Abstract

This paper reports theoretical probability density and cumulative distribution functions for glint and speckle target returns in a compact coherent laser radar. Calculator programs are given to facilitate use of these results.

© 1982 Optical Society of America

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References

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  1. J. H. Shapiro, B. A. Capron, R. C. Harney, Appl. Opt. 20, 3292 (1981).
    [CrossRef] [PubMed]
  2. Q(t) is related to the familiar error function erf(t)=2π−1/2∫0texp(−τ2)dτ as follows Q(t) = 2−1[1 − erf(2−1/2t)].
  3. S. J. Halme, B. K. Levitt, R. S. Orr, “Bounds and Approximations for Some Integral Expressions Involving Lognormal Statistics,” Quarterly Progress Report 93, Research Lab. Electron., MIT (Apr.1969), pp. 163–175.
  4. E. V. Hoversten, R. O. Harger, S. J. Halme, Proc. IEEE 58, 1626 (1970).
    [CrossRef]
  5. 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]
  6. This program uses a combination of Taylor series and asymptotic series expansions to yield accurate results for all t ≥ 0.It was previously published in J. H. Shapiro, R. C. Harney, Proc. Soc. Photo-Opt. Instrum. Eng. 295, 41 (1981).
  7. This program uses the saddle-point integration for Fr(a,0;c) of P. A. Humblet, “Two-Way Adaptive Optical Communication through the Clear Turbulent Atmosphere,” S. M. Thesis, MIT (1975).The calculator program was previously published in the paper cited in Ref. 6.

1981 (2)

This program uses a combination of Taylor series and asymptotic series expansions to yield accurate results for all t ≥ 0.It was previously published in J. H. Shapiro, R. C. Harney, Proc. Soc. Photo-Opt. Instrum. Eng. 295, 41 (1981).

J. H. Shapiro, B. A. Capron, R. C. Harney, Appl. Opt. 20, 3292 (1981).
[CrossRef] [PubMed]

1970 (1)

E. V. Hoversten, R. O. Harger, S. J. Halme, Proc. IEEE 58, 1626 (1970).
[CrossRef]

Capron, B. A.

Halme, S. J.

E. V. Hoversten, R. O. Harger, S. J. Halme, Proc. IEEE 58, 1626 (1970).
[CrossRef]

S. J. Halme, B. K. Levitt, R. S. Orr, “Bounds and Approximations for Some Integral Expressions Involving Lognormal Statistics,” Quarterly Progress Report 93, Research Lab. Electron., MIT (Apr.1969), pp. 163–175.

Harger, R. O.

E. V. Hoversten, R. O. Harger, S. J. Halme, Proc. IEEE 58, 1626 (1970).
[CrossRef]

Harney, R. C.

J. H. Shapiro, B. A. Capron, R. C. Harney, Appl. Opt. 20, 3292 (1981).
[CrossRef] [PubMed]

This program uses a combination of Taylor series and asymptotic series expansions to yield accurate results for all t ≥ 0.It was previously published in J. H. Shapiro, R. C. Harney, Proc. Soc. Photo-Opt. Instrum. Eng. 295, 41 (1981).

Hoversten, E. V.

E. V. Hoversten, R. O. Harger, S. J. Halme, Proc. IEEE 58, 1626 (1970).
[CrossRef]

Humblet, P. A.

This program uses the saddle-point integration for Fr(a,0;c) of P. A. Humblet, “Two-Way Adaptive Optical Communication through the Clear Turbulent Atmosphere,” S. M. Thesis, MIT (1975).The calculator program was previously published in the paper cited in Ref. 6.

Levitt, B. K.

S. J. Halme, B. K. Levitt, R. S. Orr, “Bounds and Approximations for Some Integral Expressions Involving Lognormal Statistics,” Quarterly Progress Report 93, Research Lab. Electron., MIT (Apr.1969), pp. 163–175.

Orr, R. S.

S. J. Halme, B. K. Levitt, R. S. Orr, “Bounds and Approximations for Some Integral Expressions Involving Lognormal Statistics,” Quarterly Progress Report 93, Research Lab. Electron., MIT (Apr.1969), pp. 163–175.

Shapiro, J. H.

J. H. Shapiro, B. A. Capron, R. C. Harney, Appl. Opt. 20, 3292 (1981).
[CrossRef] [PubMed]

This program uses a combination of Taylor series and asymptotic series expansions to yield accurate results for all t ≥ 0.It was previously published in J. H. Shapiro, R. C. Harney, Proc. Soc. Photo-Opt. Instrum. Eng. 295, 41 (1981).

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]

Appl. Opt. (1)

Proc. IEEE (1)

E. V. Hoversten, R. O. Harger, S. J. Halme, Proc. IEEE 58, 1626 (1970).
[CrossRef]

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

This program uses a combination of Taylor series and asymptotic series expansions to yield accurate results for all t ≥ 0.It was previously published in J. H. Shapiro, R. C. Harney, Proc. Soc. Photo-Opt. Instrum. Eng. 295, 41 (1981).

Other (4)

This program uses the saddle-point integration for Fr(a,0;c) of P. A. Humblet, “Two-Way Adaptive Optical Communication through the Clear Turbulent Atmosphere,” S. M. Thesis, MIT (1975).The calculator program was previously published in the paper cited in Ref. 6.

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]

Q(t) is related to the familiar error function erf(t)=2π−1/2∫0texp(−τ2)dτ as follows Q(t) = 2−1[1 − erf(2−1/2t)].

S. J. Halme, B. K. Levitt, R. S. Orr, “Bounds and Approximations for Some Integral Expressions Involving Lognormal Statistics,” Quarterly Progress Report 93, Research Lab. Electron., MIT (Apr.1969), pp. 163–175.

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Figures (4)

Fig. 1
Fig. 1

Glint target probability density function for various values of the log-amplitude variance σ χ 2.

Fig. 2
Fig. 2

Glint target cumulative distribution function for various values of the log-amplitude variance σ χ 2.

Fig. 3
Fig. 3

Speckle target probability density function for various values of the aperture-averaged log-amplitude variance σ2.

Fig. 4
Fig. 4

Speckle target cumulative distribution function for various values of the aperture-averaged log-amplitude variance σ2.

Tables (2)

Tables Icon

Table I Program a: Q(t) Calculation2

Tables Icon

Table II Program A: Fr(a,0;c) Calculation

Equations (10)

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r = y exp ( j ϕ ) + n ,
x = y 2 / y 2 ,
x = exp ( 4 χ 4 σ χ 2 )
x = υ exp ( 2 u )
p x ( X ) = { ( 32 π X 2 σ χ 2 ) 1 / 2 exp [ ( 4 1 ln X + 2 σ χ 2 ) 2 / 2 σ χ 2 ] , X 0 for a glint target , exp ( 4 σ 2 ) F r [ X exp ( 8 σ 2 ) , 0 ; σ ] , X 0 for a speckle target ,
P x ( X ) = { 1 Q [ ( 4 1 ln X + 2 σ χ 2 ) / σ χ ] , X 0 for a glint target , 1 F r [ X exp ( 4 σ 2 ) , 0 ; σ ] , X 0 for a speckle target .
Q ( t ) = t ( 2 π ) 1 / 2 exp ( τ 2 / 2 ) d τ ,
Fr ( a , 0 ; c ) = dt ( 2 π c 2 ) 1 / 2 × exp [ ( t + c 2 ) 2 / 2 c 2 ] exp ( a e 2 t ) .
SNR sat = y 2 2 / var ( y 2 ) = 1 / var ( x )
P r ( x < FdB ) = 1 P x ( 10 0.1 F ) , F > 0 ,

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