Abstract

Reliable data transmission in optical wireless communication is on the premise of the successful establishment of the optical link. In this paper, we propose an ultra-wide field-of-view (FOV) acquisition scheme, which combines the fisheye lens and Voigt anomalous dispersion optical filter (VADOF) to achieve rapid establishment of wireless optical links. Furthermore, the ultra-wide FOV signal-receiving model for this acquisition scheme is presented to analyze the receiving performance. This acquisition scheme utilizes the fisheye lens to obtain the ultra-wide FOV, not only simplifying the system architecture of the spatial acquisition, but also reducing the acquisition time; a VADOF with ultra-narrow-pass bandwidth is adopted to resist the strong background radiation induced by the ultra-wide FOV. For this ultra-wide FOV acquisition scheme, the mathematical model of long-term average acquisition probability (LTAAP) is derived based on the gamma–gamma (GG) distribution. In an atmospheric turbulence environment, the average signal count and the acquisition probability are both random variables; therefore, the probability density of the average signal count needs to be considered and LTAAP can be calculated based on the GG distribution. Comprehensive analysis and numerical results of the key parameters of this ultra-wide FOV acquisition scheme, such as LTAAP, false-alarm probability, signal-to-noise ratio, incident angle of beam, scintillation index, and acquisition threshold, provide an advantageous basis for the actual spatial acquisition system.

© 2013 Optical Society of America

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References

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  2. K. Wakamori, K. Kazaura, and M. Matsumoto, “Research and development of a next-generation free-space optical communication system,” Proc. SPIE 7234, 723404 (2009).
    [CrossRef]
  3. J. Wang and J. M. Kahn, “Acquisition in short-range free-space optical communication,” Proc. SPIE 4873, 121–131 (2002).
    [CrossRef]
  4. M. Reyes, S. Chueca, T. Viera, and Z. Sodnik, “Analysis of the preliminary optical links between ARTEMIS and the optical ground station,” Proc. SPIE 4821, 33–43 (2002).
    [CrossRef]
  5. H. J. White, D. W. Gough, R. Merry, and S. Patrick, “Demonstration of free space optical communication link incorporating a closed-loop tracking system for mobile platforms,” Proc. SPIE 5614, 119–128 (2004).
    [CrossRef]
  6. S. K. Nayar, “Omnidirectional vision,” Proceedings of the International Symposium on Robotics Research, Japan, October 1997.
  7. X. Ma and L. Liu, “Aperture-array acquisition scheme for optical links in atmospheric turbulence,” Appl. Opt. 49, 718–723 (2010).
    [CrossRef]
  8. T. H. Ho, S. D. Milner, and C. C. Davis, “Pointing, acquisition and tracking system with omnivision,” Proc. SPIE 5892, 420–431 (2005).
    [CrossRef]
  9. J. M. Kovalik, A. Biswas, J. R. Charles, and M. Regehrm, “Autonomous access links using laser communications,” Proc. SPIE 7199, 71990H (2009).
    [CrossRef]
  10. X. Song, L. Liu, and J. Tang, “High-accuracy angle detection for ultra-wide-field-of-view acquisition in wireless optical links,” Opt. Eng. 47, 025010 (2008).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  18. M. A. Al-Habash and L. C. Andrews, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40, 1554–1562 (2001).
    [CrossRef]
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2010 (1)

2009 (2)

K. Wakamori, K. Kazaura, and M. Matsumoto, “Research and development of a next-generation free-space optical communication system,” Proc. SPIE 7234, 723404 (2009).
[CrossRef]

J. M. Kovalik, A. Biswas, J. R. Charles, and M. Regehrm, “Autonomous access links using laser communications,” Proc. SPIE 7199, 71990H (2009).
[CrossRef]

2008 (2)

X. Song, L. Liu, and J. Tang, “High-accuracy angle detection for ultra-wide-field-of-view acquisition in wireless optical links,” Opt. Eng. 47, 025010 (2008).
[CrossRef]

T. A. Tsiftsis, “Performance of heterodyne wireless optical communication systems over gamma–gamma atmospheric turbulence channels,” IEEE Electron. Lett. 44, 372–373 (2008).
[CrossRef]

2005 (2)

T. H. Ho, S. D. Milner, and C. C. Davis, “Pointing, acquisition and tracking system with omnivision,” Proc. SPIE 5892, 420–431 (2005).
[CrossRef]

C. I. Moore, H. R. Burris, M. F. Stell, L. Wasiczko, M. R. Suite, R. Mahon, W. S. Rabinovich, G. C. Gilbreath, and W. J. Scharpf, “Atmospheric turbulence studies of a 16 km maritime path,” Proc. SPIE 5793, 78–88 (2005).
[CrossRef]

2004 (1)

H. J. White, D. W. Gough, R. Merry, and S. Patrick, “Demonstration of free space optical communication link incorporating a closed-loop tracking system for mobile platforms,” Proc. SPIE 5614, 119–128 (2004).
[CrossRef]

2003 (1)

2002 (2)

J. Wang and J. M. Kahn, “Acquisition in short-range free-space optical communication,” Proc. SPIE 4873, 121–131 (2002).
[CrossRef]

M. Reyes, S. Chueca, T. Viera, and Z. Sodnik, “Analysis of the preliminary optical links between ARTEMIS and the optical ground station,” Proc. SPIE 4821, 33–43 (2002).
[CrossRef]

2001 (1)

M. A. Al-Habash and L. C. Andrews, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40, 1554–1562 (2001).
[CrossRef]

1999 (1)

I. I. Kim, M. Mitchell, and R. E. Korevaar, “Measurement of scintillation for free-space laser communication at 785 nm and 1550 nm,” Proc. SPIE 3850, 49–62 (1999).
[CrossRef]

1997 (1)

1995 (1)

S. Bloom, V. Chan, and C. S. Liu, “High-elevation terrestrial validation of BMDO laser com system at 1.1  Gbit/s,” Proc. SPIE 2381, 113–128 (1995).
[CrossRef]

1994 (1)

K. Muroo, “Resonant Voigt effect spectrum of the RbD2 transition,” Opt. Soc. Am. 11, 409–414 (1994).

1991 (1)

J. M. Cotton, “Narrowband optical interference filters,” Proc. SPIE 1417, 525–536 (1991).
[CrossRef]

1989 (1)

1987 (1)

1977 (1)

1973 (1)

1964 (1)

Al-Habash, M. A.

M. A. Al-Habash and L. C. Andrews, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40, 1554–1562 (2001).
[CrossRef]

Andrews, L. C.

M. A. Al-Habash and L. C. Andrews, “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 and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005).

Arnon, S.

Bayaki, E.

E. Bayaki, R. Schober, and R. K. Mallik, “Performance analysis of free-space optical systems in gamma–gamma fading,” Globecom, New Orleans, USA, December (2008).

Bissonnette, L. R.

Biswas, A.

J. M. Kovalik, A. Biswas, J. R. Charles, and M. Regehrm, “Autonomous access links using laser communications,” Proc. SPIE 7199, 71990H (2009).
[CrossRef]

Bloom, S.

S. Bloom, V. Chan, and C. S. Liu, “High-elevation terrestrial validation of BMDO laser com system at 1.1  Gbit/s,” Proc. SPIE 2381, 113–128 (1995).
[CrossRef]

Burris, H. R.

C. I. Moore, H. R. Burris, M. F. Stell, L. Wasiczko, M. R. Suite, R. Mahon, W. S. Rabinovich, G. C. Gilbreath, and W. J. Scharpf, “Atmospheric turbulence studies of a 16 km maritime path,” Proc. SPIE 5793, 78–88 (2005).
[CrossRef]

Chan, V.

S. Bloom, V. Chan, and C. S. Liu, “High-elevation terrestrial validation of BMDO laser com system at 1.1  Gbit/s,” Proc. SPIE 2381, 113–128 (1995).
[CrossRef]

Charles, J. R.

J. M. Kovalik, A. Biswas, J. R. Charles, and M. Regehrm, “Autonomous access links using laser communications,” Proc. SPIE 7199, 71990H (2009).
[CrossRef]

Chueca, S.

M. Reyes, S. Chueca, T. Viera, and Z. Sodnik, “Analysis of the preliminary optical links between ARTEMIS and the optical ground station,” Proc. SPIE 4821, 33–43 (2002).
[CrossRef]

Churnside, J. H.

Clifford, S. F.

Cotton, J. M.

J. M. Cotton, “Narrowband optical interference filters,” Proc. SPIE 1417, 525–536 (1991).
[CrossRef]

Davis, C. C.

T. H. Ho, S. D. Milner, and C. C. Davis, “Pointing, acquisition and tracking system with omnivision,” Proc. SPIE 5892, 420–431 (2005).
[CrossRef]

Frehlich, R. G.

Gagliardi, R. M.

R. M. Gagliardi and S. Karp, Optical Communication (Wiley, 1995).

Gilbreath, G. C.

C. I. Moore, H. R. Burris, M. F. Stell, L. Wasiczko, M. R. Suite, R. Mahon, W. S. Rabinovich, G. C. Gilbreath, and W. J. Scharpf, “Atmospheric turbulence studies of a 16 km maritime path,” Proc. SPIE 5793, 78–88 (2005).
[CrossRef]

Gough, D. W.

H. J. White, D. W. Gough, R. Merry, and S. Patrick, “Demonstration of free space optical communication link incorporating a closed-loop tracking system for mobile platforms,” Proc. SPIE 5614, 119–128 (2004).
[CrossRef]

Hill, R. J.

Ho, T. H.

T. H. Ho, S. D. Milner, and C. C. Davis, “Pointing, acquisition and tracking system with omnivision,” Proc. SPIE 5892, 420–431 (2005).
[CrossRef]

Kahn, J. M.

J. Wang and J. M. Kahn, “Acquisition in short-range free-space optical communication,” Proc. SPIE 4873, 121–131 (2002).
[CrossRef]

Karp, S.

R. M. Gagliardi and S. Karp, Optical Communication (Wiley, 1995).

Kazaura, K.

K. Wakamori, K. Kazaura, and M. Matsumoto, “Research and development of a next-generation free-space optical communication system,” Proc. SPIE 7234, 723404 (2009).
[CrossRef]

Kim, I. I.

I. I. Kim, M. Mitchell, and R. E. Korevaar, “Measurement of scintillation for free-space laser communication at 785 nm and 1550 nm,” Proc. SPIE 3850, 49–62 (1999).
[CrossRef]

Korevaar, R. E.

I. I. Kim, M. Mitchell, and R. E. Korevaar, “Measurement of scintillation for free-space laser communication at 785 nm and 1550 nm,” Proc. SPIE 3850, 49–62 (1999).
[CrossRef]

Kovalik, J. M.

J. M. Kovalik, A. Biswas, J. R. Charles, and M. Regehrm, “Autonomous access links using laser communications,” Proc. SPIE 7199, 71990H (2009).
[CrossRef]

Liu, C. S.

S. Bloom, V. Chan, and C. S. Liu, “High-elevation terrestrial validation of BMDO laser com system at 1.1  Gbit/s,” Proc. SPIE 2381, 113–128 (1995).
[CrossRef]

Liu, L.

X. Ma and L. Liu, “Aperture-array acquisition scheme for optical links in atmospheric turbulence,” Appl. Opt. 49, 718–723 (2010).
[CrossRef]

X. Song, L. Liu, and J. Tang, “High-accuracy angle detection for ultra-wide-field-of-view acquisition in wireless optical links,” Opt. Eng. 47, 025010 (2008).
[CrossRef]

Ma, X.

Mahon, R.

C. I. Moore, H. R. Burris, M. F. Stell, L. Wasiczko, M. R. Suite, R. Mahon, W. S. Rabinovich, G. C. Gilbreath, and W. J. Scharpf, “Atmospheric turbulence studies of a 16 km maritime path,” Proc. SPIE 5793, 78–88 (2005).
[CrossRef]

Mallik, R. K.

E. Bayaki, R. Schober, and R. K. Mallik, “Performance analysis of free-space optical systems in gamma–gamma fading,” Globecom, New Orleans, USA, December (2008).

Matsumoto, M.

K. Wakamori, K. Kazaura, and M. Matsumoto, “Research and development of a next-generation free-space optical communication system,” Proc. SPIE 7234, 723404 (2009).
[CrossRef]

Merry, R.

H. J. White, D. W. Gough, R. Merry, and S. Patrick, “Demonstration of free space optical communication link incorporating a closed-loop tracking system for mobile platforms,” Proc. SPIE 5614, 119–128 (2004).
[CrossRef]

Milner, S. D.

T. H. Ho, S. D. Milner, and C. C. Davis, “Pointing, acquisition and tracking system with omnivision,” Proc. SPIE 5892, 420–431 (2005).
[CrossRef]

Mitchell, M.

I. I. Kim, M. Mitchell, and R. E. Korevaar, “Measurement of scintillation for free-space laser communication at 785 nm and 1550 nm,” Proc. SPIE 3850, 49–62 (1999).
[CrossRef]

Miyamoto, K.

Moore, C. I.

C. I. Moore, H. R. Burris, M. F. Stell, L. Wasiczko, M. R. Suite, R. Mahon, W. S. Rabinovich, G. C. Gilbreath, and W. J. Scharpf, “Atmospheric turbulence studies of a 16 km maritime path,” Proc. SPIE 5793, 78–88 (2005).
[CrossRef]

Muroo, K.

K. Muroo, “Resonant Voigt effect spectrum of the RbD2 transition,” Opt. Soc. Am. 11, 409–414 (1994).

Nayar, S. K.

S. K. Nayar, “Omnidirectional vision,” Proceedings of the International Symposium on Robotics Research, Japan, October 1997.

Patrick, S.

H. J. White, D. W. Gough, R. Merry, and S. Patrick, “Demonstration of free space optical communication link incorporating a closed-loop tracking system for mobile platforms,” Proc. SPIE 5614, 119–128 (2004).
[CrossRef]

Phillips, R. L.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005).

Rabinovich, W. S.

C. I. Moore, H. R. Burris, M. F. Stell, L. Wasiczko, M. R. Suite, R. Mahon, W. S. Rabinovich, G. C. Gilbreath, and W. J. Scharpf, “Atmospheric turbulence studies of a 16 km maritime path,” Proc. SPIE 5793, 78–88 (2005).
[CrossRef]

Regehrm, M.

J. M. Kovalik, A. Biswas, J. R. Charles, and M. Regehrm, “Autonomous access links using laser communications,” Proc. SPIE 7199, 71990H (2009).
[CrossRef]

Reyes, M.

M. Reyes, S. Chueca, T. Viera, and Z. Sodnik, “Analysis of the preliminary optical links between ARTEMIS and the optical ground station,” Proc. SPIE 4821, 33–43 (2002).
[CrossRef]

Scharpf, W. J.

C. I. Moore, H. R. Burris, M. F. Stell, L. Wasiczko, M. R. Suite, R. Mahon, W. S. Rabinovich, G. C. Gilbreath, and W. J. Scharpf, “Atmospheric turbulence studies of a 16 km maritime path,” Proc. SPIE 5793, 78–88 (2005).
[CrossRef]

Schober, R.

E. Bayaki, R. Schober, and R. K. Mallik, “Performance analysis of free-space optical systems in gamma–gamma fading,” Globecom, New Orleans, USA, December (2008).

Sodnik, Z.

M. Reyes, S. Chueca, T. Viera, and Z. Sodnik, “Analysis of the preliminary optical links between ARTEMIS and the optical ground station,” Proc. SPIE 4821, 33–43 (2002).
[CrossRef]

Song, X.

X. Song, L. Liu, and J. Tang, “High-accuracy angle detection for ultra-wide-field-of-view acquisition in wireless optical links,” Opt. Eng. 47, 025010 (2008).
[CrossRef]

Speck, J. P.

Stell, M. F.

C. I. Moore, H. R. Burris, M. F. Stell, L. Wasiczko, M. R. Suite, R. Mahon, W. S. Rabinovich, G. C. Gilbreath, and W. J. Scharpf, “Atmospheric turbulence studies of a 16 km maritime path,” Proc. SPIE 5793, 78–88 (2005).
[CrossRef]

Suite, M. R.

C. I. Moore, H. R. Burris, M. F. Stell, L. Wasiczko, M. R. Suite, R. Mahon, W. S. Rabinovich, G. C. Gilbreath, and W. J. Scharpf, “Atmospheric turbulence studies of a 16 km maritime path,” Proc. SPIE 5793, 78–88 (2005).
[CrossRef]

Tang, J.

X. Song, L. Liu, and J. Tang, “High-accuracy angle detection for ultra-wide-field-of-view acquisition in wireless optical links,” Opt. Eng. 47, 025010 (2008).
[CrossRef]

Titterton, P. J.

Tsiftsis, T. A.

T. A. Tsiftsis, “Performance of heterodyne wireless optical communication systems over gamma–gamma atmospheric turbulence channels,” IEEE Electron. Lett. 44, 372–373 (2008).
[CrossRef]

Viera, T.

M. Reyes, S. Chueca, T. Viera, and Z. Sodnik, “Analysis of the preliminary optical links between ARTEMIS and the optical ground station,” Proc. SPIE 4821, 33–43 (2002).
[CrossRef]

Wakamori, K.

K. Wakamori, K. Kazaura, and M. Matsumoto, “Research and development of a next-generation free-space optical communication system,” Proc. SPIE 7234, 723404 (2009).
[CrossRef]

Wang, J.

J. Wang and J. M. Kahn, “Acquisition in short-range free-space optical communication,” Proc. SPIE 4873, 121–131 (2002).
[CrossRef]

Wasiczko, L.

C. I. Moore, H. R. Burris, M. F. Stell, L. Wasiczko, M. R. Suite, R. Mahon, W. S. Rabinovich, G. C. Gilbreath, and W. J. Scharpf, “Atmospheric turbulence studies of a 16 km maritime path,” Proc. SPIE 5793, 78–88 (2005).
[CrossRef]

White, H. J.

H. J. White, D. W. Gough, R. Merry, and S. Patrick, “Demonstration of free space optical communication link incorporating a closed-loop tracking system for mobile platforms,” Proc. SPIE 5614, 119–128 (2004).
[CrossRef]

Appl. Opt. (3)

IEEE Electron. Lett. (1)

T. A. Tsiftsis, “Performance of heterodyne wireless optical communication systems over gamma–gamma atmospheric turbulence channels,” IEEE Electron. Lett. 44, 372–373 (2008).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Opt. Eng. (2)

M. A. Al-Habash and L. C. Andrews, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40, 1554–1562 (2001).
[CrossRef]

X. Song, L. Liu, and J. Tang, “High-accuracy angle detection for ultra-wide-field-of-view acquisition in wireless optical links,” Opt. Eng. 47, 025010 (2008).
[CrossRef]

Opt. Lett. (1)

Opt. Soc. Am. (1)

K. Muroo, “Resonant Voigt effect spectrum of the RbD2 transition,” Opt. Soc. Am. 11, 409–414 (1994).

Proc. SPIE (10)

I. I. Kim, M. Mitchell, and R. E. Korevaar, “Measurement of scintillation for free-space laser communication at 785 nm and 1550 nm,” Proc. SPIE 3850, 49–62 (1999).
[CrossRef]

C. I. Moore, H. R. Burris, M. F. Stell, L. Wasiczko, M. R. Suite, R. Mahon, W. S. Rabinovich, G. C. Gilbreath, and W. J. Scharpf, “Atmospheric turbulence studies of a 16 km maritime path,” Proc. SPIE 5793, 78–88 (2005).
[CrossRef]

S. Bloom, V. Chan, and C. S. Liu, “High-elevation terrestrial validation of BMDO laser com system at 1.1  Gbit/s,” Proc. SPIE 2381, 113–128 (1995).
[CrossRef]

J. M. Cotton, “Narrowband optical interference filters,” Proc. SPIE 1417, 525–536 (1991).
[CrossRef]

T. H. Ho, S. D. Milner, and C. C. Davis, “Pointing, acquisition and tracking system with omnivision,” Proc. SPIE 5892, 420–431 (2005).
[CrossRef]

J. M. Kovalik, A. Biswas, J. R. Charles, and M. Regehrm, “Autonomous access links using laser communications,” Proc. SPIE 7199, 71990H (2009).
[CrossRef]

K. Wakamori, K. Kazaura, and M. Matsumoto, “Research and development of a next-generation free-space optical communication system,” Proc. SPIE 7234, 723404 (2009).
[CrossRef]

J. Wang and J. M. Kahn, “Acquisition in short-range free-space optical communication,” Proc. SPIE 4873, 121–131 (2002).
[CrossRef]

M. Reyes, S. Chueca, T. Viera, and Z. Sodnik, “Analysis of the preliminary optical links between ARTEMIS and the optical ground station,” Proc. SPIE 4821, 33–43 (2002).
[CrossRef]

H. J. White, D. W. Gough, R. Merry, and S. Patrick, “Demonstration of free space optical communication link incorporating a closed-loop tracking system for mobile platforms,” Proc. SPIE 5614, 119–128 (2004).
[CrossRef]

Other (4)

S. K. Nayar, “Omnidirectional vision,” Proceedings of the International Symposium on Robotics Research, Japan, October 1997.

R. M. Gagliardi and S. Karp, Optical Communication (Wiley, 1995).

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005).

E. Bayaki, R. Schober, and R. K. Mallik, “Performance analysis of free-space optical systems in gamma–gamma fading,” Globecom, New Orleans, USA, December (2008).

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Figures (11)

Fig. 1.
Fig. 1.

Schematic of the ultra-wide FOV communication system.

Fig. 2.
Fig. 2.

(a) Signal-receiving model of the fisheye lens. (b) Optical imaging.

Fig. 3.
Fig. 3.

Structure diagram of VADOF.

Fig. 4.
Fig. 4.

(a) Performance of a VADOF with Tr, δλ, and ENBW. (b) Tr/ENBW versus incident angle θ.

Fig. 5.
Fig. 5.

Relationship of normalization SNR and incidence angle θ.

Fig. 6.
Fig. 6.

GG distribution diagram.

Fig. 7.
Fig. 7.

False-alarm probability Pf and LTAAP Pd versus TBNR (a) Pf versus TBNR, (b) LTAAP versus TBNR at the different σR2.

Fig. 8.
Fig. 8.

Required SNR for achieving a certain LTAAP (95%, 97%, and 99%) versus the scintillation parameter σR2.

Fig. 9.
Fig. 9.

LTAAP versus acquisition SNR at different σR2.

Fig. 10.
Fig. 10.

LTAAP versus incident angle θ at different σR2.

Fig. 11.
Fig. 11.

Required SNR for achieving a certain LTAAP (95%, 97%, and 99%) versus incident angle θ at different σR2.

Equations (23)

Equations on this page are rendered with MathJax. Learn more.

Q=(EtdS)dΩ,
W=η·Q/dS=η·(EtdS/dS)dΩ.
dS/dS=R2sinωdωdθ/rdrdθ=R2sinω/(k2f2ω).
ΔΩπD2(1+cos2θ)/(2R2),
Wrη·Et(dS/dS)ΔΩ.
Pr=η·[πD2/(k2f2)]·Pt·(1+cos2θ)·(sinθ/θ)/2.
Pr=η·[πD2/(k2f2)]·Pt·μ(θ).
ENBW=+Tr(λ)dλ/Trmax,
Ps=Tr(θ)·Pr=Tr(θ)·μ(θ)·P0,
SNR=PsPn=Tr(θ)·PrPn=Tr(θ)·μ(θ)·P0Pn.
SNR=SNR0·μ(θ),
{f1(I)=12πσsnexp[(IIsIb)22σsn2]f0(I)=12πσnexp[(IIb)22σn2],
{σsn2=σCCD2+σb-shot2+σs-shot2σn2=σCCD2+σb-shot2.
σsn2σn2.
SNR=Isσn=μ(θ)·Is0σn=μ(θ)·SNR0.
Pd(Is)=Kth12πσnexp[(KIsIb)22σn2]dK=12[1+erf(Is+IbKth2σn)],
Pd=12[1+erf(SNRTBNR2)]=12[1+erf(μ(θ)·SNR0TBNR2)],
TBNR=(KthIb)/σn.
Pf=Kth12πσnexp[(KIb)22σn2]dK=12[1erf(TBNR/2)].
Pd=0Pd(Is)fIs(Is)dIs,
f(Is)=2(αβ)α+β2Γ(α)Γ(β)msα+β21Kαβ(2αβIs),Is>0,
α={exp[0.49σI2(1+1.11σR125)76]1}1,β={exp[0.51σI2(1+0.69σR125)56]1}1.
Pd=0(αβ)α+β2Γ(α)Γ(β)xα+β21Kαβ(2αβx)[1+erf(x·μ(θ)·SNR0TBNR2)]dx.

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