Abstract

In some applications of optical communication systems, such as satellite optical communication and atmospheric optical communication, the optical beam wanders on the detector surface as a result of vibration and turbulence effects, respectively. The wandering of the beam degrades the communication system performance. In this research, we derive a mathematical model of an optical communication system with a detection matrix to improve the system performance for direct-detection pulse-position modulation. We include a centroid tracker in the communication system model. The centroid tracker tracks the center of the beam. Using the position of the beam center and an a priori model of the beam spreading, we estimate the optical power on each pixel (element) in the detection matrix. Using knowledge of the amplitudes of signal and noise in each pixel, we tune adaptively and separately the gain of each individual pixel in the detection matrix for communication signals. Tuning the gain is based on the mathematical model derived in this research. This model is defined as suboptimal, owing to some approximations in the development and is a suboptimum solution to the optimization problem of n multiplied by m free variables, where n, m are the dimensions of the detection matrix. Comparison is made between the adaptive suboptimum model and the standard model. From the mathematical analysis and the results of the comparison it is clear that this model significantly improves communication system performance.

© 1998 Optical Society of America

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References

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  1. D. Sadot, N. S. Kopeika, “Forecasting optical turbulence strength on the basis of macroscale meteorology and aerosols: models and validation,” Opt. Eng. (Bellingham) 31, 200–212 (1991).
    [CrossRef]
  2. R. L. Fante, “Electric beam propagation in turbulent media,” Proc. IEEE 63, 1669–1688 (1975).
    [CrossRef]
  3. E. C. Crittenden, A. W. Cooper, E. A. Milne, G. W. Rodeback, S. H. Kalmbach, R. L. Armstead, “Effects of turbulence on imaging through the atmosphere,” in Optical Properties of the Atmosphere, R. C. Sepucha, ed. Proc. SPIE142, 130–134 (1978).
    [CrossRef]
  4. J. H. Churnside, “Angle of arrival fluctuation of a reflected beam in atmospheric turbulence,” J. Opt. Soc. Am. A 4, 1264–1272 (1987).
    [CrossRef]
  5. M. Witting, L. van Holtz, D. E. L. Tunbridge, H. C. Vermeulen, “In orbit measurements of microaccelerations of ESA’s communication satellite OLYMPUS,” in Free Space Laser Communication Technologies II, O. L. Begley, B. D. Seery, eds. Proc. SPIE1218, 205–214 (1990).
    [CrossRef]
  6. S. Dyne, P. Collins, D. Tunbridge, “Satellite mechanical health monitoring,” Proceedings of the IEE Colloquium on Advance Vibration Measurements, Techniques and Instruments for the Early Prediction of Failure (Institute of Electrical Engineers, London, 1993), p. 4/1–8.
  7. C. C. Chen, C. S. Gardner, “Impact of random pointing and tracking errors on the design of coherent and incoherent optical intersatellite communication links,” IEEE Trans. Commun. 37, 252–260 (1989).
    [CrossRef]
  8. A. Yariv, Optical Electronics, 3rd ed. (Holt, Rinehart & Winston, New York, 1985), pp. 306–400.
  9. H. Kressel, ed., Semiconductor Devices for Optical Communication, Vol. 38 of Topics in Applied Physics (Springer-Verlag, Berlin, 1982), pp. 159–263.
  10. G. S. Mecherle, K. L. Marrs, “Description and results of satellite laser communication/tracking simulation,” in Proceedings of the 1994 IEEE Aerospace Applications Conference (IEEE, Piscataway, N.J., 1994), pp. 87–101.
  11. S. I. Green, M. P. Bobek, “Bit error rate testing of quadrant photodetectors,” in Space Sensing, Communications, and Networking, M. Ross, R. J. Temkin, eds. Proc. SPIE1059, 137–145 (1989).
    [CrossRef]
  12. S. G. Lambert, W. L. Casey, Laser Communication in Space, (Artech House, Norwood, Mass.1995), pp. 179–195.
  13. C. C. Chen, H. Ansari, J. R. Lesh, “Precision beam pointing for laser communication system using a CCD based tracker,” in Space Guidance, Control, and Tracking, G. E. Sevaston, F. Wade, eds. Proc. SPIE1949, 15–24 (1993).
    [CrossRef]
  14. Y. Bar-Shalom, H. M. Shertukde, K. R. Pattipati, “Precision target tracking for small extended objects,” Opt. Eng. (Bellingham) 29, 121–126 (1990).
    [CrossRef]
  15. D. R. Van Rheeden, R. A. Jones, “Noise effects on centroid tracker aim point estimation,” IEEE Trans. Aerosp. Electron. Syst. 24, 177–185 (1988).
    [CrossRef]
  16. Y. Bar-Shalom, H. M. Shertukde, K. R. Pattipati, “Use of measurement from an imaging sensor for precision target tracking,” IEEE Trans. Aerosp. Electron. Syst. 25, 863–872 (1989).
    [CrossRef]
  17. S. Arnon, S. Rotman, N. S. Kopeika, “Optimum transmitter optics aperture for free space satellite optical communication as a function of tracking system performance,” IEEE Trans. Aerosp. Electron. Syst. (to be published).
  18. S. Arnon, S. Rotman, N. S. Kopeika, “Beamwidth and transmitter power adaptive to tracking system performance for free-space optical communication,” Appl. Opt. 36, 6095–6101 (1997).
    [CrossRef] [PubMed]
  19. S. Arnon, N. S. Kopeika, “The performance limitations of free space optical communication satellite networks due to vibrations—analog case,” Opt. Eng. (Bellingham) 36, 175–182 (1997).
    [CrossRef]
  20. S. Arnon, S. Rotman, N. S. Kopeika, “The performance limitations of free space optical communication satellite networks due to vibrations—digital case,” Opt. Eng. (Bellingham) (to be published).
  21. R. M. Gagliardi, S. Karp, Optical Communication, 2nd ed., (Wiley, New York, 1995), pp. 201–206, 305–340.

1997 (2)

S. Arnon, N. S. Kopeika, “The performance limitations of free space optical communication satellite networks due to vibrations—analog case,” Opt. Eng. (Bellingham) 36, 175–182 (1997).
[CrossRef]

S. Arnon, S. Rotman, N. S. Kopeika, “Beamwidth and transmitter power adaptive to tracking system performance for free-space optical communication,” Appl. Opt. 36, 6095–6101 (1997).
[CrossRef] [PubMed]

1991 (1)

D. Sadot, N. S. Kopeika, “Forecasting optical turbulence strength on the basis of macroscale meteorology and aerosols: models and validation,” Opt. Eng. (Bellingham) 31, 200–212 (1991).
[CrossRef]

1990 (1)

Y. Bar-Shalom, H. M. Shertukde, K. R. Pattipati, “Precision target tracking for small extended objects,” Opt. Eng. (Bellingham) 29, 121–126 (1990).
[CrossRef]

1989 (2)

C. C. Chen, C. S. Gardner, “Impact of random pointing and tracking errors on the design of coherent and incoherent optical intersatellite communication links,” IEEE Trans. Commun. 37, 252–260 (1989).
[CrossRef]

Y. Bar-Shalom, H. M. Shertukde, K. R. Pattipati, “Use of measurement from an imaging sensor for precision target tracking,” IEEE Trans. Aerosp. Electron. Syst. 25, 863–872 (1989).
[CrossRef]

1988 (1)

D. R. Van Rheeden, R. A. Jones, “Noise effects on centroid tracker aim point estimation,” IEEE Trans. Aerosp. Electron. Syst. 24, 177–185 (1988).
[CrossRef]

1987 (1)

1975 (1)

R. L. Fante, “Electric beam propagation in turbulent media,” Proc. IEEE 63, 1669–1688 (1975).
[CrossRef]

Ansari, H.

C. C. Chen, H. Ansari, J. R. Lesh, “Precision beam pointing for laser communication system using a CCD based tracker,” in Space Guidance, Control, and Tracking, G. E. Sevaston, F. Wade, eds. Proc. SPIE1949, 15–24 (1993).
[CrossRef]

Armstead, R. L.

E. C. Crittenden, A. W. Cooper, E. A. Milne, G. W. Rodeback, S. H. Kalmbach, R. L. Armstead, “Effects of turbulence on imaging through the atmosphere,” in Optical Properties of the Atmosphere, R. C. Sepucha, ed. Proc. SPIE142, 130–134 (1978).
[CrossRef]

Arnon, S.

S. Arnon, S. Rotman, N. S. Kopeika, “Beamwidth and transmitter power adaptive to tracking system performance for free-space optical communication,” Appl. Opt. 36, 6095–6101 (1997).
[CrossRef] [PubMed]

S. Arnon, N. S. Kopeika, “The performance limitations of free space optical communication satellite networks due to vibrations—analog case,” Opt. Eng. (Bellingham) 36, 175–182 (1997).
[CrossRef]

S. Arnon, S. Rotman, N. S. Kopeika, “The performance limitations of free space optical communication satellite networks due to vibrations—digital case,” Opt. Eng. (Bellingham) (to be published).

S. Arnon, S. Rotman, N. S. Kopeika, “Optimum transmitter optics aperture for free space satellite optical communication as a function of tracking system performance,” IEEE Trans. Aerosp. Electron. Syst. (to be published).

Bar-Shalom, Y.

Y. Bar-Shalom, H. M. Shertukde, K. R. Pattipati, “Precision target tracking for small extended objects,” Opt. Eng. (Bellingham) 29, 121–126 (1990).
[CrossRef]

Y. Bar-Shalom, H. M. Shertukde, K. R. Pattipati, “Use of measurement from an imaging sensor for precision target tracking,” IEEE Trans. Aerosp. Electron. Syst. 25, 863–872 (1989).
[CrossRef]

Bobek, M. P.

S. I. Green, M. P. Bobek, “Bit error rate testing of quadrant photodetectors,” in Space Sensing, Communications, and Networking, M. Ross, R. J. Temkin, eds. Proc. SPIE1059, 137–145 (1989).
[CrossRef]

Casey, W. L.

S. G. Lambert, W. L. Casey, Laser Communication in Space, (Artech House, Norwood, Mass.1995), pp. 179–195.

Chen, C. C.

C. C. Chen, C. S. Gardner, “Impact of random pointing and tracking errors on the design of coherent and incoherent optical intersatellite communication links,” IEEE Trans. Commun. 37, 252–260 (1989).
[CrossRef]

C. C. Chen, H. Ansari, J. R. Lesh, “Precision beam pointing for laser communication system using a CCD based tracker,” in Space Guidance, Control, and Tracking, G. E. Sevaston, F. Wade, eds. Proc. SPIE1949, 15–24 (1993).
[CrossRef]

Churnside, J. H.

Collins, P.

S. Dyne, P. Collins, D. Tunbridge, “Satellite mechanical health monitoring,” Proceedings of the IEE Colloquium on Advance Vibration Measurements, Techniques and Instruments for the Early Prediction of Failure (Institute of Electrical Engineers, London, 1993), p. 4/1–8.

Cooper, A. W.

E. C. Crittenden, A. W. Cooper, E. A. Milne, G. W. Rodeback, S. H. Kalmbach, R. L. Armstead, “Effects of turbulence on imaging through the atmosphere,” in Optical Properties of the Atmosphere, R. C. Sepucha, ed. Proc. SPIE142, 130–134 (1978).
[CrossRef]

Crittenden, E. C.

E. C. Crittenden, A. W. Cooper, E. A. Milne, G. W. Rodeback, S. H. Kalmbach, R. L. Armstead, “Effects of turbulence on imaging through the atmosphere,” in Optical Properties of the Atmosphere, R. C. Sepucha, ed. Proc. SPIE142, 130–134 (1978).
[CrossRef]

Dyne, S.

S. Dyne, P. Collins, D. Tunbridge, “Satellite mechanical health monitoring,” Proceedings of the IEE Colloquium on Advance Vibration Measurements, Techniques and Instruments for the Early Prediction of Failure (Institute of Electrical Engineers, London, 1993), p. 4/1–8.

Fante, R. L.

R. L. Fante, “Electric beam propagation in turbulent media,” Proc. IEEE 63, 1669–1688 (1975).
[CrossRef]

Gagliardi, R. M.

R. M. Gagliardi, S. Karp, Optical Communication, 2nd ed., (Wiley, New York, 1995), pp. 201–206, 305–340.

Gardner, C. S.

C. C. Chen, C. S. Gardner, “Impact of random pointing and tracking errors on the design of coherent and incoherent optical intersatellite communication links,” IEEE Trans. Commun. 37, 252–260 (1989).
[CrossRef]

Green, S. I.

S. I. Green, M. P. Bobek, “Bit error rate testing of quadrant photodetectors,” in Space Sensing, Communications, and Networking, M. Ross, R. J. Temkin, eds. Proc. SPIE1059, 137–145 (1989).
[CrossRef]

Jones, R. A.

D. R. Van Rheeden, R. A. Jones, “Noise effects on centroid tracker aim point estimation,” IEEE Trans. Aerosp. Electron. Syst. 24, 177–185 (1988).
[CrossRef]

Kalmbach, S. H.

E. C. Crittenden, A. W. Cooper, E. A. Milne, G. W. Rodeback, S. H. Kalmbach, R. L. Armstead, “Effects of turbulence on imaging through the atmosphere,” in Optical Properties of the Atmosphere, R. C. Sepucha, ed. Proc. SPIE142, 130–134 (1978).
[CrossRef]

Karp, S.

R. M. Gagliardi, S. Karp, Optical Communication, 2nd ed., (Wiley, New York, 1995), pp. 201–206, 305–340.

Kopeika, N. S.

S. Arnon, N. S. Kopeika, “The performance limitations of free space optical communication satellite networks due to vibrations—analog case,” Opt. Eng. (Bellingham) 36, 175–182 (1997).
[CrossRef]

S. Arnon, S. Rotman, N. S. Kopeika, “Beamwidth and transmitter power adaptive to tracking system performance for free-space optical communication,” Appl. Opt. 36, 6095–6101 (1997).
[CrossRef] [PubMed]

D. Sadot, N. S. Kopeika, “Forecasting optical turbulence strength on the basis of macroscale meteorology and aerosols: models and validation,” Opt. Eng. (Bellingham) 31, 200–212 (1991).
[CrossRef]

S. Arnon, S. Rotman, N. S. Kopeika, “Optimum transmitter optics aperture for free space satellite optical communication as a function of tracking system performance,” IEEE Trans. Aerosp. Electron. Syst. (to be published).

S. Arnon, S. Rotman, N. S. Kopeika, “The performance limitations of free space optical communication satellite networks due to vibrations—digital case,” Opt. Eng. (Bellingham) (to be published).

Lambert, S. G.

S. G. Lambert, W. L. Casey, Laser Communication in Space, (Artech House, Norwood, Mass.1995), pp. 179–195.

Lesh, J. R.

C. C. Chen, H. Ansari, J. R. Lesh, “Precision beam pointing for laser communication system using a CCD based tracker,” in Space Guidance, Control, and Tracking, G. E. Sevaston, F. Wade, eds. Proc. SPIE1949, 15–24 (1993).
[CrossRef]

Marrs, K. L.

G. S. Mecherle, K. L. Marrs, “Description and results of satellite laser communication/tracking simulation,” in Proceedings of the 1994 IEEE Aerospace Applications Conference (IEEE, Piscataway, N.J., 1994), pp. 87–101.

Mecherle, G. S.

G. S. Mecherle, K. L. Marrs, “Description and results of satellite laser communication/tracking simulation,” in Proceedings of the 1994 IEEE Aerospace Applications Conference (IEEE, Piscataway, N.J., 1994), pp. 87–101.

Milne, E. A.

E. C. Crittenden, A. W. Cooper, E. A. Milne, G. W. Rodeback, S. H. Kalmbach, R. L. Armstead, “Effects of turbulence on imaging through the atmosphere,” in Optical Properties of the Atmosphere, R. C. Sepucha, ed. Proc. SPIE142, 130–134 (1978).
[CrossRef]

Pattipati, K. R.

Y. Bar-Shalom, H. M. Shertukde, K. R. Pattipati, “Precision target tracking for small extended objects,” Opt. Eng. (Bellingham) 29, 121–126 (1990).
[CrossRef]

Y. Bar-Shalom, H. M. Shertukde, K. R. Pattipati, “Use of measurement from an imaging sensor for precision target tracking,” IEEE Trans. Aerosp. Electron. Syst. 25, 863–872 (1989).
[CrossRef]

Rodeback, G. W.

E. C. Crittenden, A. W. Cooper, E. A. Milne, G. W. Rodeback, S. H. Kalmbach, R. L. Armstead, “Effects of turbulence on imaging through the atmosphere,” in Optical Properties of the Atmosphere, R. C. Sepucha, ed. Proc. SPIE142, 130–134 (1978).
[CrossRef]

Rotman, S.

S. Arnon, S. Rotman, N. S. Kopeika, “Beamwidth and transmitter power adaptive to tracking system performance for free-space optical communication,” Appl. Opt. 36, 6095–6101 (1997).
[CrossRef] [PubMed]

S. Arnon, S. Rotman, N. S. Kopeika, “Optimum transmitter optics aperture for free space satellite optical communication as a function of tracking system performance,” IEEE Trans. Aerosp. Electron. Syst. (to be published).

S. Arnon, S. Rotman, N. S. Kopeika, “The performance limitations of free space optical communication satellite networks due to vibrations—digital case,” Opt. Eng. (Bellingham) (to be published).

Sadot, D.

D. Sadot, N. S. Kopeika, “Forecasting optical turbulence strength on the basis of macroscale meteorology and aerosols: models and validation,” Opt. Eng. (Bellingham) 31, 200–212 (1991).
[CrossRef]

Shertukde, H. M.

Y. Bar-Shalom, H. M. Shertukde, K. R. Pattipati, “Precision target tracking for small extended objects,” Opt. Eng. (Bellingham) 29, 121–126 (1990).
[CrossRef]

Y. Bar-Shalom, H. M. Shertukde, K. R. Pattipati, “Use of measurement from an imaging sensor for precision target tracking,” IEEE Trans. Aerosp. Electron. Syst. 25, 863–872 (1989).
[CrossRef]

Tunbridge, D.

S. Dyne, P. Collins, D. Tunbridge, “Satellite mechanical health monitoring,” Proceedings of the IEE Colloquium on Advance Vibration Measurements, Techniques and Instruments for the Early Prediction of Failure (Institute of Electrical Engineers, London, 1993), p. 4/1–8.

Tunbridge, D. E. L.

M. Witting, L. van Holtz, D. E. L. Tunbridge, H. C. Vermeulen, “In orbit measurements of microaccelerations of ESA’s communication satellite OLYMPUS,” in Free Space Laser Communication Technologies II, O. L. Begley, B. D. Seery, eds. Proc. SPIE1218, 205–214 (1990).
[CrossRef]

van Holtz, L.

M. Witting, L. van Holtz, D. E. L. Tunbridge, H. C. Vermeulen, “In orbit measurements of microaccelerations of ESA’s communication satellite OLYMPUS,” in Free Space Laser Communication Technologies II, O. L. Begley, B. D. Seery, eds. Proc. SPIE1218, 205–214 (1990).
[CrossRef]

Van Rheeden, D. R.

D. R. Van Rheeden, R. A. Jones, “Noise effects on centroid tracker aim point estimation,” IEEE Trans. Aerosp. Electron. Syst. 24, 177–185 (1988).
[CrossRef]

Vermeulen, H. C.

M. Witting, L. van Holtz, D. E. L. Tunbridge, H. C. Vermeulen, “In orbit measurements of microaccelerations of ESA’s communication satellite OLYMPUS,” in Free Space Laser Communication Technologies II, O. L. Begley, B. D. Seery, eds. Proc. SPIE1218, 205–214 (1990).
[CrossRef]

Witting, M.

M. Witting, L. van Holtz, D. E. L. Tunbridge, H. C. Vermeulen, “In orbit measurements of microaccelerations of ESA’s communication satellite OLYMPUS,” in Free Space Laser Communication Technologies II, O. L. Begley, B. D. Seery, eds. Proc. SPIE1218, 205–214 (1990).
[CrossRef]

Yariv, A.

A. Yariv, Optical Electronics, 3rd ed. (Holt, Rinehart & Winston, New York, 1985), pp. 306–400.

Appl. Opt. (1)

IEEE Trans. Aerosp. Electron. Syst. (2)

D. R. Van Rheeden, R. A. Jones, “Noise effects on centroid tracker aim point estimation,” IEEE Trans. Aerosp. Electron. Syst. 24, 177–185 (1988).
[CrossRef]

Y. Bar-Shalom, H. M. Shertukde, K. R. Pattipati, “Use of measurement from an imaging sensor for precision target tracking,” IEEE Trans. Aerosp. Electron. Syst. 25, 863–872 (1989).
[CrossRef]

IEEE Trans. Commun. (1)

C. C. Chen, C. S. Gardner, “Impact of random pointing and tracking errors on the design of coherent and incoherent optical intersatellite communication links,” IEEE Trans. Commun. 37, 252–260 (1989).
[CrossRef]

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

Opt. Eng. (Bellingham) (3)

Y. Bar-Shalom, H. M. Shertukde, K. R. Pattipati, “Precision target tracking for small extended objects,” Opt. Eng. (Bellingham) 29, 121–126 (1990).
[CrossRef]

D. Sadot, N. S. Kopeika, “Forecasting optical turbulence strength on the basis of macroscale meteorology and aerosols: models and validation,” Opt. Eng. (Bellingham) 31, 200–212 (1991).
[CrossRef]

S. Arnon, N. S. Kopeika, “The performance limitations of free space optical communication satellite networks due to vibrations—analog case,” Opt. Eng. (Bellingham) 36, 175–182 (1997).
[CrossRef]

Proc. IEEE (1)

R. L. Fante, “Electric beam propagation in turbulent media,” Proc. IEEE 63, 1669–1688 (1975).
[CrossRef]

Other (12)

E. C. Crittenden, A. W. Cooper, E. A. Milne, G. W. Rodeback, S. H. Kalmbach, R. L. Armstead, “Effects of turbulence on imaging through the atmosphere,” in Optical Properties of the Atmosphere, R. C. Sepucha, ed. Proc. SPIE142, 130–134 (1978).
[CrossRef]

S. Arnon, S. Rotman, N. S. Kopeika, “The performance limitations of free space optical communication satellite networks due to vibrations—digital case,” Opt. Eng. (Bellingham) (to be published).

R. M. Gagliardi, S. Karp, Optical Communication, 2nd ed., (Wiley, New York, 1995), pp. 201–206, 305–340.

A. Yariv, Optical Electronics, 3rd ed. (Holt, Rinehart & Winston, New York, 1985), pp. 306–400.

H. Kressel, ed., Semiconductor Devices for Optical Communication, Vol. 38 of Topics in Applied Physics (Springer-Verlag, Berlin, 1982), pp. 159–263.

G. S. Mecherle, K. L. Marrs, “Description and results of satellite laser communication/tracking simulation,” in Proceedings of the 1994 IEEE Aerospace Applications Conference (IEEE, Piscataway, N.J., 1994), pp. 87–101.

S. I. Green, M. P. Bobek, “Bit error rate testing of quadrant photodetectors,” in Space Sensing, Communications, and Networking, M. Ross, R. J. Temkin, eds. Proc. SPIE1059, 137–145 (1989).
[CrossRef]

S. G. Lambert, W. L. Casey, Laser Communication in Space, (Artech House, Norwood, Mass.1995), pp. 179–195.

C. C. Chen, H. Ansari, J. R. Lesh, “Precision beam pointing for laser communication system using a CCD based tracker,” in Space Guidance, Control, and Tracking, G. E. Sevaston, F. Wade, eds. Proc. SPIE1949, 15–24 (1993).
[CrossRef]

S. Arnon, S. Rotman, N. S. Kopeika, “Optimum transmitter optics aperture for free space satellite optical communication as a function of tracking system performance,” IEEE Trans. Aerosp. Electron. Syst. (to be published).

M. Witting, L. van Holtz, D. E. L. Tunbridge, H. C. Vermeulen, “In orbit measurements of microaccelerations of ESA’s communication satellite OLYMPUS,” in Free Space Laser Communication Technologies II, O. L. Begley, B. D. Seery, eds. Proc. SPIE1218, 205–214 (1990).
[CrossRef]

S. Dyne, P. Collins, D. Tunbridge, “Satellite mechanical health monitoring,” Proceedings of the IEE Colloquium on Advance Vibration Measurements, Techniques and Instruments for the Early Prediction of Failure (Institute of Electrical Engineers, London, 1993), p. 4/1–8.

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Figures (6)

Fig. 1
Fig. 1

as a function of M.

Fig. 2
Fig. 2

Detection matrix for tracking and communication.

Fig. 3
Fig. 3

Spreading of optical laser power as a function of position on the detection matrix.

Fig. 4
Fig. 4

Gain Ai,j as a function of position on the detector matrix for σp0i,j=1 for all i, j=164.

Fig. 5
Fig. 5

Digital signal-to-noise ratio Q as function of noise σp0i,j. Solid curve, adaptive system Q; dashed curve, standard system Q.

Fig. 6
Fig. 6

BER as a function of noise σp0i,j. Solid curve, adaptive system BER; dashed curve, standard system BER.

Equations (40)

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

Ic=i=1nj=1mSijii=1nj=1mSij,
μp1i,j(Ic, Jc)=R1[spread (i-Ic, j-Jc)+P0] i=1n, j=1m,
σp1i,j(Ic, Jc)={R2[spread (i-Ic, j-Jc)+P0]+σpi,j2}1/2 i=1n, j=1m,
μ1=i=1nj=1mAi,jμp1i,j,
σ1=i=1nj=1m(Ai,jσp1i,j)21/2.
BER(Ic, Jc)=MA1-α(M)- exp-(x-μ1)22σ12×-xexp-(y-μ0)22σ02dyM-1dx,
MA=M2(M-1),
α(M)=12πσ1(2πσ0)M-1.
v=y-μ02σ0,u=x-μ12σ1.
BER(Ic, Jc)=MA1--f(u)×-βf(v)dvM-1du,
f(t)=1πexp(-t2)
β=2σ1u+μ1-μ02·σ0.
BER(Ic, Jc)=MA[1-(P1+P2+P3)],
Pi=LLULf(u)-f(v)dvM-1du,
μ1-μ02σ1.
-xminf(v)dv>0.5.
0.5>1-z3z-11/M-1.
z=erf()>1+(1/2)M-11+3(1/2)M-1,
erf(x)=2π0x exp(-y2)dy.
>inverf1+(1/2)M-11+3(1/2)M-1,
P2-f(u)-μ1-μ0/2σ0f(v)dvM-1du.
Q=μ1-μ02σ0.
Q=i=1nj=1mAi,j(μp1i,j-μp0i,j)2i=1nj=1m(Ai,jσp0i,j)21/2.
Hi,j=2Ai,jσp0i,j,
Gi,j=μp1i,j-μp0i,j2σp0i,j.
Q2=i=1nj=1mHi,jGi,j2i=1nj=1mHi,j2.
i=1nj=1m(Hi,j)Gi,j2i=1nj=1m(Hi,j)2i=1nj=1m(Gi,j)2.
const.=c=Gi,j/Hi,j i,j.
Ai,j=μp1i,j-μp0i,j2σp0i,j2
Q=i=1nj=1nμp1i,j-μp0i,jσp0i,j21/2.
spread(i-Ic, j-Jc)=exp-(i-48)2+(j-44)22(10/2)2i=164,j=164.
P2>P1+P3
xminf(v)dv>1-z3z-11/M-1,
z=f(t)dt,
xmin=σ1σ0-2+μ1/σ1-μ0/σ12.
P2>z-xminf(v)dvM-1.
P1<1-z2-xminf(v)dvM-1.
P1<1-z2.
z-xminf(v)dvM-1>1-z2-xminf(v)dvM-1+1-z2.
-xminf(v)dv>1-z3z-11/M-1.

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