Abstract

In this paper, a single-scatter propagation model is developed that expands the classical model by considering a finite receiver-aperture size for non-line-of-sight communication. The expanded model overcomes some of the difficulties with the classical model, most notably, inaccuracies in scenarios with short range and low elevation angle where significant scattering takes place near the receiver. The developed model does not approximate the receiver aperture as a point, but uses its dimensions for both field-of-view and solid-angle computations. To verify the model, a Monte Carlo simulation of photon transport in a turbid medium is applied. Simulation results for temporal responses and path losses are presented at a wavelength of 260nm that lies in the solar-blind ultraviolet region.

© 2011 Optical Society of America

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References

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  1. R. E. Huffman, Atmospheric Ultraviolet Remote Sensing(Academic, 1992).
  2. G. A. Shaw, A. M. Siegel, and M. L. Nischan, “Demonstration system and applications for compact wireless ultraviolet communications,” Proc. SPIE 5071, 241–252 (2003).
    [CrossRef]
  3. Z. Xu and B. M. Sadler, “Ultraviolet communications: potential and state-of-the-art,” IEEE Commun. Mag. 46(5), 67–73 (2008).
    [CrossRef]
  4. G. A. Shaw, A. M. Siegel, and J. Model, “Extending the range and performance of non-line-of-sight ultraviolet communication links,” Proc. SPIE 6231, 62310C (2006).
    [CrossRef]
  5. G. Chen, F. Abou-Galala, Z. Xu, and B. M. Sadler, “Experimental evaluation of LED-based solar blind NLOS communication links,” Opt. Express 16, 15059–15068 (2008).
    [CrossRef] [PubMed]
  6. G. Chen, F. Abou-Galala, H. D. Z. Xu, and B. M. Sadler, “Path loss modeling and performance trade-off study for short-range non-line-of-sight ultraviolet communications,” Opt. Express 17, 3929–3940 (2009).
    [CrossRef] [PubMed]
  7. Q. He, Z. Xu, and B. M. Sadler, “Performance of short-range non-line-of-sight LED-based ultraviolet communication receivers,” Opt. Express 18, 12226–12238 (2010).
    [CrossRef] [PubMed]
  8. G. Chen, Z. Xu, and B. M. Sadler, “Experimental demonstration of ultraviolet pulse broadening in short-range non-line-of-sight communication channels,” Opt. Express 18, 10500–10509(2010).
    [CrossRef] [PubMed]
  9. D. Kedar and S. Arnon, “Non-line-of-sight optical wireless sensor network operating in multiscattering channel,” Appl. Opt. 45, 8454–8461 (2006).
    [CrossRef] [PubMed]
  10. D. M. Reilly, “Atmospheric optical communications in the middle ultraviolet,” Master’s thesis (Massachusetts Institute of Technology, 1976).
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    [CrossRef]
  12. M. R. Luettgen, J. H. Shapiro, and D. M. Reilly, “Non-line-of-sight single-scatter propagation model,” J. Opt. Soc. Am. A 8, 1964–1972 (1991).
    [CrossRef]
  13. L. Wang, Z. Xu, and B. M. Sadler, “Non-line-of-sight ultraviolet link loss in noncoplanar geometry,” Opt. Lett. 35, 1263–1265(2010).
    [CrossRef] [PubMed]
  14. M. A. Elshimy and S. Hranilovic, “Non-line-of-sight single-scatter propagation model for noncoplanar geometries,” J. Opt. Soc. Am. A 28, 420–428 (2011).
    [CrossRef]
  15. H. Yin, S. Chang, H. Jia, J. Yang, and J. Yang, “Non-line-of-sight multiscatter propagation model,” J. Opt. Soc. Am. A 26, 2466–2469 (2009).
    [CrossRef]
  16. H. Ding, Z. Xu, and B. M. Sadler, “A path loss model for non-line-of-sight ultraviolet multiple scattering channels,” EURASIP J. Wireless Commun. Network. 2010, 598572 (2010).
    [CrossRef]
  17. R. J. Drost, T. J. Moore, and B. M. Sadler, “UV communications channel modeling incorporating multiple scattering interactions,” J. Opt. Soc. Am. A 28, 686–695 (2011).
    [CrossRef]
  18. S. A. Prahl, M. Keijzer, S. L. Jacques, and A. J. Welch, “A Monte Carlo model of light propagation in tissue,” in Dosimetry of Laser Radiation in Medicine and Biology, Vol. IS 5 of SPIE Institute Series (1989, SPIE), pp. 102–111.
  19. I. M. Sobol’, A Primer for the Monte Carlo Method (CRC Press, 1994).
  20. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley Interscience, 1983).
  21. M.Abramowitz and I.A.Stegun, eds., Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables (Dover, 1972).
  22. M. Naito, “A method of calculating the solid angle subtended by a circular aperture,” J. Phys. Soc. Jpn. 12, 1122–1129(1957).
    [CrossRef]
  23. F. Paxton, “Solid angle calculation for a circular disk,” Rev. Sci. Instrum. 30, 254–258 (1959).
    [CrossRef]
  24. I. S. Jones, “The solid angle subtended by a circular disc with application to roadsigns,” Int. J. Math. Educ. Sci. Technol. 27, 667–674 (1996).
    [CrossRef]
  25. S. Tryka, “Angular distribution of the solid angle at a point subtended by a circular disk,” Opt. Commun. 137, 317–333 (1997).
    [CrossRef]
  26. E. E. Lewis and W. F. Miller, Computational Methods of Neutron Transport (Wiley, 1984).
  27. A. F. Bielajew, Fundamentals of the Monte Carlo Method for Neutral and Charged Particle Transport (2001), http://www-personal.engin.umich.edu/bielajew/MCBook/book.pdf.
  28. H. Kahn and T. E. Harris, “Estimation of particle transmission by random sampling, Monte Carlo method,” in Vol.  12 of National Bureau of Standards Applied Mathematics Series (1951), pp. 27–30.
  29. J.F.Briesmeister, ed., “MCNP—A general Monte Carlo N-particle transport code, Version 4C,” Report LA-13709-M (Los Alamos National Laboratory, 2000).
  30. G. A. Shaw, M. Nischan, M. Iyengar, S. Kaushik, and M. K. Griffin, “NLOS UV communication for distributed sensor systems,” Proc. SPIE 4126, 83–96 (2000).
    [CrossRef]
  31. The MathWorks, Inc., “MathWorks—MATLAB and Simulink for technical computing,” http://www.mathworks.com.

2011 (2)

2010 (4)

2009 (2)

2008 (2)

2006 (2)

G. A. Shaw, A. M. Siegel, and J. Model, “Extending the range and performance of non-line-of-sight ultraviolet communication links,” Proc. SPIE 6231, 62310C (2006).
[CrossRef]

D. Kedar and S. Arnon, “Non-line-of-sight optical wireless sensor network operating in multiscattering channel,” Appl. Opt. 45, 8454–8461 (2006).
[CrossRef] [PubMed]

2003 (1)

G. A. Shaw, A. M. Siegel, and M. L. Nischan, “Demonstration system and applications for compact wireless ultraviolet communications,” Proc. SPIE 5071, 241–252 (2003).
[CrossRef]

2000 (1)

G. A. Shaw, M. Nischan, M. Iyengar, S. Kaushik, and M. K. Griffin, “NLOS UV communication for distributed sensor systems,” Proc. SPIE 4126, 83–96 (2000).
[CrossRef]

1997 (1)

S. Tryka, “Angular distribution of the solid angle at a point subtended by a circular disk,” Opt. Commun. 137, 317–333 (1997).
[CrossRef]

1996 (1)

I. S. Jones, “The solid angle subtended by a circular disc with application to roadsigns,” Int. J. Math. Educ. Sci. Technol. 27, 667–674 (1996).
[CrossRef]

1991 (1)

1979 (1)

1959 (1)

F. Paxton, “Solid angle calculation for a circular disk,” Rev. Sci. Instrum. 30, 254–258 (1959).
[CrossRef]

1957 (1)

M. Naito, “A method of calculating the solid angle subtended by a circular aperture,” J. Phys. Soc. Jpn. 12, 1122–1129(1957).
[CrossRef]

Abou-Galala, F.

Arnon, S.

Bielajew, A. F.

A. F. Bielajew, Fundamentals of the Monte Carlo Method for Neutral and Charged Particle Transport (2001), http://www-personal.engin.umich.edu/bielajew/MCBook/book.pdf.

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley Interscience, 1983).

Chang, S.

Chen, G.

Ding, H.

H. Ding, Z. Xu, and B. M. Sadler, “A path loss model for non-line-of-sight ultraviolet multiple scattering channels,” EURASIP J. Wireless Commun. Network. 2010, 598572 (2010).
[CrossRef]

Drost, R. J.

Elshimy, M. A.

Griffin, M. K.

G. A. Shaw, M. Nischan, M. Iyengar, S. Kaushik, and M. K. Griffin, “NLOS UV communication for distributed sensor systems,” Proc. SPIE 4126, 83–96 (2000).
[CrossRef]

Harris, T. E.

H. Kahn and T. E. Harris, “Estimation of particle transmission by random sampling, Monte Carlo method,” in Vol.  12 of National Bureau of Standards Applied Mathematics Series (1951), pp. 27–30.

He, Q.

Hranilovic, S.

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley Interscience, 1983).

Huffman, R. E.

R. E. Huffman, Atmospheric Ultraviolet Remote Sensing(Academic, 1992).

Iyengar, M.

G. A. Shaw, M. Nischan, M. Iyengar, S. Kaushik, and M. K. Griffin, “NLOS UV communication for distributed sensor systems,” Proc. SPIE 4126, 83–96 (2000).
[CrossRef]

Jacques, S. L.

S. A. Prahl, M. Keijzer, S. L. Jacques, and A. J. Welch, “A Monte Carlo model of light propagation in tissue,” in Dosimetry of Laser Radiation in Medicine and Biology, Vol. IS 5 of SPIE Institute Series (1989, SPIE), pp. 102–111.

Jia, H.

Jones, I. S.

I. S. Jones, “The solid angle subtended by a circular disc with application to roadsigns,” Int. J. Math. Educ. Sci. Technol. 27, 667–674 (1996).
[CrossRef]

Kahn, H.

H. Kahn and T. E. Harris, “Estimation of particle transmission by random sampling, Monte Carlo method,” in Vol.  12 of National Bureau of Standards Applied Mathematics Series (1951), pp. 27–30.

Kaushik, S.

G. A. Shaw, M. Nischan, M. Iyengar, S. Kaushik, and M. K. Griffin, “NLOS UV communication for distributed sensor systems,” Proc. SPIE 4126, 83–96 (2000).
[CrossRef]

Kedar, D.

Keijzer, M.

S. A. Prahl, M. Keijzer, S. L. Jacques, and A. J. Welch, “A Monte Carlo model of light propagation in tissue,” in Dosimetry of Laser Radiation in Medicine and Biology, Vol. IS 5 of SPIE Institute Series (1989, SPIE), pp. 102–111.

Lewis, E. E.

E. E. Lewis and W. F. Miller, Computational Methods of Neutron Transport (Wiley, 1984).

Luettgen, M. R.

Miller, W. F.

E. E. Lewis and W. F. Miller, Computational Methods of Neutron Transport (Wiley, 1984).

Model, J.

G. A. Shaw, A. M. Siegel, and J. Model, “Extending the range and performance of non-line-of-sight ultraviolet communication links,” Proc. SPIE 6231, 62310C (2006).
[CrossRef]

Moore, T. J.

Naito, M.

M. Naito, “A method of calculating the solid angle subtended by a circular aperture,” J. Phys. Soc. Jpn. 12, 1122–1129(1957).
[CrossRef]

Nischan, M.

G. A. Shaw, M. Nischan, M. Iyengar, S. Kaushik, and M. K. Griffin, “NLOS UV communication for distributed sensor systems,” Proc. SPIE 4126, 83–96 (2000).
[CrossRef]

Nischan, M. L.

G. A. Shaw, A. M. Siegel, and M. L. Nischan, “Demonstration system and applications for compact wireless ultraviolet communications,” Proc. SPIE 5071, 241–252 (2003).
[CrossRef]

Paxton, F.

F. Paxton, “Solid angle calculation for a circular disk,” Rev. Sci. Instrum. 30, 254–258 (1959).
[CrossRef]

Prahl, S. A.

S. A. Prahl, M. Keijzer, S. L. Jacques, and A. J. Welch, “A Monte Carlo model of light propagation in tissue,” in Dosimetry of Laser Radiation in Medicine and Biology, Vol. IS 5 of SPIE Institute Series (1989, SPIE), pp. 102–111.

Reilly, D. M.

Sadler, B. M.

R. J. Drost, T. J. Moore, and B. M. Sadler, “UV communications channel modeling incorporating multiple scattering interactions,” J. Opt. Soc. Am. A 28, 686–695 (2011).
[CrossRef]

L. Wang, Z. Xu, and B. M. Sadler, “Non-line-of-sight ultraviolet link loss in noncoplanar geometry,” Opt. Lett. 35, 1263–1265(2010).
[CrossRef] [PubMed]

H. Ding, Z. Xu, and B. M. Sadler, “A path loss model for non-line-of-sight ultraviolet multiple scattering channels,” EURASIP J. Wireless Commun. Network. 2010, 598572 (2010).
[CrossRef]

G. Chen, Z. Xu, and B. M. Sadler, “Experimental demonstration of ultraviolet pulse broadening in short-range non-line-of-sight communication channels,” Opt. Express 18, 10500–10509(2010).
[CrossRef] [PubMed]

Q. He, Z. Xu, and B. M. Sadler, “Performance of short-range non-line-of-sight LED-based ultraviolet communication receivers,” Opt. Express 18, 12226–12238 (2010).
[CrossRef] [PubMed]

G. Chen, F. Abou-Galala, H. D. Z. Xu, and B. M. Sadler, “Path loss modeling and performance trade-off study for short-range non-line-of-sight ultraviolet communications,” Opt. Express 17, 3929–3940 (2009).
[CrossRef] [PubMed]

Z. Xu and B. M. Sadler, “Ultraviolet communications: potential and state-of-the-art,” IEEE Commun. Mag. 46(5), 67–73 (2008).
[CrossRef]

G. Chen, F. Abou-Galala, Z. Xu, and B. M. Sadler, “Experimental evaluation of LED-based solar blind NLOS communication links,” Opt. Express 16, 15059–15068 (2008).
[CrossRef] [PubMed]

Shapiro, J. H.

Shaw, G. A.

G. A. Shaw, A. M. Siegel, and J. Model, “Extending the range and performance of non-line-of-sight ultraviolet communication links,” Proc. SPIE 6231, 62310C (2006).
[CrossRef]

G. A. Shaw, A. M. Siegel, and M. L. Nischan, “Demonstration system and applications for compact wireless ultraviolet communications,” Proc. SPIE 5071, 241–252 (2003).
[CrossRef]

G. A. Shaw, M. Nischan, M. Iyengar, S. Kaushik, and M. K. Griffin, “NLOS UV communication for distributed sensor systems,” Proc. SPIE 4126, 83–96 (2000).
[CrossRef]

Siegel, A. M.

G. A. Shaw, A. M. Siegel, and J. Model, “Extending the range and performance of non-line-of-sight ultraviolet communication links,” Proc. SPIE 6231, 62310C (2006).
[CrossRef]

G. A. Shaw, A. M. Siegel, and M. L. Nischan, “Demonstration system and applications for compact wireless ultraviolet communications,” Proc. SPIE 5071, 241–252 (2003).
[CrossRef]

Sobol’, I. M.

I. M. Sobol’, A Primer for the Monte Carlo Method (CRC Press, 1994).

Tryka, S.

S. Tryka, “Angular distribution of the solid angle at a point subtended by a circular disk,” Opt. Commun. 137, 317–333 (1997).
[CrossRef]

Wang, L.

Warde, C.

Welch, A. J.

S. A. Prahl, M. Keijzer, S. L. Jacques, and A. J. Welch, “A Monte Carlo model of light propagation in tissue,” in Dosimetry of Laser Radiation in Medicine and Biology, Vol. IS 5 of SPIE Institute Series (1989, SPIE), pp. 102–111.

Xu, H. D. Z.

Xu, Z.

Yang, J.

Yin, H.

Appl. Opt. (1)

EURASIP J. Wireless Commun. Network. (1)

H. Ding, Z. Xu, and B. M. Sadler, “A path loss model for non-line-of-sight ultraviolet multiple scattering channels,” EURASIP J. Wireless Commun. Network. 2010, 598572 (2010).
[CrossRef]

IEEE Commun. Mag. (1)

Z. Xu and B. M. Sadler, “Ultraviolet communications: potential and state-of-the-art,” IEEE Commun. Mag. 46(5), 67–73 (2008).
[CrossRef]

Int. J. Math. Educ. Sci. Technol. (1)

I. S. Jones, “The solid angle subtended by a circular disc with application to roadsigns,” Int. J. Math. Educ. Sci. Technol. 27, 667–674 (1996).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

J. Phys. Soc. Jpn. (1)

M. Naito, “A method of calculating the solid angle subtended by a circular aperture,” J. Phys. Soc. Jpn. 12, 1122–1129(1957).
[CrossRef]

Opt. Commun. (1)

S. Tryka, “Angular distribution of the solid angle at a point subtended by a circular disk,” Opt. Commun. 137, 317–333 (1997).
[CrossRef]

Opt. Express (4)

Opt. Lett. (1)

Proc. SPIE (3)

G. A. Shaw, A. M. Siegel, and J. Model, “Extending the range and performance of non-line-of-sight ultraviolet communication links,” Proc. SPIE 6231, 62310C (2006).
[CrossRef]

G. A. Shaw, A. M. Siegel, and M. L. Nischan, “Demonstration system and applications for compact wireless ultraviolet communications,” Proc. SPIE 5071, 241–252 (2003).
[CrossRef]

G. A. Shaw, M. Nischan, M. Iyengar, S. Kaushik, and M. K. Griffin, “NLOS UV communication for distributed sensor systems,” Proc. SPIE 4126, 83–96 (2000).
[CrossRef]

Rev. Sci. Instrum. (1)

F. Paxton, “Solid angle calculation for a circular disk,” Rev. Sci. Instrum. 30, 254–258 (1959).
[CrossRef]

Other (11)

E. E. Lewis and W. F. Miller, Computational Methods of Neutron Transport (Wiley, 1984).

A. F. Bielajew, Fundamentals of the Monte Carlo Method for Neutral and Charged Particle Transport (2001), http://www-personal.engin.umich.edu/bielajew/MCBook/book.pdf.

H. Kahn and T. E. Harris, “Estimation of particle transmission by random sampling, Monte Carlo method,” in Vol.  12 of National Bureau of Standards Applied Mathematics Series (1951), pp. 27–30.

J.F.Briesmeister, ed., “MCNP—A general Monte Carlo N-particle transport code, Version 4C,” Report LA-13709-M (Los Alamos National Laboratory, 2000).

The MathWorks, Inc., “MathWorks—MATLAB and Simulink for technical computing,” http://www.mathworks.com.

R. E. Huffman, Atmospheric Ultraviolet Remote Sensing(Academic, 1992).

D. M. Reilly, “Atmospheric optical communications in the middle ultraviolet,” Master’s thesis (Massachusetts Institute of Technology, 1976).

S. A. Prahl, M. Keijzer, S. L. Jacques, and A. J. Welch, “A Monte Carlo model of light propagation in tissue,” in Dosimetry of Laser Radiation in Medicine and Biology, Vol. IS 5 of SPIE Institute Series (1989, SPIE), pp. 102–111.

I. M. Sobol’, A Primer for the Monte Carlo Method (CRC Press, 1994).

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley Interscience, 1983).

M.Abramowitz and I.A.Stegun, eds., Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables (Dover, 1972).

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Figures (12)

Fig. 1
Fig. 1

Geometric example of a NLOS single-scatter propagation model.

Fig. 2
Fig. 2

Example of transmitter–receiver geometry in prolate spheroidal coordinates with a cutting plane Z C perpendicular to the z axis: (a) three-dimensional view and (b) cross section in Z C .

Fig. 3
Fig. 3

Geometry of the solid angle subtended by a circular aperture at a point q: (a)  a o a and (b)  a o > a .

Fig. 4
Fig. 4

Simulation results for temporal response h ( t ) at r = 10 m , β T = 1 ° , θ T = 0.1 ° , β R = 75 ° , and θ R = 5 ° with aperture of radius a = 5 cm . Results are generated with MC simulation (indicated by solid circle discrete points), the expanded model of Eq. (12), and the classical model [12].

Fig. 5
Fig. 5

Simulation results for temporal response h ( t ) at r = 10 m , β T = 3 ° , θ T = 0.1 ° , β R = 75 ° , and θ R = 5 ° with aperture of radius a = 5 cm . Results are generated with MC simulation (indicated by solid circle discrete points), the expanded model of Eq. (12), and the classical model [12].

Fig. 6
Fig. 6

Simulation results for temporal response h ( t ) at r = 10 m , β T = 10 ° , θ T = 0.1 ° , β R = 75 ° , and θ R = 5 ° with aperture of radius a = 5 cm . Results are generated with MC simulation (indicated by solid circle discrete points), the expanded model of Eq. (12), and the classical model [12].

Fig. 7
Fig. 7

Simulation results for temporal response h ( t ) at r = 30 m , β T = 1 ° , θ T = 0.1 ° , β R = 75 ° , and θ R = 5 ° with aperture of radius a = 5 cm . Results are generated with MC simulation (indicated by solid circle discrete points), the expanded model of Eq. (12), and the classical model [12].

Fig. 8
Fig. 8

Simulation results for temporal response h ( t ) at r = 30 m , β T = 3 ° , θ T = 0.1 ° , β R = 75 ° , and θ R = 5 ° with aperture of radius a = 5 cm . Results are generated with MC simulation (indicated by solid circle discrete points), the expanded model of Eq. (12), and the classical model [12].

Fig. 9
Fig. 9

Simulation results for temporal response h ( t ) at r = 100 m , β T = 1 ° , θ T = 0.1 ° , β R = 75 ° , and θ R = 5 ° with aperture of radius a = 5 cm . Results are generated with MC simulation (indicated by solid circle discrete points), the expanded model of Eq. (12), and the classical model [12].

Fig. 10
Fig. 10

Simulation results for channel PL, Eq. (13), for r = 10 m at β R = 75 ° , θ R = 5 ° , θ T = { 0.1 ° 1 °   with   inc   0.1 ° } , and β T in the range [ 0.1 ° , 10 ° ] with aperture of radius a = 5 cm . Results are generated with the classical model [12] and the expanded model from Eq. (12). The diamond-shaped discrete points indicate the PL at r R eq / a = 30 for the expanded model.

Fig. 11
Fig. 11

Simulation results for channel PL, Eq. (13), for r = 30 m at β R = 75 ° , θ R = 5 ° , θ T = { 0.1 ° 1 °   with   inc   0.1 ° } , and β T in the range [ 0.1 ° , 10 ° ] with aperture of radius a = 5 cm . Results are generated with the classical model [12] and the expanded model from Eq. (12). The diamond-shaped discrete points indicate the PL at r R eq / a = 30 for the expanded model.

Fig. 12
Fig. 12

Simulation results for channel PL, Eq. (13), for r = 100 m at β R = 75 ° , θ R = 5 ° , θ T = { 0.1 ° 1 °   with   inc   0.1 ° } , and β T in the range [ 0.1 ° , 10 ° ] with aperture of radius a = 5 cm . Results are generated with the classical model [12] and the expanded model from Eq. (12). The diamond-shaped discrete points indicate the PL at r R eq / a = 30 for the expanded model.

Tables (1)

Tables Icon

Table 1 Parameters of Bounding Lines for T x and R x Cones

Equations (41)

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p ( x ) δ x = Pr ( x δ x / 2 X < x + δ x / 2 ) ,
δ Ω T Ω T .
k e exp ( k e r T q ) δ r T .
k e Ω T r T q 2 exp ( k e r T q ) δ V q ,
ϖ 0 = k s / k e .
P ( Ω T · Ω R ) 4 π δ Ω R .
p s ( r R q ) = exp ( k e r R q ) ,
k s P ( Ω T · Ω R ) 4 π Ω T r T q 2 exp [ k e ( r T q + r R q ) ] δ Ω R δ V q .
r T q = r 2 ( ξ η ) ,
r R q = r 2 ( ξ + η ) .
k s r P ( Ω T · Ω R ) 8 π Ω T exp ( k e r ξ ) ( ξ + η ξ η ) δ Ω R δ ϕ δ η δ ξ ,
g ( ξ ) δ ξ = k s r exp ( k e r ξ ) 8 π Ω T η 1 ( ξ ) η 2 ( ξ ) ξ + η ξ η ( ϕ 1 ( ξ , η ) ϕ 2 ( ξ , η ) S d P ( Ω T · Ω R ) d Ω R d ϕ ) d η δ ξ .
h ( t ) = c r g ( c t r ) .
PL = 10 log 10 ( h ( t ) d t ) = 10 log 10 P A .
ξ min = [ 2 K 1 + 2 ( K 1 2 K 2 2 ) 1 / 2 ] 1 / 2 r ,
K 1 = ( m R 2 2 + 1 ) ( c T 1 c R 2 m R 2 m T 1 ) 2 + 2 m R 2 c R 2 c T 1 c R 2 m R 2 m T 1 + ( c R 2 2 + r 2 4 ) ,
K 2 = r c T 1 c R 2 m R 2 m T 1 .
η k = 2 m k c k ξ ± [ ( 2 m k c k ξ ) 2 ( m k 2 ξ 2 + ξ 2 1 ) ( 4 c k 2 + r 2 r 2 ξ 2 ) ] 1 / 2 r ( m k 2 ξ 2 + ξ 2 1 ) ,
ξ a + ( a 2 + r 2 + 2 a r sin β R ) 1 / 2 r ,
η 1 ( ξ ) = max [ η T 1 ( ξ ) , η R 1 ( ξ ) ] ,
η 2 ( ξ ) = min [ η T 2 ( ξ ) , η R 2 ( ξ ) ] .
R ( ξ , η ) = [ r 2 4 ( ξ 2 1 ) ( 1 η 2 ) ] 1 / 2 ,
A ( x x R ) 2 + y 2 + B ( x x R ) + C = 0 ,
A = cos 2 β R tan 2 θ R sin 2 β R ,
B = ( z C z R ) ( 1 + tan 2 θ R ) sin 2 β R ,
C = ( z C z R ) 2 ( sin 2 β R tan 2 θ R cos 2 β R ) ,
x 1 , 2 = ( B 2 A x R ) ± [ ( B 2 A x R ) 2 4 ( A 1 ) ( R 2 + A x R 2 B x R + C ) ] 1 / 2 2 ( A 1 ) ,
y 1 , 2 = ± ( R 2 x 1 2 ) 1 / 2 , y 3 , 4 ( ξ , η ) = ± ( R 2 x 2 2 ) 1 / 2 .
x * = { x 1 , 0 x 1 R x 2 , 0 x 2 R 0 , otherwise ,
y * 1 , 2 = ± ( R 2 x * 2 ) 1 / 2 .
ϕ R 2 , R 1 ( ξ , η ) = { tan 1 ( y * 1 , 2 x * ) , [ C ( ξ , η ) S R ( z ) ] ± π / 2 , otherwise .
ϕ 1 ( ξ , η ) = max [ ϕ T 1 ( ξ , η ) , ϕ R 1 ( ξ , η ) ] ,
ϕ 2 ( ξ , η ) = min [ ϕ T 2 ( ξ , η ) , ϕ R 2 ( ξ , η ) ] .
P r = P ( Ω T · Ω R ) 4 π π a 2 cos ζ r R q 2 ,
P r = 1 4 π ϑ min ϑ max φ min φ max P ( Ω T · Ω R ) sin ϑ U ( θ R ϑ ) d ϑ d φ ,
U ( x ) = { 0 , x < 0 1 , x 0 ,
P r = 1 2 π ϑ min ϑ max φ max sin ϑ U ( θ R ϑ ) d ϑ ,
φ max = { π , ϑ tan 1 ( a a o L ) cos 1 ( a o 2 + L 2 tan 2 ϑ a 2 2 a o L tan ϑ ) , ϑ > tan 1 ( a a o L ) ,
ϑ min = { 0 , a o a tan 1 ( a o a L ) , a o > a ,
ϑ max = tan 1 ( a + a o L ) ,
FOM = 1 a V r R q d V = r 4 16 a ξ min ξ max η 1 ( ξ ) η 2 ( ξ ) ϕ 1 ( ξ , η ) ϕ 2 ( ξ , η ) ( ξ η ) ( ξ + η ) 2 d ϕ d η d ξ ,

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