Abstract

A mathematical system model for a compact heterodyne-reception infrared radar is developed. This model incorporates the statistical effects of propagation through atmospheric turbulence, target speckle and glint, and heterodyne-reception shot noise. It is used to find the image signal-to-noise ratio of a matched-filter envelope–detector receiver and the target detection probability of the optimum likelihood ratio processor. For realistic parameter values it is shown that turbulence-induced beam spreading and coherence loss may be neglected. Target speckle and atmospheric scintillation, however, present serious limitations on single-frame imaging and target-detection performance. Experimental turbulence strength measurements are reviewed, and selected results are used in sample performance calculations for a realistic infrared radar.

© 1981 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. P. Modica, H. Kleiman, “Statistics of Global IR Atmospheric Transmission,” Project Report TT-7, Lincoln Laboratory, MIT (Mar.1976).
  2. R. J. Becherer, “System Design Study for Infrared Airborne Radar (IRAR),” Technical Note 1977-29, Lincoln Laboratory, MIT (Oct.1977).
  3. R. J. Hull, S. Marcus, “A Tactical 10.6 μm Imaging Radar,” in Proceedings, 1978 National Aerospace and Electronics Conference (IEEE, Dayton, Ohio, 1978), p. 662.
  4. R. C. Harney, “Conceptual Design of a Multifunction Infrared Radar for the Tactical Aircraft Ground Attack Scenario,” Project Report TST-25, Lincoln Laboratory, MIT (Aug.1978).
  5. R. C. Harney, R. J. Hull, “Compact Infrared Radar Technology,” Proc. Soc. Photo-Opt. Instrum. Eng. 227, 162 (1980).
  6. R. C. Harney, “Design Considerations for the Infrared Airborne Radar (IRAR) MTI Subsystem,” Project Report TST-26, Lincoln Laboratory, MIT (July1980).
  7. R. C. Harney, “Infrared Airborne Radar,” EASCON ’80 Record (IEEE, Washington, D.C., 1980), pp. 462–471.
  8. The analysis will also apply to a cw scanning transmitter for which the pixel dwell time is short compared to the atmospheric coherence time.
  9. R. M. Gagliardi, S. Karp, Optical Communications (Wiley, New York, 1976), Chap. 6.
  10. By assuming hL to be time independent we are saying that both tp and 2L/c are less than the atmospheric coherence time (which is typically 1 msec).
  11. R. L. Fante, Proc. IEEE 63, 1669 (1975).
    [Crossref]
  12. J. H. Shapiro, “Imaging and Optical Communication Through Atmospheric Turbulence,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, Ed. (Springer, Berlin, 1978).
    [Crossref]
  13. J. H. Shapiro, J. Opt. Soc. Am. 61, 492 (1971).
    [Crossref]
  14. J. H. Shapiro, IEEE Trans. Commun. Technol. COM-19, 410 (1971).
    [Crossref]
  15. J. H. Shapiro, J. Opt. Soc. Am. 65, 65 (1975).
    [Crossref]
  16. J. H. Shapiro, J. Opt. Soc. Am. 66, 460 (1976).
    [Crossref]
  17. We are assuming, for simplicity, a stationary target. A moving target will impart a Doppler frequency shift to the reflected field. Our analysis can easily be extended to include this case.
  18. A. E. Siegman, Proc. IEEE 54, 1350 (1966).
    [Crossref]
  19. Target Signature Analysis Center: Data Compilation, Eleventh Supplement: Vol. 1, Bidirectional Reflectance: Definition, Discussion, and Utilization; and Vol. 2, Bidirectional Reflectance: Graphic Data, AFAL-TR-72-226 (1972).
  20. R. S. Lawrence, J. W. Strohbehn, Proc. IEEE 58, 1523 (1970).
    [Crossref]
  21. E. Brookner, IEEE Trans. Commun. Technol. COM-18, 396 (1970).
    [Crossref]
  22. J. W. Strohbehn, Ed., Laser Beam Propagation in the Atmosphere (Springer, Berlin, 1978).
    [Crossref]
  23. H. S. Lin, “Communication Model for the Turbulent Atmosphere,” Ph.D. Thesis, Case Western Reserve U., Cleveland, Ohio, Aug.1973.
  24. R. F. Lutomirski, H. T. Yura, Appl. Opt. 10, 1652 (1971).
    [Crossref] [PubMed]
  25. H. T. Yura, Appl. Opt. 11, 1399 (1972).
    [Crossref] [PubMed]
  26. A. Kon, V. Feizulin, Radiophys. Quantum Electon. 13, 51 (1970).
    [Crossref]
  27. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chaps. 3 and 4.
  28. We have included an absorption term in (2.3) that is not usually shown in the turbulence literature. This term must be included for a realistic treatment of 10.6-μm propagation, as clear weather absorption at this wavelength is not entirely negligible. Moreover, in bad weather conditions a large portion of the extinction encountered at 10.6 μm is due to absorption. Hence the absorption term in (2.3) will be used in what follows to account for weather-dependent attenuation.
  29. Strictly speaking we should also require d to be less than the log-amplitude coherence length. When d ≪ ρ0, however, the approximations hold as stated.
  30. The ϕ(ρ¯′,0¯) term in Eqs. (2.12) and (2.14) is a target plane phase perturbation which can affect the directionality of a glint target return (see below).
  31. J. C. Dainty, Ed., Laser Speckle and Related Phenomena (Springer, Berlin, 1975).
  32. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, Oxford, 1963).
  33. Special issue on Speckle in Optics, J. Opt. Soc. Am. 66, 1145–1313 (1976).
  34. R. L. Mitchell, Proc. IEEE 62, 754 (1974).
    [Crossref]
  35. M. Skolnik, Ed., Radar Handbook (McGraw-Hill, New York, 1970).
  36. E. Brookner, Ed., Radar Technology (Artech, Dedham, 1977).
  37. It would be somewhat more general to allow for locally ellipsoidal surfaces. We shall not do so as (3.2) is sufficient for the physical arguments needed here.
  38. J. W. Goodman, Proc. IEEE 53, 1688 (1965).
    [Crossref]
  39. D. L. Fried, J. Opt. Soc. Am. 66, 1150 (1976).
    [Crossref]
  40. H. L. Van Trees, Detection, Estimation and Modulation Theory, Part 3 (Wiley, New York, 1971), Chap. 13.
  41. J. H. Shapiro, “Imaging and Target Detection with a Heterodyne-Reception Optical Radar,” Project Report TST-24, Lincoln Laboratory, MIT (Oct.1978).
  42. R. L. Mitchell, J. Opt. Soc. Am. 58, 1267 (1968).
    [Crossref]
  43. B. K. Levitt, “Detector Statistics for Optical Communication Through the Turbulent Atmosphere,” Quarterly Progress Report 99, Research Lab. Electron., MIT (Oct.1970), pp. 114–123.
  44. D. L. Fried, J. Opt. Soc. Am. 57, 169 (1967).
    [Crossref]
  45. H. L. Van Trees, Detection, Estimation, and Modulation Theory, Part I (Wiley, New York, 1968), Chap. 2.
  46. H. L. Van Trees, Detection, Estimation, and Modulation Theory, Part I (Wiley, New York, 1968), Chap. 4.
  47. J. Marcum, IEEE Trans. Inf. Theory IT-6, 59 (1960).
    [Crossref]
  48. J. Marcum, “Table of Q-Functions,” Rand Corp. Report RM-339 (1Jan.1950).
  49. B. A. Capron, R. C. Harney, J. H. Shapiro, “Turbulence Effects on the Receiver Operating Characteristics of a Heterodyne-Reception Optical Radar,” Project Report TST-33, Lincoln Laboratory, MIT (July1979).
  50. S. F. Clifford, “The Classical Theory of Wave Propagation in a Turbulent Medium,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, Ed. (Springer, Berlin, 1978).
    [Crossref]
  51. R. E. Hufnagel, “Propagation Through Atmospheric Turbulence,” in The Infrared Handbook, W. L. Wolfe, G. J. Zissis, Eds. (Environmental Research Institute of Michigan, Ann Arbor, 1979), Chap. 6.
  52. M. A. Kallistratova, D. F. Timanovskiy, Atmos. Ocean Phys. 7, 46 (1971).
  53. W. D. Neff, “Quantitative Evaluation of Acoustic Echoes from the Planetary Boundary Layer,” Technical Report ERL 322-WPL 38, National Oceanic and Atmospheric Administration (June1975).
  54. J. L. Spencer, “Long-Term Statistics of Atmospheric Turbulence Near the Ground,” Report RADC-TR-78-182, Rome Air Development Center (Aug.1978).
  55. J. C. Wyngaard, Y. Izumi, S. A. Collins, J. Opt. Soc. Am. 61, 1646 (1971).
    [Crossref]
  56. A. W. Cooper, E. C. Crittenden, A. F. Schroeder, in Digest of Topical Meeting on Optical Propagation Through Turbulence (Optical Society of America, Washington, D.C., 1974), paper WB4.

1980 (1)

R. C. Harney, R. J. Hull, “Compact Infrared Radar Technology,” Proc. Soc. Photo-Opt. Instrum. Eng. 227, 162 (1980).

1976 (3)

1975 (2)

1974 (1)

R. L. Mitchell, Proc. IEEE 62, 754 (1974).
[Crossref]

1972 (1)

1971 (5)

1970 (3)

R. S. Lawrence, J. W. Strohbehn, Proc. IEEE 58, 1523 (1970).
[Crossref]

E. Brookner, IEEE Trans. Commun. Technol. COM-18, 396 (1970).
[Crossref]

A. Kon, V. Feizulin, Radiophys. Quantum Electon. 13, 51 (1970).
[Crossref]

1968 (1)

1967 (1)

1966 (1)

A. E. Siegman, Proc. IEEE 54, 1350 (1966).
[Crossref]

1965 (1)

J. W. Goodman, Proc. IEEE 53, 1688 (1965).
[Crossref]

1960 (1)

J. Marcum, IEEE Trans. Inf. Theory IT-6, 59 (1960).
[Crossref]

Becherer, R. J.

R. J. Becherer, “System Design Study for Infrared Airborne Radar (IRAR),” Technical Note 1977-29, Lincoln Laboratory, MIT (Oct.1977).

Beckmann, P.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, Oxford, 1963).

Brookner, E.

E. Brookner, IEEE Trans. Commun. Technol. COM-18, 396 (1970).
[Crossref]

Capron, B. A.

B. A. Capron, R. C. Harney, J. H. Shapiro, “Turbulence Effects on the Receiver Operating Characteristics of a Heterodyne-Reception Optical Radar,” Project Report TST-33, Lincoln Laboratory, MIT (July1979).

Clifford, S. F.

S. F. Clifford, “The Classical Theory of Wave Propagation in a Turbulent Medium,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, Ed. (Springer, Berlin, 1978).
[Crossref]

Collins, S. A.

Cooper, A. W.

A. W. Cooper, E. C. Crittenden, A. F. Schroeder, in Digest of Topical Meeting on Optical Propagation Through Turbulence (Optical Society of America, Washington, D.C., 1974), paper WB4.

Crittenden, E. C.

A. W. Cooper, E. C. Crittenden, A. F. Schroeder, in Digest of Topical Meeting on Optical Propagation Through Turbulence (Optical Society of America, Washington, D.C., 1974), paper WB4.

Fante, R. L.

R. L. Fante, Proc. IEEE 63, 1669 (1975).
[Crossref]

Feizulin, V.

A. Kon, V. Feizulin, Radiophys. Quantum Electon. 13, 51 (1970).
[Crossref]

Fried, D. L.

Gagliardi, R. M.

R. M. Gagliardi, S. Karp, Optical Communications (Wiley, New York, 1976), Chap. 6.

Goodman, J. W.

J. W. Goodman, Proc. IEEE 53, 1688 (1965).
[Crossref]

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chaps. 3 and 4.

Harney, R. C.

R. C. Harney, R. J. Hull, “Compact Infrared Radar Technology,” Proc. Soc. Photo-Opt. Instrum. Eng. 227, 162 (1980).

R. C. Harney, “Design Considerations for the Infrared Airborne Radar (IRAR) MTI Subsystem,” Project Report TST-26, Lincoln Laboratory, MIT (July1980).

R. C. Harney, “Infrared Airborne Radar,” EASCON ’80 Record (IEEE, Washington, D.C., 1980), pp. 462–471.

R. C. Harney, “Conceptual Design of a Multifunction Infrared Radar for the Tactical Aircraft Ground Attack Scenario,” Project Report TST-25, Lincoln Laboratory, MIT (Aug.1978).

B. A. Capron, R. C. Harney, J. H. Shapiro, “Turbulence Effects on the Receiver Operating Characteristics of a Heterodyne-Reception Optical Radar,” Project Report TST-33, Lincoln Laboratory, MIT (July1979).

Hufnagel, R. E.

R. E. Hufnagel, “Propagation Through Atmospheric Turbulence,” in The Infrared Handbook, W. L. Wolfe, G. J. Zissis, Eds. (Environmental Research Institute of Michigan, Ann Arbor, 1979), Chap. 6.

Hull, R. J.

R. C. Harney, R. J. Hull, “Compact Infrared Radar Technology,” Proc. Soc. Photo-Opt. Instrum. Eng. 227, 162 (1980).

R. J. Hull, S. Marcus, “A Tactical 10.6 μm Imaging Radar,” in Proceedings, 1978 National Aerospace and Electronics Conference (IEEE, Dayton, Ohio, 1978), p. 662.

Izumi, Y.

Kallistratova, M. A.

M. A. Kallistratova, D. F. Timanovskiy, Atmos. Ocean Phys. 7, 46 (1971).

Karp, S.

R. M. Gagliardi, S. Karp, Optical Communications (Wiley, New York, 1976), Chap. 6.

Kleiman, H.

A. P. Modica, H. Kleiman, “Statistics of Global IR Atmospheric Transmission,” Project Report TT-7, Lincoln Laboratory, MIT (Mar.1976).

Kon, A.

A. Kon, V. Feizulin, Radiophys. Quantum Electon. 13, 51 (1970).
[Crossref]

Lawrence, R. S.

R. S. Lawrence, J. W. Strohbehn, Proc. IEEE 58, 1523 (1970).
[Crossref]

Levitt, B. K.

B. K. Levitt, “Detector Statistics for Optical Communication Through the Turbulent Atmosphere,” Quarterly Progress Report 99, Research Lab. Electron., MIT (Oct.1970), pp. 114–123.

Lin, H. S.

H. S. Lin, “Communication Model for the Turbulent Atmosphere,” Ph.D. Thesis, Case Western Reserve U., Cleveland, Ohio, Aug.1973.

Lutomirski, R. F.

Marcum, J.

J. Marcum, IEEE Trans. Inf. Theory IT-6, 59 (1960).
[Crossref]

J. Marcum, “Table of Q-Functions,” Rand Corp. Report RM-339 (1Jan.1950).

Marcus, S.

R. J. Hull, S. Marcus, “A Tactical 10.6 μm Imaging Radar,” in Proceedings, 1978 National Aerospace and Electronics Conference (IEEE, Dayton, Ohio, 1978), p. 662.

Mitchell, R. L.

Modica, A. P.

A. P. Modica, H. Kleiman, “Statistics of Global IR Atmospheric Transmission,” Project Report TT-7, Lincoln Laboratory, MIT (Mar.1976).

Neff, W. D.

W. D. Neff, “Quantitative Evaluation of Acoustic Echoes from the Planetary Boundary Layer,” Technical Report ERL 322-WPL 38, National Oceanic and Atmospheric Administration (June1975).

Schroeder, A. F.

A. W. Cooper, E. C. Crittenden, A. F. Schroeder, in Digest of Topical Meeting on Optical Propagation Through Turbulence (Optical Society of America, Washington, D.C., 1974), paper WB4.

Shapiro, J. H.

J. H. Shapiro, J. Opt. Soc. Am. 66, 460 (1976).
[Crossref]

J. H. Shapiro, J. Opt. Soc. Am. 65, 65 (1975).
[Crossref]

J. H. Shapiro, J. Opt. Soc. Am. 61, 492 (1971).
[Crossref]

J. H. Shapiro, IEEE Trans. Commun. Technol. COM-19, 410 (1971).
[Crossref]

J. H. Shapiro, “Imaging and Target Detection with a Heterodyne-Reception Optical Radar,” Project Report TST-24, Lincoln Laboratory, MIT (Oct.1978).

B. A. Capron, R. C. Harney, J. H. Shapiro, “Turbulence Effects on the Receiver Operating Characteristics of a Heterodyne-Reception Optical Radar,” Project Report TST-33, Lincoln Laboratory, MIT (July1979).

J. H. Shapiro, “Imaging and Optical Communication Through Atmospheric Turbulence,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, Ed. (Springer, Berlin, 1978).
[Crossref]

Siegman, A. E.

A. E. Siegman, Proc. IEEE 54, 1350 (1966).
[Crossref]

Spencer, J. L.

J. L. Spencer, “Long-Term Statistics of Atmospheric Turbulence Near the Ground,” Report RADC-TR-78-182, Rome Air Development Center (Aug.1978).

Spizzichino, A.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, Oxford, 1963).

Strohbehn, J. W.

R. S. Lawrence, J. W. Strohbehn, Proc. IEEE 58, 1523 (1970).
[Crossref]

Timanovskiy, D. F.

M. A. Kallistratova, D. F. Timanovskiy, Atmos. Ocean Phys. 7, 46 (1971).

Van Trees, H. L.

H. L. Van Trees, Detection, Estimation, and Modulation Theory, Part I (Wiley, New York, 1968), Chap. 2.

H. L. Van Trees, Detection, Estimation, and Modulation Theory, Part I (Wiley, New York, 1968), Chap. 4.

H. L. Van Trees, Detection, Estimation and Modulation Theory, Part 3 (Wiley, New York, 1971), Chap. 13.

Wyngaard, J. C.

Yura, H. T.

Appl. Opt. (2)

Atmos. Ocean Phys. (1)

M. A. Kallistratova, D. F. Timanovskiy, Atmos. Ocean Phys. 7, 46 (1971).

IEEE Trans. Commun. Technol. (2)

E. Brookner, IEEE Trans. Commun. Technol. COM-18, 396 (1970).
[Crossref]

J. H. Shapiro, IEEE Trans. Commun. Technol. COM-19, 410 (1971).
[Crossref]

IEEE Trans. Inf. Theory (1)

J. Marcum, IEEE Trans. Inf. Theory IT-6, 59 (1960).
[Crossref]

J. Opt. Soc. Am. (8)

Proc. IEEE (5)

R. L. Mitchell, Proc. IEEE 62, 754 (1974).
[Crossref]

R. S. Lawrence, J. W. Strohbehn, Proc. IEEE 58, 1523 (1970).
[Crossref]

R. L. Fante, Proc. IEEE 63, 1669 (1975).
[Crossref]

A. E. Siegman, Proc. IEEE 54, 1350 (1966).
[Crossref]

J. W. Goodman, Proc. IEEE 53, 1688 (1965).
[Crossref]

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

R. C. Harney, R. J. Hull, “Compact Infrared Radar Technology,” Proc. Soc. Photo-Opt. Instrum. Eng. 227, 162 (1980).

Radiophys. Quantum Electon. (1)

A. Kon, V. Feizulin, Radiophys. Quantum Electon. 13, 51 (1970).
[Crossref]

Other (35)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chaps. 3 and 4.

We have included an absorption term in (2.3) that is not usually shown in the turbulence literature. This term must be included for a realistic treatment of 10.6-μm propagation, as clear weather absorption at this wavelength is not entirely negligible. Moreover, in bad weather conditions a large portion of the extinction encountered at 10.6 μm is due to absorption. Hence the absorption term in (2.3) will be used in what follows to account for weather-dependent attenuation.

Strictly speaking we should also require d to be less than the log-amplitude coherence length. When d ≪ ρ0, however, the approximations hold as stated.

The ϕ(ρ¯′,0¯) term in Eqs. (2.12) and (2.14) is a target plane phase perturbation which can affect the directionality of a glint target return (see below).

J. C. Dainty, Ed., Laser Speckle and Related Phenomena (Springer, Berlin, 1975).

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, Oxford, 1963).

J. W. Strohbehn, Ed., Laser Beam Propagation in the Atmosphere (Springer, Berlin, 1978).
[Crossref]

H. S. Lin, “Communication Model for the Turbulent Atmosphere,” Ph.D. Thesis, Case Western Reserve U., Cleveland, Ohio, Aug.1973.

M. Skolnik, Ed., Radar Handbook (McGraw-Hill, New York, 1970).

E. Brookner, Ed., Radar Technology (Artech, Dedham, 1977).

It would be somewhat more general to allow for locally ellipsoidal surfaces. We shall not do so as (3.2) is sufficient for the physical arguments needed here.

R. C. Harney, “Design Considerations for the Infrared Airborne Radar (IRAR) MTI Subsystem,” Project Report TST-26, Lincoln Laboratory, MIT (July1980).

R. C. Harney, “Infrared Airborne Radar,” EASCON ’80 Record (IEEE, Washington, D.C., 1980), pp. 462–471.

The analysis will also apply to a cw scanning transmitter for which the pixel dwell time is short compared to the atmospheric coherence time.

R. M. Gagliardi, S. Karp, Optical Communications (Wiley, New York, 1976), Chap. 6.

By assuming hL to be time independent we are saying that both tp and 2L/c are less than the atmospheric coherence time (which is typically 1 msec).

A. P. Modica, H. Kleiman, “Statistics of Global IR Atmospheric Transmission,” Project Report TT-7, Lincoln Laboratory, MIT (Mar.1976).

R. J. Becherer, “System Design Study for Infrared Airborne Radar (IRAR),” Technical Note 1977-29, Lincoln Laboratory, MIT (Oct.1977).

R. J. Hull, S. Marcus, “A Tactical 10.6 μm Imaging Radar,” in Proceedings, 1978 National Aerospace and Electronics Conference (IEEE, Dayton, Ohio, 1978), p. 662.

R. C. Harney, “Conceptual Design of a Multifunction Infrared Radar for the Tactical Aircraft Ground Attack Scenario,” Project Report TST-25, Lincoln Laboratory, MIT (Aug.1978).

Target Signature Analysis Center: Data Compilation, Eleventh Supplement: Vol. 1, Bidirectional Reflectance: Definition, Discussion, and Utilization; and Vol. 2, Bidirectional Reflectance: Graphic Data, AFAL-TR-72-226 (1972).

J. H. Shapiro, “Imaging and Optical Communication Through Atmospheric Turbulence,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, Ed. (Springer, Berlin, 1978).
[Crossref]

We are assuming, for simplicity, a stationary target. A moving target will impart a Doppler frequency shift to the reflected field. Our analysis can easily be extended to include this case.

A. W. Cooper, E. C. Crittenden, A. F. Schroeder, in Digest of Topical Meeting on Optical Propagation Through Turbulence (Optical Society of America, Washington, D.C., 1974), paper WB4.

H. L. Van Trees, Detection, Estimation, and Modulation Theory, Part I (Wiley, New York, 1968), Chap. 2.

H. L. Van Trees, Detection, Estimation, and Modulation Theory, Part I (Wiley, New York, 1968), Chap. 4.

W. D. Neff, “Quantitative Evaluation of Acoustic Echoes from the Planetary Boundary Layer,” Technical Report ERL 322-WPL 38, National Oceanic and Atmospheric Administration (June1975).

J. L. Spencer, “Long-Term Statistics of Atmospheric Turbulence Near the Ground,” Report RADC-TR-78-182, Rome Air Development Center (Aug.1978).

B. K. Levitt, “Detector Statistics for Optical Communication Through the Turbulent Atmosphere,” Quarterly Progress Report 99, Research Lab. Electron., MIT (Oct.1970), pp. 114–123.

H. L. Van Trees, Detection, Estimation and Modulation Theory, Part 3 (Wiley, New York, 1971), Chap. 13.

J. H. Shapiro, “Imaging and Target Detection with a Heterodyne-Reception Optical Radar,” Project Report TST-24, Lincoln Laboratory, MIT (Oct.1978).

J. Marcum, “Table of Q-Functions,” Rand Corp. Report RM-339 (1Jan.1950).

B. A. Capron, R. C. Harney, J. H. Shapiro, “Turbulence Effects on the Receiver Operating Characteristics of a Heterodyne-Reception Optical Radar,” Project Report TST-33, Lincoln Laboratory, MIT (July1979).

S. F. Clifford, “The Classical Theory of Wave Propagation in a Turbulent Medium,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, Ed. (Springer, Berlin, 1978).
[Crossref]

R. E. Hufnagel, “Propagation Through Atmospheric Turbulence,” in The Infrared Handbook, W. L. Wolfe, G. J. Zissis, Eds. (Environmental Research Institute of Michigan, Ann Arbor, 1979), Chap. 6.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (27)

Fig. 1
Fig. 1

Coherent optical radar configuraion: (a) transmitter-to-target path, (b) target-to-receiver path.

Fig. 2
Fig. 2

Turbulence field coherence length ρ0 vs propagation path length L for conditions of weak turbulence ( C n 2 = 5 × 10 - 16 m - 2 / 3), moderate turbulence ( C n 2 = 10 - 14 m - 2 / 3), and strong turbulence ( C n 2 = 5 × 10 - 13 m - 2 / 3); 10.6-μm wavelength has been assumed throughout.

Fig. 3
Fig. 3

Log-amplitude variance σ χ 2 vs propagation path length L for weak ( C n 2 = 5 × 10 - 16 m - 2 / 3), moderate ( C n 2 = 10 - 14 m - 2 / 3), and strong ( C n 2 = 5 × 10 - 13 m - 2 / 3) turbulence; 10.6-μm wavelength is assumed, and the weak perturbation theory is employed.

Fig. 4
Fig. 4

Geometry of the planar reflection model for a polished spherical reflector; the reflected field satisfies Eq. (3.1) where for the surface shown Rc > 0.

Fig. 5
Fig. 5

Geometry for defining bidirectional reflectance ρ ( λ ; f ¯ i ; f ¯ r ). The target plane field is chosen to be a plane wave of wavelength λ propagating in the direction of the unit vector īi f ¯ i is the projection of īi on the z = L plane), and the radiance of the reflected field is measured in the direction of the unit vector īr f ¯ r is the projection of īr on the z = L plane).

Fig. 6
Fig. 6

Universal curve for normalized image signal-to-noise ratio vs normalized carrier-to-noise ratio, Eq. (4.13).

Fig. 7
Fig. 7

Atmospheric propagation/glint target saturation signal-to-noise ratio SNRgsat vs log-amplitude variance σ χ 2.

Fig. 8
Fig. 8

Atmospheric propagation/speckle target saturation signal-to-noise ratio SNRssat vs aperture averaged log-amplitude variance σ2.

Fig. 9
Fig. 9

Aperture averaged log-amplitude variance σ2 vs log-amplitude variance σ χ 2 for resolved targets with various transmitter Fresnel numbers.

Fig. 10
Fig. 10

Simulated glint target images: (a) original image; (b) σ χ 2 = 10 - 4; (c) σ χ 2 = 5 × 10 - 3; (d) σ χ 2 = 0.02; (e) σ χ 2 = 0.1; (f) σ χ 2 = 0.5.

Fig. 11
Fig. 11

Simulated speckle target images: (a) original image; (b) speckle image (σ2 = 0); (C) σ2 = 0.01; (d) σ2 = 0.1; (e) σ2 = 0.5; (f) σ2 = 2.0.

Fig. 12
Fig. 12

Glint target receiver operating characteristics for PF = 10−7 and various scintillation levels.

Fig. 13
Fig. 13

Speckle target receiver operating characteristics for PF = 10−7 and various scintillation levels.

Fig. 14
Fig. 14

Log-amplitude variance σ χ 2 vs path length L for a horizontal path at a height of 3 m assuming the RADC data for Type I turbulence obtained at 0100 EST and 1100 EST during the month of February and partial correlation along the optical path with correlation length of 100 m. For each hour curves for 〈 σ χ 2 〉 and σ χ 2 ± [ var ( σ χ 2 ) ] 1 / 2 have been plotted.

Fig. 15
Fig. 15

Geometry involved in slant path propagation calculations.

Fig. 16
Fig. 16

Log-amplitude variance 〈 σ χ 2 〉 vs slant path length L for daytime slant paths with target height h0 = 2 m and various transmitter heights assuming a typical daytime value of C n 2 ( 2 ) = 10 - 13 m - 2 / 3.

Fig. 17
Fig. 17

Log-amplitude variance 〈 σ χ 2 〉 vs slant path length L for nighttime slant paths with target height h0 = 2 m and various transmitter heights assuming a typical nighttime value of C n 2 ( 2 ) = 10 - 14 m - 2 / 3.

Fig. 18
Fig. 18

No turbulence carrier-to-noise ratio vs path length L for various absorption values; Eq. (7.1) is used with the parameter values from Table II.

Fig. 19
Fig. 19

Glint target (long dashed curves) and speckle target (short dashed curves) atmospheric propagation saturation signal-to-noise for a 2-m height horizontal path vs path length L for typical daytime ( C n 2 = 10 - 13 m - 2 / 3) and nighttime ( C n 2 = 10 - 14 m - 2 / 3) turbulence strengths.

Fig. 20
Fig. 20

Glint target (long dashed curves) and speckle target (short dashed curves) atmospheric propagation saturation signal-to-noise ratio for a slant path with 30-m transmitter height and 2-m target height vs slant path length L for typical 2-m height daytime ( C n 2 = 10 - 13 m - 2 / 3) and nighttime ( C n 2 = 10 - 14 m - 2 / 3) turbulence strengths.

Fig. 21
Fig. 21

Additional carrier-to-noise ratio ΔCNR required due to turbulence vs log-amplitude variance σ χ 2 or σ2/4 for glint targets ( σ χ 2) and speckle targets (σ2/4) with PF = 10−7 and PD = 0.90 and 0.99.

Fig. 22
Fig. 22

Detection probability PD vs log-amplitude variance σ χ 2 for glint targets at PF = 10−7 and several values of the turbulence-free propagation glint target carrier-to-noise ratio CNR g 0.

Fig. 23
Fig. 23

Detection probability PD vs log-amplitude variance σ2/4 for speckle targets at PF = 10−7 and several values of the turbulent propagation speckle target carrier-to-noise ratio CNRs.

Fig. 24
Fig. 24

Carrier-to-noise ratio CNR required to achieve PD = 0.90 and PF = 10−7 vs path length L for a ground-based radar located at a height of 3 m searching for targets on the ground along a horizontal path. Curves are provided for both speckle (short dashed lines) and glint (long dashed lines) targets in daytime, nighttime, and turbulence-free propagation conditions. The solid curve is the system CNR assuming 0.5-dB/km atmospheric attenuation.

Fig. 25
Fig. 25

Carrier-to-noise ratio CNR required to achieve PD = 0.90 and PF = 10−7 vs slant path length L for an airborne radar located at a height of 30 m searching for targets on the ground. Curves are provided for both speckle (short dashed lines) and glint (long dashed lines) targets in daytime, nighttime, and turbulence-free propagation conditions. The solid curve is the system CNR assuming 0.5-dB/km atmospheric attenuation.

Fig. 26
Fig. 26

Turbulence effects on infrared radar CNR margins. The dashed curves depict the carrier-to-noise ratio CNR required to achieve PD = 0.90 and PF = 10−7 vs path length L for a ground-based radar at a height of 3 m searching for speckle (short dashed lines) and glint (long dashed lines) targets on the ground along a horizontal path in daytime, nighttime, and turbulence-free propagation conditions. The solid curves represent the system CNR obtained for atmospheric attenuations of 0, 0.5, 1.0, 2.0, and 4.0 dB/km.

Fig. 27
Fig. 27

Seasonal averaged 10.6-μm atmospheric attenuation statistics for Germany.

Tables (2)

Tables Icon

Table I RADC Monthly Averaged Turbulence Statistics (3-m Height)

Tables Icon

Table II Typical Infrared Radar Parameters

Equations (121)

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

E T ( ρ ¯ , t ) = { ( 2 P T / c ɛ 0 ) 1 / 2 ξ T ( ρ ¯ ) , 0 t t p , 0 , elsewhere ,
ξ T ( ρ ¯ ) = ( 4 / π d 2 ) 1 / 2 exp ( j 2 π f ¯ T · ρ ¯ )
i ¯ T = [ λ f ¯ T + ( 1 - λ f ¯ T 2 ) 1 / 2 i ¯ z ] ,
E t ( ρ ¯ , t ) = d ρ ¯ E T ( ρ ¯ , t - L / c ) h L ( ρ ¯ , ρ ¯ ) .
E r ( ρ ¯ , t ) = E t ( ρ ¯ , t ) T ( ρ ¯ ) ,
E R ( ρ ¯ , t ) = d ρ ¯ E r ( ρ ¯ , t - L / c ) h L ( ρ ¯ , ρ ¯ ) .
E l ( ρ ¯ , t ) = ( 2 P l / c ɛ 0 ) 1 / 2 ξ l ( ρ ¯ ) exp ( j 2 π ν IF t ) ,
r ( t ) = Re [ r ( t ) exp ( - j 2 π ν IF t ) ] ,
r ( t ) = ( c ɛ 0 / 2 ) 1 / 2 d ρ ¯ E R ( ρ ¯ , t ) ξ l * ( ρ ¯ ) + n ( t ) .
S n n ( f ) = h ν 0 / η ,
| t p - 1 2 L / c 2 L / c + t p r ( t ) d t | 2 .
| t p - 1 2 L / c 2 L / c + t p r ( t ) d t | 2 .
E t ( ρ ¯ , t ) = d ρ ¯ E T ( ρ ¯ , t - L / c ) ( j λ L ) - 1 · exp [ j k L ( 1 + ρ ¯ - ρ ¯ 2 / 2 L 2 ) ] ,
E R ( ρ ¯ , t ) = d ρ ¯ E r ( ρ ¯ , t - L / c ) ( j λ L ) - 1 · exp [ j k L ( 1 + ρ ¯ - ρ ¯ 2 / 2 L 2 ) ] ,
h L ( ρ ¯ , ρ ¯ ) = ( j λ L ) - 1 exp [ j k L ( 1 + ρ ¯ - ρ ¯ 2 / 2 L 2 ) ] · exp ( χ ( ρ ¯ , ρ ¯ ) + j ϕ ( ρ ¯ , ρ ¯ ) - α L / 2 ) ,
D ( ρ ¯ , ρ ¯ ) = [ χ ( ρ ¯ 1 + ρ ¯ , ρ ¯ 1 + ρ ¯ ) - χ ( ρ ¯ 1 , ρ ¯ 1 ) ] 2 + [ ϕ ( ρ ¯ 1 + ρ ¯ , ρ ¯ 1 + ρ ¯ ) - ϕ ( ρ ¯ 1 , ρ ¯ 1 ) ] 2 ,
D ( ρ ¯ , ρ ¯ ) = 2.91 k 2 0 L d z C n 2 ( z ) ( ρ ¯ z + ρ ¯ ( L - z ) / L ) 5 / 3
h L ( ρ ¯ 1 + ρ ¯ , ρ ¯ 1 + ρ ¯ ) h L * ( ρ ¯ 1 , ρ ¯ 1 ) = ( λ L ) - 2 exp [ j k ( ρ ¯ 1 - ρ ¯ 1 + ρ ¯ - ρ ¯ 2 - ρ ¯ 1 - ρ ¯ 1 2 ) / 2 L ] · exp [ - D ( ρ ¯ , ρ ¯ ) / 2 - α L ] .
C χ χ ( ρ ¯ , ρ ¯ ) = [ χ ( ρ ¯ 1 + ρ ¯ , ρ ¯ 1 + ρ ¯ ) - m χ ] [ χ ( ρ ¯ 1 , ρ ¯ 1 ) - m χ ] ,
C χ χ ( ρ ¯ , ρ ¯ ) = 4 π 2 k 2 0 L d z 0 d u u C n 2 ( z ) S n ( u ) J 0 ( b u ) × sin 2 [ u 2 z ( L - z ) / 2 k L ]
σ χ 2 = 0.56 k 7 / 6 0 L d z C n 2 ( z ) ( z / L ) 5 / 6 ( L - z ) 5 / 6 ,
E t ( ρ ¯ , t ) 2 = d ρ ¯ E T ( ρ ¯ , t - L / c ) h L ( ρ ¯ , ρ ¯ ) 2
E t ( ρ ¯ , t ) 2 = d ρ ¯ 1 d ρ ¯ 2 E T ( ρ ¯ 1 , t - L / c ) E T * ( ρ ¯ 2 , t - L / c ) ( λ L ) - 2 · exp [ j k ( ρ ¯ 1 2 - ρ ¯ 2 2 ) / 2 L - j k ρ ¯ · ( ρ ¯ 1 - ρ ¯ 2 ) / L - D ( 0 ¯ , ρ ¯ 1 - ρ ¯ 2 ) / 2 - α L ] ,
D ( 0 ¯ , ρ ¯ ) = ( ρ ¯ / ρ 0 ) 5 / 3 f o r ρ 0 [ 2.91 k 2 0 L d z C n 2 ( z ) ( 1 - z / L ) 5 / 3 ] - 3 / 5 .
E t ( ρ ¯ , t ) { ( 2 P T / c ɛ 0 ) 1 / 2 ξ t ( ρ ¯ ) exp [ χ ( ρ ¯ , 0 ¯ ) + j ϕ ( ρ ¯ , 0 ¯ ) - α L / 2 ] , for L / c t L / c + t p , 0 , elsewhere ,
ξ t ( ρ ¯ ) d ρ ¯ ξ T ( ρ ¯ ) ( j λ L ) - 1 exp [ j k L ( 1 + ρ ¯ - ρ ¯ 2 / 2 L 2 ) ]
E R ( ρ ¯ , t ) d ρ ¯ E r ( ρ ¯ , t - L / c ) ( j λ L ) - 1 · exp [ j k L ( 1 + ρ ¯ - ρ ¯ 2 / 2 L 2 ) ] · exp [ χ ( ρ ¯ , 0 ¯ ) + j ϕ ( ρ ¯ , 0 ¯ ) - α L / 2 ] .
E r ( ρ ¯ , t ) = E t ( ρ ¯ , t ) Γ 1 / 2 exp ( - j k ρ ¯ - ρ ¯ c 2 / R c )
E r ( ρ ¯ , t ) = E t ( ρ ¯ , t ) Γ 1 / 2 ( ρ ¯ ) · exp [ - j k ρ ¯ - ρ ¯ c ( ρ ¯ ) 2 / R c ( ρ ¯ ) ] .
E r ( ρ ¯ , t ) = E t ( ρ ¯ , t ) T ( ρ ¯ ) .
T ( ρ ¯ ) = T g ( ρ ¯ ) exp ( j θ ) + T s ( ρ ¯ ) .
T s ( ρ ¯ ) = 0 ,
T s ( ρ ¯ 1 ) T s ( ρ ¯ 2 ) = 0 ,
T s ( ρ ¯ 1 ) T s * ( ρ ¯ 2 ) = λ 2 T s ( ρ ¯ 1 ) δ ( ρ ¯ 1 - ρ ¯ 2 ) .
ρ ( λ ; f ¯ i ; f ¯ r ) = ( λ 2 A T ) - 1 d ρ ¯ exp [ j 2 π ( f ¯ i - f ¯ r ) · ρ ¯ ] T ( ρ ¯ ) 2 ,
ρ ( λ ; f ¯ i ; f ¯ r ) = A T - 1 d ρ ¯ T s ( ρ ¯ ) + ( λ 2 A T ) - 1 d ρ ¯ 1 d ρ ¯ 2 T g ( ρ ¯ 1 ) T g * ( ρ ¯ 2 ) × exp [ j 2 π ( f ¯ i - f ¯ r ) · ( ρ ¯ 1 - ρ ¯ 2 ) ] .
( λ 2 A T ) - 1 d ρ ¯ 1 d ρ 2 T g ( ρ ¯ 1 ) T g * ( ρ ¯ 2 ) exp [ j 2 π ( f ¯ i - f ¯ r ) · ( ρ ¯ 1 - ρ ¯ 2 ) ] = ( λ 2 A T ) - 1 d ρ ¯ R g g ( ρ ¯ ) exp [ j 2 π ( f ¯ i - f ¯ r ) · ρ ¯ ] = ( λ 2 A T ) - 1 S g g ( f ¯ r - f ¯ i ) ,
R g g ( ρ ¯ ) d ρ ¯ 1 T g ( ρ ¯ 1 + ρ ¯ / 2 ) T g * ( ρ ¯ 2 - ρ ¯ / 2 )
S g g ( f ¯ ) d ρ ¯ R g g ( ρ ¯ ) exp ( - j 2 π f ¯ · ρ ¯ )
ξ T ( ρ ¯ ) = ( 4 / π d 2 ) 1 / 2 exp ( j 2 π f ¯ T · ρ ¯ ) for ρ ¯ d / 2 ,
i ¯ T = [ λ f ¯ T + ( 1 - λ f ¯ T 2 ) 1 / 2 i ¯ z ] ,
ξ t ( ρ ¯ ) ( π d 2 / 4 ) 1 / 2 ( j λ L ) - 1 exp ( j k L + j k ρ ¯ 2 / 2 L ) · J 1 ( π d ρ ¯ - λ L f ¯ T / λ L ) / ( π d ρ ¯ - λ L f ¯ T / 2 λ L ) .
( c ɛ 0 / 2 ) 1 / 2 d ρ ¯ E R ( ρ ¯ , t ) ξ l * ( ρ ¯ ) { P T 1 / 2 d ρ ¯ ξ t 2 ( ρ ¯ ) T ( ρ ¯ ) exp [ 2 χ ( ρ ¯ , 0 ¯ ) + 2 j ϕ ( ρ ¯ , 0 ¯ ) - α L ] , for 2 L / c t 2 L / c + t p , 0 , elsewhere .
r ( η / h ν 0 t p ) 1 / 2 2 L / c 2 L / c + t p r ( t ) d t ,
y = ( η P T t p / h ν 0 ) 1 / 2 d ρ ¯ ξ t 2 ( ρ ¯ ) T ( ρ ¯ ) × exp [ 2 χ ( ρ ¯ , 0 ¯ ) + 2 j ϕ ( ρ ¯ , 0 ¯ ) - α L ] ,
n ( η / h ν 0 t p ) 1 / 2 2 L / c 2 L / c + t p n ( t ) d t .
r 2 = y 2 + 1 ,
var ( r 2 ) = var ( y 2 ) + 2 y 2 + 1
SNR = ( r 2 - 1 ) 2 var ( r 2 ) .
SNR = ( y 2 ) 2 var ( y 2 ) + 2 y 2 + 1 = CNR / 2 1 + CNR var ( y 2 ) / 2 y 2 2 + ( 2 CNR ) - 1 ,
CNR y 2 / n 2 = y 2 ,
SNR CNR / 2 1 + CNR var ( y 2 ) / 2 y 2 2 ,
SNR SNR sat CNR / 2 SNR sat 1 + CNR / 2 SNR sat
SNR sat y 2 2 / var ( y 2 ) .
y ( η P T t p / h ν 0 ) 1 / 2 d ρ ¯ ξ t 2 ( ρ ¯ ) T s ( ρ ¯ ) × exp [ 2 χ ( ρ ¯ , 0 ¯ ) + 2 j ϕ ( ρ ¯ , 0 ¯ ) - α L ] + ( η P T t p / h ν 0 ) 1 / 2 d ρ ¯ ξ t 2 ( ρ ¯ ) T g ( ρ ¯ ) × exp [ 2 χ ( ρ ¯ g , 0 ¯ ) + 2 j ϕ ( ρ ¯ g , 0 ¯ ) + j θ - α L ] ,
y 2 ( η P T t p / h ν 0 ) exp ( 4 σ χ 2 - 2 α L ) [ λ 2 d ρ ¯ ξ t 2 ( ρ ¯ ) 2 T s ( ρ ¯ ) + d ρ ¯ R g g ( ρ ¯ ) ] .
R g g ( ρ ¯ ) ξ t 2 ( λ L f ¯ T ) 2 exp ( j 4 π f ¯ T · ρ ¯ ) R g g ( ρ ¯ ) ,
var ( y 2 ) ( η λ 2 P T t p / h ν 0 ) 2 × exp ( 8 σ χ 2 - 4 α L ) d ρ ¯ 1 d ρ ¯ 2 T s ( ρ ¯ 1 ) T s ( ρ ¯ 2 ) ξ t 2 ( ρ ¯ 1 ) 2 · ξ t 2 ( ρ ¯ 2 ) 2 { 2 exp [ 16 C χ χ ( ρ ¯ 1 - ρ ¯ 2 , 0 ¯ ) ] - 1 } + 2 ( η λ P T t p / h ν 0 ) 2 exp ( 8 σ χ 2 - 4 α L ) d ρ ¯ 1 T s ( ρ ¯ 1 ) ξ t 2 ( ρ ¯ 1 ) 2 . { 2 exp [ 16 C χ χ ( ρ ¯ 1 - ρ ¯ g , 0 ¯ ) ] - 1 } d ρ ¯ 2 R g g ( ρ ¯ 2 ) + ( η P T t p / h ν 0 ) 2 × exp ( 8 σ χ 2 - 4 α L ) [ d ρ ¯ R g g ( ρ ¯ ) ] 2 [ exp ( 16 σ χ 2 ) - 1 ] .
y = ( CNR g 0 ) 1 / 2 exp [ 2 χ ( ρ ¯ g . 0 ¯ ) ] ,
CNR g 0 = ( η P T t p / h ν 0 ) d ρ ¯ R g g ( ρ ¯ ) exp ( - 2 α L )
ln y = 2 - 1 ln ( CNR g 0 ) - 2 σ χ 2 ,
var ( ln y ) = 4 σ χ 2 .
var [ y ] χ ( ρ ¯ , 0 ¯ ) + j ϕ ( ρ ¯ , 0 ¯ ) ] = CNR s 0 exp ( 2 u + 4 σ χ 2 ) .
CNR s 0 = ( η P T t p / h ν 0 ) λ 2 d ρ ¯ ξ t 2 ( ρ ¯ ) 2 T s ( ρ ¯ ) exp ( - 2 α L )
exp ( 2 u ) = d ρ ¯ ξ t 2 ( ρ ¯ ) 2 T s ( ρ ¯ ) exp [ 4 χ ( ρ ¯ , 0 ¯ ) ] d ρ ¯ ξ t 2 ( ρ ¯ ) 2 T s ( ρ ¯ ) exp ( 4 σ χ 2 )
exp ( 4 σ 2 ) - 1 = ζ [ exp ( 16 σ χ 2 ) - 1
ζ = d ρ ¯ 1 d ρ ¯ 2 ξ t 2 ( ρ ¯ 1 ) 2 ξ t 2 ( ρ ¯ 2 ) 2 T s ( ρ ¯ 1 ) T s ( ρ ¯ 2 ) [ exp [ 16 C χ χ ( ρ ¯ 1 - ρ 2 , 0 ¯ ) ] - 1 } / [ exp ( 16 σ χ 2 ) - 1 ] [ d ρ ¯ ξ t 2 ( ρ ¯ ) 2 T s ( ρ ¯ ) ] 2 .
y 2 = CNR s 0 exp ( 4 σ χ 2 ) v e 2 u ,
SNR g 0 = CNR g 0 / 2 1 + ( 2 CNR g 0 ) - 1 ,
CNR g 0 = ( η P T t p / h ν 0 ) ξ t 2 ( λ L f ¯ T ) 2 λ 2 A T ρ ( λ ; f ¯ T ; - f ¯ T ) exp ( - 2 α L ) .
SNR s 0 = CNR s 0 / 2 1 + CNR s 0 / 2 + ( 2 CNR s 0 ) - 1 ,
SNR s 0 = CNR s 0 / 2 1 + CNR s 0 / 2 ,
SNR 0 = CNR 0 / 2 1 + CNR s 0 / 2 + CNR s 0 CNR g 0 / 2 CNR 0 + ( 2 CNR 0 ) - 1 ,
SNR 0 = CNR 0 / 2 1 + CNR s 0 / 2 + CNR s 0 CNR g 0 / 2 CNR 0 .
SNR sat 0 = 1 + CNR g 02 / ( CNR s 02 + 2 CNR g 0 CNR s 0 ) .
SNR g = CNR g / 2 1 + CNR g [ exp ( 16 σ χ 2 ) - 1 ] / 2 + ( 2 CNR g ) - 1 ,
SNR g = CNR g / 2 1 + CNR g / 2 SNR g sat ,
SNR g sat = [ exp ( 16 σ χ 2 ) - 1 ] - 1 .
SNR s = CNR s / 2 1 + CNR s { [ 1 + 2 [ exp ( 4 σ 2 ) - 1 ] } / 2 + 2 CNR s - 1 .
ζ = d ρ ¯ 1 d ρ ¯ 2 ξ t 2 ( ρ ¯ 1 ) 2 ξ t 2 ( ρ ¯ 2 ) 2 { exp [ 16 C χ χ ( ρ ¯ 1 - ρ 2 , 0 ) ] - 1 } [ exp ( 16 σ χ 2 ) - 1 ] [ d ρ ¯ ξ t 2 ( ρ ¯ ) 2 ] 2
ζ = ( d 2 / λ L ) / [ 1 + ( d 2 / λ L ) ]
ζ = 1 / [ 1 + ( d T 2 / λ L ) ]
SNR s = CNR s / 2 1 + CNR s / 2 SNR s sat ,
SNR s sat = { 1 + 2 [ exp ( 4 σ 2 ) - 1 ] } - 1
I g ( i , j ) = I 0 ( i , j ) exp [ 4 χ ( i , j ) ] ,
I s ( i , j ) = I 0 ( i , j ) exp ( 4 σ χ 2 ) v ( i , j ) exp [ 2 u ( i , j ) ] ,
SNR ( I ) = I · CNR / 2 1 + I · CNR / 2 SNR sat ( I ) + ( 2 CNR ) - 1
r ( t ) = n ( t ) for 2 L / c t 2 L / c + t p
r ( t ) = ( h ν 0 / η t p ) 1 / 2 y + n ( t ) for 2 L / c t 2 L / c + t p ,
0 d Y 0 2 π d ϕ Y p y ( Y ) exp [ 2 Re ( r Y * ) - Y 2 ]             say H 1 < say H 0             η ˜ ,
Y p y ( Y ) = p y ( Y ) / 2 π for 0 ϕ 2 π ,
0 d Y p y ( Y ) I 0 ( 2 r Y ) exp ( - Y 2 )             say H 1 < say H 0             η ˜ ,
r 2             say H 1 < say H 0             γ ,
P F = P r [ n 2 γ ] = exp ( - γ ) .
r 2             say H 1 < say H 0             - ln P F
P D = P r ( y + n 2 - ln P F ) .
y ( η P T t p / h ν 0 ) 1 / 2 d ρ ¯ ξ t 2 ( ρ ¯ ) T g ( ρ ¯ ) exp [ 2 χ ( ρ ¯ g , 0 ¯ ) + 2 j ϕ ( ρ ¯ g , 0 ¯ ) + j θ - α L ] .
P r [ y + n 2 - ln P F y = Y ] = Q [ 2 1 / 2 Y , ( - 2 ln P F ) 1 / 2 ] ,
Q ( α , β ) β d u u exp [ - ( u 2 + α 2 ) / 2 ] I 0 ( α u )
P D = - d χ p χ ( χ ) Q [ ( 2 CNR g 0 ) 1 / 2 exp ( 2 χ ) , ( - 2 ln P F ) 1 / 2 ] ,
P D = Q [ ( 2 CNR g 0 ) 1 / 2 , ( - 2 ln P F ) 1 / 2 ] .
y ( η P T t p / h ν 0 ) 1 / 2 d ρ ¯ ξ t 2 ( ρ ¯ ) T s ( ρ ¯ ) × exp [ 2 χ ( ρ ¯ , 0 ¯ ) + 2 j ϕ ( ρ ¯ , 0 ¯ ) - α L ]
P r [ y + n 2 - ln P F χ ( ρ ¯ , 0 ¯ ) + j ϕ ( ρ ¯ , 0 ¯ ) ] = P F [ 1 + CNR s exp ( 2 u ) ] - 1 ,
P D = - d U p u ( U ) P F [ 1 + CNR s exp ( 2 U ) ] - 1 ,
p u ( U ) = ( 2 π σ 2 ) - 1 / 2 exp [ - ( U + σ 2 ) 2 / 2 σ 2 ] , exp ( 4 σ 2 ) - 1 = ζ [ exp ( 16 σ χ 2 ) - 1 ] .
P D = P F ( 1 + CNR s 0 ) - 1 .
σ χ 2 = 0.56 k 7 / 6 0 L d z C n 2 ( z ) ( z / L ) 5 / 6 ( L - z ) 5 / 6
σ χ 2 = 0.124 k 7 / 6 C n 2 L 11 / 6 .
P r [ 5.4 × 10 - 14 m - 2 / 3 C n 2 5.4 × 10 - 13 m - 2 / 3 ] = 0.84
P r [ 5.4 × 10 - 15 m - 2 / 3 C n 2 5.4 × 10 - 14 m - 2 / 3 ] = 0.70.
C n 2 = 7 × 10 - 16 m - 2 / 3
var ( C n 2 ) = 2.2 × 10 - 30 m - 4 / 3
C n 2 = 3.0 × 10 - 15 m - 2 / 3 ,
var ( C n 2 ) = 4.0 × 10 - 28 m - 4 / 3
C n 2 = 10 - 12 C T 2 .
C n 2 ( h ) = C n 2 ( 1 ) h - 4 / 3
σ χ 2 = 0.56 k 7 / 6 C n 2 ( h 0 ) 0 L d z [ 1 + ( L - z ) ( H - h 0 ) / L h 0 ] - 4 / 3 · ( z / L ) 5 / 6 ( L - z ) 5 / 6
σ χ 2 = 0.56 k 7 / 6 C n 2 ( h 0 ) 0 L d z [ 1 + ( L - z ) ( H - h 0 ) / L h 0 ] - 2 / 3 · ( z / L ) 5 / 6 ( L - z ) 5 / 6
CNR 0 = ( P T / h ν 0 B ) ( d 2 / 4 L 2 ) ɛ ρ η exp ( - 2 α L ) ,
Δ CNR ( σ χ 2 ; P D , P F ) = CNR ( σ χ 2 ; P D , P F ) / CNR ( 0 ; P D , P F ) .
M = system CNR / CNR ( 0 ; P D , P F ) ,

Metrics