Abstract

The precision of ladar range measurements is limited by noise. The fundamental source of noise in a laser signal is the random time between photon arrivals. This phenomenon, called shot noise, is modeled as a Poisson random process. Other noise sources in the system are also modeled as Poisson processes. Under the Poisson-noise assumption, the Cramer–Rao lower bound (CRLB) on range measurements is derived. This bound on the variance of any unbiased range estimate is greater than the CRLB derived by assuming Gaussian noise of equal variance. Finally, it is shown that, for a ladar capable of dividing a fixed amount of energy into multiple laser pulses, the range precision is maximized when all energy is transmitted in a single pulse.

© 2008 Optical Society of America

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References

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    [CrossRef]
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2007

C. Gronwall, O. Steinvall, F. Gustafsson, and T. Chevalier, “Influence of laser radar sensor parameters on range-measurement and shape-fitting uncertainties,” Opt. Eng. 46, 106201 (2007).
[CrossRef]

N. Hagen, M. Kupinski, and E. Dereniak, “Gaussian profile estimation in one dimension,” Appl. Opt. 46, 5374-5383 (2007).
[CrossRef] [PubMed]

2006

2004

S. Johnson, T. Nichols, P. Gatt, and T. Klausutis, “Range precision of direct detection laser radar systems,” Proc. SPIE 5412, 72-86 (2004).
[CrossRef]

2001

P. Gatt and S. Henderson, “Laser radar detection statistics: a comparison of coherent and direct detection intensity receivers,” Proc. SPIE 4377, 251-262 (2001).
[CrossRef]

2000

1993

B. Rye and R. Hardesty, “Discrete spectral peak estimation in incoherent backscatter heterodyne lidar. I. Spectral accumulation and the Cramer-Rao lowerbound,” IEEE Trans. Geosci. Remote Sens. 31, 16-27 (1993).
[CrossRef]

1986

1981

1965

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

1947

U. Fano, “Ionization yield of radiations. II. The fluctuations of the number of ions,” Phys. Rev. 72, 26-29 (1947).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1965).

Armstrong, E.

Berger, R.

G. Casella and R. Berger, Statistical Inference (Duxbury, 2002).

Cain, S.

Capron, B.

Casella, G.

G. Casella and R. Berger, Statistical Inference (Duxbury, 2002).

Chevalier, T.

C. Gronwall, O. Steinvall, F. Gustafsson, and T. Chevalier, “Influence of laser radar sensor parameters on range-measurement and shape-fitting uncertainties,” Opt. Eng. 46, 106201 (2007).
[CrossRef]

Dereniak, E.

Fano, U.

U. Fano, “Ionization yield of radiations. II. The fluctuations of the number of ions,” Phys. Rev. 72, 26-29 (1947).
[CrossRef]

Gatt, P.

S. Johnson, T. Nichols, P. Gatt, and T. Klausutis, “Range precision of direct detection laser radar systems,” Proc. SPIE 5412, 72-86 (2004).
[CrossRef]

P. Gatt and S. Henderson, “Laser radar detection statistics: a comparison of coherent and direct detection intensity receivers,” Proc. SPIE 4377, 251-262 (2001).
[CrossRef]

Goodman, J.

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

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

Gradshteyn, I.

I. Gradshteyn and I. Ryzhik, Table of Integrals, Series, and Products (Academic, 2000).

Gronwall, C.

C. Gronwall, O. Steinvall, F. Gustafsson, and T. Chevalier, “Influence of laser radar sensor parameters on range-measurement and shape-fitting uncertainties,” Opt. Eng. 46, 106201 (2007).
[CrossRef]

Gustafsson, F.

C. Gronwall, O. Steinvall, F. Gustafsson, and T. Chevalier, “Influence of laser radar sensor parameters on range-measurement and shape-fitting uncertainties,” Opt. Eng. 46, 106201 (2007).
[CrossRef]

Hagen, N.

Hardesty, R.

B. Rye and R. Hardesty, “Discrete spectral peak estimation in incoherent backscatter heterodyne lidar. I. Spectral accumulation and the Cramer-Rao lowerbound,” IEEE Trans. Geosci. Remote Sens. 31, 16-27 (1993).
[CrossRef]

Harney, R.

Henderson, S.

P. Gatt and S. Henderson, “Laser radar detection statistics: a comparison of coherent and direct detection intensity receivers,” Proc. SPIE 4377, 251-262 (2001).
[CrossRef]

Jelalian, A.

A. Jelalian, Laser Radar Systems (Artech House, 1992).

Johnson, S.

S. Johnson, T. Nichols, P. Gatt, and T. Klausutis, “Range precision of direct detection laser radar systems,” Proc. SPIE 5412, 72-86 (2004).
[CrossRef]

Kay, S.

S. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice Hall, 1993).

Klausutis, T.

S. Johnson, T. Nichols, P. Gatt, and T. Klausutis, “Range precision of direct detection laser radar systems,” Proc. SPIE 5412, 72-86 (2004).
[CrossRef]

Kupinski, M.

Nichols, T.

S. Johnson, T. Nichols, P. Gatt, and T. Klausutis, “Range precision of direct detection laser radar systems,” Proc. SPIE 5412, 72-86 (2004).
[CrossRef]

Osche, G.

G. Osche, Optical Detection Theory (Wiley-Interscience, 2002).

Richmond, R.

Rye, B.

B. Rye and R. Hardesty, “Discrete spectral peak estimation in incoherent backscatter heterodyne lidar. I. Spectral accumulation and the Cramer-Rao lowerbound,” IEEE Trans. Geosci. Remote Sens. 31, 16-27 (1993).
[CrossRef]

Ryzhik, I.

I. Gradshteyn and I. Ryzhik, Table of Integrals, Series, and Products (Academic, 2000).

Shapiro, J.

Siegman, A.

A. Siegman, Lasers (University Science Books, 1986).

Stegun, I.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1965).

Steinvall, O.

C. Gronwall, O. Steinvall, F. Gustafsson, and T. Chevalier, “Influence of laser radar sensor parameters on range-measurement and shape-fitting uncertainties,” Opt. Eng. 46, 106201 (2007).
[CrossRef]

O. Steinvall, “Effects of target shape and reflection on laser radar cross sections,” Appl. Opt. 39, 4381-4391 (2000).
[CrossRef]

Winick, K.

Appl. Opt.

IEEE Trans. Geosci. Remote Sens.

B. Rye and R. Hardesty, “Discrete spectral peak estimation in incoherent backscatter heterodyne lidar. I. Spectral accumulation and the Cramer-Rao lowerbound,” IEEE Trans. Geosci. Remote Sens. 31, 16-27 (1993).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Eng.

C. Gronwall, O. Steinvall, F. Gustafsson, and T. Chevalier, “Influence of laser radar sensor parameters on range-measurement and shape-fitting uncertainties,” Opt. Eng. 46, 106201 (2007).
[CrossRef]

Phys. Rev.

U. Fano, “Ionization yield of radiations. II. The fluctuations of the number of ions,” Phys. Rev. 72, 26-29 (1947).
[CrossRef]

Proc. IEEE

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

Proc. SPIE

S. Johnson, T. Nichols, P. Gatt, and T. Klausutis, “Range precision of direct detection laser radar systems,” Proc. SPIE 5412, 72-86 (2004).
[CrossRef]

P. Gatt and S. Henderson, “Laser radar detection statistics: a comparison of coherent and direct detection intensity receivers,” Proc. SPIE 4377, 251-262 (2001).
[CrossRef]

Other

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1965).

G. Casella and R. Berger, Statistical Inference (Duxbury, 2002).

A. Siegman, Lasers (University Science Books, 1986).

S. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice Hall, 1993).

I. Gradshteyn and I. Ryzhik, Table of Integrals, Series, and Products (Academic, 2000).

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

A. Jelalian, Laser Radar Systems (Artech House, 1992).

G. Osche, Optical Detection Theory (Wiley-Interscience, 2002).

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Figures (3)

Fig. 1
Fig. 1

Ladar signal model.

Fig. 2
Fig. 2

Range estimation simulations.

Fig. 3
Fig. 3

CRLB as a function of number of pulses for a fixed total laser energy.

Equations (47)

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I ( t k ) = A [ 1 ( t k 2 R / c p w ) 2 ] rect ( t k 2 R / c 2 p w ) + B ,
p D ( t k ) [ d ( t k ) ; R , A , B ] = Prob [ D ( t k ) = d ( t k ) ] = 1 d ( t k ) ! I ( t k ) d ( t k ) exp [ I ( t k ) ] ,
E [ D ( t k ) ] = Var [ D ( t k ) ] = I ( t k ) .
E = h ν k = 1 K [ I ( t k ) B ] rect ( t k 2 R / c 2 p w ) ,
E h ν f s 2 R / c p w 2 R / c + p w A [ 1 ( t 2 R / c p w ) 2 ] d t .
v = t 2 R / c
E h ν f s A p w p w ( 1 v 2 p w 2 ) d v = 4 3 h ν f s A p w .
P = h ν f s B .
p D ( d ; R , A , B ) = k = 1 K 1 d ( t k ) ! I ( t k ) d ( t k ) exp [ I ( t k ) ] ,
l ( R , A , B ; d ) = k = 1 K log [ d ( t k ) ! ] + k = 1 K d ( t k ) log [ I ( t k ) ] k = 1 K I ( t k ) .
E [ l ( θ ) θ i ] = 0     θ i θ ,
θ T = [ R A B ] .
l ( θ ) θ i = k = 1 K [ d ( t k ) I ( t k ) 1 ] I ( t k ) θ i .
[ J ] i j = E [ 2 l ( θ ) θ i θ j ] .
[ J ] i i = k = 1 K 1 I ( t k ) [ I ( t k ) θ i ] 2 ,
[ J ] i j = k = 1 K 1 I ( t k ) I ( t k ) θ i I ( t k ) θ j .
I ( t k ) R 4 A c p w 2 ( t k 2 R / c ) rect ( t k 2 R / c 2 p w ) ,
I ( t k ) A [ 1 ( t k 2 R / c p w ) 2 ] rect ( t k 2 R / c 2 p w ) ,
I ( t k ) B = 1 .
J = [ J R R J R A J R B J R A J A A J A B J R B J A B J B B ] .
J R R 16 A 2 c 2 p w 4 k = 1 K ( t k 2 R / c ) 2 A [ 1 ( t k 2 R / c p w ) 2 ] + B rect ( t k 2 R / c 2 p w ) .
J R R 16 A 2 f s c 2 p w 4 2 R / c p w 2 R / c + p w ( t 2 R / c ) 2 A [ 1 ( t 2 R / c p w ) 2 ] + B d t .
u = t 2 R / c p w A A + B
J R R 16 A f s c 2 p w A + B A A / ( A + B ) A / ( A + B ) u 2 1 u 2 d u .
x 2 1 x 2 d x = arctanh ( x ) x
J R R 32 A f s c 2 p w [ A + B A arctanh ( A A + B ) 1 ] .
J R A 4 A f s c p w 2 2 R / c p w 2 R / c + p w ( t 2 R / c ) [ 1 ( t 2 R / c p w ) 2 ] A [ 1 ( t 2 R / c p w ) 2 ] + B d t = 4 A f s c p w 2 p w p w v ( 1 v 2 p w 2 ) A ( 1 v 2 p w 2 ) + B d v ,
J R A 0 .
J R B 0 .
J [ 32 A f s c 2 p w [ A + B A arctanh ( A A + B ) 1 ] 0 0 0 J A A J A B 0 J A B J B B ] .
[ A 0 0 B ] 1 = [ A 1 0 0 B 1 ] .
Var [ R ^ ] c 2 p w 32 A f s [ A + B A arctanh ( A A + B ) 1 ] ,
D g ( t k ) N [ I ( t k ) , B ] .
Var [ R ^ g ] B c 2 4 f s 0 t d [ d I ( t ) d t ] 2 d t ,
0 t d [ d I ( t ) d t ] 2 d t = 2 R / c p w 2 R / c + p w [ 2 A p w 2 ( t 2 R / c ) ] 2 d t = 4 A 2 p w 4 p w p w v 2 d v = 8 A 2 3 p w ,
Var [ R ^ g ] 3 B c 2 p w 32 A 2 f s .
arctanh ( x ) = n = 0 x 2 n + 1 2 n + 1 ,
arctanh ( x ) x 1 = n = 1 x 2 n 2 n + 1 = 1 3 n = 1 3 x 2 n 2 n + 1 < x 2 3 n = 0 x 2 n = x 2 3 ( 1 x 2 ) .
A + B A arctanh ( A A + B ) 1 < A 3 B .
c 2 p w 32 A f s [ A + B A arctanh ( A A + B ) 1 ] > 3 B c 2 p w 32 A 2 f s .
Var [ R ¯ ^ ] c 2 p w 32 A f s [ A + B N A arctanh ( A A + B N ) 1 ] ,
A + B N A arctanh ( A A + B N ) 1 = n = 1 1 2 n + 1 ( A A + B N ) n .
A A + B N < A A + B     N > 1 .
n = 1 1 2 n + 1 ( A A + B N ) n < n = 1 1 2 n + 1 ( A A + B ) n     N > 1 .
A + B N A arctanh ( A A + B N ) 1 A + B A arctanh ( A A + B ) 1
c 2 p w 32 A f s [ A + B N A arctanh ( A A + B N ) 1 ] c 2 p w 32 A f s [ A + B A arctanh ( A A + B ) 1 ] .
Var [ R ¯ ^ ] Var [ R ^ ]

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