Abstract

We consider a wireless optical communication link in which the laser source is a Gaussian Schell beam. The effects of atmospheric turbulence strength and degree of source spatial coherence on aperture averaging and average bit error rate are examined. To accomplish this, we have derived analytic expressions for the spatial covariance of irradiance fluctuations and log-intensity variance for a Gaussian beam of any degree of coherence in the weak fluctuation regime. When spatial coherence of the transmitted source beam is reduced, intensity fluctuations (scintillations) decrease, leading to a significant reduction in the bit error rate of the optical communication link. We have also identified an enhanced aperture-averaging effect that occurs in tightly focused coherent Gaussian beams and in collimated and slightly divergent partially coherent beams. The expressions derived provide a useful design tool for selecting the optimal transmitter beam size, receiver aperture size, beam spatial coherence, transmitter focusing, etc., for the anticipated atmospheric channel conditions.

© 2003 Optical Society of America

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References

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  1. A. C. Schell, “The multiple plate antenna,” Ph.D. dissertation (Massachusetts Institute of Technology, Cambridge, Mass., 1961).
  2. J. C. Ricklin, F. M. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication,” J. Opt. Soc. Am. A 19, 1794–1802 (2002).
    [CrossRef]
  3. V. A. Banach, V. M. Buldakov, V. L. Mironov, “Intensity fluctuations of a partially coherent light beam in a turbulent atmosphere,” Opt. Spektrosk. 54, 1054–1059 (1983).
  4. J. C. Ricklin, F. D. Davidson, T. Weyrauch, “Free-space laser communication using a partially coherent source,” in Optics in Atmospheric Propagation and Adaptive Systems IV, A. Kohnle, J. D. Gonglewski, T. J. Schmugge, eds., Proc. SPIE4538, 13–23 (2001).
    [CrossRef]
  5. V. I. Polejaev, J. C. Ricklin, “Controlled phase diffuser for a laser communication link,” in Artificial Turbulence for Imaging and Wave Propagation, J. D. Gonglewski, M. A. Vorontsov, eds., Proc. SPIE3432, 103–107 (1998).
    [CrossRef]
  6. J. H. Churnside, “Aperture averaging of optical scintillations in the turbulent atmosphere,” Appl. Opt. 30, 1982–1994 (1991).
    [CrossRef] [PubMed]
  7. F. M. Davidson, X. Sun, “Gaussian approximation versus nearly exact performance analysis of optical communication receivers,” IEEE Trans. Commun. 36, 1185–1192 (1988).
    [CrossRef]
  8. D. L. Snyder, Random Point Processes (Wiley Interscience, New York, 1975).
  9. R. M. Gagliardi, S. Karp, Optical Communications (Wiley Interscience, New York, 1995).
  10. R. J. McIntyre, “The distribution of gains in uniformly multiplying avalanche photodiodes,” IEEE Trans. Electron Devices 19, 703–713 (1972).
    [CrossRef]
  11. V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).
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    [CrossRef] [PubMed]
  13. J. H. Churnside, R. J. Hill, “Probability density of irradiance scintillations for strong path-integrated refractive turbulence,” J. Opt. Soc. Am. A 4, 727–733 (1987).
    [CrossRef]
  14. J. H. Churnside, S. F. Clifford, “Log-normal Rician probability-density function of optical scintillations in the turbulent atmosphere,” J. Opt. Soc. Am. A 4, 1923–1930 (1987).
    [CrossRef]
  15. S. M. Flatte, C. Bracher, G. Wang, “Probability density functions of irradiance for waves in atmospheric turbulence calculated by numerical simulation,” J. Opt. Soc. Am. A 11, 2080–2092 (1994).
    [CrossRef]
  16. R. J. Hill, R. G. Frehlich, “Probability distribution of irradiance for the onset of strong scintillation,” J. Opt. Soc. Am. A 14, 1530–1540 (1997).
    [CrossRef]
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    [CrossRef]
  18. V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  22. G. E. Homstad, J. W. Strohbehn, R. H. Berger, J. M. Heneghan, “Aperture-averaging effects for weak scintillations,” J. Opt. Soc. Am. 64, 162–165 (1974).
    [CrossRef]
  23. R. S. Iyer, J. L. Bufton, “Aperture averaging effects in stellar scintillation,” Opt. Commun. 22, 377–381 (1977).
    [CrossRef]
  24. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).
  25. R. Lutomirski, H. T. Yura, “Propagation of a finite opti-cal beam in an inhomogeneous medium,” Appl. Opt. 10, 1652–1658 (1971).
    [CrossRef] [PubMed]
  26. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).
  27. H. T. Yura, “Mutual coherence function of a finite cross section optical beam propagating in a turbulent medium,” Appl. Opt. 11, 1399–1406 (1972).
    [CrossRef] [PubMed]
  28. R. I. Joseph, Department of Electrical and Computer Engineering, Johns Hopkins University, Baltimore, Maryland 21218 (personal communication, 2001).
  29. S. J. Wang, Y. Baykal, M. A. Plonus, “Receiver-aperture averaging effects for the intensity fluctuation of a beam wave in the turbulent atmosphere,” J. Opt. Soc. Am. 73, 831–837 (1983).
    [CrossRef]
  30. D. L. Fried, J. B. Seidman, “Laser beam scintillation in the atmosphere,” J. Opt. Soc. Am. 57, 181–185 (1967).
    [CrossRef]
  31. A. Ishimaru, “Fluctuations of a beam wave propagating through a locally homogeneous medium,” Radio Sci. 4, 295–305 (1969).
    [CrossRef]
  32. F. P. Carlson, “Application of optical scintillation measurements to turbulence diagnostics,” J. Opt. Soc. Am. 59, 1343–1347 (1969).
    [CrossRef]
  33. J. R. Kerr, R. Eiss, “Transmitter-size and focus effects on scintillations,” J. Opt. Soc. Am. 62, 682–684 (1972).
    [CrossRef]
  34. W. B. Miller, J. C. Ricklin, L. C. Andrews, “Log-amplitude variance and wave structure function: a new perspective for Gaussian beams,” J. Opt. Soc. Am. A 10, 661–672 (1993).
    [CrossRef]
  35. L. C. Andrews, R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, Bellingham, Wash., 1998).
  36. W. Gander, W. Gautschi, “Adaptive quadrature—revisited,” Department Informatik Institut für Wissenschaftliches Rechnen, Eidgenüssische Technische Hochschule Zurich. Report available via anonymous ftp from ftp.inf.ethz.chasdoc/tech-rteports/1998/306.ps.

2002

1997

1994

1993

1991

1988

F. M. Davidson, X. Sun, “Gaussian approximation versus nearly exact performance analysis of optical communication receivers,” IEEE Trans. Commun. 36, 1185–1192 (1988).
[CrossRef]

1987

1983

V. A. Banach, V. M. Buldakov, V. L. Mironov, “Intensity fluctuations of a partially coherent light beam in a turbulent atmosphere,” Opt. Spektrosk. 54, 1054–1059 (1983).

S. J. Wang, Y. Baykal, M. A. Plonus, “Receiver-aperture averaging effects for the intensity fluctuation of a beam wave in the turbulent atmosphere,” J. Opt. Soc. Am. 73, 831–837 (1983).
[CrossRef]

1977

1974

1972

1971

1969

A. Ishimaru, “Fluctuations of a beam wave propagating through a locally homogeneous medium,” Radio Sci. 4, 295–305 (1969).
[CrossRef]

F. P. Carlson, “Application of optical scintillation measurements to turbulence diagnostics,” J. Opt. Soc. Am. 59, 1343–1347 (1969).
[CrossRef]

1967

1963

S. H. Reiger, “Starlight scintillation and atmospheric turbulence,” Astron. J. 68, 395–406 (1963).
[CrossRef]

1951

Andrews, L. C.

Banach, V. A.

V. A. Banach, V. M. Buldakov, V. L. Mironov, “Intensity fluctuations of a partially coherent light beam in a turbulent atmosphere,” Opt. Spektrosk. 54, 1054–1059 (1983).

Baykal, Y.

Berger, R. H.

Bracher, C.

Bufton, J. L.

R. S. Iyer, J. L. Bufton, “Aperture averaging effects in stellar scintillation,” Opt. Commun. 22, 377–381 (1977).
[CrossRef]

Buldakov, V. M.

V. A. Banach, V. M. Buldakov, V. L. Mironov, “Intensity fluctuations of a partially coherent light beam in a turbulent atmosphere,” Opt. Spektrosk. 54, 1054–1059 (1983).

Carlson, F. P.

Churnside, J. H.

Clifford, S. F.

Davidson, F. D.

J. C. Ricklin, F. D. Davidson, T. Weyrauch, “Free-space laser communication using a partially coherent source,” in Optics in Atmospheric Propagation and Adaptive Systems IV, A. Kohnle, J. D. Gonglewski, T. J. Schmugge, eds., Proc. SPIE4538, 13–23 (2001).
[CrossRef]

Davidson, F. M.

J. C. Ricklin, F. M. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication,” J. Opt. Soc. Am. A 19, 1794–1802 (2002).
[CrossRef]

F. M. Davidson, X. Sun, “Gaussian approximation versus nearly exact performance analysis of optical communication receivers,” IEEE Trans. Commun. 36, 1185–1192 (1988).
[CrossRef]

Eiss, R.

Flatte, S. M.

Frehlich, R. G.

Fried, D. L.

Gagliardi, R. M.

R. M. Gagliardi, S. Karp, Optical Communications (Wiley Interscience, New York, 1995).

Hall, J. S.

Heneghan, J. M.

Hill, R. J.

Hoag, A. A.

Homstad, G. E.

Ishimaru, A.

A. Ishimaru, “Fluctuations of a beam wave propagating through a locally homogeneous medium,” Radio Sci. 4, 295–305 (1969).
[CrossRef]

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).

Iyer, R. S.

R. S. Iyer, J. L. Bufton, “Aperture averaging effects in stellar scintillation,” Opt. Commun. 22, 377–381 (1977).
[CrossRef]

Joseph, R. I.

R. I. Joseph, Department of Electrical and Computer Engineering, Johns Hopkins University, Baltimore, Maryland 21218 (personal communication, 2001).

Karp, S.

R. M. Gagliardi, S. Karp, Optical Communications (Wiley Interscience, New York, 1995).

Keister, M. P.

Kerr, J. R.

Lutomirski, R.

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).

McIntyre, C. M.

McIntyre, R. J.

R. J. McIntyre, “The distribution of gains in uniformly multiplying avalanche photodiodes,” IEEE Trans. Electron Devices 19, 703–713 (1972).
[CrossRef]

Meyers, G. E.

Mikesell, H.

Miller, W. B.

Mironov, V. L.

V. A. Banach, V. M. Buldakov, V. L. Mironov, “Intensity fluctuations of a partially coherent light beam in a turbulent atmosphere,” Opt. Spektrosk. 54, 1054–1059 (1983).

Phillips, R. L.

L. C. Andrews, R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, Bellingham, Wash., 1998).

Plonus, M. A.

Polejaev, V. I.

V. I. Polejaev, J. C. Ricklin, “Controlled phase diffuser for a laser communication link,” in Artificial Turbulence for Imaging and Wave Propagation, J. D. Gonglewski, M. A. Vorontsov, eds., Proc. SPIE3432, 103–107 (1998).
[CrossRef]

Reiger, S. H.

S. H. Reiger, “Starlight scintillation and atmospheric turbulence,” Astron. J. 68, 395–406 (1963).
[CrossRef]

Ricklin, J. C.

J. C. Ricklin, F. M. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication,” J. Opt. Soc. Am. A 19, 1794–1802 (2002).
[CrossRef]

W. B. Miller, J. C. Ricklin, L. C. Andrews, “Log-amplitude variance and wave structure function: a new perspective for Gaussian beams,” J. Opt. Soc. Am. A 10, 661–672 (1993).
[CrossRef]

V. I. Polejaev, J. C. Ricklin, “Controlled phase diffuser for a laser communication link,” in Artificial Turbulence for Imaging and Wave Propagation, J. D. Gonglewski, M. A. Vorontsov, eds., Proc. SPIE3432, 103–107 (1998).
[CrossRef]

J. C. Ricklin, F. D. Davidson, T. Weyrauch, “Free-space laser communication using a partially coherent source,” in Optics in Atmospheric Propagation and Adaptive Systems IV, A. Kohnle, J. D. Gonglewski, T. J. Schmugge, eds., Proc. SPIE4538, 13–23 (2001).
[CrossRef]

Schell, A. C.

A. C. Schell, “The multiple plate antenna,” Ph.D. dissertation (Massachusetts Institute of Technology, Cambridge, Mass., 1961).

Seidman, J. B.

Snyder, D. L.

D. L. Snyder, Random Point Processes (Wiley Interscience, New York, 1975).

Strohbehn, J. W.

Sun, X.

F. M. Davidson, X. Sun, “Gaussian approximation versus nearly exact performance analysis of optical communication receivers,” IEEE Trans. Commun. 36, 1185–1192 (1988).
[CrossRef]

Tatarskii, V. I.

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971).

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).

Wang, G.

Wang, S. J.

Weyrauch, T.

J. C. Ricklin, F. D. Davidson, T. Weyrauch, “Free-space laser communication using a partially coherent source,” in Optics in Atmospheric Propagation and Adaptive Systems IV, A. Kohnle, J. D. Gonglewski, T. J. Schmugge, eds., Proc. SPIE4538, 13–23 (2001).
[CrossRef]

Wolf, E.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).

Yura, H. T.

Appl. Opt.

Astron. J.

S. H. Reiger, “Starlight scintillation and atmospheric turbulence,” Astron. J. 68, 395–406 (1963).
[CrossRef]

IEEE Trans. Commun.

F. M. Davidson, X. Sun, “Gaussian approximation versus nearly exact performance analysis of optical communication receivers,” IEEE Trans. Commun. 36, 1185–1192 (1988).
[CrossRef]

IEEE Trans. Electron Devices

R. J. McIntyre, “The distribution of gains in uniformly multiplying avalanche photodiodes,” IEEE Trans. Electron Devices 19, 703–713 (1972).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

R. S. Iyer, J. L. Bufton, “Aperture averaging effects in stellar scintillation,” Opt. Commun. 22, 377–381 (1977).
[CrossRef]

Opt. Spektrosk.

V. A. Banach, V. M. Buldakov, V. L. Mironov, “Intensity fluctuations of a partially coherent light beam in a turbulent atmosphere,” Opt. Spektrosk. 54, 1054–1059 (1983).

Radio Sci.

A. Ishimaru, “Fluctuations of a beam wave propagating through a locally homogeneous medium,” Radio Sci. 4, 295–305 (1969).
[CrossRef]

Other

L. C. Andrews, R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, Bellingham, Wash., 1998).

W. Gander, W. Gautschi, “Adaptive quadrature—revisited,” Department Informatik Institut für Wissenschaftliches Rechnen, Eidgenüssische Technische Hochschule Zurich. Report available via anonymous ftp from ftp.inf.ethz.chasdoc/tech-rteports/1998/306.ps.

J. C. Ricklin, F. D. Davidson, T. Weyrauch, “Free-space laser communication using a partially coherent source,” in Optics in Atmospheric Propagation and Adaptive Systems IV, A. Kohnle, J. D. Gonglewski, T. J. Schmugge, eds., Proc. SPIE4538, 13–23 (2001).
[CrossRef]

V. I. Polejaev, J. C. Ricklin, “Controlled phase diffuser for a laser communication link,” in Artificial Turbulence for Imaging and Wave Propagation, J. D. Gonglewski, M. A. Vorontsov, eds., Proc. SPIE3432, 103–107 (1998).
[CrossRef]

D. L. Snyder, Random Point Processes (Wiley Interscience, New York, 1975).

R. M. Gagliardi, S. Karp, Optical Communications (Wiley Interscience, New York, 1995).

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).

A. C. Schell, “The multiple plate antenna,” Ph.D. dissertation (Massachusetts Institute of Technology, Cambridge, Mass., 1961).

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).

R. I. Joseph, Department of Electrical and Computer Engineering, Johns Hopkins University, Baltimore, Maryland 21218 (personal communication, 2001).

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Figures (13)

Fig. 1
Fig. 1

Direct-detection optical communication link consisting of a laser transmitter, log-normal atmospheric channel, and maximum-likelihood receiver.

Fig. 2
Fig. 2

Comparison of the Churnside data6 for the aperture-averaging factor A with the aperture-averaging factor calculated with Eq. (26) (“new theory”), as a function of normalized receiver aperture diameter ( kD 2 / 4 L ) 1 / 2 for a strongly divergent beam ( r ˆ = 330 ) in the weak fluctuation regime ( σ 1 2 = 0.232 ,   σ 1 2 z ^ rec 5 / 6 = 3.4 × 10 - 4 ,   C n 2 = 1.46 ± 1.18 × 10 - 13   m - 2 / 3 ) .

Fig. 3
Fig. 3

Aperture-averaging factor as a function of atmospheric turbulence strength for a convergent (focused) beam ( r ˆ = 0.1 ) for coherent ( ζ S = 1 ) and almost incoherent ( ζ S = 1000 ) beams. Two receiver diameters D are considered.

Fig. 4
Fig. 4

Same as Fig. 3 but for a collimated beam ( r ˆ = 1 ) and 10-cm diameter receiver. Three source coherence levels are considered: coherent beam ( ζ S = 1 ) , slight partial coherence ( ζ S = 3 ) , and an almost incoherent beam ( ζ S = 1000 ) .

Fig. 5
Fig. 5

Same as Fig. 4 but for a divergent beam ( r ˆ = 5 ) .

Fig. 6
Fig. 6

Aperture-averaging factor as a function of the transmitter focusing parameter r ˆ . Three source coherence states are considered, from a coherent beam ( ζ S = 1 ) to an almost incoherent beam ( ζ S = 10 , 000 ) .

Fig. 7
Fig. 7

Log-intensity variance as a function of radial distance ρ from the beam center for slightly divergent ( r ˆ = 5 ) source beams ranging from coherent ( ζ S = 1 ) to almost incoherent ( ζ S = 1000 ) . The beam size at the receiver increases as the source beam loses coherence: (1) ζ S = 1 , w ζ ( z ) = 12.7   cm ; (2) ζ S = 20 , w ζ ( z ) = 15.8   cm ; (3) ζ S = 50 , w ζ ( z ) = 19.2   cm ; (4) ζ S = 1000 , w ζ ( z ) = 64.5   cm .

Fig. 8
Fig. 8

Log-intensity variances given in Fig. 7 showing the effects of aperture averaging over a 10-cm-diameter receiver aperture.

Fig. 9
Fig. 9

Effects of aperture averaging the log-intensity variance over 1-, 3-, and 10-cm-diameter receiver apertures for a slightly incoherent divergent beam ( r ˆ = 5 ) . Each curve stops at the radial size of the specified aperture.

Fig. 10
Fig. 10

Aperture-averaged log-intensity variance as a function of the focusing parameter r ˆ . Values given are for the log-intensity variance at the edge of the receiver aperture ( ρ = 5   cm ) .

Fig. 11
Fig. 11

Aperture-averaged log-intensity variance showing the effect of atmospheric turbulence strength ( r ˆ = 5 ) . Values given are for the log-intensity variance at the edge of the receiver aperture ( ρ = 5   cm ) .

Fig. 12
Fig. 12

BER as a function of received optical power in decibel milliwatts ( 1   mW = 0   dBm ) . The focusing parameter r ˆ is adjusted so that, regardless of source coherence, the 5-cm diameter transmitted beam always has a receiver beam size of w ζ ( z ) = 50   cm (beam footprint of 1 m). Left to right: (1) coherent beam in free space (best possible performance); (2) r ˆ = 1 , ζ S = 2500 , A σ ln   Z 2 = 0.013 ; (3) r ˆ = 15.45 , ζ S = 1000 , A σ ln   Z 2 = 0.017 ; (4) r ˆ = 20 , ζ S = 1 , A σ ln   Z 2 = 0.021 (coherent source beam).

Fig. 13
Fig. 13

Effect of atmospheric turbulence strength on BER as a function of received optical power for collimated ( r ˆ = 1 ) coherent ( ζ S = 1 ) and partially coherent ( ζ S = 10 ) beams. Left to right: (1) coherent beam in free space (best possible performance); (2) ζ S = 10 , A σ ln   Z 2 = 0.013 , w ζ ( z ) = 6.9   cm ; (3) ζ S = 10 , A σ ln   Z 2 = 0.05 , w ζ ( z ) = 7.9   cm ; (4) ζ S = 1 , A σ ln   Z 2 = 0.045 , w ζ ( z ) = 3.2   cm ; (5) ζ S = 1 , A σ ln   Z 2 = 0.118 , w ζ ( z ) = 3.2   cm .

Equations (38)

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F = k eff G + ( 1 - k eff ) ( 2 - 1 / G ) ,
p ( y T b ) = 1 2 π σ y T b 2 exp - ( y T b - y T b ) 2 2 σ y T b 2 .
y T b = e η hf   GP opt = eGn ϕ ,
σ y T b 2 = 2 B e 2 η hf   FG 2 P opt + 2 K B T R L ,
Z P opt P opt ,
p ( Z ) dZ = 1 2 π σ ln   Z 2 exp - ( ln   Z + 1 2 σ ln   Z 2 ) 2 2 σ ln   Z 2 1 Z   dZ .
P opt 1 = Z P opt
P opt 0 = Z P opt ,
y T b 1 = eGZ   η hf   P opt = eGZn ϕ ,
σ 1 2 var ( y T b 1 ) = 2 B e 2 η hf   FG 2 Z P opt + 2 K B T R L ,
y T b 0 = eG Z   η hf   P opt = eG Zn ϕ ,
σ 0 2 var ( y T b 0 ) = 2 B e 2 η hf   FG 2 Z P opt + 2 K B T R L .
p ( y T b | H 1 ) Decide H 1 > < Decide H 0 p ( y T b | H 0 ) .
σ 1 2 σ 0 2 - 1 y Th 2 + 2 y T b 1 - σ 1 2 σ 0 2   y T b 0 y Th - σ 1 2 ln σ 1 2 σ 0 2
+ σ 1 2 σ 0 2   y T b 0 2 - y T b 1 2 = 0 .
Pr ( error | no fading ) = 1 2   Q y Th - y T b 0 2 σ 0 + 1 2   Q y Th - y T b 1 2 σ 1 ,
Q ( x ) = 1 2 π x exp - t 2 2 d t = 1 2 erfc x 2 .
p ˆ ( y T b | H i ) = 0 1 2 π σ y T b 2 ( z ) × exp - [ y T b - y T b i ( z ) ] 2 2 σ i 2 ( z ) p ( z ) d z ,
Z = I ( ρ ,   z ) I ( ρ ,   z ) ,
σ ln   Z 2 ln I ( ρ ,   z ) I ( ρ ,   z ) 2 - ln I ( ρ ,   z ) I ( ρ ,   z ) 2 .
σ ln   Z 2 = A σ ln   Z 2 ,
A = 16 π 0 1 x d xb I ( ρ ,   z ) [ cos - 1 ( x ) - x ( 1 - x 2 ) 1 / 2 ] .
B I ( ρ 1 ,   ρ 2 ,   z ) = I ( ρ 1 ,   z ) I ( ρ 2 ,   z ) - I ( ρ 1 ,   z ) × I ( ρ 2 ,   z ) .
I ( ρ 1 ,   z ) I ( ρ 2 ,   z ) = I ( ρ 1 ,   z ) I ( ρ 2 ,   z ) + | Γ ( ρ 1 ,   ρ 2 ,   z ) | 2 ,
B I ( ρ 1 ,   ρ 2 ,   z ) = | Γ ( ρ 1 ,   ρ 2 ,   z ) | 2 .
Γ ( ρ 1 ,   ρ 2 ,   z ) W ( ρ 1 ,   ρ 2 ,   z ) = 1 ( λ z ) 2 d 2 r 1 d 2 r 2 W o ( r 1 ,   r 2 ,   0 ) × exp [ Ψ * ( r 1 ,   ρ 1 ) + Ψ ( r 2 ,   ρ 2 ) ] × exp jk 2 z   [ ( ρ 1 - r 1 ) 2 - ( ρ 2 - r 2 ) 2 ] .
exp [ Ψ * ( r 1 ,   ρ 1 ) + Ψ ( r 2 ,   ρ 2 ) ]
exp - 1 ρ o 2   ( r d 2 + r d ρ d + ρ d 2 ) ,
W o ( r 1 ,   r 2 ,   0 ) = I o exp - | r 1 - r 2 | 2 2 σ g 2 .
b 1 ( ρ ,   z ) = B I ( ρ ,   z ) B I ( 0 ,   z ) = exp - ρ 2 ρ o 2 2 + ρ o 2 w o 2 z ^ 2 - ρ o 2 ϕ 2 w ζ 2 ( z ) = μ ( 2 ρ ,   z ) .
A = 16 π 0 1 x d x   exp - D 2 x 2 ρ o 2 2 + ρ o 2 w o 2 z ^ 2 - ρ o 2 ϕ 2 w ζ 2 ( z ) [ cos - 1 ( x ) - x ( 1 - x 2 ) 1 / 2 ] ,
A = 4 p 2 1 - exp - 1 2   p 2 I o 1 2   p 2 + I 1 1 2   p 2 ,
p 2 D 2 ρ o 2 2 + ρ o 2 w o 2 z ^ 2 - ρ o 2 ϕ 2 w ζ 2 ( z ) .
σ ln   Z 2 4.42 σ 1 2 z ^ rec 5 / 6 ρ 2 w ζ 2 ( z ) + 3.86 σ 1 2 0.4 [ ( 1 + 2 r ^ rec ) 2 + 4 z ^ rec 2 ] 5 / 12 cos 5 6 tan - 1 1 + 2 r ^ rec 2 z ^ rec - 11 16   z ^ rec 5 / 6 ,
r ^ rec ( z ) = R ζ ( z ) + z R ζ ( z ) , z ^ rec ( z ) = z 0.5 kw ζ 2 ( z ) .
R ζ ( z ) = z ( r ^ 2 + ζ S z ^ 2 ) ϕ z ˆ - ζ S z ^ 2 - r ^ 2 , ϕ r ˆ z ˆ - z ˆ   w o 2 ρ o 2 ,
σ 1 2 < 1 , σ 1 2 z ^ rec 5 / 6 < 1 .
P opt dBm = 10   log 10 ( P opt mW ) .

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