Abstract

A thorough investigation of the impact of nonzero boresight pointing errors on the ergodic capacity of multiple-input/multiple-output (MIMO) free-space optical (FSO) systems with equal gain combining (EGC) reception under different turbulence models, which are modeled as statistically independent, but not necessarily identically distributed (i.n.i.d.) is addressed in this paper. Novel closed-form asymptotic expressions at high signal-to-noise ratio (SNR) for the ergodic capacity of MIMO FSO systems are derived when different geometric arrangements of the receive apertures at the receiver are considered in order to reduce the effect of nonzero inherent boresight displacement, which is inevitably present when more than one receive aperture is considered. As a result, the asymptotic ergodic capacity of MIMO FSO systems is evaluated over log-normal (LN), gamma-gamma (GG) and exponentiated Weibull (EW) atmospheric turbulence in order to study different turbulence conditions, different sizes of receive apertures as well as different aperture averaging conditions. It is concluded that the use of single-input/multiple-output (SIMO) and MIMO techniques can significantly increase the ergodic capacity respect to the direct path link when the inherent boresight displacement takes small values, i.e. when the spacing among receive apertures is not too big. The effect of nonzero additional boresight errors, which is due to the thermal expansion of the building, is evaluated in multiple-input/single-output (MISO) and single-input/single-output (SISO) FSO systems. Simulation results are further included to confirm the analytical results.

© 2016 Optical Society of America

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2016 (2)

I. Ansari, F. Yilmaz, and M. Alouini, “Performance Analysis of Free-Space Optical Links Over Malaga (M) Turbulence channels with pointing errors,” IEEE Trans. Wireless Commun. 15(1), 91–102 (2016).
[Crossref]

R. Boluda-Ruiz, A. Garcia-Zambrana, B. Castillo-Vazquez, and C. Castillo-Vazquez, “Ergodic capacity analysis of decode-and-forward relay-assisted FSO systems over alpha-mu fading channels considering pointing errors,” IEEE Photonics J. 8(1), 1–11 (2016).
[Crossref]

2015 (10)

R. Boluda-Ruiz, A. García-Zambrana, B. Castillo-Vázquez, and C. Castillo-Vázquez, “Ergodic capacity analysis for DF strategies in cooperative FSO systems,” Opt. Express 23(17), 21565–21584 (2015).
[Crossref] [PubMed]

X. Yi and M. Yao, “Free-space communications over exponentiated Weibull turbulence channels with nonzero boresight pointing errors,” Opt. Express 23(3), 2904–2917 (2015).
[Crossref] [PubMed]

I. Ansari, M.-S. Alouini, and J. Cheng, “Ergodic capacity analysis of free-space optical links with nonzero bore-sight pointing errors,” IEEE Trans. Wireless Commun. 14(8), 4248–4264 (2015).
[Crossref]

P. Wang, T. Cao, L. Guo, R. Wang, and Y. Yang, “Performance Analysis of Multihop Parallel Free-Space Optical Systems Over Exponentiated Weibull Fading Channels,” IEEE Photonics J. 7(1), 1–17 (2015).

P. Wang, J. Zhang, L. Guo, T. Shang, T. Cao, R. Wang, and Y. Yang, “Performance analysis for relay-aided multihop BPPM FSO communication system over exponentiated Weibull fading channels with pointing error impairments,” IEEE Photonics J. 7(4), 1–20 (2015).

P. Kaur, V. Jain, and S. Kar, “Performance analysis of free space optical links using multi-input multi-output and aperture averaging in presence of turbulence and various weather conditions,” IET Commun. 9(8), 1104–1109 (2015).
[Crossref]

S. Jiang, G. Yang, Y. Wei, M. Bi, Y. Lu, X. Zhou, M. Hu, and Q. Li, “Performance analysis of space-diversity free-space optical links over exponentiated Weibull channels,” IEEE Photonics Technol. Lett. 27(21), 2250–2252 (2015).
[Crossref]

A. García-Zambrana, R. Boluda-Ruiz, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Novel space-time trellis codes for free-space optical communications using transmit laser selection,” Opt. Express 23(19), 24195–24211 (2015).
[Crossref]

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. 33(9), 1925–1934 (2015).
[Crossref]

R. Boluda-Ruiz, A. García-Zambrana, B. Castillo-Vázquez, and C. Castillo-Vázquez, “On the capacity of MISO FSO systems over gamma-gamma and misalignment fading channels,” Opt. Express 23(17), 22371–22385 (2015).
[Crossref] [PubMed]

2014 (7)

L. Yang, M. O. Hasna, and X. Gao, “Asymptotic BER analysis of FSO with multiple receive apertures over ℳ-distributed turbulence channels with pointing errors,” Opt. Express 22(15), 18238–18245 (2014).
[Crossref] [PubMed]

A. García-Zambrana, R. Boluda-Ruiz, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Transmit alternate laser selection with time diversity for FSO communications,” Opt. Express 22(20), 23861–23874 (2014).
[Crossref] [PubMed]

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]

F. Yang, J. Cheng, and T. Tsiftsis, “Free-space optical communication with nonzero boresight pointing errors,” IEEE Trans. Commun. 62(2), 713–725 (2014).
[Crossref]

J.-Y. Wang, J.-B. Wang, M. Chen, Y. Tang, and Y. Zhang, “Outage analysis for relay-aided free-space optical communications over turbulence channels with nonzero boresight pointing errors,” IEEE Photonics J. 6(4), 1–15 (2014).

P. Wang, L. Zhang, L. Guo, F. Huang, T. Shang, R. Wang, and Y. Yang, “Average BER of subcarrier intensity modulated free space optical systems over the exponentiated Weibull fading channels,” Opt. Express 22(17), 20828–20841 (2014).
[Crossref] [PubMed]

M. Cheng, Y. Zhang, J. Gao, F. Wang, and F. Zhao, “Average capacity for optical wireless communication systems over exponentiated Weibull distribution non-Kolmogorov turbulent channels,” Appl. Opt. 53(18), 4011–4017 (2014).
[Crossref] [PubMed]

2013 (3)

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

P. Deng, M. Kavehrad, Z. Liu, Z. Zhou, and X. Yuan, “Capacity of MIMO free space optical communications using multiple partially coherent beams propagation through non-Kolmogorov strong turbulence,” Opt. Express 21(13), 15213–15229 (2013).
[Crossref] [PubMed]

M. Bhatnagar, “Differential Decoding of SIM DPSK over FSO MIMO Links,” IEEE Commun. Lett. 17(1), 79–82 (2013).
[Crossref]

2012 (4)

2011 (2)

2010 (1)

2009 (5)

S. D. Lyke, D. G. Voelz, and M. C. Roggemann, “Probability density of aperture-averaged irradiance fluctuations for long range free space optical communication links,” Appl. Opt. 48(33), 6511–6527 (2009).
[Crossref] [PubMed]

E. Bayaki, R. Schober, and R. K. Mallik, “Performance analysis of MIMO free-space optical systems in gamma-gamma fading,” IEEE Trans. Commun. 57(11), 3415–3424 (2009).
[Crossref]

T. A. Tsiftsis, H. G. Sandalidis, G. K. Karagiannidis, and M. Uysal, “Optical wireless links with spatial diversity over strong atmospheric turbulence channels,” IEEE Trans. Wireless Commun. 8(2), 951–957 (2009).
[Crossref]

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. 27(18), 3965–3973 (2009).
[Crossref]

A. Lapidoth, S. Moser, and M. Wigger, “On the Capacity of Free-Space Optical Intensity Channels,” IEEE Trans. Inf. Theory 55(10), 4449–4461 (2009).
[Crossref]

2007 (3)

2005 (1)

S. Nadarajah and A. K. Gupta, #x0201C;On the moments of the exponentiated Weibull distribution,” Commun. Statistics-Theory Methods 34(2), 253–256 (2005).
[Crossref]

2004 (1)

E. J. Lee and V. W. S. Chan, “Part 1: optical communication over the clear turbulent atmospheric channel using diversity,” IEEE J. Sel. Areas Commun. 22(9), 1896–1906 (2004).
[Crossref]

2003 (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]

2000 (1)

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, “Aperture averaging of optical scintillations: power fluctuations and the temporal spectrum,” Waves Random Media 10(1), 53–70 (2000).
[Crossref]

1999 (1)

Abou-Rjeily, C.

C. Abou-Rjeily, “On the optimality of the selection transmit diversity for MIMO-FSO links with feedback,” IEEE Commun. Lett. 15(6), 641–643 (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]

L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. A 16(6), 1417–1429 (1999).
[Crossref]

Alouini, M.

I. Ansari, F. Yilmaz, and M. Alouini, “Performance Analysis of Free-Space Optical Links Over Malaga (M) Turbulence channels with pointing errors,” IEEE Trans. Wireless Commun. 15(1), 91–102 (2016).
[Crossref]

Alouini, M.-S.

I. Ansari, M.-S. Alouini, and J. Cheng, “Ergodic capacity analysis of free-space optical links with nonzero bore-sight pointing errors,” IEEE Trans. Wireless Commun. 14(8), 4248–4264 (2015).
[Crossref]

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

Aminikashani, M.

M. Aminikashani, M. Uysal, and M. Kavehrad, “On the Performance of MIMO FSO Communications over Double Generalized Gamma Fading Channels,” arXiv preprintarXiv:1502.00365 (2015).

Andrews, L.

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]

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, “Aperture averaging of optical scintillations: power fluctuations and the temporal spectrum,” Waves Random Media 10(1), 53–70 (2000).
[Crossref]

L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. A 16(6), 1417–1429 (1999).
[Crossref]

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, vol. 1, (SPIE, 2005).
[Crossref]

Anguita, J. A.

J. A. Anguita, M. A. Neifeld, and B. V. Vasic, “Spatial correlation and irradiance statistics in a multiple-beam terrestrial free-space optical communication link,” Appl. Opt. 46(26), 6561–6571 (2007).
[Crossref] [PubMed]

J. A. Anguita and J. E. Cisternas, “Experimental evaluation of transmitter and receiver diversity in a terrestrial FSO link,” in 2010 IEEE GLOBECOM Workshops (GC Wkshps), (IEEE, 2010), pp. 1005–1009.

Ansari, I.

I. Ansari, F. Yilmaz, and M. Alouini, “Performance Analysis of Free-Space Optical Links Over Malaga (M) Turbulence channels with pointing errors,” IEEE Trans. Wireless Commun. 15(1), 91–102 (2016).
[Crossref]

I. Ansari, M.-S. Alouini, and J. Cheng, “Ergodic capacity analysis of free-space optical links with nonzero bore-sight pointing errors,” IEEE Trans. Wireless Commun. 14(8), 4248–4264 (2015).
[Crossref]

Arnon, S.

Bansal, A.

P. Sharma, A. Bansal, P. Garg, T. Tsiftsis, and R. Barrios, “Performance of FSO links under exponentiated Weibull turbulence fading with misalignment errors,” in 2015 IEEE International Conference on Communications (ICC) (IEEE, 2015), pp. 5110–5114.

Barrios, R.

R. Barrios and F. Dios, “Exponentiated Weibull model for the irradiance probability density function of a laser beam propagating through atmospheric turbulence,” Optics 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(12), 13055–13064 (2012).
[Crossref] [PubMed]

P. Sharma, A. Bansal, P. Garg, T. Tsiftsis, and R. Barrios, “Performance of FSO links under exponentiated Weibull turbulence fading with misalignment errors,” in 2015 IEEE International Conference on Communications (ICC) (IEEE, 2015), pp. 5110–5114.

Bayaki, E.

E. Bayaki, R. Schober, and R. K. Mallik, “Performance analysis of MIMO free-space optical systems in gamma-gamma fading,” IEEE Trans. Commun. 57(11), 3415–3424 (2009).
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S. Jiang, G. Yang, Y. Wei, M. Bi, Y. Lu, X. Zhou, M. Hu, and Q. Li, “Performance analysis of space-diversity free-space optical links over exponentiated Weibull channels,” IEEE Photonics Technol. Lett. 27(21), 2250–2252 (2015).
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Borah, D. K.

Cao, T.

P. Wang, J. Zhang, L. Guo, T. Shang, T. Cao, R. Wang, and Y. Yang, “Performance analysis for relay-aided multihop BPPM FSO communication system over exponentiated Weibull fading channels with pointing error impairments,” IEEE Photonics J. 7(4), 1–20 (2015).

P. Wang, T. Cao, L. Guo, R. Wang, and Y. Yang, “Performance Analysis of Multihop Parallel Free-Space Optical Systems Over Exponentiated Weibull Fading Channels,” IEEE Photonics J. 7(1), 1–17 (2015).

Castillo-Vazquez, B.

R. Boluda-Ruiz, A. Garcia-Zambrana, B. Castillo-Vazquez, and C. Castillo-Vazquez, “Ergodic capacity analysis of decode-and-forward relay-assisted FSO systems over alpha-mu fading channels considering pointing errors,” IEEE Photonics J. 8(1), 1–11 (2016).
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Castillo-Vazquez, C.

R. Boluda-Ruiz, A. Garcia-Zambrana, B. Castillo-Vazquez, and C. Castillo-Vazquez, “Ergodic capacity analysis of decode-and-forward relay-assisted FSO systems over alpha-mu fading channels considering pointing errors,” IEEE Photonics J. 8(1), 1–11 (2016).
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Castillo-Vázquez, C.

Chan, V. W. S.

E. J. Lee and V. W. S. Chan, “Part 1: optical communication over the clear turbulent atmospheric channel using diversity,” IEEE J. Sel. Areas Commun. 22(9), 1896–1906 (2004).
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Chen, M.

J.-Y. Wang, J.-B. Wang, M. Chen, Y. Tang, and Y. Zhang, “Outage analysis for relay-aided free-space optical communications over turbulence channels with nonzero boresight pointing errors,” IEEE Photonics J. 6(4), 1–15 (2014).

Cheng, J.

I. Ansari, M.-S. Alouini, and J. Cheng, “Ergodic capacity analysis of free-space optical links with nonzero bore-sight pointing errors,” IEEE Trans. Wireless Commun. 14(8), 4248–4264 (2015).
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F. Yang, J. Cheng, and T. Tsiftsis, “Free-space optical communication with nonzero boresight pointing errors,” IEEE Trans. Commun. 62(2), 713–725 (2014).
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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).
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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. 33(9), 1925–1934 (2015).
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Dios, F.

R. Barrios and F. Dios, “Exponentiated Weibull model for the irradiance probability density function of a laser beam propagating through atmospheric turbulence,” Optics Laser Technol. 45, 13–20 (2013).
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R. Barrios and F. Dios, “Exponentiated Weibull distribution family under aperture averaging for Gaussian beam waves,” Opt. Express 20(12), 13055–13064 (2012).
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Farid, A.

A. Farid and S. Hranilovic, “Diversity gain and outage probability for MIMO free-space optical links with misalignment,” IEEE Trans. Commun. 60(2), 479–487 (2012).
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Gao, J.

Gao, X.

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R. Boluda-Ruiz, A. Garcia-Zambrana, B. Castillo-Vazquez, and C. Castillo-Vazquez, “Ergodic capacity analysis of decode-and-forward relay-assisted FSO systems over alpha-mu fading channels considering pointing errors,” IEEE Photonics J. 8(1), 1–11 (2016).
[Crossref]

García-Zambrana, A.

Garg, P.

P. Sharma, A. Bansal, P. Garg, T. Tsiftsis, and R. Barrios, “Performance of FSO links under exponentiated Weibull turbulence fading with misalignment errors,” in 2015 IEEE International Conference on Communications (ICC) (IEEE, 2015), pp. 5110–5114.

Garrido-Balsells, J. M.

A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, and A. Puerta-Notario, “A Unifying Statistical Model for Atmospheric Optical Scintillation,” in Numerical Simulations of Physical and Engineering Processes, J. Awrejcewicz, ed. (Intech, 2011).
[Crossref]

Ghassemlooy, Z.

M. Bhatnagar and Z. Ghassemlooy, “Performance evaluation of FSO MIMO links in Gamma-Gamma fading with pointing errors,” in 2015 IEEE International Conference on Communications (ICC) (IEEE, 2015), pp. 5084–5090.

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

Guo, L.

P. Wang, J. Zhang, L. Guo, T. Shang, T. Cao, R. Wang, and Y. Yang, “Performance analysis for relay-aided multihop BPPM FSO communication system over exponentiated Weibull fading channels with pointing error impairments,” IEEE Photonics J. 7(4), 1–20 (2015).

P. Wang, T. Cao, L. Guo, R. Wang, and Y. Yang, “Performance Analysis of Multihop Parallel Free-Space Optical Systems Over Exponentiated Weibull Fading Channels,” IEEE Photonics J. 7(1), 1–17 (2015).

P. Wang, L. Zhang, L. Guo, F. Huang, T. Shang, R. Wang, and Y. Yang, “Average BER of subcarrier intensity modulated free space optical systems over the exponentiated Weibull fading channels,” Opt. Express 22(17), 20828–20841 (2014).
[Crossref] [PubMed]

P. Wang, J. Qin, L. Guo, and Y. Yang, “BER Performance of FSO Limited by Shot and Thermal Noise over Exponentiated Weibull Fading Channels,” to be accepted in “IEEE Photonics Technol. Lett.” (2015).

Gupta, A. K.

S. Nadarajah and A. K. Gupta, #x0201C;On the moments of the exponentiated Weibull distribution,” Commun. Statistics-Theory Methods 34(2), 253–256 (2005).
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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. 33(9), 1925–1934 (2015).
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Hasna, M. O.

Hopen, C.

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

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, “Aperture averaging of optical scintillations: power fluctuations and the temporal spectrum,” Waves Random Media 10(1), 53–70 (2000).
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L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. A 16(6), 1417–1429 (1999).
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Hranilovic, S.

A. Farid and S. Hranilovic, “Diversity gain and outage probability for MIMO free-space optical links with misalignment,” IEEE Trans. Commun. 60(2), 479–487 (2012).
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A. A. Farid and S. Hranilovic, “Outage capacity optimization for free-space optical links with pointing errors,” J. Lightwave Technol. 25(7), 1702–1710 (2007).
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S. Jiang, G. Yang, Y. Wei, M. Bi, Y. Lu, X. Zhou, M. Hu, and Q. Li, “Performance analysis of space-diversity free-space optical links over exponentiated Weibull channels,” IEEE Photonics Technol. Lett. 27(21), 2250–2252 (2015).
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Jain, V.

P. Kaur, V. Jain, and S. Kar, “Performance analysis of free space optical links using multi-input multi-output and aperture averaging in presence of turbulence and various weather conditions,” IET Commun. 9(8), 1104–1109 (2015).
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P. Kaur, V. K. Jain, and S. Kar, “Capacity of free space optical links with spatial diversity and aperture averaging,” in 2014 27th Biennial Symposium on Communications (QBSC) (IEEE, 2014), pp. 14–18.

Jiang, S.

S. Jiang, G. Yang, Y. Wei, M. Bi, Y. Lu, X. Zhou, M. Hu, and Q. Li, “Performance analysis of space-diversity free-space optical links over exponentiated Weibull channels,” IEEE Photonics Technol. Lett. 27(21), 2250–2252 (2015).
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A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, and A. Puerta-Notario, “A Unifying Statistical Model for Atmospheric Optical Scintillation,” in Numerical Simulations of Physical and Engineering Processes, J. Awrejcewicz, ed. (Intech, 2011).
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P. Kaur, V. Jain, and S. Kar, “Performance analysis of free space optical links using multi-input multi-output and aperture averaging in presence of turbulence and various weather conditions,” IET Commun. 9(8), 1104–1109 (2015).
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P. Kaur, V. K. Jain, and S. Kar, “Capacity of free space optical links with spatial diversity and aperture averaging,” in 2014 27th Biennial Symposium on Communications (QBSC) (IEEE, 2014), pp. 14–18.

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T. A. Tsiftsis, H. G. Sandalidis, G. K. Karagiannidis, and M. Uysal, “Optical wireless links with spatial diversity over strong atmospheric turbulence channels,” IEEE Trans. Wireless Commun. 8(2), 951–957 (2009).
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Kashani, M.

M. Kashani, M. Uysal, and M. Kavehrad, “On the performance of MIMO FSO communications over Double Generalized Gamma fading channels,” in 2015 IEEE International Conference on Communications (ICC)(IEEE, 2015), pp. 5144–5149.

M. Kashani, M. Uysal, and M. Kavehrad, “A novel statistical model for turbulence-induced fading in free-space optical systems,” in 2013 15th International Conference on Transparent Optical Networks (ICTON) (IEEE, 2013), pp. 1–5.

Kaur, P.

P. Kaur, V. Jain, and S. Kar, “Performance analysis of free space optical links using multi-input multi-output and aperture averaging in presence of turbulence and various weather conditions,” IET Commun. 9(8), 1104–1109 (2015).
[Crossref]

P. Kaur, V. K. Jain, and S. Kar, “Capacity of free space optical links with spatial diversity and aperture averaging,” in 2014 27th Biennial Symposium on Communications (QBSC) (IEEE, 2014), pp. 14–18.

Kavehrad, M.

P. Deng, M. Kavehrad, Z. Liu, Z. Zhou, and X. Yuan, “Capacity of MIMO free space optical communications using multiple partially coherent beams propagation through non-Kolmogorov strong turbulence,” Opt. Express 21(13), 15213–15229 (2013).
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M. Kashani, M. Uysal, and M. Kavehrad, “On the performance of MIMO FSO communications over Double Generalized Gamma fading channels,” in 2015 IEEE International Conference on Communications (ICC)(IEEE, 2015), pp. 5144–5149.

M. Kashani, M. Uysal, and M. Kavehrad, “A novel statistical model for turbulence-induced fading in free-space optical systems,” in 2013 15th International Conference on Transparent Optical Networks (ICTON) (IEEE, 2013), pp. 1–5.

M. Aminikashani, M. Uysal, and M. Kavehrad, “On the Performance of MIMO FSO Communications over Double Generalized Gamma Fading Channels,” arXiv preprintarXiv:1502.00365 (2015).

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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).
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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 (ISOP, 2001), pp. 26–37.

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 (ISOP, 2001), pp. 26–37.

Lapidoth, A.

A. Lapidoth, S. Moser, and M. Wigger, “On the Capacity of Free-Space Optical Intensity Channels,” IEEE Trans. Inf. Theory 55(10), 4449–4461 (2009).
[Crossref]

Lee, E. J.

E. J. Lee and V. W. S. Chan, “Part 1: optical communication over the clear turbulent atmospheric channel using diversity,” IEEE J. Sel. Areas Commun. 22(9), 1896–1906 (2004).
[Crossref]

Li, Q.

S. Jiang, G. Yang, Y. Wei, M. Bi, Y. Lu, X. Zhou, M. Hu, and Q. Li, “Performance analysis of space-diversity free-space optical links over exponentiated Weibull channels,” IEEE Photonics Technol. Lett. 27(21), 2250–2252 (2015).
[Crossref]

Liu, Z.

Lu, Y.

S. Jiang, G. Yang, Y. Wei, M. Bi, Y. Lu, X. Zhou, M. Hu, and Q. Li, “Performance analysis of space-diversity free-space optical links over exponentiated Weibull channels,” IEEE Photonics Technol. Lett. 27(21), 2250–2252 (2015).
[Crossref]

Luong, D. A.

D. A. Luong, T. C. Thang, and A. T. Pham, “Average capacity of MIMO/FSO systems with equal gain combining over log-normal channels,” in 2013 Fifth International Conference on Ubiquitous and Future Networks (ICUFN) (IEEE, 2013), pp. 306–309.

D. A. Luong and A. T. Pham, “Average capacity of MIMO free-space optical gamma-gamma fading channel,” in 2014 IEEE International Conference on Communications (ICC) (IEEE, 2014), pp. 3354–3358.

Lyke, S. D.

Majumdar, A. K.

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

Mallik, R. K.

E. Bayaki, R. Schober, and R. K. Mallik, “Performance analysis of MIMO free-space optical systems in gamma-gamma fading,” IEEE Trans. Commun. 57(11), 3415–3424 (2009).
[Crossref]

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 (ISOP, 2001), pp. 26–37.

Moser, S.

A. Lapidoth, S. Moser, and M. Wigger, “On the Capacity of Free-Space Optical Intensity Channels,” IEEE Trans. Inf. Theory 55(10), 4449–4461 (2009).
[Crossref]

Nadarajah, S.

S. Nadarajah and A. K. Gupta, #x0201C;On the moments of the exponentiated Weibull distribution,” Commun. Statistics-Theory Methods 34(2), 253–256 (2005).
[Crossref]

Neifeld, M. A.

Paris, J. F.

A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, and A. Puerta-Notario, “A Unifying Statistical Model for Atmospheric Optical Scintillation,” in Numerical Simulations of Physical and Engineering Processes, J. Awrejcewicz, ed. (Intech, 2011).
[Crossref]

Pham, A. T.

D. A. Luong and A. T. Pham, “Average capacity of MIMO free-space optical gamma-gamma fading channel,” in 2014 IEEE International Conference on Communications (ICC) (IEEE, 2014), pp. 3354–3358.

D. A. Luong, T. C. Thang, and A. T. Pham, “Average capacity of MIMO/FSO systems with equal gain combining over log-normal channels,” in 2013 Fifth International Conference on Ubiquitous and Future Networks (ICUFN) (IEEE, 2013), pp. 306–309.

Phillips, R.

L. Andrews, R. Phillips, and C. Hopen, Laser Beam Scintillation with Applications, vol. 99 (SPIE, 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]

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, “Aperture averaging of optical scintillations: power fluctuations and the temporal spectrum,” Waves Random Media 10(1), 53–70 (2000).
[Crossref]

L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. A 16(6), 1417–1429 (1999).
[Crossref]

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, vol. 1, (SPIE, 2005).
[Crossref]

Puerta-Notario, A.

A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, and A. Puerta-Notario, “A Unifying Statistical Model for Atmospheric Optical Scintillation,” in Numerical Simulations of Physical and Engineering Processes, J. Awrejcewicz, ed. (Intech, 2011).
[Crossref]

Qin, J.

P. Wang, J. Qin, L. Guo, and Y. Yang, “BER Performance of FSO Limited by Shot and Thermal Noise over Exponentiated Weibull Fading Channels,” to be accepted in “IEEE Photonics Technol. Lett.” (2015).

Recolons, J.

Ricklin, J. C.

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

Roggemann, M. C.

Ryzhik, I. M.

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

Sandalidis, H. G.

T. A. Tsiftsis, H. G. Sandalidis, G. K. Karagiannidis, and M. Uysal, “Optical wireless links with spatial diversity over strong atmospheric turbulence channels,” IEEE Trans. Wireless Commun. 8(2), 951–957 (2009).
[Crossref]

Schober, R.

E. Bayaki, R. Schober, and R. K. Mallik, “Performance analysis of MIMO free-space optical systems in gamma-gamma fading,” IEEE Trans. Commun. 57(11), 3415–3424 (2009).
[Crossref]

Shang, T.

P. Wang, J. Zhang, L. Guo, T. Shang, T. Cao, R. Wang, and Y. Yang, “Performance analysis for relay-aided multihop BPPM FSO communication system over exponentiated Weibull fading channels with pointing error impairments,” IEEE Photonics J. 7(4), 1–20 (2015).

P. Wang, L. Zhang, L. Guo, F. Huang, T. Shang, R. Wang, and Y. Yang, “Average BER of subcarrier intensity modulated free space optical systems over the exponentiated Weibull fading channels,” Opt. Express 22(17), 20828–20841 (2014).
[Crossref] [PubMed]

Sharma, P.

P. Sharma, A. Bansal, P. Garg, T. Tsiftsis, and R. Barrios, “Performance of FSO links under exponentiated Weibull turbulence fading with misalignment errors,” in 2015 IEEE International Conference on Communications (ICC) (IEEE, 2015), pp. 5110–5114.

Tang, Y.

J.-Y. Wang, J.-B. Wang, M. Chen, Y. Tang, and Y. Zhang, “Outage analysis for relay-aided free-space optical communications over turbulence channels with nonzero boresight pointing errors,” IEEE Photonics J. 6(4), 1–15 (2014).

Thang, T. C.

D. A. Luong, T. C. Thang, and A. T. Pham, “Average capacity of MIMO/FSO systems with equal gain combining over log-normal channels,” in 2013 Fifth International Conference on Ubiquitous and Future Networks (ICUFN) (IEEE, 2013), pp. 306–309.

Tsiftsis, T.

F. Yang, J. Cheng, and T. Tsiftsis, “Free-space optical communication with nonzero boresight pointing errors,” IEEE Trans. Commun. 62(2), 713–725 (2014).
[Crossref]

P. Sharma, A. Bansal, P. Garg, T. Tsiftsis, and R. Barrios, “Performance of FSO links under exponentiated Weibull turbulence fading with misalignment errors,” in 2015 IEEE International Conference on Communications (ICC) (IEEE, 2015), pp. 5110–5114.

Tsiftsis, T. A.

T. A. Tsiftsis, H. G. Sandalidis, G. K. Karagiannidis, and M. Uysal, “Optical wireless links with spatial diversity over strong atmospheric turbulence channels,” IEEE Trans. Wireless Commun. 8(2), 951–957 (2009).
[Crossref]

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]

T. A. Tsiftsis, H. G. Sandalidis, G. K. Karagiannidis, and M. Uysal, “Optical wireless links with spatial diversity over strong atmospheric turbulence channels,” IEEE Trans. Wireless Commun. 8(2), 951–957 (2009).
[Crossref]

M. Kashani, M. Uysal, and M. Kavehrad, “On the performance of MIMO FSO communications over Double Generalized Gamma fading channels,” in 2015 IEEE International Conference on Communications (ICC)(IEEE, 2015), pp. 5144–5149.

M. Kashani, M. Uysal, and M. Kavehrad, “A novel statistical model for turbulence-induced fading in free-space optical systems,” in 2013 15th International Conference on Transparent Optical Networks (ICTON) (IEEE, 2013), pp. 1–5.

M. Aminikashani, M. Uysal, and M. Kavehrad, “On the Performance of MIMO FSO Communications over Double Generalized Gamma Fading Channels,” arXiv preprintarXiv:1502.00365 (2015).

Vasic, B. V.

Vetelino, F. S.

Voelz, D. G.

Wang, F.

Wang, J.-B.

J.-Y. Wang, J.-B. Wang, M. Chen, Y. Tang, and Y. Zhang, “Outage analysis for relay-aided free-space optical communications over turbulence channels with nonzero boresight pointing errors,” IEEE Photonics J. 6(4), 1–15 (2014).

Wang, J.-Y.

J.-Y. Wang, J.-B. Wang, M. Chen, Y. Tang, and Y. Zhang, “Outage analysis for relay-aided free-space optical communications over turbulence channels with nonzero boresight pointing errors,” IEEE Photonics J. 6(4), 1–15 (2014).

Wang, N.

Wang, P.

P. Wang, T. Cao, L. Guo, R. Wang, and Y. Yang, “Performance Analysis of Multihop Parallel Free-Space Optical Systems Over Exponentiated Weibull Fading Channels,” IEEE Photonics J. 7(1), 1–17 (2015).

P. Wang, J. Zhang, L. Guo, T. Shang, T. Cao, R. Wang, and Y. Yang, “Performance analysis for relay-aided multihop BPPM FSO communication system over exponentiated Weibull fading channels with pointing error impairments,” IEEE Photonics J. 7(4), 1–20 (2015).

P. Wang, L. Zhang, L. Guo, F. Huang, T. Shang, R. Wang, and Y. Yang, “Average BER of subcarrier intensity modulated free space optical systems over the exponentiated Weibull fading channels,” Opt. Express 22(17), 20828–20841 (2014).
[Crossref] [PubMed]

P. Wang, J. Qin, L. Guo, and Y. Yang, “BER Performance of FSO Limited by Shot and Thermal Noise over Exponentiated Weibull Fading Channels,” to be accepted in “IEEE Photonics Technol. Lett.” (2015).

Wang, R.

P. Wang, T. Cao, L. Guo, R. Wang, and Y. Yang, “Performance Analysis of Multihop Parallel Free-Space Optical Systems Over Exponentiated Weibull Fading Channels,” IEEE Photonics J. 7(1), 1–17 (2015).

P. Wang, J. Zhang, L. Guo, T. Shang, T. Cao, R. Wang, and Y. Yang, “Performance analysis for relay-aided multihop BPPM FSO communication system over exponentiated Weibull fading channels with pointing error impairments,” IEEE Photonics J. 7(4), 1–20 (2015).

P. Wang, L. Zhang, L. Guo, F. Huang, T. Shang, R. Wang, and Y. Yang, “Average BER of subcarrier intensity modulated free space optical systems over the exponentiated Weibull fading channels,” Opt. Express 22(17), 20828–20841 (2014).
[Crossref] [PubMed]

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. 33(9), 1925–1934 (2015).
[Crossref]

Wei, Y.

S. Jiang, G. Yang, Y. Wei, M. Bi, Y. Lu, X. Zhou, M. Hu, and Q. Li, “Performance analysis of space-diversity free-space optical links over exponentiated Weibull channels,” IEEE Photonics Technol. Lett. 27(21), 2250–2252 (2015).
[Crossref]

Wigger, M.

A. Lapidoth, S. Moser, and M. Wigger, “On the Capacity of Free-Space Optical Intensity Channels,” IEEE Trans. Inf. Theory 55(10), 4449–4461 (2009).
[Crossref]

Yang, F.

F. Yang, J. Cheng, and T. Tsiftsis, “Free-space optical communication with nonzero boresight pointing errors,” IEEE Trans. Commun. 62(2), 713–725 (2014).
[Crossref]

Yang, G.

S. Jiang, G. Yang, Y. Wei, M. Bi, Y. Lu, X. Zhou, M. Hu, and Q. Li, “Performance analysis of space-diversity free-space optical links over exponentiated Weibull channels,” IEEE Photonics Technol. Lett. 27(21), 2250–2252 (2015).
[Crossref]

Yang, L.

Yang, Y.

P. Wang, J. Zhang, L. Guo, T. Shang, T. Cao, R. Wang, and Y. Yang, “Performance analysis for relay-aided multihop BPPM FSO communication system over exponentiated Weibull fading channels with pointing error impairments,” IEEE Photonics J. 7(4), 1–20 (2015).

P. Wang, T. Cao, L. Guo, R. Wang, and Y. Yang, “Performance Analysis of Multihop Parallel Free-Space Optical Systems Over Exponentiated Weibull Fading Channels,” IEEE Photonics J. 7(1), 1–17 (2015).

P. Wang, L. Zhang, L. Guo, F. Huang, T. Shang, R. Wang, and Y. Yang, “Average BER of subcarrier intensity modulated free space optical systems over the exponentiated Weibull fading channels,” Opt. Express 22(17), 20828–20841 (2014).
[Crossref] [PubMed]

P. Wang, J. Qin, L. Guo, and Y. Yang, “BER Performance of FSO Limited by Shot and Thermal Noise over Exponentiated Weibull Fading Channels,” to be accepted in “IEEE Photonics Technol. Lett.” (2015).

Yao, M.

Yi, X.

Yilmaz, F.

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Figures (7)

Fig. 1
Fig. 1 Block diagram of the considered MIMO FSO communications system.
Fig. 2
Fig. 2 Different geometric arrangement for the receiver from the juxtaposition of equilateral triangles.
Fig. 3
Fig. 3 Asymptotic ergodic capacity of MIMO FSO systems when (a) LN and (b) GG atmospheric turbulence models are assumed for different values of normalized beam width and normalized jitter of (ωz/r, σs/r) = {(5,1),(10,2)} as well as different values of normalized spacing among receive apertures of d/r ={6,8}.
Fig. 4
Fig. 4 Asymptotic ergodic capacity of MIMO FSO systems over EW atmospheric turbulence for different values of normalized beam width and normalized jitter of (ωz/r, σs/r) = {(5,1),(10,2)}.
Fig. 5
Fig. 5 Receive apertures.
Fig. 6
Fig. 6 Comparison among different geometric arrangements for the receiver over EW atmospheric turbulence when different values of spacing among receive apertures of d′ = {12 mm, 24 mm}.
Fig. 7
Fig. 7 (a) Disadvantage as a function of the horizontal displacement of the normalized additional boresight error when different values of normalized beam width and normalized jitter of (ωz/r,σs/r) = {(5,1),(10,2)} are considered and, (b) performance over gamma-gamma atmospheric turbulence.

Equations (43)

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s k = d R k p c 2 + Δ x 2 + Δ y 2 2 ( Δ x x k + Δ y y k ) , k = 1 N
f I a L N ( i ) = 1 i 8 π σ X 2 exp ( ( ln ( i ) + 2 σ X 2 ) 2 8 σ X 2 ) , i 0
f I a G G ( i ) = 2 ( α β ) ( α + β ) / 2 Γ ( α ) Γ ( β ) i ( ( α + β ) / 2 ) 1 K α β ( 2 α β i ) , i 0
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 a E W ( i ) = a b c ( i c ) b 1 exp ( ( i c ) b ) { 1 exp ( ( i c ) b ) } a 1 , i 0
a = 7.220 σ I 2 / 3 Γ ( 2.487 σ I 2 / 6 ) 0.104 , b = 1.012 ( a σ I 2 ) 13 / 25 + 0.142 , c = 1 a Γ ( 1 + 1 / b ) g 1 ( a , b ) ,
g n ( a , b ) = k = 0 ( 1 ) k Γ ( a ) k ! ( k + 1 ) 1 + n / b Γ ( a k ) .
f I p N Z B ( i ) = φ 2 i φ 2 1 A 0 φ 2 exp ( s 2 2 σ s 2 ) I 0 ( s σ s 2 φ 2 ln ( i A 0 ) ) , 0 i A 0
f I p Z B ( i ) = φ 2 A 0 φ 2 i φ 2 1 . 0 i A 0
Y = X M N k = 1 M l = 1 N I k l + Z EGC , X { 0 , 2 P opt } , Z EGC ~ N ( 0 , N 0 / 2 ) ,
γ MIMO = 1 2 ( 2 P o p t / M N ) 2 N 0 / 2 ( k = 1 M l = 1 N I k l ) 2 = 4 γ 0 M 2 N 2 ( k = 1 M l = 1 N I k l ) 2 ,
C MIMO = B 2 ln ( 2 ) 0 0 MN-fold ln ( 1 + 4 γ 0 M 2 N 2 ( k = 1 M l = 1 N i k l ) 2 ) k = 1 M l = 1 N f I k l ( i k l ) d i k l ,
k = 1 M l = 1 N I k l M N F k = 1 M l = 1 N I k l M N .
F = E [ k = 1 M l = 1 N I k l ] M N ( M N ) M N E [ k = 1 M l = 1 N I k l M N ] M N .
C MIMO B ln ( 4 ) 0 0 MN-fold ln ( 1 + 4 γ 0 F 2 M N ( k = 1 M l = 1 N i k l ) 2 M N ) k = 1 M l = 1 N f I k l ( i k l ) d i k l .
C MIMO B ln ( 4 ) 0 0 MN-fold ln ( 4 γ 0 F 2 M N ( k = 1 M l = 1 N i k l ) 2 M N ) k = 1 M l = 1 N f I k l ( i k l ) d i k l .
ln ( 4 γ 0 F 2 M N ( k = 1 M l = 1 N i k l ) 2 M N ) = ln ( 4 γ 0 ) + 2 M N ln ( F ) + 2 M N k = 1 M l = 1 N ln ( i k l ) ,
C MIMO B ln ( 4 γ 0 ) ln ( 4 ) + B ln ( F ) M N ln ( 2 ) + B M N ln ( 2 ) k = 1 M l = 1 N 0 ln ( i k l ) f I k l ( i k l ) d i k l I N T .
I N T = 0 ln ( i k l ) f I k l ( i k l ) d i k l = 0 0 A 0 k l ln ( L k l i k l a i k l p ) f I k l p ( i k l a ) f I k l p ( i k l p ) d i k l a d i k l p = ln ( L k l ) + 0 ln ( i k l a ) f I k l a ( i k l a ) d i k l a I N T 1 + 0 A 0 k l ln ( i k l p ) f I k l p ( i k l p ) d i k l p I N T 1 + ln ( L k l ) + I N T 1 + I N T 2 .
I N T 2 = 0 A 0 k l ln ( i k l P ) f I k l P ( i k l P ) d i k l P = ln ( A 0 k l ) 1 φ k l 2 s k l 2 1 σ s k l 2 2 φ k l 2 .
C MIMO LN B ln ( 4 γ 0 ) ln ( 4 ) + B ln ( F ) M N ln ( 2 ) + B M N ln ( 2 ) k = 1 M l = 1 N ln ( L k l ) σ R k l 2 2 + ln ( A 0 k l ) 1 φ k l 2 s k l 2 2 σ s k l 2 φ k l 2 .
I N T 1 G G = ln ( 1 α k l β k l ) + ψ ( α k l ) + ψ ( β k l ) ,
C MIMO GG B ln ( 4 γ 0 ) ln ( 4 ) + B ln ( F ) M N ln ( 2 ) + B M N ln ( 2 ) k = 1 M l = 1 N ln ( L k l α k l β k l ) + ψ ( α k l ) + ψ ( β k l ) + ln ( A 0 k l ) 1 φ k l 2 s k l 2 2 σ s k l 2 φ k l 2 .
I N T 1 E W = a b c 0 ln ( i ) ( i c ) b 1 exp ( ( i c ) b ) { 1 exp ( ( i c ) b ) } a 1 d i .
( 1 + e x ) a = k = 1 Γ ( α + 1 ) e x k k ! Γ ( α k + 1 ) .
I N T 1 E W = b Γ ( a + 1 ) c b k = o ( 1 ) k k ! Γ ( a k ) 0 ln ( i ) i b 1 exp ( ( k + 1 ) ( i c ) b ) d i .
C MIMO EW B ln ( 4 γ 0 ) ln ( 4 ) + B ln ( F ) M N ln ( 2 ) + B M N ln ( 2 ) × k = 1 M l = 1 N ln ( L k l ) + g ( a k l , b k l , c k l ) + ln ( A 0 k l ) 1 φ k l 2 s k l 2 2 σ s k l 2 φ k l 2 ,
g ( a k l , b k l , c k l ) = Γ ( a k l + 1 ) b k l i = 0 ( 1 ) i ( ψ ( 1 ) ln ( ( i + 1 ) ( 1 c k l ) b k l ) ) Γ ( i + 2 ) Γ ( a k l i ) ,
γ MIMO t h [ d B ] = 20 ln ( 10 ) ( ln ( F ) M N + ln ( 2 ) + 1 M N k = 1 M L = 1 N I N T 1 + ln ( A 0 k l ) 1 φ k l 2 s k l 2 2 σ s k l 2 φ k l 2 ) ,
D A B [ d B ] = γ t h N Z A B [ d B ] γ t h N Z I B [ d B ] = 20 ln ( F N Z I B / F N Z A B ) M N ln ( 10 ) + 10 M N ln ( 10 ) σ s 2 φ 2 k = 1 M l = 1 N s A B 2 2 ( Δ x x k l + Δ y y k l ) ,
F = E [ k = 1 M l = 1 N I k l ] M N ( M N ) M N E [ k = 1 M l = 1 N I k l M N ] M N = ( k = 1 M l = 1 N E [ I k l ] ) M N ( M N ) M N ( k = 1 M l = 1 N E [ I k l M N ] ) M N .
E [ I n ] = E [ ( L I a I p ) n ] = L n E [ ( I a ) n ] E [ ( I p ) n ] = L n 0 x n f I a ( x ) d x 0 A 0 y n f I p N Z B ( y ) d y .
E [ ( I p N Z B ) n ] = A 0 φ 2 n + φ 2 exp ( n s 2 ( n + φ 2 ) 2 σ s 2 ) .
E [ ( I a L N ) n ] = exp ( n σ R 2 2 ( n 1 ) ) .
E [ ( I a G G ) n ] = ( 1 α β ) n Γ ( n + α ) Γ ( n + β ) Γ ( α ) Γ ( β ) .
E [ ( I a E W ) n ] = a c n Γ ( 1 + n b ) g n ( a , b ) .
φ 2 exp ( s 2 2 σ s 2 ) A φ 2 0 A 0 ln ( x ) x φ 2 1 I 0 ( s σ s 2 φ 2 ln ( x A 0 ) ) d x .
φ 2 exp ( s 2 2 σ s 2 ) A φ 2 0 A 0 ln ( x ) x φ 2 1 k = 0 ( 2 φ 2 ) k ( s 2 2 σ s ) 2 k k ! Γ ( k + 1 ) ln ( x / A 0 ) k d x .
φ 2 exp ( s 2 2 σ s 2 ) A φ 2 k = 0 ( 2 φ 2 ) k ( s 2 σ s ) 2 k k ! Γ ( k + 1 ) 0 A 0 ln ( x ) x φ 2 1 ln ( x / A 0 ) k d x = φ 2 exp ( s 2 2 σ s 2 ) k = 0 ( 2 φ 2 ) k ( s 2 σ s ) 2 k k ! Γ ( k + 1 ) 0 1 ln ( x A 0 ) x φ 2 1 ln ( x ) k d x .
0 1 ln ( x A 0 ) x φ 2 1 ln ( x ) k d x = ln ( A 0 ) 0 1 x φ 2 1 ln ( x ) k d x + 0 1 x φ 2 1 ln ( x ) k + 1 d x .
exp ( s 2 2 σ s 2 ) k = 0 ( s 2 2 σ s 2 ) k k ! ( ln ( A 0 ) 1 k φ 2 ) exp ( s 2 2 σ s 2 ) { ( ln ( A 0 ) 1 φ 2 ) k = 0 ( s 2 2 σ s 2 ) k k ! k = 0 k ( s 2 2 σ s 2 ) k φ 2 k ! } exp ( s 2 2 σ s 2 ) { ( ln ( A 0 ) 1 φ 2 ) k = 0 ( s 2 2 σ s 2 ) k k ! s 2 2 φ 2 σ s 2 k = 1 ( s 2 2 σ s 2 ) k 1 ( k 1 ) ! } .
exp ( s 2 2 σ s 2 ) { ( ln ( A 0 ) 1 φ 2 ) exp ( s 2 2 σ s 2 ) s 2 2 φ 2 σ s 2 exp ( s 2 2 σ s 2 ) } = ln ( A 0 ) 1 φ 2 s 2 2 φ 2 σ s 2 .

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