Abstract

Expressions for the minimum number of photons required to measure the position of a light pattern on a noncoherent detector array with a desired accuracy are obtained from the Cramér–Rao lower bounds. The dependence of the Cramér–Rao bound on the shape, on the statistical properties of the intensity, on spatial sampling by the detector array, and on the array area is examined quantitatively. Examples of linear and nonlinear algorithms for measuring the pattern position with accuracies approaching the Cramér–Rao bound are presented. The analysis is carried out for photon statistics modeled as conditional Poisson processes.

© 1983 Optical Society of America

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References

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  1. J. H. Shapiro, “Imaging and optical communication through atmospheric turbulence,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, Berlin, 1978), pp. 183–193.
  2. C. McIntyre, W. N. Peters, C. Chi, and H. F. Wischnia, “Optical components and technology in laser space communications systems,” Proc. IEEE 58, 1491–1503 (1970).
    [Crossref]
  3. M. Elbaum, P. Diament, M. King, and B. Edelson, “Maximum angular accuracy of pulsed laser radar in photocounting limit,” Appl. Opt. 16, 1982–1992 (1977).
    [Crossref] [PubMed]
  4. D. L. Snyder, Random Point Processes (Wiley, New York, 1975), p. 450.
  5. M. Elbaum and M. Greenebaum, “Angular apertures for angular tracking,” Appl. Opt. 16, 2438–2440 (1977).
    [Crossref] [PubMed]
  6. J. F. Walkup and J. W. Goodman, “Limitations of fringe parameter estimation at low light levels,” J. Opt. Soc. Am. 63, 399–407 (1973).
    [Crossref]
  7. M. Elbaum and P. Diament, “Estimation of image centroid size, and orientation with laser radar,” Appl. Opt. 16, 2433–2437 (1977).
    [Crossref] [PubMed]
  8. M. Elbaum, N. Orenstein, and J. MacEachin, “Algorithms for estimating image position,” Proc. Soc. Photo-Opt. Instrum. Eng. 252, 98–104 (1980).
  9. P. M. Solomon and T. A. Glavich, “Image signal processing in sub-pixel accuracy star trackers,” Proc. Soc. Photo-Opt. Instrum. Eng. 252, 64–74 (1980).
  10. L. Mandel, “Photoelectric counting measurements as a test for the existence of photons,” J. Opt. Soc. Am. 67, 1101–1104 (1977).
    [Crossref]
  11. L. Mandel, “Fluctuations of photon beams: distribution of the photo-electrons,” Proc. Phys. Soc. London 74, 233–243 (1959).
    [Crossref]
  12. H. Cramér, Mathematical Methods of Statistics (Princeton U. Press, Princeton, N.J., 1946).
  13. J. Nowakowski and M. Elbaum, “Fundamental limits in estimating light pattern position,” , Vol. II of Studies of Electro-Optical Sensors for Laser Applications (Riverside Research Institute, New York, February1982).
  14. J. Nowakowski, “Barankin bound and Cramér–Rao bound: an interpretation,” (Riverside Research Institute, New York, December1981).
  15. J. Nowakowski, “Barankin type bounds and estimators,” (Riverside Research Institute, New York, December1981).

1980 (2)

M. Elbaum, N. Orenstein, and J. MacEachin, “Algorithms for estimating image position,” Proc. Soc. Photo-Opt. Instrum. Eng. 252, 98–104 (1980).

P. M. Solomon and T. A. Glavich, “Image signal processing in sub-pixel accuracy star trackers,” Proc. Soc. Photo-Opt. Instrum. Eng. 252, 64–74 (1980).

1977 (4)

1973 (1)

1970 (1)

C. McIntyre, W. N. Peters, C. Chi, and H. F. Wischnia, “Optical components and technology in laser space communications systems,” Proc. IEEE 58, 1491–1503 (1970).
[Crossref]

1959 (1)

L. Mandel, “Fluctuations of photon beams: distribution of the photo-electrons,” Proc. Phys. Soc. London 74, 233–243 (1959).
[Crossref]

Chi, C.

C. McIntyre, W. N. Peters, C. Chi, and H. F. Wischnia, “Optical components and technology in laser space communications systems,” Proc. IEEE 58, 1491–1503 (1970).
[Crossref]

Cramér, H.

H. Cramér, Mathematical Methods of Statistics (Princeton U. Press, Princeton, N.J., 1946).

Diament, P.

Edelson, B.

Elbaum, M.

M. Elbaum, N. Orenstein, and J. MacEachin, “Algorithms for estimating image position,” Proc. Soc. Photo-Opt. Instrum. Eng. 252, 98–104 (1980).

M. Elbaum and P. Diament, “Estimation of image centroid size, and orientation with laser radar,” Appl. Opt. 16, 2433–2437 (1977).
[Crossref] [PubMed]

M. Elbaum and M. Greenebaum, “Angular apertures for angular tracking,” Appl. Opt. 16, 2438–2440 (1977).
[Crossref] [PubMed]

M. Elbaum, P. Diament, M. King, and B. Edelson, “Maximum angular accuracy of pulsed laser radar in photocounting limit,” Appl. Opt. 16, 1982–1992 (1977).
[Crossref] [PubMed]

J. Nowakowski and M. Elbaum, “Fundamental limits in estimating light pattern position,” , Vol. II of Studies of Electro-Optical Sensors for Laser Applications (Riverside Research Institute, New York, February1982).

Glavich, T. A.

P. M. Solomon and T. A. Glavich, “Image signal processing in sub-pixel accuracy star trackers,” Proc. Soc. Photo-Opt. Instrum. Eng. 252, 64–74 (1980).

Goodman, J. W.

Greenebaum, M.

King, M.

MacEachin, J.

M. Elbaum, N. Orenstein, and J. MacEachin, “Algorithms for estimating image position,” Proc. Soc. Photo-Opt. Instrum. Eng. 252, 98–104 (1980).

Mandel, L.

L. Mandel, “Photoelectric counting measurements as a test for the existence of photons,” J. Opt. Soc. Am. 67, 1101–1104 (1977).
[Crossref]

L. Mandel, “Fluctuations of photon beams: distribution of the photo-electrons,” Proc. Phys. Soc. London 74, 233–243 (1959).
[Crossref]

McIntyre, C.

C. McIntyre, W. N. Peters, C. Chi, and H. F. Wischnia, “Optical components and technology in laser space communications systems,” Proc. IEEE 58, 1491–1503 (1970).
[Crossref]

Nowakowski, J.

J. Nowakowski, “Barankin bound and Cramér–Rao bound: an interpretation,” (Riverside Research Institute, New York, December1981).

J. Nowakowski, “Barankin type bounds and estimators,” (Riverside Research Institute, New York, December1981).

J. Nowakowski and M. Elbaum, “Fundamental limits in estimating light pattern position,” , Vol. II of Studies of Electro-Optical Sensors for Laser Applications (Riverside Research Institute, New York, February1982).

Orenstein, N.

M. Elbaum, N. Orenstein, and J. MacEachin, “Algorithms for estimating image position,” Proc. Soc. Photo-Opt. Instrum. Eng. 252, 98–104 (1980).

Peters, W. N.

C. McIntyre, W. N. Peters, C. Chi, and H. F. Wischnia, “Optical components and technology in laser space communications systems,” Proc. IEEE 58, 1491–1503 (1970).
[Crossref]

Shapiro, J. H.

J. H. Shapiro, “Imaging and optical communication through atmospheric turbulence,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, Berlin, 1978), pp. 183–193.

Snyder, D. L.

D. L. Snyder, Random Point Processes (Wiley, New York, 1975), p. 450.

Solomon, P. M.

P. M. Solomon and T. A. Glavich, “Image signal processing in sub-pixel accuracy star trackers,” Proc. Soc. Photo-Opt. Instrum. Eng. 252, 64–74 (1980).

Walkup, J. F.

Wischnia, H. F.

C. McIntyre, W. N. Peters, C. Chi, and H. F. Wischnia, “Optical components and technology in laser space communications systems,” Proc. IEEE 58, 1491–1503 (1970).
[Crossref]

Appl. Opt. (3)

J. Opt. Soc. Am. (2)

Proc. IEEE (1)

C. McIntyre, W. N. Peters, C. Chi, and H. F. Wischnia, “Optical components and technology in laser space communications systems,” Proc. IEEE 58, 1491–1503 (1970).
[Crossref]

Proc. Phys. Soc. London (1)

L. Mandel, “Fluctuations of photon beams: distribution of the photo-electrons,” Proc. Phys. Soc. London 74, 233–243 (1959).
[Crossref]

Proc. Soc. Photo-Opt. Instrum. Eng. (2)

M. Elbaum, N. Orenstein, and J. MacEachin, “Algorithms for estimating image position,” Proc. Soc. Photo-Opt. Instrum. Eng. 252, 98–104 (1980).

P. M. Solomon and T. A. Glavich, “Image signal processing in sub-pixel accuracy star trackers,” Proc. Soc. Photo-Opt. Instrum. Eng. 252, 64–74 (1980).

Other (6)

J. H. Shapiro, “Imaging and optical communication through atmospheric turbulence,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, Berlin, 1978), pp. 183–193.

D. L. Snyder, Random Point Processes (Wiley, New York, 1975), p. 450.

H. Cramér, Mathematical Methods of Statistics (Princeton U. Press, Princeton, N.J., 1946).

J. Nowakowski and M. Elbaum, “Fundamental limits in estimating light pattern position,” , Vol. II of Studies of Electro-Optical Sensors for Laser Applications (Riverside Research Institute, New York, February1982).

J. Nowakowski, “Barankin bound and Cramér–Rao bound: an interpretation,” (Riverside Research Institute, New York, December1981).

J. Nowakowski, “Barankin type bounds and estimators,” (Riverside Research Institute, New York, December1981).

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Figures (11)

Fig. 1
Fig. 1

Dependence of normalized CRB on the pattern displacement Δx; two-detector array; pattern h1 and h2 are defined by Eqs. (3.16).

Fig. 2
Fig. 2

Dependence of normalized CRB on detector array area At, where AT = 4lT2; detectors with small apertures; patterns h1 and h2 are defined by Eqs. (3.16).

Fig. 3
Fig. 3

Dependence of NCRB(L; AT) on the number of detectors L and detector array area AT. Patterns h1 and h2 are defined by Eqs. (3.16). Detector array area is AT =(6.6)2 for pattern h1. For pattern h2, AT is equal to (2π)2, (4π)2, (8π)2, and (20π)2 for cases (a), (b), (c), and (d), respectively.

Fig. 4
Fig. 4

Dependence of NCRB on the four-quadrant detector area AT = (2lT)2. Patterns h1 and h2 are defined by Eqs. (3.16).

Fig. 5
Fig. 5

Dependence of NCRB(4, ρ/Ns, AT) on the four-quadrant detector area AT = (2lT)2 in presence of background noise. Patterns h1 and h2 are defined by Eqs. (3.16). The value of ρ/Ns is selected to be 10−1. Nonfading signal.

Fig. 6
Fig. 6

Optimal detector size lopt for the four-quadrant detector as a function of the ratio of background-noise density ρ to signal value Ns. Patterns h1 and h2 are defined by Eqs. (3.16). Nonfading signal.

Fig. 7
Fig. 7

Dependence of NCRB(4, ρ/Ns, Aopt) for the four-quadrant detector of optimal size as a function of the ratio of background noise density ρ to signal value Ns. Patterns h1 and h2 are defined by Eqs. (3.16). Nonfading signal.

Fig. 8
Fig. 8

Dependence of NCRB(∞, ρ/Ns, AT) on the ratio of background noise density ρ to signal value Ns. Patterns h1 and h2 are defined by Eqs. (3.16). Detector array area is AT = (6.6)2 for pattern h1; for curve (a) AT = (2π)2 and for curve (b) AT = (6π)2 for pattern h2. Detectors have small apertures. Nonfading signal.

Fig. 9
Fig. 9

Ratio of CRB’s Δ(ρ/Ns) for optimal-size four-quadrant detector to the CRB for a large number of detectors as a function of the ratio of background-noise density ρ to signal value Ns. Patterns h1 and h2 are defined by Eqs. (3.16). Nonfading signal.

Fig. 10
Fig. 10

Signal Ns necessary to obtain the CRB of 0.01 as a function of background-noise density ρ. Four-quadrant detector of optimal aperture area. Patterns h1 and h2 are defined by Eqs. (3.16). Nonfading signal.

Fig. 11
Fig. 11

Signal Ns necessary to obtain the CRB of 0.01 as a function of background-noise density ρ. Detectors have small apertures. Patterns h1 and h2 are defined by Eqs. (3.16). Nonfading signal.

Equations (84)

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W = t t + T d t A d 2 x I ( x , t ) ,
p 1 ( n ) = 0 d N ( N n ) exp ( - N ) P ( N ) / n ! ,
- d 2 x h ( x ) = 1 ,             S k = A k d 2 x h ( x ) .
h ( x 1 , x 2 ) = h ( x 2 , x 1 ) = h ( - x 1 , x 2 ) = h ( x 1 , - x 2 ) .
S k ( Δ x ) = A k d 2 x h ( x - Δ x ) .
k = 1 L S k ( Δ x ) = H ( Δ x ) ,
p s ( n Δ x ) = 0 d N P ( N ) q s ( n N , Δ x ) ,
q s ( n N , Δ x ) = k = 1 L [ N S k ( Δ x ) ] n k exp [ - N S k ( Δ x ) ] / n k ! .
p ( m Δ x ) = 0 d N P ( N ) q ( m N , Δ x ) ,
q ( m N , Δ x ) = k = 1 L [ N k ( Δ x ) ] m k exp [ - N k ( Δ x ) ] / m k !
N k ( Δ x ) = N S k ( Δ x ) + A k ρ .
g ( ξ , ξ Δ x ) = p ( m ξ ) p ( m ξ ) p ( m Δ x ) p ( m Δ x ) m < ,
J i j ( Δ x ) = J i j = 2 ξ i ξ j g ( ξ , ξ Δ x ) ξ = ξ = Δ x ,
J ˜ i j = ( J - 1 ) i j ,
J ˜ 11 = J 22 / det J ,             J ˜ 22 = J 11 / det J ,             var ( Δ x ^ i ) J ˜ i i = CRB ,
g ( ξ , ξ Δ x ) = m 1 = 0 m L = 0 × ( { k = 1 L 1 m k ! } k = 1 L [ N k ( ξ ) ] m k exp [ - N k ( ξ ) ] N × k = 1 L [ N k ( ξ ) ] m k exp [ - N k ( ξ ) ] N k = 1 L [ N k ( Δ x ) ] m k exp [ - N k ( Δ x ) ] N ) ,
g ( ξ , ξ Δ x ) = m = 0 a m ,
a m = 1 m ! k 1 = 1 L k m = 1 L ( { exp [ - N T ( ξ ) ] i = 1 m N k i ( ξ ) N } × exp [ - N T ( ξ ) ] i = 1 m N k i ( ξ ) N exp [ - N T ( Δ x ) ] i = 1 m N k i ( Δ x ) N ) , N T ( ξ ) = k = 1 L N k ( ξ ) = N H ( ξ ) + ρ A T ,
a m = N m exp [ - NH ( ξ ) ] N N m exp [ - N H ( ξ ) ] N N m exp [ - N H ( Δ x ) ] N × { 1 m ! [ k = 1 1 S k ( ξ ) S k ( ξ ) S k ( Δ x ) ] m } .
J i j = N S k = 1 L S k ( Δ x ) Δ x i S k ( Δ x ) Δ x j 1 S k ( Δ x ) + H ( Δ x ) Δ x i H ( Δ x ) Δ x j × { m = 0 [ H ( Δ x ) ] m m ! N m + 1 exp [ - N H ( Δ x ) ] N 2 N m exp [ - N H ( Δ x ) ] N - N 2 N } .
J i j = N S k = 1 L S k ( Δ x ) Δ x i S k ( Δ x ) Δ x j 1 S k ( Δ x ) .
S 1 ( Δ x ) = - 0 h ( ξ - Δ x ) d ξ = 1 / 2 - 0 Δ x h ( ξ ) d ξ , S 2 ( Δ x ) = 0 h ( ξ - Δ x ) d ξ = 1 / 2 + 0 Δ x h ( ξ ) d ξ , S i ( Δ x ) Δ x = ( - 1 ) i h ( Δ x ) ;             i = 1 , 2 ,
var Δ x ^ CRB ( Δ x ; 2 ) = 1 N s 1 / 4 - [ 0 Δ x h ( ξ ) d ξ ] 2 h 2 ( Δ x ) .
var Δ x ^ Δ x = 0 1 N s h 2 ( 0 ) × { 1 / 4 - ( Δ x ) 2 [ h 2 ( 0 ) + h ( 0 ) 4 h ( 0 ) ] + } .
CRB ( 0 ; 2 ) = 1 N s 1 4 h 2 ( 0 ) .
var Δ x ^ i Δ x = 0 CRB ( 0 ; 4 ) = 1 16 N s 1 [ 0 h ( 0 , ξ ) d ξ ] 2 .
h ( x ) = 1 2 π σ 2 exp [ - ( x 1 2 + x 2 2 ) / 2 σ 2 ] ,             pattern h 1 , h ( x ) = sin 2 a x 1 π a x 1 2 sin 2 a x 2 π a x 2 2 ,             pattern h 2 ,
h 1 ( x ) = h 1 ( x 1 ) h 1 ( x 2 ) = 1 2 π exp [ - ( x 1 2 + x 2 2 ) / 2 ] ,             pattern h 1 , h 2 ( x ) = h 2 ( x 2 ) h 2 ( x 2 ) = sin 2 x 1 π x 1 2 sin 2 x 2 π x 2 2 ,             pattern h 2 .
var Δ x ^ i | Δ x = 0 CRB ( 0 , L ) = 1 N s 2 k = 1 L [ S k ( Δ x ) ] 2 / S k ( Δ x ) | Δ x = 0 ,
S k ( Δ x ) h ( x k - Δ x ) A k ,
J i j N s k A k [ h ( x k - Δ x ) Δ x i h ( x k - Δ x ) Δ x j h ( x k - Δ x ) ] .
J 12 = J 21 = 0 ,             J 11 = J 22 .
var Δ x ^ i CRB ( ) = 1 N s 2 - d 2 x h ( x ) 2 h ( x ) ,
h 2 = [ h ( x ) x 1 ] 2 + [ h ( x ) x 2 ] 2 .
CRB ( A T ) = 1 N s 2 A T d 2 x [ h ( x ) 2 / h ( x ) ] .
CRB ( 4 ; 4 T ) = 1 4 N s ( 1 π - sin 2 l T π l T 2 ) 2 ,
J 12 = J 21 = 0 , J 22 = N s k = 1 L [ S k ( Δ x ) Δ x 2 ] 2 1 S k ( Δ x ) | Δ x 2 = 0 , J 11 = N 2 k = 1 L [ S k ( Δ x ) Δ x 1 ] 2 1 S k ( Δ x ) + [ H ( Δ x ) Δ x 1 ] 2 { m = 0 exp [ - H ( Δ x ) N ] N m + 1 N 2 exp [ - H ( Δ x ) N ] N m N × [ H ( Δ x ) ] m m ! - N 2 N } | Δ x 2 = 0 .
var Δ x ^ 1 1 J 11 , var Δ x ^ 2 1 J 22 ,
J 11 J 11 N F = N s k = 1 L [ S k ( Δ x ) Δ x 1 ] 2 1 S k ( Δ x ) ,
a m N F = 1 m ! exp { - N s [ H ( ξ ) + H ( ξ ) - H ( Δ x ) ] - A T ρ } [ k = 1 L N k ( ξ ) N k ( ξ ) N k ( Δ x ) ] m ,
J i j N F = N s k = 1 L [ S k ( Δ x ) Δ x i ] [ S k ( Δ x ) Δ x j ] 1 S k ( Δ x ) + ( ρ / N s ) A k .
var Δ ^ x i Δ x = 0 CRB N F ( Δ x = 0 ; N s ; ρ ; L ; A T ) = 1 N s 1 k = 1 L [ S k ( Δ x ) Δ x i ] 2 1 S k ( Δ x ) + ( ρ / N s ) A k | Δ x = 0 .
var Δ ^ x i Δ x = 0 CRB N F ( Δ x = 0 ; N s ; ρ ; L = 4 ; A T ) = 1 4 N s ρ N s l T 2 + 0 l T 0 l T h ( x ) d 2 x { 0 l T [ h ( 0 , ξ ) - h ( l T , ξ ) ] d ξ } 2 .
var Δ ^ x i Δ x = 0 CRB N F ( Δ x = 0 ; N s ; ρ ; L = ρ ; A T ) = 1 N s 2 A T h ( x ) 2 d 2 x h ( x ) + ρ / N s .
var Δ ^ x i CRB N F ( Δ x ; N s ; ρ ; L = ; A T = ) = 1 N s 2 - h ( x ) 2 d 2 x h ( x ) + ρ / N s .
J i j = J δ i j ,             J = exp ( - ρ A T ) m = 0 b m , b m = 1 m ! ( k = 1 L ( 1 / 2 ) { [ S k ( Δ x ) Δ x 1 ] 2 + [ S k ( Δ x ) Δ x 2 ] 2 } | Δ x = 0 × k 1 = 1 L k m = 1 L exp ( - N H ) N i = 1 m N k i ( 0 ) N 2 exp ( - N H ) N k ( 0 ) i = 1 m N k i ( 0 ) N ) ,
J i j = J δ i j ,             J = lim A T exp ( - ρ A T ) m = 0 b m , b m = 1 m ! A T 1 / 2 h ( x ) 2 d 2 x A T d 2 x 1 A T d 2 x m N exp ( - N ) i = 1 m N x i N 2 exp ( - N ) N ( x ) i = 1 m N x 1 N , N x i = N h ( x i ) + ρ .
CRB 0 L CRB k L CRB CBR k u CBR 0 u , CRB k L k CRB ,             CRB k u k CRB .
CRB 0 L = 1 k = 1 L ( 1 / 2 ) { [ S k ( Δ x ) Δ x 1 ] 1 + [ S k ( Δ x ) Δ x 2 ] 2 } N 2 N S k ( Δ x ) + ρ A N | Δ x = 0 CRB 1 k = 1 L ( 1 / 2 ) { [ S k ( Δ x ) Δ x 1 ] 2 + [ S k ( Δ x ) Δ x 2 ] 2 } N s 2 N s S k ( Δ x ) + ρ A | Δ x = 0             = CRB 0 u = CRB N F .
CRB 0 L ( 4 ) = 1 4 { 0 l T [ h ( 0 , ξ ) - h ( l T , ξ ) ] d ξ } 2 N 2 N 00 l T l T h ( x ) d 2 x + ρ l T 2 N CRB ( 4 ) 1 4 N ρ N s l T 2 + 0 l T 0 l T h ( x ) d 2 x { 0 L T [ h ( 0 , ξ ) - h ( l T , ξ ) ] d ξ } 2 = CRB 0 u ( 4 ) = CRB N F ( 4 ) .
CRB 0 L ( ) = 2 - h ( x ) 2 N 2 N h ( x ) + ρ N d 2 x CRB ( ) 2 - h ( x ) N s 2 d 2 x N s h ( x ) + ρ = CRB 0 u ( ) = CRB N F ( ) .
var Δ x ^ i CRB k L .
N s 2 N 2 N CRB 0 L CRB N F CRB CRB N F 1.
Δ x ^ = k m k σ k = [ k m k σ 1 ( x k ) , k m k σ 2 ( x k ) ]
σ k = 1 N s ln [ h ( x ) + ρ / N s ] x = x k - d 2 x h ( x ) 2 h ( x ) + ρ / N s .
b ( Δ x ) = Δ x ^ - Δ x = - [ h ( x + Δ x ) + ρ / N s ] ln [ h ( x ) + ρ / N s ] d 2 x - d 2 x h ( x ) 2 [ h ( x ) + ρ / N s ] - Δ x .
var Δ x ^ Δ x = 0 = 1 N s 1 - d 2 x h ( x ) 2 h ( x ) + ρ / N s .
σ ( x k ) = 1 N s ln [ S k ( Δ x ) + ( ρ / N s ) A ] Δ x = 0 k = 1 L { [ S k ( Δ x ) Δ x 1 ] 2 + [ S k ( Δ x ) Δ x 2 ] 2 } 1 S k ( Δ x ) + ( ρ / N s ) A | Δ x = 0 .
Δ x ^ = k m k σ k ,
Δ x ^ 1 = ( n 1 + n 2 - n 3 - n 4 ) 1 N s 1 0 h ( 0 , ξ ) d ξ , Δ x ^ 2 = ( n 1 + n 4 - n 2 - n 3 ) 1 N s 1 4 0 h ( η , 0 ) d η ,
Δ x ^ = k σ k m k k m k .
Δ x ^ i = j = 1 2 [ ξ j ln p ( m ξ ) J ˜ i j ( 0 ) ] ξ = 0 .
a 2 b 2 a b 2 ,
( α a α 2 ) ( α b α 2 ) ( α a α b α ) 2 ,
A 4 B 2 B 2 A 2 2 ,
α A α 4 B α 2 α B α 2 ( α A α 2 ) 2 .
A 2 2 B 2 A 4 B 2 ,
α A α 4 B α 2 ( α A α 2 ) 2 α B α 2 .
J i j = J δ i j ,             m = 0 b m exp ( - ρ A T ) , b m = k ( 1 / 2 ) { [ S k ( Δ x ) Δ x 1 ] 2 + [ S k ( Δ x ) Δ x 2 ] 2 } | Δ x = 0 C m k , C m k = 1 m ! k 1 = 1 L k m = 1 L × exp [ - N H ( 0 ) ] N i = 1 m N k i ( 0 ) N 2 exp [ - N H ( 0 ) ] N k ( 0 ) i = 1 m N k i ( 0 ) N ,
A 2 = exp [ - N H ( 0 ) ] N i = 1 m N k i ( 0 ) , B 2 = exp [ - N H ( 0 ) ] N K ( 0 ) i = 1 m N k i ( 0 ) .
exp [ - N H ( 0 ) ] N i = 1 m N k i ( 0 ) N 2 exp [ - N H ( 0 ) N k ( 0 ) i = 1 m N k i ( 0 ) ] N exp [ - N H ( 0 ) N 2 N k ( 0 ) i = 1 m N k i ( 0 ) ] N
C m k k i = 1 L k m = 1 L 1 m ! exp [ - N H ( 0 ) ] N 2 N k ( 0 ) i = 1 m N k i ( 0 ) N = 1 m ! exp [ - N H ( 0 ) ] N 2 N k ( 0 ) [ N T ( 0 ) ] m N ,
J exp ( - ρ A T ) k = 1 L ( 1 / 2 ) { [ S k ( Δ x 1 ) Δ x 1 ] 2 + [ S k ( Δ x ) Δ x 2 ] 2 } | Δ x = 0 × exp [ - N H ( 0 ) { m = 0 1 m ! [ N T ( 0 ) ] m } N 2 N k ( 0 ) N = k = 1 L ( 1 / 2 ) { [ S k ( Δ x 1 ) Δ x 1 ] 2 + [ S k ( Δ x ) Δ x 2 ] 2 } | Δ x = 0 × N 2 N k ( 0 ) N = J 0 u .
1 J 0 u = CRB 0 L CRB .
A α 2 = exp [ - N H ( 0 ) ] N i = 1 m N k i ( 0 ) , B α 2 = exp [ - N H ( 0 ) ] N k ( 0 ) i = 1 m N k i ( 0 ) .
( k 1 , k 2 , k m ) exp [ - N H ( 0 ) ] N i = 1 m N k i ( 0 ) N 2 exp [ - N H ( 0 ) ] N k ( 0 ) i = 1 m N k i ( 0 ) k 1 , k 2 , , k m exp [ - N H ] ( 0 ) N i = 1 m N k i ( 0 ) N 2 k 1 , k 2 , k m exp [ - N H ( 0 ) ] N k ( 0 ) i = 1 m N k i ( 0 ) N = exp [ - N H ( 0 ) N ( N T ) m ] N 2 exp [ - N H ( 0 [ N k ( 0 ) N T m N
J J ˜ 0 L = exp ( - ρ A T ) k = 1 L ( 1 / 2 ) { [ S k ( Δ x ) Δ x 1 ] 2 + [ S k ( Δ x ) Δ x 2 ] 2 } | Δ x = 0 × m = 0 1 m ! exp [ - N H ( 0 ) N N T m N 2 exp [ - N H ( 0 ) N k ( 0 ) N T m N .
CRB 1 J ˜ 0 L = C R ˜ B 0 u .
m = 0 1 m ! exp [ - N H ( 0 ) ] N N T m N 2 exp [ - N H ( 0 ) ] N k ( 0 ) N T m N { m = 0 1 m ! exp [ - N H ( 0 ) ] N N T m T } 2 { m = 0 1 m ! exp [ - N H ( 0 ) ] N k ( 0 ) N T m N } = N S 2 exp ( 2 ρ A T ) N k ( 0 ) exp ( ρ A T ) = N S 1 S k ( 0 ) + ( ρ / N s ) A exp ( ρ A T ) ,
J J ˜ 0 L J 0 L = N s k = 1 L ( 1 / 2 ) { ( S k ( Δ x ) Δ x 1 ) 2 + [ S k ( Δ x ) Δ x 2 ] 2 } | Δ x = 0 1 S k ( 0 ) + ( ρ / N s ) A = J N F ,
CRB 1 J 0 L = 1 J N F = CRB 0 u = CRB N F ,
Z = m = 0 exp [ - N H ( Δ x ) ] N m + 1 N 2 exp [ - N H ( Δ x ) ] N m N [ H ( Δ x ) ] m m ! - N 2 N < 0.
exp [ - N H ( Δ x ) ] N m + 1 N 2 exp [ - N H ( Δ x ) ] N m N exp [ - N H ( Δ x ) ] N 2 + m N ,
Z m = 0 [ H ( Δ x ) ] m m ! exp [ - N H ( Δ x ) ] N 2 + m N - N 2 N = N 2 N - N 2 N = 0 ,