Yang, X. Gao, and M.-S. Alouini, “Performance analysis of free-space optical communication systems with multiuser diversity over stmospheric turbulence channels,” IEEE Photonics J. 6(2), 7901217 (2014).
[Crossref]

Yanikomeroglu, H.

I. S. Ansari, S. Al-Ahmadi, F. Yilmaz, M.-S. Alouini, and H. Yanikomeroglu, “A new formula for the BER of binary modulations with dual-branch selection over generalized-K,” IEEE Trans. Commun. 59(10), 2654–2658 (2011).
[Crossref]

Yao, Y.

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,” Chinese Opt. Lett. 8(6), 537–540 (2010).
[Crossref]

Yilmaz, F.

I. S. Ansari, S. Al-Ahmadi, F. Yilmaz, M.-S. Alouini, and H. Yanikomeroglu, “A new formula for the BER of binary modulations with dual-branch selection over generalized-K,” IEEE Trans. Commun. 59(10), 2654–2658 (2011).
[Crossref]

F. Yilmaz and M.-S. Alouini, “Novel asymptotic results on the high-order statistics of the channel capacity over generalized fading channels,” in Signal Processing Advances in Wireless Communications (SPAWC), 2012 IEEE 13th International Workshop on, pp. 389–393 (IEEE, 2012).
[Crossref]

I. Ansari, F. Yilmaz, and M. Alouini, “A unified performance of free-space optical links over Gamma-Gamma turbulence channels with pointing errors,” submitted to IEEE Trans. Communications, technical report available at http://hdl.handle.net/10754/305353 (2015).

Yu, M.

M. Uysal, J. Li, and M. Yu, “Error rate performance analysis of coded free-space optical links over gamma-gamma atmospheric turbulence channels,” IEEE Trans. Wireless Commun. 5(6), 1229–1233 (2006).
[Crossref]

Zedini, E.

E. Zedini, I. S. Ansari, and M.-S. Alouini, “Performance analysis of mixed Nakagami-and Gamma–Gamma Dual-Hop FSO Transmission Systems,” IEEE Photonics J. 7(1), 1–20 (2015).

Zhang, J.

J. Zhang, L. Dai, Y. Han, Y. Zhang, and Z. Wang, “On the Ergodic Capacity of MIMO Free-Space Optical Systems over Turbulence Channels,” IEEE J. Sel. Areas Commun. (to be published) (2015).
[Crossref]

Zhang, Y.

J. Zhang, L. Dai, Y. Han, Y. Zhang, and Z. Wang, “On the Ergodic Capacity of MIMO Free-Space Optical Systems over Turbulence Channels,” IEEE J. Sel. Areas Commun. (to be published) (2015).
[Crossref]

Zhao, X.

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,” Chinese Opt. Lett. 8(6), 537–540 (2010).
[Crossref]

Chinese Opt. Lett. (1)

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,” Chinese Opt. Lett. 8(6), 537–540 (2010).
[Crossref]

Electron. Lett. (1)

H. Sandalidis and T. Tsiftsis, “Outage probability and ergodic capacity of free-space optical links over strong turbulence,” Electron. Lett. 44(1), 46–47 (2008).
[Crossref]

IEEE Commun. Lett. (1)

F. Benkhelifa, Z. Rezki, and M. Alouini, “Low SNR Capacity of FSO Links over Gamma-Gamma Atmospheric Turbulence Channels,” IEEE Commun. Lett. 17(6), 1264–1267 (2013).
[Crossref]

IEEE Communications Surveys Tutorials (1)

M. A. Khalighi and M. Uysal, “Survey on free space optical communication: A communication theory perspective,” IEEE Communications Surveys Tutorials 16(4), 2231–2258 (2014).
[Crossref]

IEEE Photon. Technol. Lett. (3)

C. del Castillo-Vazquez, R. Boluda-Ruiz, B. del Castillo-Vazquez, and A. Garcia-Zambrana, “Outage performance of DF relay-assisted FSO communications using time-diversity,” IEEE Photon. Technol. Lett. 27(11), 1149–1152 (2015).

M. Aggarwal, P. Garg, and P. Puri, “Ergodic capacity of SIM based DF relayed optical wireless communication systems,” IEEE Photon. Technol. Lett. 27(10), 1104–1107 (2015).
[Crossref]

M. Aggarwal, P. Garg, and P. Puri, “Exact capacity of amplify-and-forward relayed optical wireless communication systems,” IEEE Photon. Technol. Lett. 27(8), 903–906 (2015).
[Crossref]

IEEE Photonics J. (2)

E. Zedini, I. S. Ansari, and M.-S. Alouini, “Performance analysis of mixed Nakagami-and Gamma–Gamma Dual-Hop FSO Transmission Systems,” IEEE Photonics J. 7(1), 1–20 (2015).

L. Yang, X. Gao, and M.-S. Alouini, “Performance analysis of free-space optical communication systems with multiuser diversity over stmospheric turbulence channels,” IEEE Photonics J. 6(2), 7901217 (2014).
[Crossref]

IEEE Trans. Commun. (3)

I. S. Ansari, S. Al-Ahmadi, F. Yilmaz, M.-S. Alouini, and H. Yanikomeroglu, “A new formula for the BER of binary modulations with dual-branch selection over generalized-K,” IEEE Trans. Commun. 59(10), 2654–2658 (2011).
[Crossref]

N. Letzepis and A. Guillen i Fabregas, “Outage probability of the Gaussian MIMO free-space optical channel with PPM,” IEEE Trans. Commun. 57(12), 3682–3690 (2009).
[Crossref]

C. Abou-Rjeily, “Achievable Diversity Orders of Decode-and-Forward Cooperative Protocols over Gamma-Gamma Fading FSO Links,” IEEE Trans. Commun. 61(9), 3919–3930 (2013).
[Crossref]

IEEE Trans. Inf. Theory (1)

E. Biglieri, J. Proakis, and S. Shamai, “Fading channels: information-theoretic and communications aspects,” IEEE Trans. Inf. Theory 44(6), 2619–2692 (1998).
[Crossref]

IEEE Trans. Wireless Commun. (2)

M. Safari and M. Uysal, “Relay-assisted free-space optical communication,” IEEE Trans. Wireless Commun. 7(12), 5441–5449 (2008).
[Crossref]

M. Uysal, J. Li, and M. Yu, “Error rate performance analysis of coded free-space optical links over gamma-gamma atmospheric turbulence channels,” IEEE Trans. Wireless Commun. 5(6), 1229–1233 (2006).
[Crossref]

IEEE Wireless Commun. Lett. (1)

K. P. Peppas, “A new formula for the average bit error probability of dual-hop amplify-and-forward relaying systems over generalized shadowed fading channels,” IEEE Wireless Commun. Lett. 1(2), 85–88 (2012).
[Crossref]

IET Communications (1)

M. Karimi and M. Nasiri-Kenari, “Outage analysis of relay-assisted free-space optical communications,” IET Communications 4(12), 1423–1432 (2010).
[Crossref]

J. Light-wave Technol. (1)

M. Karimi and M. Nasiri-Kenari, “BER analysis of cooperative systems in free-space optical networks,” J. Light-wave Technol. 27(24), 5639–5647 (2009).
[Crossref]

J. Lightwave Technol. (5)

J. Opt. Commun. Netw. (2)

J. Opt. Netw. (1)

Opt. Eng. (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, 8 (2001).
[Crossref]

Opt. Express (6)

Opt. Lett. (1)

Other (15)

M. R. Bhatnagar, “Average BER analysis of relay selection based decode-and-forward cooperative communication over Gamma-Gamma fading FSO links,” in Communications (ICC), 2013 IEEE International Conference on, pp. 3142–3147 (IEEE, 2013).
[Crossref]

J.-Y. Wang, J.-B. Wang, M. Chen, Q.-S. Hu, N. Huang, R. Guan, and L. Jia, “Free-space optical communications using all-optical relays over weak turbulence channels with pointing errors,” in Wireless Communications & Signal Processing (WCSP), 2013 International Conference on, pp. 1–6 (IEEE, 2013).

L. Andrews, R. Phillips, and C. Hopen, Laser Beam Scintillation with Applications, vol. 99 (SPIE Press, 2001).
[Crossref]

A. K. Majumdar and J. C. Ricklin, Free-Space Laser Communications: Principles and Advances, vol. 2 (Springer Science & Business Media, 2010).

I. I. Kim, B. McArthur, and E. J. Korevaar, “Comparison of laser beam propagation at 785 nm and 1550 nm in fog and haze for optical wireless communications,” in Information Technologies 2000, pp. 26–37 (International Society for Optics and Photonics, 2001).

M. Petkovic and G. Dordevic, “Effects of pointing errors on average capacity of FSO links over gamma-gamma turbulence channel,” in Telecommunication in Modern Satellite, Cable and Broadcasting Services (TELSIKS), 2013 11th International Conference on, vol. 02, pp. 481–484 (2013).
[Crossref]

I. Ansari, F. Yilmaz, and M. Alouini, “A unified performance of free-space optical links over Gamma-Gamma turbulence channels with pointing errors,” submitted to IEEE Trans. Communications, technical report available at http://hdl.handle.net/10754/305353 (2015).

J. Zhang, L. Dai, Y. Han, Y. Zhang, and Z. Wang, “On the Ergodic Capacity of MIMO Free-Space Optical Systems over Turbulence Channels,” IEEE J. Sel. Areas Commun. (to be published) (2015).
[Crossref]

M. K. Simon and M.-S. Alouini, Digital Communications Over Fading Channels, 2nd ed. (Wiley-IEEE Press, New Jersey, 2005).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 7th ed. (Academic Press Inc., 2007).

S. Anees and M. R. Bhatnagar, “On the capacity of decode-and-forward dual-hop free space optical communication systems,” in Wireless Communications and Networking Conference (WCNC), 2014 IEEE, pp. 18–23 (IEEE, 2014).
[Crossref]

J. Galambos and I. Simonelli, Products of Random Variables: Applications to Problems of Physics and to Arithmetical Functions (CRC Press, 2004).

A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and series Volume 3: More Special Functions, vol. 3 (Gordon and Breach Science Publishers, 1999).

Wolfram Research Inc., “The Wolfram functions site,” URL http://functions.wolfram.com .

F. Yilmaz and M.-S. Alouini, “Novel asymptotic results on the high-order statistics of the channel capacity over generalized fading channels,” in Signal Processing Advances in Wireless Communications (SPAWC), 2012 IEEE 13th International Workshop on, pp. 389–393 (IEEE, 2012).
[Crossref]

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Figures (4)

Fig. 1
Fig. 1 Block diagram of the considered 3-way FSO communication system, where dSD is the S-D link distance and (xR, yR) represents the location of the node R.
Fig. 2
Fig. 2 Ergodic capacity at high SNR for a source-destination link distance of dSD = 3 km when different weather condition and values of of normalized beam width and normalized jitter of (ωz/r, σs/r) = (5, 1) and (ωz/r, σs/r) = (5, 3) are assumed.
Fig. 3
Fig. 3 Gain, G[dB], for a source-destination link distance of dSD = 3 km when different weather condition and values of of normalized beam width and normalized jitter of (ωz/r, σs/r) = (5, 1) and (ωz/r, σs/r) = (5, 3) are assumed.
Fig. 4
Fig. 4 Gain, G[dB], for a source-destination link distance of dSD = 3 km when different weather condition and values of of normalized beam width and normalized jitter of (ωz/r, σs/r) = (7, 1) and (ωz/r, σs/r) = (7, 3) are assumed.

Equations (58)

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

γ = 1 2 d E 2 I m 2 σ m 2 = 4 P opt 2 T b ξ I m 2 N 0 = 4 γ 0 ξ I m 2 ,
α = [ exp ( 0.49 σ R 2 / ( 1 + 1.11 σ R 12 / 5 ) 7 / 6 ) 1 ] 1 ,
β = [ exp ( 0.51 σ R 2 / ( 1 + 0.69 σ R 12 / 5 ) 5 / 6 ) 1 ] 1 ,
f I m ( i ) = φ m 2 i 1 Γ ( α m ) Γ ( β m ) G 1 , 3 3 , 0 ( α m β m i A m L m | φ m 2 + 1 φ m 2 , α m , β m ) ,
F I m ( i ) = φ m 2 Γ ( α m ) Γ ( β m ) G 2 , 4 3 , 1 ( α m β m i A m L m | 1 , φ m 2 + 1 φ m 2 , α m , β m , 0 ) .
Y BDF = 1 2 X I SD + Z SD + X * I RD + Z RD , X { 0 , d E } , Z SD , Z RD ~ N ( 0 , N 0 / 2 )
C BDF = C 0 ( 1 P b SR ) + C 1 P b SR = C 0 ( C 0 C 1 ) P b SR ,
C BDF C 0 = B 2 ln ( 2 ) 0 0 ln ( 1 + γ 0 ξ 2 ( i 1 + 2 i 2 ) 2 ) f I SD ( i 1 ) f I RD ( i 2 ) d i 1 d i 2 ,
C BDF B ln ( 4 ) 0 ln ( 1 + γ 0 ξ 2 i 2 ) f I T ( i ) d i ,
AM GM ,
I SD + 2 I RD 8 I SD I RD = 8 I T LB .
C BDF B ln ( 4 ) 0 ln ( 1 + 4 γ 0 ξ Fi ) f I T LB ( i ) d i .
f I T LB ( i ) = φ SD 2 φ RD 2 i 1 G 2 , 6 6 , 0 ( α SD β SD α RD β RD A SD L SD A RD L RD i | φ SD 2 + 1 , φ RD 2 + 1 φ SD 2 , α SD , β SD , φ RD 2 , α RD , β RD ) Γ ( α SD ) Γ ( β SD ) Γ ( α RD ) Γ ( β RD ) .
C BDF β φ SD 2 φ RD 2 G 4 , 8 8 , 1 ( α SD β SD α RD β RD γ 0 ξ 4 F A SD L SD A RD L RD | 0 , 1 , φ SD 2 + 1 , φ RD 2 + 1 φ SD 2 , α SD , β SD , φ RD 2 , α RD , β RD , 0 , 0 ) ln ( 4 ) Γ ( α SD ) Γ ( β SD ) Γ ( α RD ) Γ ( β RD ) .
C BDF B ln ( 4 γ 0 ξ F ) ln ( 4 ) + B ln ( 4 ) 0 ln ( i ) f I T LB ( i ) d i .
ln ( z ) = z ln ( z ) z 1 ln ( z ) z 1 .
C BDF B ln ( 4 γ 0 ξ F ) ln ( 4 ) + B ln ( 4 ) ( 0 i G 2 , 2 2 , 2 ( i | 0 , 0 0 , 0 ) f I T LB ( i ) d i 0 G 2 , 2 2 , 2 ( i | 0 , 0 0 , 0 ) f I T LB ( i ) d i ) .
C BDF H B ln ( 4 γ 0 ξ F ) ln ( 4 ) + B φ SD 2 φ RD 2 G 4 , 8 8 , 2 ( α SD β SD α RD β RD A SD L SD A RD L RD | 0 , 0 , φ SD 2 + 1 , φ RD 2 + 1 φ SD 2 , α SD , β SD , φ RD 2 , α RD , β RD , 0 , 0 ) ln ( 4 ) Γ ( α SD ) Γ ( β SD ) Γ ( α RD ) Γ ( β RD ) B φ SD 2 φ RD 2 G 4 , 8 8 , 2 ( α SD β SD α RD β RD A SD L SD A RD L RD | 1 , 1 , φ SD 2 + 1 , φ RD 2 + 1 φ SD 2 , α SD , β SD , φ RD 2 , α RD , β RD , 1 , 1 ) ln ( 4 ) Γ ( α SD ) Γ ( β SD ) Γ ( α RD ) Γ ( β RD ) .
C BDF H B ln ( γ 0 ξ 4 F ) ln ( 4 ) + B ln ( 4 ) ( 1 φ SD 2 1 φ RD 2 ) + B ln ( 4 ) ( ψ ( α SD ) + ψ ( β SD ) + ψ ( α RD ) + ψ ( β RD ) + ln ( A SD L SD A RD L RD α SD β SD α RD β RD ) ) ,
Y BDF = 1 2 X I SD + X * I RD + Z SD + Z RD , I RD > I SD
Y DT = X I SD + Z SD , I RD < I SD
C ADF B 2 ln ( 2 ) 0 0 i 2 ln ( 1 + γ 0 ξ 2 ( i 1 + 2 i 2 ) 2 ) f I SD ( i 1 ) f I RD ( i 2 ) d i 1 d i 2 + B 2 ln ( 2 ) 0 ln ( 1 + 4 γ 0 ξ i 2 ) F I RD ( i ) f I SD ( i ) d i .
C ADF B ln ( 4 ) 0 0 i 2 ln ( 1 + 4 γ 0 ξ F i 1 i 2 ) f I SD ( i 1 ) f I RD ( i 2 ) d i 1 d i 2 + B ln ( 4 ) 0 ln ( 1 + 4 γ 0 ξ i 2 ) F I RD ( i ) f I SD ( i ) d i = C BDF + C SD ,
C ADF H B ln ( 4 γ 0 ξ F ) ln ( 4 ) 0 F I SD ( i ) f I RD ( i ) d i + B ln ( 4 ) 0 0 i 2 ln ( i 1 i 2 ) f I SD ( i 1 ) f I RD ( i 2 ) d i 1 d i 2 + B ln ( 4 γ 0 ξ ) ln ( 4 ) 0 F I RD ( i ) f I SD ( i ) d i + B ln ( 2 ) 0 ln ( i ) F I RD ( i ) f I SD ( i ) d i = C BDF H + C SD H .
P 0 F I SD ( i ) f I RD ( i ) d i .
P = φ RD 2 φ SD 2 G 5 , 4 4 , 3 ( α RD β RD A SD L SD α SD β SD A RD L RD | 1 φ SD 2 , 1 α SD , 1 β SD , 1 , φ RD 2 + 1 α RD 2 , α RD , β RD , 0 φ SD 2 ) Γ ( α RD ) Γ ( α SD ) Γ ( β RD ) Γ ( β SD ) .
C ADF H B ( ln ( 4 γ 0 ξ ) + P ln ( F ) + I 1 + 2 I 2 ) ln ( 4 ) ,
I 1 = 0 0 i 2 ln ( i 1 i 2 ) f I SD ( i 1 ) f I RD ( i 2 ) d i 1 d i 2 ,
I 2 = 0 ln ( i ) F I RD ( i ) f I SD ( i ) d i .
I 2 = φ SD 2 φ RD 2 Γ ( α SD ) Γ ( β SD ) Γ ( α RD ) Γ ( β RD ) × G 2 , 2 : 2 , 4 : 1 , 3 2 , 2 : 3 , 1 : 3 , 0 ( 1 , 1 1 , 1 | 1 , φ RD 2 + 1 φ RD 2 , α RD , β RD | φ SD 2 + 1 φ SD 2 , α SD , β SD | α RD β RD A RD L RD , α SD β SD A SD L SD ) .
C DT = B 2 ln ( 2 ) 0 ln ( 1 + 4 γ 0 ξ i 2 ) f I SD ( i ) d i .
C DT = B φ SD 2 2 α SD + β SD 4 π ln ( 2 ) Γ ( α SD ) Γ ( β SD ) × G 8 , 4 1 , 8 ( 64 A SD 2 L SD 2 γ 0 ξ α SD 2 β SD 2 | 1 , 1 , 1 α SD 2 , 2 α SD 2 , 1 β SD 2 , 2 β SD 2 , 1 φ SD 2 2 , 2 φ SD 2 2 1 , 0 , φ SD 2 2 , 1 φ SD 2 2 ) .
C DT B ln ( 4 γ 0 ξ ) ln ( 4 ) + B ln ( 2 ) 0 i G 2 , 2 2 , 2 ( i | 0 , 0 0 , 0 ) f I SD ( i ) d i B ln ( 2 ) 0 G 2 , 2 2 , 2 ( i | 0 , 0 0 , 0 ) f I SD ( i ) d i .
C DT H B ln ( 4 γ 0 ξ ) ln ( 4 ) + B φ SD 2 G 3 , 5 5 , 2 ( α SD β SD A SD L SD | 0 , 0 , φ SD 2 + 1 φ SD 2 , α SD , β SD , 0 , 0 ) ln ( 2 ) Γ ( α SD ) Γ ( β SD ) B φ SD 2 G 3 , 5 5 , 2 ( α SD β SD A SD L SD | 1 , 1 , φ SD 2 + 1 φ SD 2 , α SD , β SD , 1 , 1 ) ln ( 2 ) Γ ( α SD ) Γ ( β SD ) .
C DT H B ln ( 4 γ 0 ξ ) ln ( 4 ) + B ln ( 2 ) ( ψ ( α SD ) + ψ ( β SD ) 1 φ SD 2 ln ( α SD β SD A SD L SD ) ) .
γ BDF th [ d B ] = 10 ln ( 4 F ξ ) ln ( 10 ) + 10 ln ( 10 ) ( 1 φ SD 2 + 1 φ RD 2 ) + 10 ln ( 10 ) ( ln ( α SD β SD α RD β RD A SD L SD A RD L RD ) ψ ( α SD ) ψ ( β SD ) ψ ( α RD ) ψ ( β RD ) ) .
γ ADF th [ d B ] = 10 ln ( 10 ) ( I 1 + 2 I 2 + P ln ( F ) + ln ( 4 ξ ) ) ) .
γ DT th [ d B ] = 10 ln ( 4 ξ ) ln ( 10 ) + 20 ln ( 10 ) ( ln ( α SD β SD A SD L SD ) + 1 φ SD 2 ψ ( α SD ) ψ ( β SD ) ) .
G BDF [ d B ] = γ DT th [ d B ] γ BDF th [ d B ] .
G ADF [ d B ] = γ DT th [ d B ] γ ADF th [ d B ] .
𝔼 [ I T ] = 8 F 𝔼 [ I T LB ] .
F = 𝔼 [ I T ] 2 8 𝔼 [ I T LB ] 2 .
𝔼 [ I m ] = 0 i f I m ( i ) d i .
𝔼 [ I m ] = A m L m φ m 2 1 + φ m 2 .
𝔼 [ I T LB ] = 0 i 1 / 2 f I T LB ( i ) d i .
𝔼 [ I T LB ] = A SD A RD L SD L RD α SD β SD α RD β RD × 4 φ SD 2 Γ ( α SD + 1 / 2 ) Γ ( β SD + 1 / 2 ) φ RD 2 Γ ( α RD + 1 / 2 ) Γ ( β RD + 1 / 2 ) ( 1 + 2 φ SD 2 ) Γ ( α SD ) Γ ( β SD ) ( 1 + 2 φ RD 2 ) Γ ( α RD ) Γ ( β RD ) .
F = ( A SD L SD φ SD 2 1 + φ SD 2 + 2 A RD L RD φ RD 2 1 + φ RD 2 ) 2 α SD β SD ( 1 + 2 φ SD 2 ) 2 Γ ( α SD ) 2 Γ ( β SD ) 2 128 A SD L SD φ SD 4 Γ ( α SD + 1 / 2 ) 2 Γ ( β SD + 1 / 2 ) 2 × α RD β RD ( 1 + 2 φ RD 2 ) 2 Γ ( α RD ) 2 Γ ( β RD ) 2 A RD L RD φ RD 4 Γ ( α RD + 1 / 2 ) 2 Γ ( β RD + 1 / 2 ) 2 .
F = 𝔼 [ I SD + 2 I RD ] 2 8 𝔼 [ I SD I RD ] 2 , I RD > I SD .
E [ I SD + 2 I RD ] = 0 0 i 2 ( i 1 + 2 i 2 ) f I SD ( i 1 ) f I RD ( i 2 ) d i 1 d i 2 = A + 2 B = 0 [ 0 i 2 i 1 f I SD ( i 1 ) d i 1 ] f I RD ( i 2 ) d i 2 + 2 0 i 2 F I SD ( i 2 ) F I RD ( i 2 ) d i 2 .
A = φ RD 2 φ SD 2 A SD L SD G 5 , 5 4 , 3 ( α RD β RD A SD L SD α SD β SD A RD L RD | φ SD 2 , α SD , β SD , 1 , φ RD 2 + 1 α RD 2 , α RD , β RD , 0 , φ SD 2 1 ) α SD β SD Γ ( α RD ) Γ ( α SD ) Γ ( β RD ) Γ ( β SD ) .
B = 2 φ RD 2 φ SD 2 A SD L SD G 5 , 5 4 , 3 ( α RD β RD A SD L SD α SD β SD A RD L RD | φ SD 2 , α SD , β SD , 0 , φ RD 2 + 1 α RD 2 , α RD , β RD , 1 , φ SD 2 1 ) α SD β SD Γ ( α RD ) Γ ( α SD ) Γ ( β RD ) Γ ( β SD ) .
𝔼 [ I SD I RD ] = 0 0 i 2 i 1 i 2 f I SD ( i 1 ) f I RD ( i 2 ) d i 1 d i 2 = C = 0 i 2 [ 0 i 2 i 1 f I SD ( i 1 ) d i 1 ] f I RD ( i 2 ) d i 2 .
C = φ RD 2 φ SD 2 A SD L SD G 5 , 5 4 , 3 ( α RD β RD A SD L SD α SD β SD A RD L RD | φ SD 2 , α SD , β SD , 1 2 , φ RD 2 + 1 α RD 2 , α RD , β RD , 1 2 , φ SD 2 1 ) α SD β SD Γ ( α RD ) Γ ( α SD ) Γ ( β RD ) Γ ( β SD ) .
F = ( A + 2 B ) 2 8 C 2 .
S = B φ SD 2 ( M 1 M 2 ) ln ( 2 ) Γ ( α SD ) Γ ( β SD ) = B φ SD 2 ln ( 2 ) Γ ( α SD ) Γ ( β SD ) × ( G 3 , 5 5 , 2 ( α SD β SD A SD L SD | 0 , 0 , φ SD 2 + 1 φ SD 2 , α SD , β SD , 0 , 0 ) G 3 , 5 5 , 2 ( α SD β SD A SD L SD | 1 , 1 , φ SD 2 + 1 φ SD 2 , α SD , β SD , 1 , 1 ) ) .
M 2 = ( α SD β SD A SD L SD ) φ SD 2 π 4 csc [ π φ SD 2 ] 2 csc [ π ( α SD φ SD 2 ) ] csc [ π ( β SD φ SD 2 ) ] Γ ( 1 α SD + φ SD 2 ) Γ ( 1 β SD φ SD 2 ) + ( α SD β SD A SD L SD ) α SD k = 0 ( α SD β SD A SD L SD ) k π 3 csc [ π α SD ] 2 csc [ π ( α SD β SD ) ] k ! ( k + α SD φ SD 2 ) Γ ( 1 + k + α SD β SD ) ( α SD β SD A SD L SD ) β SD k = 0 ( α SD β SD A SD L SD ) k π 3 csc [ π β SD ] 2 csc [ π ( α SD β SD ) ] k ! ( k + β SD φ SD 2 ) Γ ( 1 + k + β SD α SD ) + k = 1 Γ ( α SD k ) Γ ( β SD k ) φ SD 2 k ( ψ ( α SD k ) + ψ ( β SD k ) + 1 k φ SD 2 ln ( α SD β SD A SD L SD ) ) .
M 1 = M 2 + Γ ( α SD ) Γ ( β SD ) φ SD 2 ( ψ ( α SD ) + ψ ( β SD ) 1 φ SD 2 ln ( α SD β SD A SD L SD ) ) .
S = B ln ( 2 ) ( ψ ( α SD ) + ψ ( β SD ) 1 φ SD 2 ln ( α SD β SD A SD L SD ) ) .

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