Abstract

The probability of detection of optically rough targets with pulsed LADAR systems that use direct detection is considered. It is assumed that the LADAR operates under conditions of both unintentional pointing offset bias (i.e., bore-sight error) and jitter. Under these conditions the probabilities of detection of targets in both the near field and the far field of the collecting aperture (i.e., for resolved, partially resolved, and unresolved targets) and for both large and small photoelectron counts are derived, and in many cases of practical interest accurate, elementary analytic approximations that are useful for parametric system studies are obtained. A number of technical references are appended, in which some of the key results are derived. In particular, an interesting new mathematical result involving the complementary incomplete gamma function and an analytic expression for the probability distribution function of a signal photoelectron count obeying Bose–Einstein statistics (such as that arising from unresolved targets) immersed in Poisson noise is derived.

© 1994 Optical Society of America

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References

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  1. R. H. Kingston, Detection of Optical and Infrared Radiation (Springer-Verlag, New York, 1979).
  2. C. G. Bachman, Laser Radar Systems and Techniques (Artech, Dedham, Mass., 1979), Chap. 2.
  3. J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 9.
  4. J. W. Goodman, “Some effects of target-induced scintillation on optical radar performance,” Proc. IEEE 53, 1688–1700 (1965).
    [CrossRef]
  5. L. C. Andrews, Special Functions for Engineers and Applied Mathematicians (Macmillian, New York, 1985), Chap. 2.
  6. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), Chap. 7.
  7. P. Beckman, Probability in Communication Engineering (Harcourt, Brace & World, New York, 1967), Chap. 4.
  8. A. P. Prudnikov, Yu A. Brychov, O. I. Marichev, Elementary Functions, Vol. 1 of Integrals and Series (Gordon & Breach, New York, 1986), Chap. 4.

1965

J. W. Goodman, “Some effects of target-induced scintillation on optical radar performance,” Proc. IEEE 53, 1688–1700 (1965).
[CrossRef]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), Chap. 7.

Andrews, L. C.

L. C. Andrews, Special Functions for Engineers and Applied Mathematicians (Macmillian, New York, 1985), Chap. 2.

Bachman, C. G.

C. G. Bachman, Laser Radar Systems and Techniques (Artech, Dedham, Mass., 1979), Chap. 2.

Beckman, P.

P. Beckman, Probability in Communication Engineering (Harcourt, Brace & World, New York, 1967), Chap. 4.

Brychov, Yu A.

A. P. Prudnikov, Yu A. Brychov, O. I. Marichev, Elementary Functions, Vol. 1 of Integrals and Series (Gordon & Breach, New York, 1986), Chap. 4.

Goodman, J. W.

J. W. Goodman, “Some effects of target-induced scintillation on optical radar performance,” Proc. IEEE 53, 1688–1700 (1965).
[CrossRef]

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 9.

Kingston, R. H.

R. H. Kingston, Detection of Optical and Infrared Radiation (Springer-Verlag, New York, 1979).

Marichev, O. I.

A. P. Prudnikov, Yu A. Brychov, O. I. Marichev, Elementary Functions, Vol. 1 of Integrals and Series (Gordon & Breach, New York, 1986), Chap. 4.

Prudnikov, A. P.

A. P. Prudnikov, Yu A. Brychov, O. I. Marichev, Elementary Functions, Vol. 1 of Integrals and Series (Gordon & Breach, New York, 1986), Chap. 4.

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), Chap. 7.

Proc. IEEE

J. W. Goodman, “Some effects of target-induced scintillation on optical radar performance,” Proc. IEEE 53, 1688–1700 (1965).
[CrossRef]

Other

L. C. Andrews, Special Functions for Engineers and Applied Mathematicians (Macmillian, New York, 1985), Chap. 2.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), Chap. 7.

P. Beckman, Probability in Communication Engineering (Harcourt, Brace & World, New York, 1967), Chap. 4.

A. P. Prudnikov, Yu A. Brychov, O. I. Marichev, Elementary Functions, Vol. 1 of Integrals and Series (Gordon & Breach, New York, 1986), Chap. 4.

R. H. Kingston, Detection of Optical and Infrared Radiation (Springer-Verlag, New York, 1979).

C. G. Bachman, Laser Radar Systems and Techniques (Artech, Dedham, Mass., 1979), Chap. 2.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 9.

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Figures (18)

Fig. 1
Fig. 1

Parameter m, the mean number of speckles contained within the collection aperture, for a wavelength of 0.532 μm and a collecting-aperture diameter of 0.2 m.

Fig. 2
Fig. 2

Mean number of signal photoelectron counts as a function of range for various values of laser beam divergence. Note that the beam angles indicated here correspond to the 1 / e angular beam radius. The usual 1/e2 full-angle beam divergence is equal to 4 times the angles indicated above.

Fig. 3
Fig. 3

Mean number of signal photoelectron counts as function of range for various values of the laser spot area at the target.

Fig. 4
Fig. 4

Probability of detection for the case in which m = 1.5 and b = 0. The dashed curves are the exact numerical results, and the solid curves are the approximate numerical results [given by Eq. (13)].

Fig. 5
Fig. 5

Probability of detection for the case in which m = 1.5 and b = 0.5. The dashed curves are the numerical results, and solid curves are the approximate numerical results [given by Eq. (13)].

Fig. 6
Fig. 6

Probability of detection for the case in which m = 1.5 and b = 1. The dashed curves are the numerical results, and solid curves are the approximate numerical results [given by Eq. (13)].

Fig. 7
Fig. 7

Probability of detection for the case in which m = 15 and b = 0. The dashed curves are the numerical results, and the solid curves are the approximate numerical results [given by Eq. (13)].

Fig. 8
Fig. 8

Probability of detection for the case in which m = 15 and b = 0.5. The dashed curves are the numerical results, and the solid curves are the approximate numerical results [given by Eq. (13)].

Fig. 9
Fig. 9

Probability of detection for the case in which m = 15 and b = 1. The dashed curves are the numerical results, and the solid curves are the approximate numerical results [given by Eq. (13)].

Fig. 10
Fig. 10

Probability of deletion for a large number of photoelectron counts, large m (≥ 5–10), and b = 0. The dashed and the solid curves are the numerical and the approximate results, respectively.

Fig. 11
Fig. 11

Probability of deletion for a large number of photoelectron counts, large m (≥ 5–10), and b = 0.5. The dashed and the solid curves are the numerical and the approximate results, respectively.

Fig. 12
Fig. 12

Probability of deletion for a large number of photoelectron counts, large m (≥ 5–10), and b = 1. The dashed and the solid curves are the numerical and the approximate results, respectively.

Fig. 13
Fig. 13

Detection probability for m ≈ 1, b = 0, and large SNR and TNR. The dashed and the solid curves are the numerical and the approximate results, respectively.

Fig. 14
Fig. 14

Detection probability for m ≈ 1, b = 0.5, and large SNR and TNR. The dashed and the solid curves are the numerical and the approximate results, respectively.

Fig. 15
Fig. 15

Detection probability for m ≈ 1, b = 1, and large SNR and TNR. The dashed and the solid curves are the numerical and the approximate results, respectively.

Fig. 16
Fig. 16

Signal-noise-limited detection: a comparison of the detection probability as a function of the signal-to-threshold ratio. The dashed curves are based on Eq. (24), and the solid curves are based on Eq. (27).

Fig. 17
Fig. 17

Probability distribution of a single obeying Bose–Einstein statistics immersed in Poisson noise, where the mean signal count is (a) 1, (b) 10, (c) 100, and (d) 1000 for various values of the noise count. For convenience, solid curves are drawn between integer count values.

Fig. 18
Fig. 18

Quantity G(K, n) = Γ(K + 1, n)/Γ(K + 1) for photoelectron counts near 1000 and various values of n.

Tables (2)

Tables Icon

Table 1 Definitions of Key Parameters, with a Reference to Their Equations

Tables Icon

Table 2 Detection Probability per Pulse, Averaged over the Pointing Statistics in the High-Photoelectron-Count Regime

Equations (93)

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P ( K ) = K s K K ! exp ( - K s ) ,             K = 0 , 1 , 2 ,
K s = ( η h ν ) E s ,
E s = ( E t A spot ) ( ρ A T π ) ( A R R 2 ) T 2 ,
I ( θ ) = I 0 exp ( - θ 2 / 2 θ 0 2 ) ,
A spot = 2 π ( R θ 0 ) 2 .
P ( K ) = Γ ( K + m ) Γ ( K + 1 ) Γ ( m ) ( 1 + m K s ) - K ( 1 + K s m ) - m ,
E ( K ) = K s ,
Var ( K ) = K s + K s 2 m ,
m 1 + A R A T λ 2 R 2 .
i = q K τ ,
P d = 1 - P ( 0 ) = 1 - ( 1 + K s m ) - m ,
P d ( N ) = 1 - ( 1 - P d ) N = 1 - P ( 0 ) N = 1 - ( 1 + K s m ) - N m .
P pt ( k ) = [ a 2 exp ( - a 2 b 2 / 2 ) K s ] ( k / K s ) a 2 - 1 × J 0 { a 2 b [ 2 ln ( k / K s ) ] 1 / 2 } ,             0 k K s , = 0 otherwise ,
a = θ 0 σ j ,
b = θ pt θ 0 ,
P ¯ d = 0 K s P d ( k ) P pt ( k ) d k ,
P ¯ d P d ( K ¯ s ) = 1 - ( 1 + K ¯ s m ) - m ,
K ¯ s = 0 k P pt ( k ) d k K s ( a 2 a 2 + 1 ) exp [ - b 2 2 ( a 2 a 2 + 1 ) ] .
P n ( i ) = 1 ( 2 π σ n 2 ) 1 / 2 exp ( - i 2 2 σ n 2 ) ,
σ n 2 = q 2 T 2 K B + σ dk 2 + σ th 2 ,
P fa = 1 2 { 1 - erf [ ( TNR ) 0 2 ] } ,
( TNR ) 0 = i T σ n .
( TNR ) 0 = 0.513 - 1.02 log P fa - 0.0539 ( log P fa ) 2 .
P s ( i ) 1 ( 2 π σ s 2 ) 1 / 2 exp [ - ( i - i s ) 2 2 σ s 2 ] ,
i s = q K s τ ,
σ s 2 = q i s τ + i s 2 m .
P sn ( i ) = 1 ( 2 π σ sn 2 ) 1 / 2 exp [ - ( i - i s ) 2 2 σ sn 2 ] ,
σ sn 2 = σ s 2 + σ n 2 .
P d = 1 2 [ 1 + erf ( i s - i T 2 σ sn ) ] .
P sn ( K ) = ( m m + K s ) m exp ( - n ) Γ ( m ) × j = 0 K Γ ( K - j + m ) Γ ( j + 1 ) Γ ( K - j + 1 ) n j ( K s m + K s ) K - j ,
P sn ( K ) = 1 Γ ( m ) ( m K s ) m exp ( - m K / K s ) K m - 1 ,
P d = Γ ( m , ɛ m ) Γ ( m ) ,
ɛ m = m i T i s ,
Γ ( a , z ) = z exp ( - t ) t a - 1 d t .
P d = exp ( - i T / i s ) ,
P d = exp ( - ɛ 2 ) ( ɛ 2 + 1 ) ,
P d = exp ( - ɛ 3 ) 2 ( ɛ 3 2 + 2 ɛ 3 + 2 ) ,
P d = exp ( - ɛ 4 ) 6 ( ɛ 4 3 + 3 ɛ 4 2 + 6 ɛ 4 + 6 ) ,
P d = exp ( - ɛ 5 ) 24 ( ɛ 5 4 + 4 ɛ 5 3 + 12 ɛ 5 2 + 24 ɛ 5 + 24 ) ,
P d = exp ( - ɛ 6 ) 120 ( ɛ 6 5 + 5 ɛ 6 4 + 20 ɛ 6 3 + 60 ɛ 6 2 + 120 ɛ + 120 ) .
P sn ( K ) = exp [ - ( K - n ) / K s ] K s Γ ( K + 1 , n ) Γ ( K + 1 ) exp [ - ( K - n ) / K s ] K s U ( K - n ) ,
P sn ( i ) = exp [ - ( i - i n ) / i s ] i s U ( i - i n ) ,
P d exp ( - i T / i s ) ,
i ¯ s i s = ( a 2 a 2 + 1 ) exp [ - b 2 2 ( a 2 a 2 + 1 ) ] .
SNR = i s σ sn ,
TNR = i T σ sn ,
erf ( x ) = 1 - ( a 1 t + a 2 t 2 + a 3 t 3 ) exp ( - x 2 ) + ɛ ( x ) ,
ɛ ( x ) 2.5 × 10 - 5 , t = 1 1 + p x ,
P ¯ d = Γ ( m , ɛ ¯ m ) Γ ( m ) ,
ɛ ¯ m = m i T i ¯ s = m i T i s ( i s / i ¯ s ) ,
P d ( N ) = 1 - ( 1 - P d ) N ,
( SNR ) 0 = i s σ n
P ¯ d ( N ) = 1 - ( 1 + α K s m ) - m N ,             m 1 ,
α = ( a 2 a 2 + 1 ) exp [ - b 2 2 ( a 2 a 2 + 1 ) ] .
P pt ( θ ) = θ σ j 2 exp ( - θ 2 + θ pt 2 2 σ j 2 ) I 0 ( θ θ pt σ j 2 ) ,
P pt ( I ) = | d θ d I | P pt ( θ ) .
I ( θ ) = I 0 exp ( - θ 2 / 2 θ 0 2 ) ,
θ = [ - 2 θ 0 2 l n ( I / I 0 ) ] 1 / 2 ,
P pt ( I ) = [ a 2 exp ( - a 2 b 2 / 2 ) I 0 ] ( I / I 0 ) a 2 - 1 × J 0 { a 2 b [ 2 ln ( I / I 0 ) ] 1 / 2 } ,             0 I I 0 , = 0 otherwise ,
P ¯ d = 1 - F 2 1 ( a 2 , m ; 1 + a 2 ; - K s / m ) ,
P s ( K ) = 1 2 π exp [ f ( K ) ] ,
f ( K ) = ( K + m - 1 2 ) ln ( K + m ) - ( K + 1 2 ) ln ( K ) - ( m - 1 2 ) ln ( m ) - K ln ( K s + m K s ) - m ln ( K s + m m ) .
K = K s .
f ( K ) f ( K s ) + ( K - K s ) f ( K s ) + 1 2 ( K - K s ) 2 f ( K s ) = - 1 2 ln σ K 2 - ( K - K s ) 2 2 σ K 2 ,
σ K 2 = K s + K s 2 m .
P s ( K ) = 1 ( 2 π σ K 2 ) 1 / 2 exp [ - ( K - K s ) 2 2 σ K 2 ] .
P s ( K ) = 1 1 + K s ( K s 1 + K s ) K ,             m = 1.
P s ( K ) exp ( - K / K s ) K s ,
P s ( K ) = 1 K ! lim m [ Γ ( K + m ) Γ ( m ) ( 1 + m K s ) - K ( 1 + K s m ) - m ] = 1 K ! m K ( K s m ) K exp ( - K s ) = K s K exp ( - K s ) K ! ,
P ( K ) = 1 2 π exp [ g ( K ) ] ,
g ( K ) = K + K ln K s - ( K + 1 2 ) ln K - K s .
g ( K ) - 1 2 ln K s - ( K - K s ) 2 2 K s .
P ( K ) = 1 ( 2 π K s ) 1 / 2 exp [ - ( K - K s ) 2 2 K s ] ,
P s ( K ) = Γ ( K + m + 1 ) Γ ( m ) Γ ( K ) ( 1 + m K s ) - K ( 1 + K s m ) - m .
Γ ( K + m - 1 ) K m - 1 Γ ( K ) , ( 1 + m K s ) - K exp ( - m K K s ) .
P s ( K ) 1 Γ ( m ) ( m K s ) m K m - 1 exp ( - m K K s ) .
P d = Γ ( m , m ɛ ) Γ ( m ) = 1 Γ ( m ) m ɛ exp ( - t ) t m - 1 d t .
exp [ f ( t ) ] = t m - 1 exp ( - t ) = exp [ ( m - 1 ) l n t - t ] ,
f ( t ) = ( m - 1 ) l n t - t , f ( t ) = m - 1 t - 1 ,             f = 0 for t = m - 1 , f = - ( m - 1 ) t 2 ;
P d = ( m - 1 ) m - 1 exp [ - ( m - 1 ) ] Γ ( m ) × m ɛ exp { - [ t - ( m - 1 ) ] 2 2 ( m - 1 ) } d t = [ π / 2 ( m - 1 ) ] 1 / 2 ( m - 1 ) m - 1 exp [ - ( m - 1 ) ] Γ ( m ) × { 1 + erf [ ( m - 1 ) 1 / 2 ( 1 - ɛ ) 2 ] } .
P d = 1 2 { 1 + erf [ m ( 1 - ɛ ) 2 ] } ,
lim m Γ ( m , m ɛ ) Γ ( m ) = { 1 , 0 ɛ < 1 1 2 , ɛ = 1 0 , ɛ > 1 .
P sn = ( 1 1 + K s ) ( 1 + 1 K s ) - K exp ( - n ) j = 0 K β j j ! ,
β = n ( 1 + K s - 1 ) .
j = 0 K β j j ! = e β Γ ( K + 1 , β ) Γ ( K + 1 ) ,
P sn ( K ) = ( 1 + 1 K s ) - K 1 + K s exp ( n / K s ) Γ ( K + 1 , n + n K s ) K ! ,             K = 0 , 1 , 2 , ,
P sn ( K ) exp [ - ( K - n ) / K s ] K s G ( K , n ) ,
G ( K , n ) = Γ ( K + 1 , n ) Γ ( K + 1 ) .
G { 0 K < n 1 2 K = n 1 K > n .
P sn exp [ - ( K - n ) / K s ] K s U ( K - n ) ,
P d = K T P sn ( K ) d K = n + Δ K T P sn ( K ) d K = exp ( - Δ K T / K s ) ,
Γ ( a , z ) exp ( - z ) z a - 1 ( 1 + a - 1 z + ) ,
P sn ( 1 + K s - 1 ) - K 1 + K s exp ( n / K s ) exp ( - n - n / K s ) × n K ( 1 + K s - 1 ) K ( 1 + K K s / n ) K ! = ( 1 1 + K s ) exp ( - n ) n K ( 1 + K K s / n ) K ! = exp [ - ( n + K s ) ] ( n + K s ) K K !

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