Abstract

When laser pulses are reflected by targets that have range spreads larger than the transmitted pulse width, the width of the received pulses will be longer than the correlation length of the speckle-induced fluctuations. As a consequence, speckle will cause random small-scale fluctuations within the received pulse that will distort its shape. This phenomenon is called time-resolved speckle. In laser ranging and altimetry, the random pulse distortion caused by time-resolved speckle can seriously degrade the timing accuracy of the receivers. In this paper, we study the statistical properties of time-resolved speckle and the problem of estimating the arrival times of laser pulses in its presence. The maximum-likelihood (ML) estimator of the pulse arrival time is derived, and its performance is evaluated for pulse reflections from flat diffuse targets. The performance of the ML estimator is compared with the performance of several suboptimal estimators. When the signal level is high, speckle noise places a fundamental limit on the accuracy of the suboptimal estimators. It is shown that the ML estimator performs considerably better than the suboptimal estimators and that its accuracy improves as the width of the receiver observation interval increases.

© 1985 Optical Society of America

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References

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  1. J. W. Goodman, “Statistical properties of laser speckle pattern,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1975), pp. 9–65.
    [CrossRef]
  2. J. Opt. Soc. Am. 66(11) (1976).
  3. J. W. Goodman, in Remote Techniques for Capillary Wave Measurement, K. S. Krishnan, N. A. Peppers, eds. (Stanford Research Institute Report, Stanford, Calif., 1973).
  4. J. W. Goodman, “Some effects of target-induced scintillation on optical radar performance,” Proc. IEEE 53, 1688 (1965).
    [CrossRef]
  5. C. S. Gardner, “Target signatures for laser altimeters: an analysis,” Appl. Opt. 21, 3932 (1982).
    [CrossRef] [PubMed]
  6. B. M. Tsai, C. S. Gardner, “Remote sensing of sea state using laser altimeters,” Appl. Opt. 21, 3932 (1982).
    [CrossRef] [PubMed]
  7. I. Bar-David, “Communication under the Poisson regime,”IEEE Trans. Inform. Theory IT-15, 31 (1969).
    [CrossRef]
  8. I. Bar-David, “Minimum-mean-square-error estimation of photon pulse delay,”IEEE Trans. Inf. Theory IT-21, 326 (1975).
    [CrossRef]
  9. M. Elbaum, P. Diament, “Estimation of image centroid, size and orientation with laser radar,” Appl. Opt. 16, 2433 (1977).
    [CrossRef] [PubMed]
  10. C. W. Helmstrom, Statistical Theory of Signal Detection (Pergamon, London, 1960).

1982

1977

1976

J. Opt. Soc. Am. 66(11) (1976).

1975

I. Bar-David, “Minimum-mean-square-error estimation of photon pulse delay,”IEEE Trans. Inf. Theory IT-21, 326 (1975).
[CrossRef]

1969

I. Bar-David, “Communication under the Poisson regime,”IEEE Trans. Inform. Theory IT-15, 31 (1969).
[CrossRef]

1965

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

Bar-David, I.

I. Bar-David, “Minimum-mean-square-error estimation of photon pulse delay,”IEEE Trans. Inf. Theory IT-21, 326 (1975).
[CrossRef]

I. Bar-David, “Communication under the Poisson regime,”IEEE Trans. Inform. Theory IT-15, 31 (1969).
[CrossRef]

Diament, P.

Elbaum, M.

Gardner, C. S.

Goodman, J. W.

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

J. W. Goodman, “Statistical properties of laser speckle pattern,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1975), pp. 9–65.
[CrossRef]

J. W. Goodman, in Remote Techniques for Capillary Wave Measurement, K. S. Krishnan, N. A. Peppers, eds. (Stanford Research Institute Report, Stanford, Calif., 1973).

Helmstrom, C. W.

C. W. Helmstrom, Statistical Theory of Signal Detection (Pergamon, London, 1960).

Tsai, B. M.

Appl. Opt.

IEEE Trans. Inf. Theory

I. Bar-David, “Minimum-mean-square-error estimation of photon pulse delay,”IEEE Trans. Inf. Theory IT-21, 326 (1975).
[CrossRef]

IEEE Trans. Inform. Theory

I. Bar-David, “Communication under the Poisson regime,”IEEE Trans. Inform. Theory IT-15, 31 (1969).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. 66(11) (1976).

Proc. IEEE

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

Other

J. W. Goodman, “Statistical properties of laser speckle pattern,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1975), pp. 9–65.
[CrossRef]

J. W. Goodman, in Remote Techniques for Capillary Wave Measurement, K. S. Krishnan, N. A. Peppers, eds. (Stanford Research Institute Report, Stanford, Calif., 1973).

C. W. Helmstrom, Statistical Theory of Signal Detection (Pergamon, London, 1960).

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Figures (11)

Fig. 1
Fig. 1

Mean received waveform for normal incidence upon an infinite flat diffuse target (z = 500 km, f = 0.5 cm, AR = 100 cm2, λ = 1 μm).

Fig. 2
Fig. 2

Speckle variance for normal incidence on an infinite flat diffuse target (z = 500 km, f = 0.5 cm, AR = 100 cm2, λ = 1 μm).

Fig. 3
Fig. 3

Signal-to-noise (S/N) ratio of speckle for normal incidence on an infinite flat diffuse target (z = 500 km, f = 0.5 cm, AR = 100 cm2, λ = 1 μm).

Fig. 4
Fig. 4

Rms ranging error using the ML estimator τ ^ SPEC for normal incidence upon an infinite flat diffuse target (z = 500 km, f = 0.5 cm, AR = 100 cm2, λ = 1 μm).

Fig. 5
Fig. 5

Rms ranging error using the suboptimal estimator τ ^ SHOT for normal incidence upon an infinite flat diffuse target (z = 500 km, f = 0.5 cm, AR = 100 cm2, λ = 1 μm).

Fig. 6
Fig. 6

Comparison of the rms ranging errors using τ ^ SPEC and τ ^ SHOT for normal incidence upon an infinite flat diffuse target (z = 500 km, f = 0.5 cm, AR = 100 cm2, λ = 1 μm).

Fig. 7
Fig. 7

Mean received waveform for nonnormal incidence upon an infinite flat diffuse target (z = 500 km, f = 5.0 cm, AR = 100 cm2, λ = 1 νm).

Fig. 8
Fig. 8

Speckle variance for nonnormal incidence upon an infinite flat diffuse target (z = 500 km, f = 0.5 cm, AR = 100 cm2, λ = 1 μm).

Fig. 9
Fig. 9

Rms ranging error using the ML estimator τ ^ SPEC for non- normal incidence upon an infinite flat diffuse target (z = 500 km, f = 0.5 cm, AR = 100 cm2, λ = 1 μm).

Fig. 10
Fig. 10

Rms ranging error using τ ^ SHOT for nonnormal incidence upon an infinite flat diffuse target (z = 500 km, f = 0.5 cm, AR = 100cm2, λ = 1 μm).

Fig. 11
Fig. 11

Comparison of the rms ranging errors using τ ^ SPEC and τ ^ SHOT for nonnormal incidence upon an infinite flat diffuse target (z = 500 km, f = 0.5 cm, AR = 100 cm2, λ = 1 μm).

Tables (2)

Tables Icon

Table 1 Summary of Laser-Pulse Arrival-Time Estimators and Their Variances

Tables Icon

Table 2 Summary of the Timing Variance for the Flat Tilted Target

Equations (56)

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k = ( k 1 , k 2 , , k n ) ,
p ( W i ) = [ Γ ( M i ) W i ] - 1 ( M i W i / W ¯ i ) M i exp ( - M i W i / W ¯ i ) ,
M i = W ¯ i 2 / var ( W i ) .
W ¯ i = Q ( i - 1 ) T i T d t d 2 ρ b 2 ( ρ , z ) f ( t - ψ ) 2
var ( W i ) = Q 2 K s - 1 ( i - 1 ) T i T d t 1 ( i - 1 ) T i T d t 2 d 2 ρ b 4 ( ρ , z ) × f ( t 1 - ψ ) 2 f ( t 2 - ψ ) 2 ,
b l ( ρ , z ) = a ( ρ , z ) l β r l / 2 ( ρ ) d 2 ρ a ( ρ , z ) l β r l / 2 ( ρ ) ,
K s = A R [ d 2 ρ a ( ρ , z ) 2 β r ( ρ ) ] 2 / λ 2 z 2 × d 2 ρ a ( ρ , z ) 4 β r 2 ( ρ ) ,
ψ = 2 z / c + ρ 2 / c z - 2 ξ ( ρ ) / c .
W ¯ i Q F 1 ( i T ) T
var ( W i ) Q 2 K s - 1 F 2 ( i T ) T ,
F 1 ( t ) = d 2 ρ b 2 ( ρ , z ) G ( σ f , t - ψ ) ,
F 2 ( t ) = d 2 ρ b 4 ( ρ , z ) G ( σ f / 2 , t - ψ ) ,
G ( σ , t ) = ( 2 π σ ) - 1 exp ( - t 2 / 2 σ 2 ) .
M i = K s F 1 2 ( i T ) T / F 2 ( i T ) .
M i K s F 1 ( i T ) T .
p ( k i ) = Γ ( k i + M i ) Γ ( k i + 1 ) Γ ( M i ) ( 1 + M i k ¯ i ) - k i ( 1 + k ¯ i M i ) - M i ,
k ¯ i = η W ¯ i / h ν = N F 1 ( i T ) T .
var ( k i ) = k ¯ i + ( k ¯ i 2 ) / ( M i ) .
p ( k ) = i = 1 n p ( k i ) .
p ( k τ ) = i = 1 n Γ [ k i + M i ( τ ) ] Γ ( k i + 1 ) Γ [ M i ( τ ) ] ( 1 + M i ( τ ) k ¯ i ( τ ) ) - k i × ( 1 + k ¯ i ( τ ) M i ( τ ) ) - M i ( τ ) .
τ ^ SPEC = arg max τ [ ln p ( k τ ) ] ,
τ ^ SPEC = arg max τ [ H ( τ ) ] ,
H ( τ ) = i = 1 n [ k i + M i ( τ ) - 1 / 2 ] ln [ k i + M i ( τ ) ] - i = 1 n k i ln ( 1 + M i ( τ ) k ¯ i ( τ ) ) .
ln Γ ( x ) ( x - 1 / 2 ) ln x - x + ( 1 / 2 ) ln 2 π ,             x 1.
E ( Δ τ ^ SPEC ) = E ( τ ^ SPEC - τ 0 ) = - E [ H ˙ ( τ 0 ) ] / E [ H ¨ ( τ 0 ) ]
var ( τ ^ SPEC ) = var [ H ˙ ( τ 0 ) ] / E [ H ¨ ( τ 0 ) ] 2 .
var [ H ˙ ( τ 0 ) ] = i = 1 n { k ¯ ˙ i 2 [ k ¯ i ( 1 + k ¯ i M i ) ] } ,
E [ H ¨ ( τ 0 ) ] = - i = 1 n { k ¯ ˙ i 2 / [ k ¯ i ( 1 + k ¯ i M i ) ] } .
var ( τ ^ SPEC ) { i = 1 n k ¯ ˙ i 2 / [ k ¯ i ( 1 + k ¯ i M i ) ] } - 1 .
var ( τ ^ SPEC ) [ i = 1 n F ˙ 1 2 T / ( F 1 N + F 2 K s ) ] - 1 .
τ ^ SHOT = arg max τ [ i = 1 n k i ln k ¯ i ( τ ) ] .
var ( τ ^ SHOT ) = ( i = 1 n k ¯ i 2 k ¯ i ) - 1 + i = 1 n k ¯ i 2 M i / ( i = 1 n k ¯ i 2 k ¯ i ) 2 = 1 N ( i = 1 n F ˙ 1 2 F 1 T ) - 1 + 1 K s i = 1 n ( F ˙ 1 F 1 ) 2 F 2 T / ( i = 1 n F ˙ 1 2 F 1 T ) 2 .
var ( τ ^ SPEC ) { i = 1 n k ¯ ˙ i 2 k ¯ i ( 1 - k ¯ i M i ) } - 1 ( i = 1 n k ¯ ˙ i 2 k ¯ i ) - 1 + i = 1 n k ¯ i 2 M i / ( i = 1 n k ¯ i 2 k ¯ i ) 2 .
var ( τ ^ SPEC ) = var ( τ ^ SHOT ) = ( 1 + N K s ) { i = 1 n 4 A 2 } - 1 ( 1 N + 1 K s ) 1 4 B 2 ,
A ( t ) = [ k ¯ ( t ) ] 1 / 2
B 2 = - d t A ˙ 2 ( t ) - d t A 2 ( t ) = - d ω ω 2 A ^ ( ω ) 2 - d ω A ^ ( ω ) 2 .
τ ^ CENT = i = 1 n i T k i / i = 1 n k i .
E { Δ τ ^ CENT } = E { τ ^ CENT - τ 0 } i ¯ T - τ 0 ,
i ¯ = i = 1 n i k ¯ i / i = 1 n k ¯ i .
var ( τ ^ CENT ) = 1 N 2 i = 1 n ( i - i ¯ ) 2 T 2 ( k ¯ i + k ¯ i 2 M i ) = i = 1 n ( i - i ¯ ) 2 T 3 ( F 1 N + F 2 K s ) .
K s = 4 π A R tan 2 θ / λ 2 .
W ¯ i = Q c T 4 z tan 2 θ exp [ c 2 σ f 2 8 z 2 tan 4 θ - c ( i T - 2 z / c ) 2 z tan 2 θ ] × erfc ( c σ f 2 2 z tan 2 θ - i T - 2 z / c 2 σ f )
var ( W i ) = Q 2 c T 2 K s z tan 2 θ exp [ c 2 σ f 2 4 z 2 tan 4 θ - c ( i T - 2 z / c ) z tan 2 θ ] × erfc ( c σ f 2 z tan 2 θ - i T - 2 z / c σ f ) .
M i = K s c T 8 z tan 2 θ erfc 2 ( c σ f 2 2 z tan 2 θ - i T - 2 z / c 2 σ f ) / erfc ( c σ f 2 z tan 2 θ - i T - 2 z / c σ f ) ,
M i π A R c T / λ 2 z ,
W ¯ i = Q G ( σ T , i T - 2 z / c ) T
var ( W i ) = Q 2 K s - 1 G ( σ T / 2 , i T - 2 z / c ) T ,
σ T = ( σ f 2 + 4 z 2 tan 2 θ tan 2 ϕ c 2 ) 1 / 2 .
M i = K s T / 2 π σ T .
var ( τ ^ SPEC ) σ T 2 N 1 g ( N ) ,
g ( N ) = 1 2 π - ( n T / 2 σ T ) ( n T / 2 σ T ) d y × y 2 exp ( - y 2 / 2 ) 1 + K s - 1 2 N exp ( - y 2 / 2 ) .
var ( τ ^ SPEC ) σ T 2 N ,             N K s .
var ( τ ^ SPEC ) 6 π ( 2 σ T n T ) 3 σ T 2 2 K s ,             N K s ,
var ( τ ^ SHOT ) = var ( τ ^ CENT ) σ T 2 N + σ T 2 2 K s .
var ( τ ^ SHOT ) σ T 2 N ,             N K s ,
var ( τ ^ SHOT ) σ T 2 2 K s ,             N K s .

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