Abstract

We investigate the performance of coherent free-space optical (FSO) system in terms of bit error rate (BER) evaluation by adopting the modified Rician distribution based coherent channel model which allows taking into consideration the composite effects of both Rician turbulence, including amplitude fluctuation and optical phase distortion, and pointing errors (PEs). By expanding the Rician distribution, a mathematically traceable expression of the probability density function (PDF) for the composite channel is derived in the form of the Meijer-G function. Based on the composite channel PDF, the exact BER expression is obtained, allowing the analysis of BER performance for single-input single-output (SISO) links. This analysis is extended to single-input multi-output (SIMO) links with maximal ratio combining (MRC). With the help of the moment generating function (MGF), the exact BER expression can be simplified into a single integral, facilitating the analysis with high accuracy and reducing calculation complexity. Engineering insights including high-SNR approximated channel PDF, asymptotic BER expression, coding and diversity gains, are investigated and cross-validated for both SISO and SIMO links. Through both analytical and numerical verifications, the impairment due to PEs as well as the effect of modal compensation on the BER performance are discussed in detail, unveiling the fact that their inner relations should be taken into account for optimization. These verify the effectiveness of our models for both SISO and SIMO links with a wide range of different conditions and can be feasibly applied for different types of coherent FSO links.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (4)

2017 (5)

E. Zedini, H. Soury, and M.-S. Alouini, “Dual-hop FSO transmission systems over gamma–gamma turbulence with pointing errors,” IEEE Trans. Wirel. Commun. 16(2), 784–796 (2017).
[Crossref]

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[Crossref]

Z. Zeng, S. Fu, H. Zhang, Y. Dong, and J. Cheng, “A survey of underwater optical wireless communications,” IEEE Commun. Surv. Tut. 19(1), 204–238 (2017).
[Crossref]

H. Kaushal and G. Kaddoum, “Optical communication in space: challenges and mitigation techniques,” IEEE Commun. Surv. Tut. 19(1), 57–96 (2017).
[Crossref]

A. Mansour, R. Mesleh, and M. Abaza, “New challenges in wireless and free space optical communications,” Opt. Lasers Eng. 89, 95–108 (2017).
[Crossref]

2016 (2)

M. R. Bhatnagar and Z. Ghassemlooy, “Performance analysis of gamma–gamma fading FSO MIMO links with pointing errors,” J. Lightwave Technol. 34(9), 2158–2169 (2016).
[Crossref]

J. Park, E. Lee, C.-B. Chae, and G. Yoon, “Impact of pointing errors on the performance of coherent free-space optical systems,” IEEE Photonics Technol. Lett. 28(2), 181–184 (2016).
[Crossref]

2015 (3)

M. Li and M. Cvijetic, “Coherent free space optics communications over the maritime atmosphere with use of adaptive optics for beam wavefront correction,” Appl. Opt. 54(6), 1453–1462 (2015).
[Crossref]

J. Park, E. Lee, C.-B. Chae, and G. Yoon, “Outage probability analysis of a coherent fso amplify-and-forward relaying system,” IEEE Photonics Technol. Lett. 27(11), 1204–1207 (2015).
[Crossref]

J. Park, E. Lee, C.-B. Chae, and G. Yoon, “Performance analysis of coherent free-space optical systems with multiple receivers,” IEEE Photonics Technol. Lett. 27(9), 1010–1013 (2015).
[Crossref]

2014 (2)

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

C. Liu, S. Chen, X. Li, and H. Xian, “Performance evaluation of adaptive optics for atmospheric coherent laser communications,” Opt. Express 22(13), 15554–15563 (2014).
[Crossref]

2013 (1)

C. Tang, R. Li, Y. Shao, N. Chi, J. Yu, Z. Dong, and G. Chang, “Experimental demonstration for 40 km fiber and 2 m wireless transmission of 4 Gb/s OOK signals at 100 GHz carrier,” Chin. Opt. Lett. 11(2), 20608–20610 (2013).
[Crossref]

2012 (3)

2011 (1)

2010 (1)

2009 (2)

2008 (1)

2007 (1)

2003 (1)

Z. Wang and G. B. Giannakis, “A simple and general parameterization quantifying performance in fading channels,” IEEE Trans. Commun. 51(8), 1389–1398 (2003).
[Crossref]

Abaza, M.

A. Mansour, R. Mesleh, and M. Abaza, “New challenges in wireless and free space optical communications,” Opt. Lasers Eng. 89, 95–108 (2017).
[Crossref]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of mathematical functions: with formulas, graphs, and mathematical tables, vol. 55 (Courier Corporation, 1965).

Adamchik, V.

V. Adamchik and O. Marichev, “The algorithm for calculating integrals of hypergeometric type functions and its realization in REDUCE system,” in Proceedings of the international symposium on Symbolic and algebraic computation, (ACM, 1990), pp.212–224.

Aghajanzadeh, S.

Alouini, M.-S.

E. Zedini, H. Soury, and M.-S. Alouini, “Dual-hop FSO transmission systems over gamma–gamma turbulence with pointing errors,” IEEE Trans. Wirel. Commun. 16(2), 784–796 (2017).
[Crossref]

M. K. Simon and M.-S. Alouini, Digital communication over fading channels (John Wiley & Sons, 2005).

Amirabadi, M. A.

M. A. Amirabadi and V. T. Vakili, “A new optimization problem in FSO communication system,” IEEE Commun. Lett. 22(7), 1442–1445 (2018).
[Crossref]

Andrews, L. C.

L. C. Andrews and R. L. Phillips, Laser beam propagation through random media (SPIE, Bellingham, WA, 2005).

Belmonte, A.

Bhatnagar, M. R.

Bian, Y.

Castillo-Vázquez, B.

Castillo-Vázquez, C.

Chae, C.-B.

J. Park, E. Lee, C.-B. Chae, and G. Yoon, “Impact of pointing errors on the performance of coherent free-space optical systems,” IEEE Photonics Technol. Lett. 28(2), 181–184 (2016).
[Crossref]

J. Park, E. Lee, C.-B. Chae, and G. Yoon, “Performance analysis of coherent free-space optical systems with multiple receivers,” IEEE Photonics Technol. Lett. 27(9), 1010–1013 (2015).
[Crossref]

J. Park, E. Lee, C.-B. Chae, and G. Yoon, “Outage probability analysis of a coherent fso amplify-and-forward relaying system,” IEEE Photonics Technol. Lett. 27(11), 1204–1207 (2015).
[Crossref]

Chang, G.

C. Tang, R. Li, Y. Shao, N. Chi, J. Yu, Z. Dong, and G. Chang, “Experimental demonstration for 40 km fiber and 2 m wireless transmission of 4 Gb/s OOK signals at 100 GHz carrier,” Chin. Opt. Lett. 11(2), 20608–20610 (2013).
[Crossref]

Chen, S.

Cheng, J.

Z. Zeng, S. Fu, H. Zhang, Y. Dong, and J. Cheng, “A survey of underwater optical wireless communications,” IEEE Commun. Surv. Tut. 19(1), 204–238 (2017).
[Crossref]

M. Niu, X. Song, J. Cheng, and J. F. Holzman, “Performance analysis of coherent wireless optical communications with atmospheric turbulence,” Opt. Express 20(6), 6515–6520 (2012).
[Crossref]

Chi, N.

C. Tang, R. Li, Y. Shao, N. Chi, J. Yu, Z. Dong, and G. Chang, “Experimental demonstration for 40 km fiber and 2 m wireless transmission of 4 Gb/s OOK signals at 100 GHz carrier,” Chin. Opt. Lett. 11(2), 20608–20610 (2013).
[Crossref]

Cvijetic, M.

Czichy, R.

R. Lange, B. Smutny, B. Wandernoth, R. Czichy, and D. Giggenbach, “142 km, 5.625 Gbps free-space optical link based on homodyne bpsk modulation,” in Free-Space Laser Communication Technologies XVIII, vol. 6105 (International Society for Optics and Photonics, 2006), p. 61050A.

Dhungana, Y.

Y. Dhungana and C. Tellambura, “New simple approximations for error probability and outage in fading,” IEEE Commun. Lett. 16(11), 1760–1763 (2012).
[Crossref]

Dong, Y.

Z. Zeng, S. Fu, H. Zhang, Y. Dong, and J. Cheng, “A survey of underwater optical wireless communications,” IEEE Commun. Surv. Tut. 19(1), 204–238 (2017).
[Crossref]

Z. Zhu, H. Zhou, W. Xie, J. Qin, and Y. Dong, “10 Gb/s homodyne receiver based on costas loop with enhanced dynamic performance,” in 16th International Conference on Optical Communications and Networks (ICOCN), (IEEE, 2017), pp. 1–3.

Dong, Z.

C. Tang, R. Li, Y. Shao, N. Chi, J. Yu, Z. Dong, and G. Chang, “Experimental demonstration for 40 km fiber and 2 m wireless transmission of 4 Gb/s OOK signals at 100 GHz carrier,” Chin. Opt. Lett. 11(2), 20608–20610 (2013).
[Crossref]

Farid, A. A.

Fu, S.

Z. Zeng, S. Fu, H. Zhang, Y. Dong, and J. Cheng, “A survey of underwater optical wireless communications,” IEEE Commun. Surv. Tut. 19(1), 204–238 (2017).
[Crossref]

García-Zambrana, A.

Ghassemlooy, Z.

Giannakis, G. B.

Z. Wang and G. B. Giannakis, “A simple and general parameterization quantifying performance in fading channels,” IEEE Trans. Commun. 51(8), 1389–1398 (2003).
[Crossref]

Giggenbach, D.

R. Lange, B. Smutny, B. Wandernoth, R. Czichy, and D. Giggenbach, “142 km, 5.625 Gbps free-space optical link based on homodyne bpsk modulation,” in Free-Space Laser Communication Technologies XVIII, vol. 6105 (International Society for Optics and Photonics, 2006), p. 61050A.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products (Elsevier, 2014).

Guo, H.

Holzman, J. F.

Hong, X.

Hranilovic, S.

Ian, P. G.

Jian, W.

Kaddoum, G.

H. Kaushal and G. Kaddoum, “Optical communication in space: challenges and mitigation techniques,” IEEE Commun. Surv. Tut. 19(1), 57–96 (2017).
[Crossref]

Kahn, J. M.

Karagiannidis, G. K.

Kaushal, H.

H. Kaushal and G. Kaddoum, “Optical communication in space: challenges and mitigation techniques,” IEEE Commun. Surv. Tut. 19(1), 57–96 (2017).
[Crossref]

Khalighi, M. A.

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

Lange, R.

R. Lange, B. Smutny, B. Wandernoth, R. Czichy, and D. Giggenbach, “142 km, 5.625 Gbps free-space optical link based on homodyne bpsk modulation,” in Free-Space Laser Communication Technologies XVIII, vol. 6105 (International Society for Optics and Photonics, 2006), p. 61050A.

Lee, E.

J. Park, E. Lee, C.-B. Chae, and G. Yoon, “Impact of pointing errors on the performance of coherent free-space optical systems,” IEEE Photonics Technol. Lett. 28(2), 181–184 (2016).
[Crossref]

J. Park, E. Lee, C.-B. Chae, and G. Yoon, “Outage probability analysis of a coherent fso amplify-and-forward relaying system,” IEEE Photonics Technol. Lett. 27(11), 1204–1207 (2015).
[Crossref]

J. Park, E. Lee, C.-B. Chae, and G. Yoon, “Performance analysis of coherent free-space optical systems with multiple receivers,” IEEE Photonics Technol. Lett. 27(9), 1010–1013 (2015).
[Crossref]

Li, M.

Li, R.

C. Tang, R. Li, Y. Shao, N. Chi, J. Yu, Z. Dong, and G. Chang, “Experimental demonstration for 40 km fiber and 2 m wireless transmission of 4 Gb/s OOK signals at 100 GHz carrier,” Chin. Opt. Lett. 11(2), 20608–20610 (2013).
[Crossref]

Li, W.

Li, X.

Li, Y.

Liu, C.

Mansour, A.

A. Mansour, R. Mesleh, and M. Abaza, “New challenges in wireless and free space optical communications,” Opt. Lasers Eng. 89, 95–108 (2017).
[Crossref]

Marichev, O.

V. Adamchik and O. Marichev, “The algorithm for calculating integrals of hypergeometric type functions and its realization in REDUCE system,” in Proceedings of the international symposium on Symbolic and algebraic computation, (ACM, 1990), pp.212–224.

Mesleh, R.

A. Mansour, R. Mesleh, and M. Abaza, “New challenges in wireless and free space optical communications,” Opt. Lasers Eng. 89, 95–108 (2017).
[Crossref]

Niu, M.

Odeyemi, K. O.

K. O. Odeyemi, P. A. Owolawi, and V. M. Srivastava, “Optical spatial modulation over gamma–gamma turbulence and pointing error induced fading channels,” Optik 147, 214–223 (2017).
[Crossref]

Owolawi, P. A.

K. O. Odeyemi, P. A. Owolawi, and V. M. Srivastava, “Optical spatial modulation over gamma–gamma turbulence and pointing error induced fading channels,” Optik 147, 214–223 (2017).
[Crossref]

Park, J.

J. Park, E. Lee, C.-B. Chae, and G. Yoon, “Impact of pointing errors on the performance of coherent free-space optical systems,” IEEE Photonics Technol. Lett. 28(2), 181–184 (2016).
[Crossref]

J. Park, E. Lee, C.-B. Chae, and G. Yoon, “Performance analysis of coherent free-space optical systems with multiple receivers,” IEEE Photonics Technol. Lett. 27(9), 1010–1013 (2015).
[Crossref]

J. Park, E. Lee, C.-B. Chae, and G. Yoon, “Outage probability analysis of a coherent fso amplify-and-forward relaying system,” IEEE Photonics Technol. Lett. 27(11), 1204–1207 (2015).
[Crossref]

Phillips, R. L.

L. C. Andrews and R. L. Phillips, Laser beam propagation through random media (SPIE, Bellingham, WA, 2005).

Qin, J.

Z. Zhu, H. Zhou, W. Xie, J. Qin, and Y. Dong, “10 Gb/s homodyne receiver based on costas loop with enhanced dynamic performance,” in 16th International Conference on Optical Communications and Networks (ICOCN), (IEEE, 2017), pp. 1–3.

Qiu, J.

Rose, T.

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products (Elsevier, 2014).

Sandalidis, H. G.

Shao, Y.

C. Tang, R. Li, Y. Shao, N. Chi, J. Yu, Z. Dong, and G. Chang, “Experimental demonstration for 40 km fiber and 2 m wireless transmission of 4 Gb/s OOK signals at 100 GHz carrier,” Chin. Opt. Lett. 11(2), 20608–20610 (2013).
[Crossref]

Simon, M. K.

M. K. Simon and M.-S. Alouini, Digital communication over fading channels (John Wiley & Sons, 2005).

Smutny, B.

R. Lange, B. Smutny, B. Wandernoth, R. Czichy, and D. Giggenbach, “142 km, 5.625 Gbps free-space optical link based on homodyne bpsk modulation,” in Free-Space Laser Communication Technologies XVIII, vol. 6105 (International Society for Optics and Photonics, 2006), p. 61050A.

Song, X.

Song, Z.

Soury, H.

E. Zedini, H. Soury, and M.-S. Alouini, “Dual-hop FSO transmission systems over gamma–gamma turbulence with pointing errors,” IEEE Trans. Wirel. Commun. 16(2), 784–796 (2017).
[Crossref]

Srivastava, V. M.

K. O. Odeyemi, P. A. Owolawi, and V. M. Srivastava, “Optical spatial modulation over gamma–gamma turbulence and pointing error induced fading channels,” Optik 147, 214–223 (2017).
[Crossref]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of mathematical functions: with formulas, graphs, and mathematical tables, vol. 55 (Courier Corporation, 1965).

Tang, C.

C. Tang, R. Li, Y. Shao, N. Chi, J. Yu, Z. Dong, and G. Chang, “Experimental demonstration for 40 km fiber and 2 m wireless transmission of 4 Gb/s OOK signals at 100 GHz carrier,” Chin. Opt. Lett. 11(2), 20608–20610 (2013).
[Crossref]

Tellambura, C.

Y. Dhungana and C. Tellambura, “New simple approximations for error probability and outage in fading,” IEEE Commun. Lett. 16(11), 1760–1763 (2012).
[Crossref]

Tsiftsis, T. A.

Uysal, M.

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

S. Aghajanzadeh and M. Uysal, “Diversity–multiplexing trade-off in coherent free-space optical systems with multiple receivers,” J. Opt. Commun. Netw. 2(12), 1087–1094 (2010).
[Crossref]

Vakili, V. T.

M. A. Amirabadi and V. T. Vakili, “A new optimization problem in FSO communication system,” IEEE Commun. Lett. 22(7), 1442–1445 (2018).
[Crossref]

Wandernoth, B.

R. Lange, B. Smutny, B. Wandernoth, R. Czichy, and D. Giggenbach, “142 km, 5.625 Gbps free-space optical link based on homodyne bpsk modulation,” in Free-Space Laser Communication Technologies XVIII, vol. 6105 (International Society for Optics and Photonics, 2006), p. 61050A.

Wang, Z.

Z. Wang and G. B. Giannakis, “A simple and general parameterization quantifying performance in fading channels,” IEEE Trans. Commun. 51(8), 1389–1398 (2003).
[Crossref]

Weijun, T.

Xian, H.

Xie, W.

Z. Zhu, H. Zhou, W. Xie, J. Qin, and Y. Dong, “10 Gb/s homodyne receiver based on costas loop with enhanced dynamic performance,” in 16th International Conference on Optical Communications and Networks (ICOCN), (IEEE, 2017), pp. 1–3.

Xu, G.

Yang, C.

Yoon, G.

J. Park, E. Lee, C.-B. Chae, and G. Yoon, “Impact of pointing errors on the performance of coherent free-space optical systems,” IEEE Photonics Technol. Lett. 28(2), 181–184 (2016).
[Crossref]

J. Park, E. Lee, C.-B. Chae, and G. Yoon, “Performance analysis of coherent free-space optical systems with multiple receivers,” IEEE Photonics Technol. Lett. 27(9), 1010–1013 (2015).
[Crossref]

J. Park, E. Lee, C.-B. Chae, and G. Yoon, “Outage probability analysis of a coherent fso amplify-and-forward relaying system,” IEEE Photonics Technol. Lett. 27(11), 1204–1207 (2015).
[Crossref]

Yu, J.

C. Tang, R. Li, Y. Shao, N. Chi, J. Yu, Z. Dong, and G. Chang, “Experimental demonstration for 40 km fiber and 2 m wireless transmission of 4 Gb/s OOK signals at 100 GHz carrier,” Chin. Opt. Lett. 11(2), 20608–20610 (2013).
[Crossref]

Yura, H.

Zedini, E.

E. Zedini, H. Soury, and M.-S. Alouini, “Dual-hop FSO transmission systems over gamma–gamma turbulence with pointing errors,” IEEE Trans. Wirel. Commun. 16(2), 784–796 (2017).
[Crossref]

Zeng, Z.

Z. Zeng, S. Fu, H. Zhang, Y. Dong, and J. Cheng, “A survey of underwater optical wireless communications,” IEEE Commun. Surv. Tut. 19(1), 204–238 (2017).
[Crossref]

Zhang, H.

Z. Zeng, S. Fu, H. Zhang, Y. Dong, and J. Cheng, “A survey of underwater optical wireless communications,” IEEE Commun. Surv. Tut. 19(1), 204–238 (2017).
[Crossref]

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Figures (3)

Fig. 1.
Fig. 1. BER performance of coherent FSO-SISO link in the composite channel with Rician turbulence and PEs, under (a) weak turbulence $SI = 0.1$, (b) strong turbulence $SI = 1.6$. To represent the impacts of PEs from negligible to severe, the $\xi$ is varied as $\xi = 6.7,1.4,0.7,0.4$. Analytical results (solid lines), asymptotic results (dashed lines), and numerical simulation (*) results are both included.
Fig. 2.
Fig. 2. BER performance under various compensation modes $(J = 1, 3, 6)$ over the composite Rician turbulence and PE fading channel. The $J = 1, 3, 6$ describes compensation for piston, tip-tilt, astigmatism, respectively. To represent general turbulence, moderate turbulence of $SI = 0.33$ is considered.
Fig. 3.
Fig. 3. Comparison of BER performance for MRC-SIMO link in cases of different number of receiver branches $(L = 1, 2, 4)$ under moderate turbulence $(SI = 0.33)$ with varied PEs. Two situations are involved, (a) without and (b) with modal compensation $(J = 3)$. The BER for the SISO link $(L = 1)$ is depicted as a benchmark.

Equations (29)

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i ( t ) = i a c ( t ) + i n ( t )
P h a ( h a ) = 1 + K h a e K exp ( 1 + K h a h a ) I 0 [ 2 ( 1 + K ) K h a h a ]
f h p ( h p ) = ξ 2 A 0 ξ 2 h p ξ 2 1 0 h p A 0
f h ( h ) = f h h a ( h | h a ) f h a ( h a ) d h a
f h | h a ( h | h a ) = 1 h a h l ξ 2 A 0 ξ 2 ( h h a h l ) ξ 2 1 , 0 h A 0 h a h l
P h a ( h a ) = i = 1 N α i h a i 1 exp ( 1 + K h a h a )
f h ( h ) = ξ 2 h ξ 2 1 ( A 0 h l ) ξ 2 i = 1 N α i h A 0 h i h a ξ 2 + i 1 e 1 + K h a h a d h a
f h ( h ) = ξ 2 ( A 0 h l ) ξ 2 h ξ 2 1 i = 1 N α i ( 1 + K h a ) ξ 2 i Γ ( ξ 2 + i , 1 + K A 0 h l h a h )
f h ( h ) = ξ 2 A 0 h l i = 1 N α i ( 1 + K h a ) 1 i G 1 , 2 2 , 0 [ 1 + K A 0 h l h a h | ξ 2 ξ 2 1 , i 1 ]
P e = 0 + P e ( γ 0 | h ) f h ( h ) d h
B E R = ξ 2 2 π A 0 h l 1 γ 0 i = 1 N α i ( 1 + K h a ) 1 i G 3 , 3 2 , 2 [ 1 + K A 0 h l h a γ 0 | 0 , 0.5 , ξ 2 ξ 2 1 , i 1 , 1 ]
B E R = 1 2 i = 1 N α i { ζ 2 Γ ( i + 0.5 ) ( i ζ 2 ) i π ( A 0 γ 0 ) i [ 3 F 2 ( i ζ 2 , i , i + 0.5 ; i ζ 2 + 1 , i + 1 ; 1 A 0 γ 0 1 + K h a ) ] + Γ ( i ζ 2 ) Γ ( ζ 2 + 0.5 ) π ( A 0 γ 0 ) ζ 2 ( 1 + K h a ) ζ 2 i }
G 1 , 2 2 , 0 [ z | b 1 , b 2 a 1 ] Γ ( b 2 b 1 ) Γ ( a 1 b 1 ) z b 1 + Γ ( b 1 b 2 ) Γ ( a 1 b 2 ) z b 2 + o ( z )
a = { ξ 2 e K ( ξ 2 1 ) ( 1 + K ) A 0 h l h a , ξ > 1 ξ 2 e K Γ ( 1 ξ 2 ) ( 1 + K A 0 h l h a ) ξ 2 , ξ < 1
b = { 0 , ξ > 1 ξ 2 1 , ξ < 1
f h 0 ( h ) = { ξ 2 e K ( ξ 2 1 ) ( 1 + K ) A 0 h 1 h a ,   ξ > 1 ξ 2 e r Γ ( 1 ξ 2 ) ( 1 + K A 0 h l h a ) ξ 2 h ξ 2 1 , ξ < 1
B E R S I S O , a s y m = { Γ ( 1.5 ) e K ξ 2 2 π ( ξ 2 1 ) ( 1 + K ) ( A 0 h l h a ) γ 0 1 ,   ξ > 1 e K Γ ( 1 ξ 2 ) Γ ( ξ 2 + 0.5 ) 2 π ( 1 + K A 0 h l h a ) ξ 2 γ 0 ξ 2 , ξ < 1
G c , S I S O = { ( e K ξ 2 Γ ( 1.5 ) 2 π ( ξ 2 1 ) ( 1 + K ) ( A 0 h l h a ) ) 1 ,   ξ > 1 ( e K Γ ( 1 ξ 2 ) 2 π ( 1 + K A 0 h l h a ) ξ 2 ) 1 / ξ 2 , ξ < 1
G d , S I S O = { 1 , ξ > 1 ξ 2 , ξ < 1
Loss ( dB ) = 10 log 10 ( ζ 2 A 0 ( ζ 2 1 ) ) , ξ > 1
B E R = 1 π 0 π 2 MGF M R C ( s = 1 sin 2 θ ) d θ = 1 π 0 π 2 l = 1 L MGF γ l ( s = 1 sin 2 θ ) d θ
f γ i ( γ l ) = ξ 2 A 0 h l γ o i = 1 N α i ( ( 1 + K ) h a ) 1 i G 1 , 2 2 , 0 [ ( 1 + K ) γ l A 0 h l h a γ o | ξ 2 ξ 2 1 , i 1 ]
M G F γ i ( s ) = ξ 2 s A 0 h l γ o i = 1 N α i ( ( 1 + K ) h a ) 1 i G 2 , 2 2 , 1 [ ( 1 + K ) s A 0 h l h a γ o | 0 , ξ 2 ξ 2 1 , i 1 ]
B E R = 1 π 0 π 2 l = 1 L { ξ 2 A 0 h l γ o i = 1 N α i ( ( 1 + K ) h a ) 1 i G 2 , 2 2 , 1 [ ( 1 + K ) sin 2 θ A 0 h l h a γ o | 0 , ξ 2 ξ 2 1 , i 1 ] } d θ
M G F γ ( s ) = { [ ξ 2 e K ( 1 + K ) ( ξ 2 1 ) A 0 h 1 h a γ o ] L s L + o ( s ) , ξ > 1 e r Γ ( 1 ξ 2 ) Γ ( 1 + ξ 2 ) ( ( 1 + K ) A 0 h l h a γ o ) ξ 2 ] L s L ξ 2 + o ( s ) , ξ < 1
f M R C , a s y m ( h ) = { 1 ( L 1 ) ! [ ξ 2 e K ( 1 + K ) ( ξ 2 1 ) A 0 h l h a ] L h ( L 1 ) , ξ > 1 1 Γ ( L ξ 2 ) [ ξ 2 e r Γ ( 1 ξ 2 ) Γ ( ξ 2 ) ( ( 1 + K ) A 0 h l h a γ o ) ξ 2 ] L h L ξ 2 1 , ξ < 1
B E R M R C , a s y m = { 1 2 π L ! [ ξ 2 e K ( 1 + K ) ( ξ 2 1 ) A 0 h l h a ] L Γ ( L + 0.5 ) ( γ 0 ) L , ξ > 1 Γ ( L ξ 2 + 0.5 ) 2 π Γ ( L ξ 2 + 1 ) [ e r Γ ( 1 ξ 2 ) Γ ( ξ 2 + 1 ) ( ( 1 + K ) A 0 h l h a γ o ) ξ 2 ] L γ 0 L ξ 2 ,   ξ < 1
G c , M R C = { ( Γ ( L + 0.5 ) 2 π L ! [ ξ 2 e K ( 1 + K ) ( ξ 2 1 ) A 0 h l h a ] 1 L , ξ > 1 ( Γ ( L ξ 2 + 0.5 ) 2 π Γ ( L ξ 2 + 1 ) [ e r Γ ( 1 ξ 2 ) Γ ( ξ 2 + 1 ) ( ( 1 + K ) A 0 h l h a γ o ) ξ 2 ] 1 / ( L ξ 2 ) , ξ < 1
G d , M R C = { L , ξ > 1 L ξ 2 , ξ < 1

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