Abstract

The propagation of a free-space optical communications signal through atmospheric turbulence experiences random fluctuations in intensity, including signal fades, which negatively impact the performance of the communications link. The gamma–gamma probability density function is commonly used to model the scintillation of a single beam. One proposed method to reduce the occurrence of scintillation-induced fades at the receiver plane involves the use of multiple beams propagating through independent paths, resulting in a sum of independent gamma–gamma random variables. Recently an analytical model for the probability distribution of irradiance from the sum of multiple independent beams was developed. Because truly independent beams are practically impossible to create, we present here a more general but approximate model for the distribution of beams traveling through partially correlated paths. This model compares favorably with wave-optics simulations and highlights the reduced scintillation as the number of transmitted beams is increased. Additionally, a pulse-position modulation scheme is used to reduce the impact of signal fades when they occur. Analytical and simulated results showed significantly improved performance when compared to fixed threshold on/off keying.

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  1. J. A. Louthain and J. D. Schmidt, “Anisoplanatism in airborne laser communication,” Opt. Express 16, 10769–10785 (2008).
    [CrossRef] [PubMed]
  2. J. A. Louthain and J. D. Schmidt, “Synergy of adaptive thresholds and multiple transmitters in free-space optical communication,” Opt. Express 18, 8948–8962 (2010).
    [CrossRef] [PubMed]
  3. J. A. Louthain and J. D. Schmidt, “Anisoplanatic approach to airborne laser communication,” presented at the 2007 MSS Active E-O Systems, Atlanta, Georgia, USA 24–27 Sept. 2007.
  4. J. Ma, Y. Jiang, S. Yu, L. Tan, and W. Du, “Packet error rate analysis of OOK, DPIM and PPM modulation schemes for ground-to-satellite optical communications,” Opt. Commun. 283, 237–242 (2010).
    [CrossRef]
  5. J. A. Tellez and J. D. Schmidt, “Multibeam scintillation cumulative distribution function,” Opt. Lett. 36, 286–288(2011).
    [CrossRef] [PubMed]
  6. 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, 1554–1562 (2001).
    [CrossRef]
  7. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE, 2001).
    [CrossRef]
  8. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005).
    [CrossRef]
  9. N. D. Chatzidiamantis, G. K. Karagiannidis, and D. S. Michalopoulos, “On the distribution of the sum of gamma–gamma variates and application in MIMO optical wireless systems,” in Proceedings of IEEE Global Telecommunications Conference (IEEE, 2009), pp. 1–6.
  10. J. D. Schmidt, Numerical Simulation of Optical Wave Propagation With Examples in MATLAB (SPIE, 2010).
  11. S. Coy, “Choosing mesh spacings and mesh dimensions for wave optics simulation,” Proc. SPIE 5894, 589405 (2005).
    [CrossRef]
  12. M. C. Roggemann and B. Welsh, Imaging through Turbulence (CRC Press, 1996).
  13. J. A. Louthain, “Atmospheric turbulence scintillation effects of wavefront tilt estimation,” Master’s thesis (Air Force Institute of Technology, 1997).
  14. D. L. Fried, “Anisoplanatism in adaptive optics,” J. Opt. Soc. Am. 72, 52–52 (1982).
    [CrossRef]
  15. J. A. Tellez and J. D. Schmidt, “Multi-beam transmitter geometries for free-space optical communications,” Proc. SPIE 7588, 758803 (2010).
    [CrossRef]
  16. N. Tahir, N. M. Saad, B. B. Samir, V. K. Jain, and S. A. Aljunid, “Binary pulse position modulation simulation system in free space optical communication systems,” in Proceedings of IEEE International Conference on Intelligent and Advanced Systems (IEEE, 2010), pp. 1–4.
  17. H. R. Burris, A. E. Reed, N. M. Namazi, M. J. Vilcheck, and M. Ferraro, “Use of Kalman filtering in data detection in optical communication systems with multiplicative noise,” in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, 2001), pp. 2685–2688.

2011 (1)

2010 (3)

J. A. Louthain and J. D. Schmidt, “Synergy of adaptive thresholds and multiple transmitters in free-space optical communication,” Opt. Express 18, 8948–8962 (2010).
[CrossRef] [PubMed]

J. Ma, Y. Jiang, S. Yu, L. Tan, and W. Du, “Packet error rate analysis of OOK, DPIM and PPM modulation schemes for ground-to-satellite optical communications,” Opt. Commun. 283, 237–242 (2010).
[CrossRef]

J. A. Tellez and J. D. Schmidt, “Multi-beam transmitter geometries for free-space optical communications,” Proc. SPIE 7588, 758803 (2010).
[CrossRef]

2008 (1)

2005 (1)

S. Coy, “Choosing mesh spacings and mesh dimensions for wave optics simulation,” Proc. SPIE 5894, 589405 (2005).
[CrossRef]

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, 1554–1562 (2001).
[CrossRef]

1982 (1)

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, 1554–1562 (2001).
[CrossRef]

Aljunid, S. A.

N. Tahir, N. M. Saad, B. B. Samir, V. K. Jain, and S. A. Aljunid, “Binary pulse position modulation simulation system in free space optical communication systems,” in Proceedings of IEEE International Conference on Intelligent and Advanced Systems (IEEE, 2010), pp. 1–4.

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, 1554–1562 (2001).
[CrossRef]

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

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005).
[CrossRef]

Burris, H. R.

H. R. Burris, A. E. Reed, N. M. Namazi, M. J. Vilcheck, and M. Ferraro, “Use of Kalman filtering in data detection in optical communication systems with multiplicative noise,” in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, 2001), pp. 2685–2688.

Chatzidiamantis, N. D.

N. D. Chatzidiamantis, G. K. Karagiannidis, and D. S. Michalopoulos, “On the distribution of the sum of gamma–gamma variates and application in MIMO optical wireless systems,” in Proceedings of IEEE Global Telecommunications Conference (IEEE, 2009), pp. 1–6.

Coy, S.

S. Coy, “Choosing mesh spacings and mesh dimensions for wave optics simulation,” Proc. SPIE 5894, 589405 (2005).
[CrossRef]

Du, W.

J. Ma, Y. Jiang, S. Yu, L. Tan, and W. Du, “Packet error rate analysis of OOK, DPIM and PPM modulation schemes for ground-to-satellite optical communications,” Opt. Commun. 283, 237–242 (2010).
[CrossRef]

Ferraro, M.

H. R. Burris, A. E. Reed, N. M. Namazi, M. J. Vilcheck, and M. Ferraro, “Use of Kalman filtering in data detection in optical communication systems with multiplicative noise,” in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, 2001), pp. 2685–2688.

Fried, D. L.

Hopen, C. Y.

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

Jain, V. K.

N. Tahir, N. M. Saad, B. B. Samir, V. K. Jain, and S. A. Aljunid, “Binary pulse position modulation simulation system in free space optical communication systems,” in Proceedings of IEEE International Conference on Intelligent and Advanced Systems (IEEE, 2010), pp. 1–4.

Jiang, Y.

J. Ma, Y. Jiang, S. Yu, L. Tan, and W. Du, “Packet error rate analysis of OOK, DPIM and PPM modulation schemes for ground-to-satellite optical communications,” Opt. Commun. 283, 237–242 (2010).
[CrossRef]

Karagiannidis, G. K.

N. D. Chatzidiamantis, G. K. Karagiannidis, and D. S. Michalopoulos, “On the distribution of the sum of gamma–gamma variates and application in MIMO optical wireless systems,” in Proceedings of IEEE Global Telecommunications Conference (IEEE, 2009), pp. 1–6.

Louthain, J. A.

J. A. Louthain and J. D. Schmidt, “Synergy of adaptive thresholds and multiple transmitters in free-space optical communication,” Opt. Express 18, 8948–8962 (2010).
[CrossRef] [PubMed]

J. A. Louthain and J. D. Schmidt, “Anisoplanatism in airborne laser communication,” Opt. Express 16, 10769–10785 (2008).
[CrossRef] [PubMed]

J. A. Louthain and J. D. Schmidt, “Anisoplanatic approach to airborne laser communication,” presented at the 2007 MSS Active E-O Systems, Atlanta, Georgia, USA 24–27 Sept. 2007.

J. A. Louthain, “Atmospheric turbulence scintillation effects of wavefront tilt estimation,” Master’s thesis (Air Force Institute of Technology, 1997).

Ma, J.

J. Ma, Y. Jiang, S. Yu, L. Tan, and W. Du, “Packet error rate analysis of OOK, DPIM and PPM modulation schemes for ground-to-satellite optical communications,” Opt. Commun. 283, 237–242 (2010).
[CrossRef]

Michalopoulos, D. S.

N. D. Chatzidiamantis, G. K. Karagiannidis, and D. S. Michalopoulos, “On the distribution of the sum of gamma–gamma variates and application in MIMO optical wireless systems,” in Proceedings of IEEE Global Telecommunications Conference (IEEE, 2009), pp. 1–6.

Namazi, N. M.

H. R. Burris, A. E. Reed, N. M. Namazi, M. J. Vilcheck, and M. Ferraro, “Use of Kalman filtering in data detection in optical communication systems with multiplicative noise,” in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, 2001), pp. 2685–2688.

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, 1554–1562 (2001).
[CrossRef]

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

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005).
[CrossRef]

Reed, A. E.

H. R. Burris, A. E. Reed, N. M. Namazi, M. J. Vilcheck, and M. Ferraro, “Use of Kalman filtering in data detection in optical communication systems with multiplicative noise,” in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, 2001), pp. 2685–2688.

Roggemann, M. C.

M. C. Roggemann and B. Welsh, Imaging through Turbulence (CRC Press, 1996).

Saad, N. M.

N. Tahir, N. M. Saad, B. B. Samir, V. K. Jain, and S. A. Aljunid, “Binary pulse position modulation simulation system in free space optical communication systems,” in Proceedings of IEEE International Conference on Intelligent and Advanced Systems (IEEE, 2010), pp. 1–4.

Samir, B. B.

N. Tahir, N. M. Saad, B. B. Samir, V. K. Jain, and S. A. Aljunid, “Binary pulse position modulation simulation system in free space optical communication systems,” in Proceedings of IEEE International Conference on Intelligent and Advanced Systems (IEEE, 2010), pp. 1–4.

Schmidt, J. D.

J. A. Tellez and J. D. Schmidt, “Multibeam scintillation cumulative distribution function,” Opt. Lett. 36, 286–288(2011).
[CrossRef] [PubMed]

J. A. Tellez and J. D. Schmidt, “Multi-beam transmitter geometries for free-space optical communications,” Proc. SPIE 7588, 758803 (2010).
[CrossRef]

J. A. Louthain and J. D. Schmidt, “Synergy of adaptive thresholds and multiple transmitters in free-space optical communication,” Opt. Express 18, 8948–8962 (2010).
[CrossRef] [PubMed]

J. A. Louthain and J. D. Schmidt, “Anisoplanatism in airborne laser communication,” Opt. Express 16, 10769–10785 (2008).
[CrossRef] [PubMed]

J. A. Louthain and J. D. Schmidt, “Anisoplanatic approach to airborne laser communication,” presented at the 2007 MSS Active E-O Systems, Atlanta, Georgia, USA 24–27 Sept. 2007.

J. D. Schmidt, Numerical Simulation of Optical Wave Propagation With Examples in MATLAB (SPIE, 2010).

Tahir, N.

N. Tahir, N. M. Saad, B. B. Samir, V. K. Jain, and S. A. Aljunid, “Binary pulse position modulation simulation system in free space optical communication systems,” in Proceedings of IEEE International Conference on Intelligent and Advanced Systems (IEEE, 2010), pp. 1–4.

Tan, L.

J. Ma, Y. Jiang, S. Yu, L. Tan, and W. Du, “Packet error rate analysis of OOK, DPIM and PPM modulation schemes for ground-to-satellite optical communications,” Opt. Commun. 283, 237–242 (2010).
[CrossRef]

Tellez, J. A.

J. A. Tellez and J. D. Schmidt, “Multibeam scintillation cumulative distribution function,” Opt. Lett. 36, 286–288(2011).
[CrossRef] [PubMed]

J. A. Tellez and J. D. Schmidt, “Multi-beam transmitter geometries for free-space optical communications,” Proc. SPIE 7588, 758803 (2010).
[CrossRef]

Vilcheck, M. J.

H. R. Burris, A. E. Reed, N. M. Namazi, M. J. Vilcheck, and M. Ferraro, “Use of Kalman filtering in data detection in optical communication systems with multiplicative noise,” in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, 2001), pp. 2685–2688.

Welsh, B.

M. C. Roggemann and B. Welsh, Imaging through Turbulence (CRC Press, 1996).

Yu, S.

J. Ma, Y. Jiang, S. Yu, L. Tan, and W. Du, “Packet error rate analysis of OOK, DPIM and PPM modulation schemes for ground-to-satellite optical communications,” Opt. Commun. 283, 237–242 (2010).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Commun. (1)

J. Ma, Y. Jiang, S. Yu, L. Tan, and W. Du, “Packet error rate analysis of OOK, DPIM and PPM modulation schemes for ground-to-satellite optical communications,” Opt. Commun. 283, 237–242 (2010).
[CrossRef]

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, 1554–1562 (2001).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Proc. SPIE (2)

J. A. Tellez and J. D. Schmidt, “Multi-beam transmitter geometries for free-space optical communications,” Proc. SPIE 7588, 758803 (2010).
[CrossRef]

S. Coy, “Choosing mesh spacings and mesh dimensions for wave optics simulation,” Proc. SPIE 5894, 589405 (2005).
[CrossRef]

Other (9)

M. C. Roggemann and B. Welsh, Imaging through Turbulence (CRC Press, 1996).

J. A. Louthain, “Atmospheric turbulence scintillation effects of wavefront tilt estimation,” Master’s thesis (Air Force Institute of Technology, 1997).

N. Tahir, N. M. Saad, B. B. Samir, V. K. Jain, and S. A. Aljunid, “Binary pulse position modulation simulation system in free space optical communication systems,” in Proceedings of IEEE International Conference on Intelligent and Advanced Systems (IEEE, 2010), pp. 1–4.

H. R. Burris, A. E. Reed, N. M. Namazi, M. J. Vilcheck, and M. Ferraro, “Use of Kalman filtering in data detection in optical communication systems with multiplicative noise,” in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, 2001), pp. 2685–2688.

J. A. Louthain and J. D. Schmidt, “Anisoplanatic approach to airborne laser communication,” presented at the 2007 MSS Active E-O Systems, Atlanta, Georgia, USA 24–27 Sept. 2007.

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

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005).
[CrossRef]

N. D. Chatzidiamantis, G. K. Karagiannidis, and D. S. Michalopoulos, “On the distribution of the sum of gamma–gamma variates and application in MIMO optical wireless systems,” in Proceedings of IEEE Global Telecommunications Conference (IEEE, 2009), pp. 1–6.

J. D. Schmidt, Numerical Simulation of Optical Wave Propagation With Examples in MATLAB (SPIE, 2010).

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Figures (11)

Fig. 1
Fig. 1

Analytical and wave-optics generated CDF for an aperture-averaged, single-beam irradiance.

Fig. 2
Fig. 2

Analytical CDF (solid lines) and wave-optics results (dashed lines) plotted for one through seven beams.

Fig. 3
Fig. 3

Normalized amplitude and phase structure functions. When the structure function no longer increases with increased separation, d, then points separated by that distance are uncorrelated. The vertical lines show the separations due to the isoplanatic angle θ 0 , the tilt isoplanatic angle θ TA , and the chosen separation of twice the Fresnel zone size 2 ρ c .

Fig. 4
Fig. 4

Configuration used for multiple transmitters. Each configuration is a subset of a hexagonal close-pack grid. The top row shows one, two, three, and four transmitters from left to right, while the bottom row shows five, six, and seven transmitters. In all cases, the total transmitted power is held constant at P = 1 W .

Fig. 5
Fig. 5

Correlation coefficient versus beam separation for two Gaussian beams propagated over 100 km path with σ 1 2 = 1.0 .

Fig. 6
Fig. 6

Exponent argument versus beam separation for two Gaussian beams propagated over 100 km path with σ 1 2 = 1.0 .

Fig. 7
Fig. 7

Normalized CDFs for integrated irradiance with receivers placed at various off-axis distances.

Fig. 8
Fig. 8

Mean integrated irradiance plotted as a function of receiver distance from beam center.

Fig. 9
Fig. 9

Comparison of wave-optics results with analytical CDF for one to seven spatially separated beams.

Fig. 10
Fig. 10

BER for multiple beams using OOK with fixed threshold. The markers indicate results from the Monte Carlo simulation of the communications link. The dashed lines are the “analytical” results using the data from the wave-optics simulation. The solid line is the analytical result obtained from the newly derived model of the received intensity. Results are shown for one, two, four, and seven transmitters.

Fig. 11
Fig. 11

BER for multiple beams using PPM. The markers indicate results from the Monte Carlo simulation of the communications link. The dashed lines are the “analytical” results using the data from the wave-optics simulation. The solid line is the analytical result obtained from the newly derived model of the received intensity. Results are shown for one, two, four, and seven transmitters.

Equations (37)

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p ( 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 ,
P ( I I T ) = 0 I T p ( I ) d I = π sin [ π ( α β ) ] Γ ( α ) Γ ( β ) × { ( α β I T ) β β Γ ( β 1 ) × F 2 1 ( β ; β + 1 , β 1 ; α β I T ) ( α β I T ) α α Γ ( α 1 ) × F 2 1 ( α ; α + 1 , α 1 ; α β I T ) } ,
σ ln x 2 ( D ) 0.49 σ 1 2 ( Ω G Λ 1 Ω G + Λ 1 ) 2 ( 1 3 1 2 Θ ¯ 1 + 1 5 Θ ¯ 1 2 ) × [ η x 1 + 0.40 η x ( 2 Θ ¯ 1 ) / ( Λ 1 + Ω G ) ] 7 / 6 ,
η x = ( 1 3 1 2 Θ ¯ 1 + 1 5 Θ ¯ 1 2 ) 6 / 7 ( σ B / σ 1 ) 12 / 7 1 + 0.56 σ B 12 / 5 .
σ ln y 2 ( D ) 1.27 σ 1 2 η y 5 / 6 1 + 0.40 η y / ( Λ 1 + Ω G ) , η y 1 ,
η y = 3 ( σ 1 σ B ) 12 / 5 ( 1 + 0.69 σ B 12 / 5 ) .
σ B 2 3.86 σ 1 2 { 0.40 [ ( 1 + 2 Θ 1 ) 2 + 4 Λ 1 2 ] 5 / 12 × cos [ 5 6 tan 1 ( 1 + 2 Θ 1 2 Λ 1 ) ] 11 16 Λ 1 5 / 6 } ,
Θ 1 = Θ 0 Θ 0 2 + Λ 0 2 ,
Λ 1 = Λ 0 Θ 0 2 + Λ 0 2 .
Θ 0 = 1 L F 0 , Λ 0 = 2 L k W 0 2 ,
I N = i = 1 N x i y i ,
I N = 1 N ( i = 1 N x i ) ( i = 1 N y i ) + 1 N i = 1 N 1 j = i + 1 N ( x i x j ) ( y i y j ) .
ϵ = 1 N i = 1 N 1 j = i + 1 N ( x i x j ) ( y i y j ) .
α N = N α + ϵ N ,
ϵ N = ( N 1 ) 0.127 0.95 α 0.0058 β 1 + 0.00124 α + 0.98 β .
P R = P T 0 2 π 0 D / 2 2 π W e exp ( 2 r 2 W e 2 ) r d r d θ = P T [ 1 exp ( D 2 2 W e 2 ) ] ,
W e W 1 ( 1 + 1.63 σ 1 12 / 5 Λ 1 ) 1 / 2
W 1 = W 0 ( Θ 0 2 + Λ 0 2 ) 1 / 2 .
Φ n ( κ , z ) = 0.033 C n 2 ( z ) ( κ 2 + κ 0 2 ) 11 / 6 ,
D χ ( d ) = 3.089 ( L 0 r 0 ) 0 [ 1 J 0 ( κ d L 0 ) ] [ 1 2 π L 0 2 λ L κ 2 sin ( λ L κ 2 2 π L 0 2 ) ] κ d κ ( κ 2 + 4 π 2 ) 11 / 6
D ψ ( d ) = 3.089 ( L 0 r 0 ) 0 [ 1 J 0 ( κ d L 0 ) ] [ 1 + 2 π L 0 2 λ L κ 2 sin ( λ L κ 2 2 π L 0 2 ) ] κ d κ ( κ 2 + 4 π 2 ) 11 / 6
θ 0 = 0.949 ( k 2 C n 2 L 8 / 3 ) 3 / 5 ,
θ TA = 0.319 λ D 1 / 6 C n L 3 / 2 ,
α N = α [ N α + ϵ α ] f ( ρ ( d ) ) ,
β N = β N f ( ρ ( d ) ) ,
ρ ( d ) = ( I A ( d / 2 , 0 ) μ I A ( d / 2 , 0 ) ) ( I A ( d / 2 , 0 ) μ I A ( d / 2 , 0 ) ) σ I A ( d / 2 , 0 ) σ I A ( d / 2 , 0 ) ,
I A ( x , y ) = A I ( ξ x , η y ) d η d ξ
= D / 2 D / 2 ( D / 2 ) 2 ( ξ x ) 2 ( D / 2 ) 2 ( ξ x ) 2 I ( ξ x , η y ) d η d ξ ,
ρ ( d ) exp ( 0.6875 d / ρ c ) .
f ( ρ ) ( 1 ρ ) 1.4894 .
f ( ρ ( d ) ) [ 1 exp ( 0.6875 d / ρ c ) ] 1.4894 .
P B = P [ s 1 ] P [ H 2 | s 1 ] + P [ s 2 ] P [ H 1 | s 2 ] ,
P [ H 2 | s 1 ] = γ T p ( z | s 1 ) d z = γ T 1 σ 0 2 π exp [ 1 2 ( z a 1 σ 0 ) 2 ] d z = Q ( γ T μ 1 σ 0 ) ,
P [ H 1 | s 2 ] = Q ( μ 2 γ T σ 0 ) ,
Q ( x ) = 1 2 π x exp ( u 2 2 ) d u .
γ Th = μ 1 σ 2 2 μ 2 σ 1 2 σ 2 2 σ 1 2 + σ 1 σ 2 σ 2 2 σ 1 2 ( μ 2 μ 1 ) 2 + 2 ( σ 2 2 σ 1 2 ) ln ( σ 2 σ 1 ) ,
P B = 0 P [ γ ] p ( γ ) d γ ,

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