Abstract

A novel accurate and useful approximation of the well-known Beckmann distribution is presented here, which is used to model generalized pointing errors in the context of free-space optical (FSO) communication systems. We derive an approximate closed-form probability density function (PDF) for the composite gamma-gamma (GG) atmospheric turbulence with the pointing error model using the proposed approximation of the Beckmann distribution, which is valid for most practical terrestrial FSO links. This approximation takes into account the effect of the beam width, different jitters for the elevation and the horizontal displacement and the simultaneous effect of nonzero boresight errors for each axis at the receiver plane. Additionally, the proposed approximation allows us to delimit two different FSO scenarios. The first of them is when atmospheric turbulence is the dominant effect in relation to generalized pointing errors, and the second one when generalized pointing error is the dominant effect in relation to atmospheric turbulence. The second FSO scenario has not been studied in-depth by the research community. Moreover, the accuracy of the method is measured both visually and quantitatively using curve-fitting metrics. Simulation results are further included to confirm the analytical results.

© 2016 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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  27. N. B. Mehta, J. Wu, A. F. Molisch, and J. Zhang, “Approximating a sum of random variables with a lognormal,” IEEE Trans. Wireless Commun. 6(7), 2690–2699 (2007).
    [Crossref]

2016 (3)

H. AlQuwaiee, H. C. Yang, and M. Alouini, “On the asymptotic capacity of dual-aperture FSO systems with a generalized pointing error model,” IEEE Transactions on Wireless Commun. 99, 1 (2016).
[Crossref]

R. Boluda-Ruiz, A. García-Zambrana, B. Castillo-Vázquez, and C. Castillo-Vázquez, “MISO Relay-Assisted FSO Systems Over Gamma-Gamma Fading Channels With Pointing Errors,” IEEE Photonics Technol. Lett. 28(3), 229–232 (2016).
[Crossref]

R. Boluda-Ruiz, A. García-Zambrana, B. Castillo-Vázquez, and C. Castillo-Vázquez, “Impact of nonzero boresight pointing error on ergodic capacity of MIMO FSO communication systems,” Opt. Express 24(4), 3513–3534 (2016).
[Crossref] [PubMed]

2015 (1)

2014 (2)

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]

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

2011 (2)

W. Gappmair, S. Hranilovic, and E. Leitgeb, “OOK performance for terrestrial FSO links in turbulent atmosphere with pointing errors modeled by Hoyt distributions,” IEEE Commun. Lett. 15(8), 875–877 (2011).
[Crossref]

A. García-Zambrana, B. Castillo-Vázquez, and C. Castillo-Vázquez, “Asymptotic error-rate analysis of FSO links using transmit laser selection over gamma-gamma atmospheric turbulence channels with pointing errors,” Opt. Express 19(24), 2096–2109 (2011).

2010 (1)

N. Wang and J. Cheng, “Moment-based estimation for the shape parameters of the gamma-gamma atmospheric turbulence model,” Opt. Express 18(12), 12,824–12,831 (2010).
[Crossref]

2009 (2)

2007 (2)

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).
[Crossref]

N. B. Mehta, J. Wu, A. F. Molisch, and J. Zhang, “Approximating a sum of random variables with a lognormal,” IEEE Trans. Wireless Commun. 6(7), 2690–2699 (2007).
[Crossref]

2003 (3)

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)

I. I. Kim, R. Stieger, J. A. Koontz, C. Moursund, M. Barclay, P. Adhikari, J. Schuster, E. Korevaar, R. Ruigrok, and C. DeCusatis, “Wireless optical transmission of fast ethernet, FDDI, ATM, and ESCON protocol data using the TerraLink laser communication system,” Opt. Eng. 37(12), 3143–3155 (1998).
[Crossref]

1982 (1)

S. C. Schwartz and Y.-S. Yeh, “On the Distribution Function and Moments of Power Sums With Log-Normal Components,” Bell System Technical Journal 61(7), 1441–1462 (1982).
[Crossref]

Adhikari, P.

I. I. Kim, R. Stieger, J. A. Koontz, C. Moursund, M. Barclay, P. Adhikari, J. Schuster, E. Korevaar, R. Ruigrok, and C. DeCusatis, “Wireless optical transmission of fast ethernet, FDDI, ATM, and ESCON protocol data using the TerraLink laser communication system,” Opt. Eng. 37(12), 3143–3155 (1998).
[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.

H. AlQuwaiee, H. C. Yang, and M. Alouini, “On the asymptotic capacity of dual-aperture FSO systems with a generalized pointing error model,” IEEE Transactions on Wireless Commun. 99, 1 (2016).
[Crossref]

Alouini, M.-S.

M. K. Simon and M.-S. Alouini, Digital Communications over Fading Channels, 2nd ed. (Wiley, 2005).

AlQuwaiee, H.

H. AlQuwaiee, H. C. Yang, and M. Alouini, “On the asymptotic capacity of dual-aperture FSO systems with a generalized pointing error model,” IEEE Transactions on Wireless Commun. 99, 1 (2016).
[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]

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

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

Arnon, S.

Barclay, M.

I. I. Kim, R. Stieger, J. A. Koontz, C. Moursund, M. Barclay, P. Adhikari, J. Schuster, E. Korevaar, R. Ruigrok, and C. DeCusatis, “Wireless optical transmission of fast ethernet, FDDI, ATM, and ESCON protocol data using the TerraLink laser communication system,” Opt. Eng. 37(12), 3143–3155 (1998).
[Crossref]

Bloom, S.

Boluda-Ruiz, R.

Borah, D. K.

Brychkov, Y. A.

A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and series Volume 2: Special Functions, vol. 2, 1st ed. (Gordon and Breach Science Publishers, 1986).

Castillo-Vázquez, B.

R. Boluda-Ruiz, A. García-Zambrana, B. Castillo-Vázquez, and C. Castillo-Vázquez, “MISO Relay-Assisted FSO Systems Over Gamma-Gamma Fading Channels With Pointing Errors,” IEEE Photonics Technol. Lett. 28(3), 229–232 (2016).
[Crossref]

R. Boluda-Ruiz, A. García-Zambrana, B. Castillo-Vázquez, and C. Castillo-Vázquez, “Impact of nonzero boresight pointing error on ergodic capacity of MIMO FSO communication systems,” Opt. Express 24(4), 3513–3534 (2016).
[Crossref] [PubMed]

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] [PubMed]

A. García-Zambrana, B. Castillo-Vázquez, and C. Castillo-Vázquez, “Asymptotic error-rate analysis of FSO links using transmit laser selection over gamma-gamma atmospheric turbulence channels with pointing errors,” Opt. Express 19(24), 2096–2109 (2011).

Castillo-Vázquez, C.

R. Boluda-Ruiz, A. García-Zambrana, B. Castillo-Vázquez, and C. Castillo-Vázquez, “Impact of nonzero boresight pointing error on ergodic capacity of MIMO FSO communication systems,” Opt. Express 24(4), 3513–3534 (2016).
[Crossref] [PubMed]

R. Boluda-Ruiz, A. García-Zambrana, B. Castillo-Vázquez, and C. Castillo-Vázquez, “MISO Relay-Assisted FSO Systems Over Gamma-Gamma Fading Channels With Pointing Errors,” IEEE Photonics Technol. Lett. 28(3), 229–232 (2016).
[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] [PubMed]

A. García-Zambrana, B. Castillo-Vázquez, and C. Castillo-Vázquez, “Asymptotic error-rate analysis of FSO links using transmit laser selection over gamma-gamma atmospheric turbulence channels with pointing errors,” Opt. Express 19(24), 2096–2109 (2011).

Cheng, J.

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]

N. Wang and J. Cheng, “Moment-based estimation for the shape parameters of the gamma-gamma atmospheric turbulence model,” Opt. Express 18(12), 12,824–12,831 (2010).
[Crossref]

DeCusatis, C.

I. I. Kim, R. Stieger, J. A. Koontz, C. Moursund, M. Barclay, P. Adhikari, J. Schuster, E. Korevaar, R. Ruigrok, and C. DeCusatis, “Wireless optical transmission of fast ethernet, FDDI, ATM, and ESCON protocol data using the TerraLink laser communication system,” Opt. Eng. 37(12), 3143–3155 (1998).
[Crossref]

Farid, A. A.

Gappmair, W.

W. Gappmair, S. Hranilovic, and E. Leitgeb, “OOK performance for terrestrial FSO links in turbulent atmosphere with pointing errors modeled by Hoyt distributions,” IEEE Commun. Lett. 15(8), 875–877 (2011).
[Crossref]

García-Zambrana, A.

R. Boluda-Ruiz, A. García-Zambrana, B. Castillo-Vázquez, and C. Castillo-Vázquez, “MISO Relay-Assisted FSO Systems Over Gamma-Gamma Fading Channels With Pointing Errors,” IEEE Photonics Technol. Lett. 28(3), 229–232 (2016).
[Crossref]

R. Boluda-Ruiz, A. García-Zambrana, B. Castillo-Vázquez, and C. Castillo-Vázquez, “Impact of nonzero boresight pointing error on ergodic capacity of MIMO FSO communication systems,” Opt. Express 24(4), 3513–3534 (2016).
[Crossref] [PubMed]

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] [PubMed]

A. García-Zambrana, B. Castillo-Vázquez, and C. Castillo-Vázquez, “Asymptotic error-rate analysis of FSO links using transmit laser selection over gamma-gamma atmospheric turbulence channels with pointing errors,” Opt. Express 19(24), 2096–2109 (2011).

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]

Gradshteyn, I. S.

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

Hopen, C.

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

Hranilovic, S.

W. Gappmair, S. Hranilovic, and E. Leitgeb, “OOK performance for terrestrial FSO links in turbulent atmosphere with pointing errors modeled by Hoyt distributions,” IEEE Commun. Lett. 15(8), 875–877 (2011).
[Crossref]

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).
[Crossref]

Karagiannidis, G. K.

Khalighi, M. A.

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

Kim, I. I.

I. I. Kim, R. Stieger, J. A. Koontz, C. Moursund, M. Barclay, P. Adhikari, J. Schuster, E. Korevaar, R. Ruigrok, and C. DeCusatis, “Wireless optical transmission of fast ethernet, FDDI, ATM, and ESCON protocol data using the TerraLink laser communication system,” Opt. Eng. 37(12), 3143–3155 (1998).
[Crossref]

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).

Koontz, J. A.

I. I. Kim, R. Stieger, J. A. Koontz, C. Moursund, M. Barclay, P. Adhikari, J. Schuster, E. Korevaar, R. Ruigrok, and C. DeCusatis, “Wireless optical transmission of fast ethernet, FDDI, ATM, and ESCON protocol data using the TerraLink laser communication system,” Opt. Eng. 37(12), 3143–3155 (1998).
[Crossref]

Korevaar, E.

S. Bloom, E. Korevaar, J. Schuster, and H. Willebrand, “Understanding the performance of free-space optics,” J. Opt. Netw. 2(6), 178–200 (2003).

I. I. Kim, R. Stieger, J. A. Koontz, C. Moursund, M. Barclay, P. Adhikari, J. Schuster, E. Korevaar, R. Ruigrok, and C. DeCusatis, “Wireless optical transmission of fast ethernet, FDDI, ATM, and ESCON protocol data using the TerraLink laser communication system,” Opt. Eng. 37(12), 3143–3155 (1998).
[Crossref]

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).

Leitgeb, E.

W. Gappmair, S. Hranilovic, and E. Leitgeb, “OOK performance for terrestrial FSO links in turbulent atmosphere with pointing errors modeled by Hoyt distributions,” IEEE Commun. Lett. 15(8), 875–877 (2011).
[Crossref]

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).

Marichev, O. I.

A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and series Volume 2: Special Functions, vol. 2, 1st ed. (Gordon and Breach Science Publishers, 1986).

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).

Mehta, N. B.

N. B. Mehta, J. Wu, A. F. Molisch, and J. Zhang, “Approximating a sum of random variables with a lognormal,” IEEE Trans. Wireless Commun. 6(7), 2690–2699 (2007).
[Crossref]

Molisch, A. F.

N. B. Mehta, J. Wu, A. F. Molisch, and J. Zhang, “Approximating a sum of random variables with a lognormal,” IEEE Trans. Wireless Commun. 6(7), 2690–2699 (2007).
[Crossref]

Moursund, C.

I. I. Kim, R. Stieger, J. A. Koontz, C. Moursund, M. Barclay, P. Adhikari, J. Schuster, E. Korevaar, R. Ruigrok, and C. DeCusatis, “Wireless optical transmission of fast ethernet, FDDI, ATM, and ESCON protocol data using the TerraLink laser communication system,” Opt. Eng. 37(12), 3143–3155 (1998).
[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 and R. L. Phillips, Laser Beam Propagation through Random Media, vol. 1 (SPIE, 2005).
[Crossref]

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

Prudnikov, A. P.

A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and series Volume 2: Special Functions, vol. 2, 1st ed. (Gordon and Breach Science Publishers, 1986).

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).

Ruigrok, R.

I. I. Kim, R. Stieger, J. A. Koontz, C. Moursund, M. Barclay, P. Adhikari, J. Schuster, E. Korevaar, R. Ruigrok, and C. DeCusatis, “Wireless optical transmission of fast ethernet, FDDI, ATM, and ESCON protocol data using the TerraLink laser communication system,” Opt. Eng. 37(12), 3143–3155 (1998).
[Crossref]

Ryzhik, I. M.

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

Sandalidis, H. G.

Schuster, J.

S. Bloom, E. Korevaar, J. Schuster, and H. Willebrand, “Understanding the performance of free-space optics,” J. Opt. Netw. 2(6), 178–200 (2003).

I. I. Kim, R. Stieger, J. A. Koontz, C. Moursund, M. Barclay, P. Adhikari, J. Schuster, E. Korevaar, R. Ruigrok, and C. DeCusatis, “Wireless optical transmission of fast ethernet, FDDI, ATM, and ESCON protocol data using the TerraLink laser communication system,” Opt. Eng. 37(12), 3143–3155 (1998).
[Crossref]

Schwartz, S. C.

S. C. Schwartz and Y.-S. Yeh, “On the Distribution Function and Moments of Power Sums With Log-Normal Components,” Bell System Technical Journal 61(7), 1441–1462 (1982).
[Crossref]

Simon, M. K.

M. K. Simon and M.-S. Alouini, Digital Communications over Fading Channels, 2nd ed. (Wiley, 2005).

Stieger, R.

I. I. Kim, R. Stieger, J. A. Koontz, C. Moursund, M. Barclay, P. Adhikari, J. Schuster, E. Korevaar, R. Ruigrok, and C. DeCusatis, “Wireless optical transmission of fast ethernet, FDDI, ATM, and ESCON protocol data using the TerraLink laser communication system,” Opt. Eng. 37(12), 3143–3155 (1998).
[Crossref]

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]

Tsiftsis, T. A.

Uysal, M.

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

Voelz, D. G.

Wang, N.

N. Wang and J. Cheng, “Moment-based estimation for the shape parameters of the gamma-gamma atmospheric turbulence model,” Opt. Express 18(12), 12,824–12,831 (2010).
[Crossref]

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]

Willebrand, H.

Wu, J.

N. B. Mehta, J. Wu, A. F. Molisch, and J. Zhang, “Approximating a sum of random variables with a lognormal,” IEEE Trans. Wireless Commun. 6(7), 2690–2699 (2007).
[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, H. C.

H. AlQuwaiee, H. C. Yang, and M. Alouini, “On the asymptotic capacity of dual-aperture FSO systems with a generalized pointing error model,” IEEE Transactions on Wireless Commun. 99, 1 (2016).
[Crossref]

Yeh, Y.-S.

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Communications Surveys and Tutorials, IEEE (1)

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Figures (8)

Fig. 1
Fig. 1

Beam footprint with generalized pointing errors on the receiver aperture plane.

Fig. 2
Fig. 2

Outage performance over GG atmospheric turbulence and generalized misalignment fading channels, when different weather conditions (a) C n 2 = 1.7 × 10 14 m 2 / 3 and (b) C n 2 = 8 × 10 14 m 2 / 3 are assumed for a link distance of dSD = 3 km.

Fig. 3
Fig. 3

BER performance over GG atmospheric turbulence and generalized misalignment fading channels, when different weather conditions (a) C n 2 = 1.7 × 10 14 m 2 / 3 and (b) C n 2 = 8 × 10 14 m 2 / 3 are assumed for a link distance of dSD = 3 km.

Fig. 4
Fig. 4

Coding gain disadvantage, Dpe[dB], as a function of the q parameter for a link distance of dSD = 3 km and different normalized boresight error values.

Fig. 5
Fig. 5

Impact of nonzero boresight error as a function of the normalized horizontal boresight error μx/a for a link distance of dSD = 3 km under different normalized vertical boresight error values of μy/a = {1, 2, 3}.

Fig. 6
Fig. 6

Minimum normalized beam width as a function of the q parameter for a link distance of dSD = 3 km and different normalized boresight error values.

Fig. 7
Fig. 7

Accuracy metric, MOP, as a function of the q parameter for the outage probability when a source-destination link distance of dSD = 3 km is assumed.

Fig. 8
Fig. 8

Accuracy metric, MOP, as a function of the q parameter for the outage probability when a source-destination link distance of dSD = 5 km is assumed.

Equations (28)

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Y = η X I T + Z ,
f I a ( 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 ,
I p ( r ; z ) A 0 exp ( 2 r 2 ω z eq 2 ) , r 0 ,
f r ( r ) = r 2 π σ x σ y 0 2 π exp ( ( r cos θ μ x ) 2 2 σ x 2 ( r sin θ μ y ) 2 2 σ y 2 ) d θ , r 0 .
f r 2 ( u ) = 1 2 σ mod 2 exp ( u 2 σ mod 2 ) , u 0 .
Ω 3 r 2 = 8 σ x 4 ( 3 μ x 2 + σ x 2 ) + 8 σ y 4 ( 3 μ y 2 + σ y 2 ) ,
Ω 3 u = 16 σ mod 6 .
σ mod 2 = ( 3 μ x 2 σ x 4 + 3 μ y 2 σ y 4 + σ x 6 + σ y 6 2 ) 1 / 3 .
f r ( r ) r σ mod 2 exp ( r 2 2 σ mod 2 ) , r 0 .
f I p ( i ) φ mod 2 ( A 0 G ) φ mod 2 i φ mod 2 1 , 0 i A 0 G
𝔼 [ ln ( I p ) ] = ln ( A 0 ) ( 2 / ω z eq 2 ) 𝔼 [ r 2 ] .
𝔼 [ ln ( I p ) ] = ln ( A 0 ) 2 ( μ x 2 + μ y 2 + σ x 2 + σ y 2 ) / ω z eq 2 .
𝔼 [ ln ( I p ) ] = ln ( A 0 ) + ln ( G ) 1 / φ mod 2 .
G = exp ( 1 φ mod 2 1 2 φ x 2 1 2 φ y 2 μ x 2 2 σ x 2 φ x 2 μ y 2 2 σ y 2 φ y 2 ) .
f I T ( i ) α β φ mod 2 i 1 A mod L Γ ( α ) Γ ( β ) G 1 , 3 3 , 0 ( α β A mod L i | φ mod 2 + 1 φ mod 2 , α , β ) , i 0 ,
F I T ( i ) φ mod 2 Γ ( α ) Γ ( β ) G 2 , 4 3 , 1 ( α β A mod L i | 1 , φ mod 2 + 1 φ mod 2 , α , β , 0 ) . i 0 .
f I T ( i ) a T i b T 1 = { φ mod 2 ( α β ) β Γ ( α β ) ( A mod L ) β Γ ( α ) Γ ( β ) ( φ mod 2 β ) i β 1 , φ mod 2 > β φ mod 2 ( α β ) φ mod 2 Γ ( α φ mod 2 ) Γ ( β φ mod 2 ) ( A mod L ) φ mod 2 Γ ( α ) Γ ( β ) i φ mod 2 1 . φ mod 2 < β
P out : = P ( γ T γ t h ) = 0 γ t h f γ T ( i ) d i .
P out = P ( 4 γ i 2 γ t h ) = 0 γ t h / 4 γ f I T ( i ) d i = f I T ( γ t h 4 γ ) .
P b = 0 Q ( 2 γ ξ i ) f I T ( i ) d i .
P b a T Γ ( ( b T + 1 ) / 2 ) 2 b T π γ b T / 2 .
D pe [ d B ] ( 20 / β ) log 10 ( φ mod 2 / ( A mod L ) β ( φ mod 2 β ) ) .
ω z eq 2 = ω z 2 π erf ( v ) 2 v exp ( v 2 ) ω z 2 + 3 2 2 .
φ mod 2 = ω z eq 2 4 σ mod 2 2 2 ω z 2 + 3 8 2 σ mod 2 .
ω z min / a 2 3 / 4 ( 2 1 / 6 8 β ( 3 μ x 2 σ x 4 + 3 μ y 2 σ y 4 + σ x 6 + σ y 6 ) 1 / 3 3 ) 1 / 2 .
M OP = k = 0 N e k | F OP ( γ k ) F ^ OP ( γ k ) | F OP ( γ k ) ,

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