Abstract

Utilizing an empirical path loss model proposed in the first paper of a two-part series, the bit error rate performance of short-range non-line-of-sight ultraviolet communication receivers is analyzed. Typical photodetector models and modulation formats are considered. Our results provide semi-analytical prediction of the achievable communication performance as a function of system and channel parameters, and serve as a basis for system design.

© 2010 OSA

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References

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  1. M. Shatalov, J. Zhang, A. S. Chitnis, V. Adivarahan, J. Yang, G. Simin, and M. A. Kahn, “Deep ultraviolet light-emitting diodes using quaternary AlInGaN multiple quantum wells,” IEEE J. Sel. Top. Quantum Electron. 8(2), 302–309 (2002).
    [CrossRef]
  2. V. Adivarahan, Q. Fareed, S. Srivastava, T. Katona, M. Gaevski, and A. Khan, “Robust 285 nm deep UV light emitting diodes over metal organic hydride vapor phase epitaxially grown AlN/sapphire templates,” Jpn. J. Appl. Phys. 46(23), L537– L539 (2007).
    [CrossRef]
  3. W. S. Ross and R. S. Kennedy, “An investigation of atmospheric optically scattered non-line-of-sight communication links,” Army Research Office Project Report, Research Triangle Park, NC, January 1980.
  4. M. Shatalov, J. Zhang, A. S. Chitnis, V. Adivarahan, J. Yang, G. Simin, and M. A. Kahn, “Deep ultraviolet light-emitting diodes using quaternary AlInGaN multiple quantum wells,” IEEE J. Sel. Top. Quantum Electron. 8(2), 302–309 (2002).
    [CrossRef]
  5. Z. Xu and B. M. Sadler, “Ultraviolet communications: potential and state-of-the-art,” IEEE Commun. Mag. 46(5), 67–73 (2008).
    [CrossRef]
  6. M. R. Luettgen, J. H. Shapiro, and D. M. Reilly, “Non-line-of-sight single-scatter propagation model,” J. Opt. Soc. Am. A 8(12), 1964–1972 (1991).
    [CrossRef]
  7. G. A. Shaw, A. M. Siegel, J. Model, and M. L. Nischan, “Field testing and evaluation of a solar-blind UV communication link for unattended ground sensors,” Proc. SPIE 5417, 250–261 (2004).
    [CrossRef]
  8. Z. Xu, “Approximate performance analysis of wireless ultraviolet links,” in Proceedings of IEEE Intl. Conf. on Acoustics, Speech, and Signal Proc. (IEEE, 2007).
  9. Z. Xu, H. Ding, B. M. Sadler, and G. Chen, “Analytical performance study of solar blind non-line-of-sight ultraviolet short-range communication links,” Opt. Lett. 33(16), 1860–1862 (2008).
    [CrossRef] [PubMed]
  10. H. Ding, G. Chen, A. Majumdar, B. M. Sadler, and Z. Xu, “Modeling of non-line-of-sight ultraviolet scattering channels for communication,” IEEE J. Sel. Areas Comm. 27(9), 1535–1544 (2009).
    [CrossRef]
  11. G. Chen, Z. Xu, H. Ding, and B. M. Sadler, “Path loss modeling and performance trade-off study for short-range non-line-of-sight ultraviolet communications,” Opt. Express 17(5), 3929–3940 (2009).
    [CrossRef] [PubMed]
  12. G. Chen, F. Abou-Galala, Z. Xu, and B. M. Sadler, “Experimental evaluation of LED-based solar blind NLOS communication links,” Opt. Express 16(19), 15059–15068 (2008).
    [CrossRef] [PubMed]
  13. R. M. Gagliardi and S. Karp, Optical Communications, 2nd ed. (John Wiley & Sons, 1995).
  14. Q. He, B. M. Sadler, and Z. Xu, “On the achievable performance of non-line of sight ultraviolet communications,” Proceedings of OSA Optics & Photonics Congress: Applications of Lasers for Sensing and Free Space Communications (OSA, 2010).
  15. S. Karp and R. Gagliardi, “The design of a pulse-position modulated optical communication system,” IEEE Trans. Commun. Technol. 17(6), 670–676 (1969).
    [CrossRef]
  16. R. McIntyre, “The distribution of gains in uniformly multiplying avalanche photodiodes: theory,” IEEE Trans. Electron. Dev. 19(6), 703–713 (1972).
    [CrossRef]
  17. J. Conradi, “The distribution of gains in uniformly multiplying avalanche photodiodes: experimental,” IEEE Trans. Electron. Dev. 19(6), 713–718 (1972).
    [CrossRef]
  18. K. Kiasaleh, “Performance of APD-based, PPM free-space optical communication systems in atmospheric turbulence,” IEEE Trans. Commun. 53(9), 1455–1461 (2005).
    [CrossRef]
  19. S. Alexander, Optical Communication Receiver Design (SPIE Optical Engineering Press, 1997.

2009 (2)

H. Ding, G. Chen, A. Majumdar, B. M. Sadler, and Z. Xu, “Modeling of non-line-of-sight ultraviolet scattering channels for communication,” IEEE J. Sel. Areas Comm. 27(9), 1535–1544 (2009).
[CrossRef]

G. Chen, Z. Xu, H. Ding, and B. M. Sadler, “Path loss modeling and performance trade-off study for short-range non-line-of-sight ultraviolet communications,” Opt. Express 17(5), 3929–3940 (2009).
[CrossRef] [PubMed]

2008 (3)

2007 (1)

V. Adivarahan, Q. Fareed, S. Srivastava, T. Katona, M. Gaevski, and A. Khan, “Robust 285 nm deep UV light emitting diodes over metal organic hydride vapor phase epitaxially grown AlN/sapphire templates,” Jpn. J. Appl. Phys. 46(23), L537– L539 (2007).
[CrossRef]

2005 (1)

K. Kiasaleh, “Performance of APD-based, PPM free-space optical communication systems in atmospheric turbulence,” IEEE Trans. Commun. 53(9), 1455–1461 (2005).
[CrossRef]

2004 (1)

G. A. Shaw, A. M. Siegel, J. Model, and M. L. Nischan, “Field testing and evaluation of a solar-blind UV communication link for unattended ground sensors,” Proc. SPIE 5417, 250–261 (2004).
[CrossRef]

2002 (2)

M. Shatalov, J. Zhang, A. S. Chitnis, V. Adivarahan, J. Yang, G. Simin, and M. A. Kahn, “Deep ultraviolet light-emitting diodes using quaternary AlInGaN multiple quantum wells,” IEEE J. Sel. Top. Quantum Electron. 8(2), 302–309 (2002).
[CrossRef]

M. Shatalov, J. Zhang, A. S. Chitnis, V. Adivarahan, J. Yang, G. Simin, and M. A. Kahn, “Deep ultraviolet light-emitting diodes using quaternary AlInGaN multiple quantum wells,” IEEE J. Sel. Top. Quantum Electron. 8(2), 302–309 (2002).
[CrossRef]

1991 (1)

1972 (2)

R. McIntyre, “The distribution of gains in uniformly multiplying avalanche photodiodes: theory,” IEEE Trans. Electron. Dev. 19(6), 703–713 (1972).
[CrossRef]

J. Conradi, “The distribution of gains in uniformly multiplying avalanche photodiodes: experimental,” IEEE Trans. Electron. Dev. 19(6), 713–718 (1972).
[CrossRef]

1969 (1)

S. Karp and R. Gagliardi, “The design of a pulse-position modulated optical communication system,” IEEE Trans. Commun. Technol. 17(6), 670–676 (1969).
[CrossRef]

Abou-Galala, F.

Adivarahan, V.

V. Adivarahan, Q. Fareed, S. Srivastava, T. Katona, M. Gaevski, and A. Khan, “Robust 285 nm deep UV light emitting diodes over metal organic hydride vapor phase epitaxially grown AlN/sapphire templates,” Jpn. J. Appl. Phys. 46(23), L537– L539 (2007).
[CrossRef]

M. Shatalov, J. Zhang, A. S. Chitnis, V. Adivarahan, J. Yang, G. Simin, and M. A. Kahn, “Deep ultraviolet light-emitting diodes using quaternary AlInGaN multiple quantum wells,” IEEE J. Sel. Top. Quantum Electron. 8(2), 302–309 (2002).
[CrossRef]

M. Shatalov, J. Zhang, A. S. Chitnis, V. Adivarahan, J. Yang, G. Simin, and M. A. Kahn, “Deep ultraviolet light-emitting diodes using quaternary AlInGaN multiple quantum wells,” IEEE J. Sel. Top. Quantum Electron. 8(2), 302–309 (2002).
[CrossRef]

Chen, G.

Chitnis, A. S.

M. Shatalov, J. Zhang, A. S. Chitnis, V. Adivarahan, J. Yang, G. Simin, and M. A. Kahn, “Deep ultraviolet light-emitting diodes using quaternary AlInGaN multiple quantum wells,” IEEE J. Sel. Top. Quantum Electron. 8(2), 302–309 (2002).
[CrossRef]

M. Shatalov, J. Zhang, A. S. Chitnis, V. Adivarahan, J. Yang, G. Simin, and M. A. Kahn, “Deep ultraviolet light-emitting diodes using quaternary AlInGaN multiple quantum wells,” IEEE J. Sel. Top. Quantum Electron. 8(2), 302–309 (2002).
[CrossRef]

Conradi, J.

J. Conradi, “The distribution of gains in uniformly multiplying avalanche photodiodes: experimental,” IEEE Trans. Electron. Dev. 19(6), 713–718 (1972).
[CrossRef]

Ding, H.

Fareed, Q.

V. Adivarahan, Q. Fareed, S. Srivastava, T. Katona, M. Gaevski, and A. Khan, “Robust 285 nm deep UV light emitting diodes over metal organic hydride vapor phase epitaxially grown AlN/sapphire templates,” Jpn. J. Appl. Phys. 46(23), L537– L539 (2007).
[CrossRef]

Gaevski, M.

V. Adivarahan, Q. Fareed, S. Srivastava, T. Katona, M. Gaevski, and A. Khan, “Robust 285 nm deep UV light emitting diodes over metal organic hydride vapor phase epitaxially grown AlN/sapphire templates,” Jpn. J. Appl. Phys. 46(23), L537– L539 (2007).
[CrossRef]

Gagliardi, R.

S. Karp and R. Gagliardi, “The design of a pulse-position modulated optical communication system,” IEEE Trans. Commun. Technol. 17(6), 670–676 (1969).
[CrossRef]

Kahn, M. A.

M. Shatalov, J. Zhang, A. S. Chitnis, V. Adivarahan, J. Yang, G. Simin, and M. A. Kahn, “Deep ultraviolet light-emitting diodes using quaternary AlInGaN multiple quantum wells,” IEEE J. Sel. Top. Quantum Electron. 8(2), 302–309 (2002).
[CrossRef]

M. Shatalov, J. Zhang, A. S. Chitnis, V. Adivarahan, J. Yang, G. Simin, and M. A. Kahn, “Deep ultraviolet light-emitting diodes using quaternary AlInGaN multiple quantum wells,” IEEE J. Sel. Top. Quantum Electron. 8(2), 302–309 (2002).
[CrossRef]

Karp, S.

S. Karp and R. Gagliardi, “The design of a pulse-position modulated optical communication system,” IEEE Trans. Commun. Technol. 17(6), 670–676 (1969).
[CrossRef]

Katona, T.

V. Adivarahan, Q. Fareed, S. Srivastava, T. Katona, M. Gaevski, and A. Khan, “Robust 285 nm deep UV light emitting diodes over metal organic hydride vapor phase epitaxially grown AlN/sapphire templates,” Jpn. J. Appl. Phys. 46(23), L537– L539 (2007).
[CrossRef]

Khan, A.

V. Adivarahan, Q. Fareed, S. Srivastava, T. Katona, M. Gaevski, and A. Khan, “Robust 285 nm deep UV light emitting diodes over metal organic hydride vapor phase epitaxially grown AlN/sapphire templates,” Jpn. J. Appl. Phys. 46(23), L537– L539 (2007).
[CrossRef]

Kiasaleh, K.

K. Kiasaleh, “Performance of APD-based, PPM free-space optical communication systems in atmospheric turbulence,” IEEE Trans. Commun. 53(9), 1455–1461 (2005).
[CrossRef]

Luettgen, M. R.

Majumdar, A.

H. Ding, G. Chen, A. Majumdar, B. M. Sadler, and Z. Xu, “Modeling of non-line-of-sight ultraviolet scattering channels for communication,” IEEE J. Sel. Areas Comm. 27(9), 1535–1544 (2009).
[CrossRef]

McIntyre, R.

R. McIntyre, “The distribution of gains in uniformly multiplying avalanche photodiodes: theory,” IEEE Trans. Electron. Dev. 19(6), 703–713 (1972).
[CrossRef]

Model, J.

G. A. Shaw, A. M. Siegel, J. Model, and M. L. Nischan, “Field testing and evaluation of a solar-blind UV communication link for unattended ground sensors,” Proc. SPIE 5417, 250–261 (2004).
[CrossRef]

Nischan, M. L.

G. A. Shaw, A. M. Siegel, J. Model, and M. L. Nischan, “Field testing and evaluation of a solar-blind UV communication link for unattended ground sensors,” Proc. SPIE 5417, 250–261 (2004).
[CrossRef]

Reilly, D. M.

Sadler, B. M.

Shapiro, J. H.

Shatalov, M.

M. Shatalov, J. Zhang, A. S. Chitnis, V. Adivarahan, J. Yang, G. Simin, and M. A. Kahn, “Deep ultraviolet light-emitting diodes using quaternary AlInGaN multiple quantum wells,” IEEE J. Sel. Top. Quantum Electron. 8(2), 302–309 (2002).
[CrossRef]

M. Shatalov, J. Zhang, A. S. Chitnis, V. Adivarahan, J. Yang, G. Simin, and M. A. Kahn, “Deep ultraviolet light-emitting diodes using quaternary AlInGaN multiple quantum wells,” IEEE J. Sel. Top. Quantum Electron. 8(2), 302–309 (2002).
[CrossRef]

Shaw, G. A.

G. A. Shaw, A. M. Siegel, J. Model, and M. L. Nischan, “Field testing and evaluation of a solar-blind UV communication link for unattended ground sensors,” Proc. SPIE 5417, 250–261 (2004).
[CrossRef]

Siegel, A. M.

G. A. Shaw, A. M. Siegel, J. Model, and M. L. Nischan, “Field testing and evaluation of a solar-blind UV communication link for unattended ground sensors,” Proc. SPIE 5417, 250–261 (2004).
[CrossRef]

Simin, G.

M. Shatalov, J. Zhang, A. S. Chitnis, V. Adivarahan, J. Yang, G. Simin, and M. A. Kahn, “Deep ultraviolet light-emitting diodes using quaternary AlInGaN multiple quantum wells,” IEEE J. Sel. Top. Quantum Electron. 8(2), 302–309 (2002).
[CrossRef]

M. Shatalov, J. Zhang, A. S. Chitnis, V. Adivarahan, J. Yang, G. Simin, and M. A. Kahn, “Deep ultraviolet light-emitting diodes using quaternary AlInGaN multiple quantum wells,” IEEE J. Sel. Top. Quantum Electron. 8(2), 302–309 (2002).
[CrossRef]

Srivastava, S.

V. Adivarahan, Q. Fareed, S. Srivastava, T. Katona, M. Gaevski, and A. Khan, “Robust 285 nm deep UV light emitting diodes over metal organic hydride vapor phase epitaxially grown AlN/sapphire templates,” Jpn. J. Appl. Phys. 46(23), L537– L539 (2007).
[CrossRef]

Xu, Z.

Yang, J.

M. Shatalov, J. Zhang, A. S. Chitnis, V. Adivarahan, J. Yang, G. Simin, and M. A. Kahn, “Deep ultraviolet light-emitting diodes using quaternary AlInGaN multiple quantum wells,” IEEE J. Sel. Top. Quantum Electron. 8(2), 302–309 (2002).
[CrossRef]

M. Shatalov, J. Zhang, A. S. Chitnis, V. Adivarahan, J. Yang, G. Simin, and M. A. Kahn, “Deep ultraviolet light-emitting diodes using quaternary AlInGaN multiple quantum wells,” IEEE J. Sel. Top. Quantum Electron. 8(2), 302–309 (2002).
[CrossRef]

Zhang, J.

M. Shatalov, J. Zhang, A. S. Chitnis, V. Adivarahan, J. Yang, G. Simin, and M. A. Kahn, “Deep ultraviolet light-emitting diodes using quaternary AlInGaN multiple quantum wells,” IEEE J. Sel. Top. Quantum Electron. 8(2), 302–309 (2002).
[CrossRef]

M. Shatalov, J. Zhang, A. S. Chitnis, V. Adivarahan, J. Yang, G. Simin, and M. A. Kahn, “Deep ultraviolet light-emitting diodes using quaternary AlInGaN multiple quantum wells,” IEEE J. Sel. Top. Quantum Electron. 8(2), 302–309 (2002).
[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]

IEEE J. Sel. Areas Comm. (1)

H. Ding, G. Chen, A. Majumdar, B. M. Sadler, and Z. Xu, “Modeling of non-line-of-sight ultraviolet scattering channels for communication,” IEEE J. Sel. Areas Comm. 27(9), 1535–1544 (2009).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (2)

M. Shatalov, J. Zhang, A. S. Chitnis, V. Adivarahan, J. Yang, G. Simin, and M. A. Kahn, “Deep ultraviolet light-emitting diodes using quaternary AlInGaN multiple quantum wells,” IEEE J. Sel. Top. Quantum Electron. 8(2), 302–309 (2002).
[CrossRef]

M. Shatalov, J. Zhang, A. S. Chitnis, V. Adivarahan, J. Yang, G. Simin, and M. A. Kahn, “Deep ultraviolet light-emitting diodes using quaternary AlInGaN multiple quantum wells,” IEEE J. Sel. Top. Quantum Electron. 8(2), 302–309 (2002).
[CrossRef]

IEEE Trans. Commun. (1)

K. Kiasaleh, “Performance of APD-based, PPM free-space optical communication systems in atmospheric turbulence,” IEEE Trans. Commun. 53(9), 1455–1461 (2005).
[CrossRef]

IEEE Trans. Commun. Technol. (1)

S. Karp and R. Gagliardi, “The design of a pulse-position modulated optical communication system,” IEEE Trans. Commun. Technol. 17(6), 670–676 (1969).
[CrossRef]

IEEE Trans. Electron. Dev. (2)

R. McIntyre, “The distribution of gains in uniformly multiplying avalanche photodiodes: theory,” IEEE Trans. Electron. Dev. 19(6), 703–713 (1972).
[CrossRef]

J. Conradi, “The distribution of gains in uniformly multiplying avalanche photodiodes: experimental,” IEEE Trans. Electron. Dev. 19(6), 713–718 (1972).
[CrossRef]

J. Opt. Soc. Am. A (1)

Jpn. J. Appl. Phys. (1)

V. Adivarahan, Q. Fareed, S. Srivastava, T. Katona, M. Gaevski, and A. Khan, “Robust 285 nm deep UV light emitting diodes over metal organic hydride vapor phase epitaxially grown AlN/sapphire templates,” Jpn. J. Appl. Phys. 46(23), L537– L539 (2007).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Proc. SPIE (1)

G. A. Shaw, A. M. Siegel, J. Model, and M. L. Nischan, “Field testing and evaluation of a solar-blind UV communication link for unattended ground sensors,” Proc. SPIE 5417, 250–261 (2004).
[CrossRef]

Other (5)

Z. Xu, “Approximate performance analysis of wireless ultraviolet links,” in Proceedings of IEEE Intl. Conf. on Acoustics, Speech, and Signal Proc. (IEEE, 2007).

W. S. Ross and R. S. Kennedy, “An investigation of atmospheric optically scattered non-line-of-sight communication links,” Army Research Office Project Report, Research Triangle Park, NC, January 1980.

R. M. Gagliardi and S. Karp, Optical Communications, 2nd ed. (John Wiley & Sons, 1995).

Q. He, B. M. Sadler, and Z. Xu, “On the achievable performance of non-line of sight ultraviolet communications,” Proceedings of OSA Optics & Photonics Congress: Applications of Lasers for Sensing and Free Space Communications (OSA, 2010).

S. Alexander, Optical Communication Receiver Design (SPIE Optical Engineering Press, 1997.

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Figures (7)

Fig. 1
Fig. 1

Non-line-of-sight UV communications system model.

Fig. 2
Fig. 2

Achievable rate vs. baseline range for OOK (left) and 4-PPM (right) modulations, high noise case.

Fig. 3
Fig. 3

Achievable rate vs. baseline range for OOK (left) and PPM (right) modulations, and no background noise.

Fig. 4
Fig. 4

Error probability vs. data rate under small (left) and large (right) pointing angles.

Fig. 5
Fig. 5

Error probability vs. baseline range for OOK modulation, Pt = 100mW, PMT (left) and APD (right).

Fig. 6
Fig. 6

Error probability vs. baseline range for 4-PPM modulation, Pt = 100mW, PMT (left) and APD (right).

Fig. 7
Fig. 7

Error probability vs. multiplication gain, Pt = 100mW, PMT (left) and APD (right).

Tables (3)

Tables Icon

Table 1 Path loss factor ξ of the NLOS UV channel model

Tables Icon

Table 2 Path loss exponent α of the NLOS UV channel model

Tables Icon

Table 3 Typical UV communication system parameters

Equations (24)

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

L = ξ r α ,
P k 1 ( j ) = λ j e λ j ! ,
P e _ O O K = 1 2 k = 0 m T ( λ s + λ b ) k e ( λ s + λ b ) k ! + 1 2 k = m T + 1 λ b k e λ b k ! .
r = η P t ξ R b h ν ln ( 2 P e _ O O K ) α ,   or    r R b 1 / α = C ˜ ,    where    C ˜ = η P t ξ h ν ln ( 2 P e _ O O K ) α .
P e _ P P M = M 2 ( M 1 ) { 1 e [ ( λ s + M λ b ) ] M r = 0 M 1 ( M 1 ) ! r ! ( M 1 r ) ! ( r + 1 )                       × k = 1 ( λ s + λ b ) k e ( λ s + λ b ) k ! [ λ b k e λ b k ! ] r [ j = 0 k 1 λ b j e λ b j ! ] M 1 r } .
P k 2 ( k 2 ) = k 1 = 0 P k 2 ( k 2 | k 1 ) P k 1 ( k 1 ) ,
P k 2 ( k 2 ) = C exp [ ( k 2 A λ ) 2 2 ( ζ A λ ) 2 ] .
P k 2 ( k 2 ) = 1 ( 2 π C 1 2 ) 1 / 2 [ 1 1 + ( k 2 A λ ) 3 / 2 C 1 C 2 ] exp { ( k 2 A λ ) 2 2 C 1 2 [ 1 + ( k 2 A λ ) C 1 C 2 ] } ,
C 1 = ( A λ ) 2 F 1 ,           C 2 = A ( λ F ) 1 / 2 / ( F 1 ) ,           F = γ A + ( 2 1 A ) ( 1 γ ) ,
z = v + n ,
σ n 2 = ( 2 k e T o / R L ) T p ,
μ = k 2 e ,
p z ( z | λ ) = j = 0 P k 2 ( j | λ ) G ( z , j e , σ n 2 ) .
z = v + n = i = 1 k 1 A i e + n ,
p z ( z | λ ) = j = 0 P k 1 ( j | λ ) G ( z , j A e , σ 2 ) .
σ 2 = σ p d 2 + σ n 2 = j ( F 1 ) ( A e ) 2 + ( 2 k e T o / R L ) T p .
σ P M T 2 = j ( ζ A e ) 2 + ( 2 k e T o / R L ) T p ,     σ A P D 2 = j [ γ A + 2 ( 1 γ ) 1 γ A 1 ] ( A e ) 2 + ( 2 k e T o / R L ) T p .
P e _ O O K = P 1 z t h p z ( z | λ S + λ b ) d z + P 0 z t h p z ( z | λ b ) d z .
p z ( z t h | λ S + λ b ) = p z ( z t h | λ b ) .
z t h p z ( z | λ S + λ b ) d z = 1 z t h p z ( z | λ S + λ b ) d z ,
P e _ O O K = 1 2 1 2 j = 0 [ ( λ S + λ b ) j j ! e ( λ S + λ b ) λ b j j ! e λ b ] Q ( z t h j A e σ ) .
P D = p z ( z | λ S + λ b ) [ z p z ( y | λ b ) d y ] M 1 d z ,
P D = 1 2 M 1 j = 0 ( λ s + λ b ) j j ! e ( λ s + λ b ) G ( z , j A e , σ 2 ) { 1 + k = 0 λ b k e λ b k ! e r f ( z k A e 2 σ ) } M 1 d z ,
e r f ( x ) = 2 π 0 x e u 2 d u .

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