Abstract

We describe a coded power-efficient transmission scheme based on repetition MIMO principle suitable for communication over the atmospheric turbulence channel, and determine its channel capacity. The proposed scheme employs the Q-ary pulse-position modulation. We further study how to approach the channel capacity limits using low-density parity-check (LDPC) codes. Component LDPC codes are designed using the concept of pairwise-balanced designs. Contrary to the several recent publications, bit-error rates and channel capacities are reported assuming that p.i.n. photodetectors are used instead of ideal photon-counting receivers. The atmospheric turbulence channel is modeled using the Gamma-Gamma distribution function due to Al-Habash et al. Excellent bit-error rate performance improvement, over uncoded case, is found.

© 2007 Optical Society of America

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References

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  1. H. Willebrand and B.S. Ghuman, Free-Space Optics: Enabling Optical Connectivity in Today’s Networks: Sams Publishing, 2002.
  2. S. G. Wilson, M. Brandt-Pearce, Q. Cao, and J.J.H. Leveque, III, “Free-space optical MIMO transmission with Q-ary PPM,” IEEE Trans. Commun. 53, 1402–1412 (2005).
    [Crossref]
  3. S. G. Wilson, M. Brandt-Pearce, Q. Cao, and M. Baedke, “Optical repetition MIMO transmission with multipulse PPM,” IEEE Selected Areas Comm. 23, 1901–1910 (2005).
    [Crossref]
  4. 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]
  5. I. B. Djordjevic, B. Vasic, and M. A. Neifeld, “Power efficient LDPC-coded modulation for free-space optical communication over the atmospheric turbulence channel,” in Proc. OFC 2007, Paper no. JThA46, March 25–29, 2007, Anaheim, CA, USA.
  6. I. B. Djordjevic, B. Vasic, and M. A. Neifeld, “Multilevel coding in free-space optical MIMO transmission with Q-ary PPM over the atmospheric turbulence channel,” IEEE Photon. Tehnol. Lett. 18, 1491–1493 (2006).
    [Crossref]
  7. N. Cvijetic, S.G. Wilson, and M. Brandt-Pearce, “Receiver optimization in turbulent free-space optical MIMO channels with APDs and Q-ary PPM,” IEEE Photon. Tehnol. Lett. 19, 1491–1493 (2007).
  8. G. Ungerboeck, “Channel coding with multilevel/phase signals,” IEEE. Trans Inf. Theory 28, 55–67 (1982).
    [Crossref]
  9. I. B. Djordjevic, S. Sankaranarayanan, S. K. Chilappagari, and B. Vasic, “Low-density parity-check codes for 40 Gb/s optical transmission systems,” IEEE J. Sel. Top. Quantum Electron. 12, 555–562 (2006).
    [Crossref]
  10. J. A. Anguita, I. B. Djordjevic, M.A. Neifeld, and B. V. Vasic, “Shannon capacities and error-correction codes for optical atmospheric turbulent channels,” J. Opt. Networking 4, 586–601 (2005).
    [Crossref]
  11. I. Anderson, Combinatorial Designs and Tournaments:Oxford University Press, 1997.
  12. G. Caire, G. Tarrico, and E. Biglieri, “Bit-interleaved coded modulation,” IEEE. Trans Inf. Theory 44, 927–946 (1998).
    [Crossref]
  13. H. Imai and S. Hirakawa, “A new multilevel coding method using error correcting codes,” IEEE. Trans Inf. Theory IT-23, 371–377 (1977).
    [Crossref]
  14. K. Simon and V.A Vilnrotter, “Alamouti-type space-time coding for free-space optical communication with direct detection,” IEEE Trans. Wireless Comm. 4, 35–39 (2005).
    [Crossref]
  15. V. Tarokh, H. Jafarkani, and A. R. Calderbank, “Space-time block codes from orthogonal designs,” IEEE. Trans Inf. Theory 45, 1456–1467 (1999).
    [Crossref]
  16. I. B. Djordjevic, B. Vasic, and M. A. Neifeld, “LDPC coded OFDM over the atmospheric turbulence channel,” Opt. Express 15, 6332–6346 (2007).
  17. I. B. Djordjevic, S. Denic, J. Anguita, B. Vasic, and M. A. Neifeld, “LDPC-Coded MIMO optical communication over the atmospheric turbulence channel,” accepted for presentation at Globecom 2007.

2007 (2)

N. Cvijetic, S.G. Wilson, and M. Brandt-Pearce, “Receiver optimization in turbulent free-space optical MIMO channels with APDs and Q-ary PPM,” IEEE Photon. Tehnol. Lett. 19, 1491–1493 (2007).

I. B. Djordjevic, B. Vasic, and M. A. Neifeld, “LDPC coded OFDM over the atmospheric turbulence channel,” Opt. Express 15, 6332–6346 (2007).

2006 (2)

I. B. Djordjevic, B. Vasic, and M. A. Neifeld, “Multilevel coding in free-space optical MIMO transmission with Q-ary PPM over the atmospheric turbulence channel,” IEEE Photon. Tehnol. Lett. 18, 1491–1493 (2006).
[Crossref]

I. B. Djordjevic, S. Sankaranarayanan, S. K. Chilappagari, and B. Vasic, “Low-density parity-check codes for 40 Gb/s optical transmission systems,” IEEE J. Sel. Top. Quantum Electron. 12, 555–562 (2006).
[Crossref]

2005 (4)

J. A. Anguita, I. B. Djordjevic, M.A. Neifeld, and B. V. Vasic, “Shannon capacities and error-correction codes for optical atmospheric turbulent channels,” J. Opt. Networking 4, 586–601 (2005).
[Crossref]

S. G. Wilson, M. Brandt-Pearce, Q. Cao, and J.J.H. Leveque, III, “Free-space optical MIMO transmission with Q-ary PPM,” IEEE Trans. Commun. 53, 1402–1412 (2005).
[Crossref]

S. G. Wilson, M. Brandt-Pearce, Q. Cao, and M. Baedke, “Optical repetition MIMO transmission with multipulse PPM,” IEEE Selected Areas Comm. 23, 1901–1910 (2005).
[Crossref]

K. Simon and V.A Vilnrotter, “Alamouti-type space-time coding for free-space optical communication with direct detection,” IEEE Trans. Wireless Comm. 4, 35–39 (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]

1999 (1)

V. Tarokh, H. Jafarkani, and A. R. Calderbank, “Space-time block codes from orthogonal designs,” IEEE. Trans Inf. Theory 45, 1456–1467 (1999).
[Crossref]

1998 (1)

G. Caire, G. Tarrico, and E. Biglieri, “Bit-interleaved coded modulation,” IEEE. Trans Inf. Theory 44, 927–946 (1998).
[Crossref]

1982 (1)

G. Ungerboeck, “Channel coding with multilevel/phase signals,” IEEE. Trans Inf. Theory 28, 55–67 (1982).
[Crossref]

1977 (1)

H. Imai and S. Hirakawa, “A new multilevel coding method using error correcting codes,” IEEE. Trans Inf. Theory IT-23, 371–377 (1977).
[Crossref]

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]

Anderson, I.

I. Anderson, Combinatorial Designs and Tournaments:Oxford University Press, 1997.

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]

Anguita, J.

I. B. Djordjevic, S. Denic, J. Anguita, B. Vasic, and M. A. Neifeld, “LDPC-Coded MIMO optical communication over the atmospheric turbulence channel,” accepted for presentation at Globecom 2007.

Anguita, J. A.

J. A. Anguita, I. B. Djordjevic, M.A. Neifeld, and B. V. Vasic, “Shannon capacities and error-correction codes for optical atmospheric turbulent channels,” J. Opt. Networking 4, 586–601 (2005).
[Crossref]

Baedke, M.

S. G. Wilson, M. Brandt-Pearce, Q. Cao, and M. Baedke, “Optical repetition MIMO transmission with multipulse PPM,” IEEE Selected Areas Comm. 23, 1901–1910 (2005).
[Crossref]

Biglieri, E.

G. Caire, G. Tarrico, and E. Biglieri, “Bit-interleaved coded modulation,” IEEE. Trans Inf. Theory 44, 927–946 (1998).
[Crossref]

Brandt-Pearce, M.

N. Cvijetic, S.G. Wilson, and M. Brandt-Pearce, “Receiver optimization in turbulent free-space optical MIMO channels with APDs and Q-ary PPM,” IEEE Photon. Tehnol. Lett. 19, 1491–1493 (2007).

S. G. Wilson, M. Brandt-Pearce, Q. Cao, and J.J.H. Leveque, III, “Free-space optical MIMO transmission with Q-ary PPM,” IEEE Trans. Commun. 53, 1402–1412 (2005).
[Crossref]

S. G. Wilson, M. Brandt-Pearce, Q. Cao, and M. Baedke, “Optical repetition MIMO transmission with multipulse PPM,” IEEE Selected Areas Comm. 23, 1901–1910 (2005).
[Crossref]

Caire, G.

G. Caire, G. Tarrico, and E. Biglieri, “Bit-interleaved coded modulation,” IEEE. Trans Inf. Theory 44, 927–946 (1998).
[Crossref]

Calderbank, A. R.

V. Tarokh, H. Jafarkani, and A. R. Calderbank, “Space-time block codes from orthogonal designs,” IEEE. Trans Inf. Theory 45, 1456–1467 (1999).
[Crossref]

Cao, Q.

S. G. Wilson, M. Brandt-Pearce, Q. Cao, and M. Baedke, “Optical repetition MIMO transmission with multipulse PPM,” IEEE Selected Areas Comm. 23, 1901–1910 (2005).
[Crossref]

S. G. Wilson, M. Brandt-Pearce, Q. Cao, and J.J.H. Leveque, III, “Free-space optical MIMO transmission with Q-ary PPM,” IEEE Trans. Commun. 53, 1402–1412 (2005).
[Crossref]

Chilappagari, S. K.

I. B. Djordjevic, S. Sankaranarayanan, S. K. Chilappagari, and B. Vasic, “Low-density parity-check codes for 40 Gb/s optical transmission systems,” IEEE J. Sel. Top. Quantum Electron. 12, 555–562 (2006).
[Crossref]

Cvijetic, N.

N. Cvijetic, S.G. Wilson, and M. Brandt-Pearce, “Receiver optimization in turbulent free-space optical MIMO channels with APDs and Q-ary PPM,” IEEE Photon. Tehnol. Lett. 19, 1491–1493 (2007).

Denic, S.

I. B. Djordjevic, S. Denic, J. Anguita, B. Vasic, and M. A. Neifeld, “LDPC-Coded MIMO optical communication over the atmospheric turbulence channel,” accepted for presentation at Globecom 2007.

Djordjevic, I. B.

I. B. Djordjevic, B. Vasic, and M. A. Neifeld, “LDPC coded OFDM over the atmospheric turbulence channel,” Opt. Express 15, 6332–6346 (2007).

I. B. Djordjevic, S. Sankaranarayanan, S. K. Chilappagari, and B. Vasic, “Low-density parity-check codes for 40 Gb/s optical transmission systems,” IEEE J. Sel. Top. Quantum Electron. 12, 555–562 (2006).
[Crossref]

I. B. Djordjevic, B. Vasic, and M. A. Neifeld, “Multilevel coding in free-space optical MIMO transmission with Q-ary PPM over the atmospheric turbulence channel,” IEEE Photon. Tehnol. Lett. 18, 1491–1493 (2006).
[Crossref]

J. A. Anguita, I. B. Djordjevic, M.A. Neifeld, and B. V. Vasic, “Shannon capacities and error-correction codes for optical atmospheric turbulent channels,” J. Opt. Networking 4, 586–601 (2005).
[Crossref]

I. B. Djordjevic, B. Vasic, and M. A. Neifeld, “Power efficient LDPC-coded modulation for free-space optical communication over the atmospheric turbulence channel,” in Proc. OFC 2007, Paper no. JThA46, March 25–29, 2007, Anaheim, CA, USA.

I. B. Djordjevic, S. Denic, J. Anguita, B. Vasic, and M. A. Neifeld, “LDPC-Coded MIMO optical communication over the atmospheric turbulence channel,” accepted for presentation at Globecom 2007.

Ghuman, B.S.

H. Willebrand and B.S. Ghuman, Free-Space Optics: Enabling Optical Connectivity in Today’s Networks: Sams Publishing, 2002.

Hirakawa, S.

H. Imai and S. Hirakawa, “A new multilevel coding method using error correcting codes,” IEEE. Trans Inf. Theory IT-23, 371–377 (1977).
[Crossref]

Imai, H.

H. Imai and S. Hirakawa, “A new multilevel coding method using error correcting codes,” IEEE. Trans Inf. Theory IT-23, 371–377 (1977).
[Crossref]

Jafarkani, H.

V. Tarokh, H. Jafarkani, and A. R. Calderbank, “Space-time block codes from orthogonal designs,” IEEE. Trans Inf. Theory 45, 1456–1467 (1999).
[Crossref]

Leveque, III, J.J.H.

S. G. Wilson, M. Brandt-Pearce, Q. Cao, and J.J.H. Leveque, III, “Free-space optical MIMO transmission with Q-ary PPM,” IEEE Trans. Commun. 53, 1402–1412 (2005).
[Crossref]

Neifeld, M. A.

I. B. Djordjevic, B. Vasic, and M. A. Neifeld, “LDPC coded OFDM over the atmospheric turbulence channel,” Opt. Express 15, 6332–6346 (2007).

I. B. Djordjevic, B. Vasic, and M. A. Neifeld, “Multilevel coding in free-space optical MIMO transmission with Q-ary PPM over the atmospheric turbulence channel,” IEEE Photon. Tehnol. Lett. 18, 1491–1493 (2006).
[Crossref]

I. B. Djordjevic, B. Vasic, and M. A. Neifeld, “Power efficient LDPC-coded modulation for free-space optical communication over the atmospheric turbulence channel,” in Proc. OFC 2007, Paper no. JThA46, March 25–29, 2007, Anaheim, CA, USA.

I. B. Djordjevic, S. Denic, J. Anguita, B. Vasic, and M. A. Neifeld, “LDPC-Coded MIMO optical communication over the atmospheric turbulence channel,” accepted for presentation at Globecom 2007.

Neifeld, M.A.

J. A. Anguita, I. B. Djordjevic, M.A. Neifeld, and B. V. Vasic, “Shannon capacities and error-correction codes for optical atmospheric turbulent channels,” J. Opt. Networking 4, 586–601 (2005).
[Crossref]

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]

Sankaranarayanan, S.

I. B. Djordjevic, S. Sankaranarayanan, S. K. Chilappagari, and B. Vasic, “Low-density parity-check codes for 40 Gb/s optical transmission systems,” IEEE J. Sel. Top. Quantum Electron. 12, 555–562 (2006).
[Crossref]

Simon, K.

K. Simon and V.A Vilnrotter, “Alamouti-type space-time coding for free-space optical communication with direct detection,” IEEE Trans. Wireless Comm. 4, 35–39 (2005).
[Crossref]

Tarokh, V.

V. Tarokh, H. Jafarkani, and A. R. Calderbank, “Space-time block codes from orthogonal designs,” IEEE. Trans Inf. Theory 45, 1456–1467 (1999).
[Crossref]

Tarrico, G.

G. Caire, G. Tarrico, and E. Biglieri, “Bit-interleaved coded modulation,” IEEE. Trans Inf. Theory 44, 927–946 (1998).
[Crossref]

Ungerboeck, G.

G. Ungerboeck, “Channel coding with multilevel/phase signals,” IEEE. Trans Inf. Theory 28, 55–67 (1982).
[Crossref]

Vasic, B.

I. B. Djordjevic, B. Vasic, and M. A. Neifeld, “LDPC coded OFDM over the atmospheric turbulence channel,” Opt. Express 15, 6332–6346 (2007).

I. B. Djordjevic, S. Sankaranarayanan, S. K. Chilappagari, and B. Vasic, “Low-density parity-check codes for 40 Gb/s optical transmission systems,” IEEE J. Sel. Top. Quantum Electron. 12, 555–562 (2006).
[Crossref]

I. B. Djordjevic, B. Vasic, and M. A. Neifeld, “Multilevel coding in free-space optical MIMO transmission with Q-ary PPM over the atmospheric turbulence channel,” IEEE Photon. Tehnol. Lett. 18, 1491–1493 (2006).
[Crossref]

I. B. Djordjevic, B. Vasic, and M. A. Neifeld, “Power efficient LDPC-coded modulation for free-space optical communication over the atmospheric turbulence channel,” in Proc. OFC 2007, Paper no. JThA46, March 25–29, 2007, Anaheim, CA, USA.

I. B. Djordjevic, S. Denic, J. Anguita, B. Vasic, and M. A. Neifeld, “LDPC-Coded MIMO optical communication over the atmospheric turbulence channel,” accepted for presentation at Globecom 2007.

Vasic, B. V.

J. A. Anguita, I. B. Djordjevic, M.A. Neifeld, and B. V. Vasic, “Shannon capacities and error-correction codes for optical atmospheric turbulent channels,” J. Opt. Networking 4, 586–601 (2005).
[Crossref]

Vilnrotter, V.A

K. Simon and V.A Vilnrotter, “Alamouti-type space-time coding for free-space optical communication with direct detection,” IEEE Trans. Wireless Comm. 4, 35–39 (2005).
[Crossref]

Willebrand, H.

H. Willebrand and B.S. Ghuman, Free-Space Optics: Enabling Optical Connectivity in Today’s Networks: Sams Publishing, 2002.

Wilson, S. G.

S. G. Wilson, M. Brandt-Pearce, Q. Cao, and M. Baedke, “Optical repetition MIMO transmission with multipulse PPM,” IEEE Selected Areas Comm. 23, 1901–1910 (2005).
[Crossref]

S. G. Wilson, M. Brandt-Pearce, Q. Cao, and J.J.H. Leveque, III, “Free-space optical MIMO transmission with Q-ary PPM,” IEEE Trans. Commun. 53, 1402–1412 (2005).
[Crossref]

Wilson, S.G.

N. Cvijetic, S.G. Wilson, and M. Brandt-Pearce, “Receiver optimization in turbulent free-space optical MIMO channels with APDs and Q-ary PPM,” IEEE Photon. Tehnol. Lett. 19, 1491–1493 (2007).

IEEE J. Sel. Top. Quantum Electron. (1)

I. B. Djordjevic, S. Sankaranarayanan, S. K. Chilappagari, and B. Vasic, “Low-density parity-check codes for 40 Gb/s optical transmission systems,” IEEE J. Sel. Top. Quantum Electron. 12, 555–562 (2006).
[Crossref]

IEEE Photon. Tehnol. Lett. (2)

I. B. Djordjevic, B. Vasic, and M. A. Neifeld, “Multilevel coding in free-space optical MIMO transmission with Q-ary PPM over the atmospheric turbulence channel,” IEEE Photon. Tehnol. Lett. 18, 1491–1493 (2006).
[Crossref]

N. Cvijetic, S.G. Wilson, and M. Brandt-Pearce, “Receiver optimization in turbulent free-space optical MIMO channels with APDs and Q-ary PPM,” IEEE Photon. Tehnol. Lett. 19, 1491–1493 (2007).

IEEE Selected Areas Comm. (1)

S. G. Wilson, M. Brandt-Pearce, Q. Cao, and M. Baedke, “Optical repetition MIMO transmission with multipulse PPM,” IEEE Selected Areas Comm. 23, 1901–1910 (2005).
[Crossref]

IEEE Trans. Commun. (1)

S. G. Wilson, M. Brandt-Pearce, Q. Cao, and J.J.H. Leveque, III, “Free-space optical MIMO transmission with Q-ary PPM,” IEEE Trans. Commun. 53, 1402–1412 (2005).
[Crossref]

IEEE Trans. Wireless Comm. (1)

K. Simon and V.A Vilnrotter, “Alamouti-type space-time coding for free-space optical communication with direct detection,” IEEE Trans. Wireless Comm. 4, 35–39 (2005).
[Crossref]

IEEE. Trans Inf. Theory (4)

V. Tarokh, H. Jafarkani, and A. R. Calderbank, “Space-time block codes from orthogonal designs,” IEEE. Trans Inf. Theory 45, 1456–1467 (1999).
[Crossref]

G. Caire, G. Tarrico, and E. Biglieri, “Bit-interleaved coded modulation,” IEEE. Trans Inf. Theory 44, 927–946 (1998).
[Crossref]

H. Imai and S. Hirakawa, “A new multilevel coding method using error correcting codes,” IEEE. Trans Inf. Theory IT-23, 371–377 (1977).
[Crossref]

G. Ungerboeck, “Channel coding with multilevel/phase signals,” IEEE. Trans Inf. Theory 28, 55–67 (1982).
[Crossref]

J. Opt. Networking (1)

J. A. Anguita, I. B. Djordjevic, M.A. Neifeld, and B. V. Vasic, “Shannon capacities and error-correction codes for optical atmospheric turbulent channels,” J. Opt. Networking 4, 586–601 (2005).
[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 (1)

I. B. Djordjevic, B. Vasic, and M. A. Neifeld, “LDPC coded OFDM over the atmospheric turbulence channel,” Opt. Express 15, 6332–6346 (2007).

Other (4)

I. B. Djordjevic, S. Denic, J. Anguita, B. Vasic, and M. A. Neifeld, “LDPC-Coded MIMO optical communication over the atmospheric turbulence channel,” accepted for presentation at Globecom 2007.

I. B. Djordjevic, B. Vasic, and M. A. Neifeld, “Power efficient LDPC-coded modulation for free-space optical communication over the atmospheric turbulence channel,” in Proc. OFC 2007, Paper no. JThA46, March 25–29, 2007, Anaheim, CA, USA.

H. Willebrand and B.S. Ghuman, Free-Space Optics: Enabling Optical Connectivity in Today’s Networks: Sams Publishing, 2002.

I. Anderson, Combinatorial Designs and Tournaments:Oxford University Press, 1997.

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Figures (3)

Fig. 1.
Fig. 1.

(a) Atmospheric optical MLMD system with Q-ary PPM and BICM, (b) transmitter side, (c) mth transmitter – receiver array configuration, and (d) processor configuration.

Fig. 2.
Fig. 2.

Channel capacity for different number of lasers (M), photodetectors (N) and number of slots (Q) in strong turbulence regime (σ R=3.0).

Fig. 3.
Fig. 3.

BER performance of bit-interleaved LDPC-coded modulation against MLC for different MLMD configurations.

Equations (7)

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

λ ( q ) = n = 1 N ( Z n , q E s M m = 1 M I n , m ) 2 σ 2 n = 1 N l = 1 , l q Q Z n , l σ 2 ,
f ( 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 .
α = 1 exp [ 0.49 σ R 2 ( 1 + 1.11 σ R 12 5 ) 7 6 ] 1 , β = 1 exp [ 0.51 σ R 2 ( 1 + 0.69 σ R 12 5 ) 5 6 ] 1 ,
σ R 2 = 1.23 C n 2 k 7 6 L 11 6 .
L ( c j ) = log c : c j = 0 exp [ λ ( q ) ] c : c j = 1 exp [ λ ( q ) ] ,
C = log 2 Q E I ¯ n E Z | I ¯ n { log 2 q = 1 Q exp [ n = 1 N ( Z n , q I ¯ n E s ) 2 ( Z n , q 0 I ¯ n E s ) 2 2 σ 2 ] } ,
p ( Z n , q | I ¯ n ) = 1 σ 2 π exp [ ( Z n , q I ¯ n ) 2 2 σ 2 ] .

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