Abstract

We study the imaging sensitivity of a ground-based optical array of n apertures in which the beams are combined pairwise, as in radio-interferometric arrays, onto n(n − 1)/2 detectors, the so-called nC2 interferometer. Ground-based operation forces the use of the fringe power and the bispectrum phasor as the primary observables rather than the simpler and superior observable, the Michelson fringe phasor. At high photon rates we find that bispectral imaging suffers no loss of sensitivity compared with an ideal array (space based) that directly uses the Michelson fringe phasor. In the opposite limit, when the number of photons per spatial coherence area per coherence time drops below unity, the sensitivity of the array drops rapidly relative to an ideal array. In this regime the sensitivity is independent of n, and hence it may be efficient to have many smaller arrays, each operating separately and simultaneously.

© 1991 Optical Society of America

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  1. S. Prasad, S. R. Kulkarni, “Noise in optical synthesis images. I. Ideal Michelson interferometer,” J. Opt. Soc. Am. A 6, 1702–1714 (1989).
    [CrossRef]
  2. S. R. Kulkarni, in JPL Workshop on Space Interferometry, M. Shao, S. R. Kulkarni, eds. (NASA, Washington, D.C., to be published).
  3. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1970), p. 510.
  4. T. J. Pearson, A. C. S. Readhead, “Image formation by self-calibration in radio astronomy,” Annu. Rev. Astron. Astrophys. 22, 97–130 (1984).
    [CrossRef]
  5. A. R. Thompson, B. G. Clark, C. M. Wade, P. J. Napier, “The very large array,” Astrophys. J. Suppl. Ser. 44, 151–167 (1980).
    [CrossRef]
  6. T. Nakajima, S. R. Kulkarni, “Noise in optical synthesis images. IV. Effects of atmospheric disturbances,” submitted to J. Opt. Soc. Am. A.
  7. J. C. Dainty, A. H. Greenaway, “Estimation of spatial power spectra in speckle interferometry,”J. Opt. Soc. Am. 69, 786–790 (1979).
    [CrossRef]
  8. B. Wirnitzer, “Bispectral analysis at low light levels and astronomical speckle masking,” J. Opt. Soc. Am. A 2, 14–21 (1985).
    [CrossRef]
  9. G. R. Ayers, M. J. Northcott, J. C. Dainty, “Knox–Thompson and triple-correlation imaging through turbulence,” J. Opt. Soc. Am. A 5, 963–985 (1988).
    [CrossRef]
  10. T. Nakajima, “Signal-to-noise ratio of the bispectral analysis of speckle interferometry,” J. Opt. Soc. Am. A 5, 1477–1491 (1988).
    [CrossRef]
  11. J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 5.
  12. T. J. Cornwell, “Radio-interferometric imaging of weak objects in conditions of poor phase stability: the relationship between speckle masking and phase closure methods,” Astron. Astrophys. 180, 269–274 (1987).
  13. P. W. Gorham, A. M. Ghez, S. R. Kulkarni, T. Nakajima, G. Neugebauer, J. B. Oke, T. A. Prince, “Diffraction-limited imaging. III. 30 mas closure phase imaging of six binary stars with the Hale 5 m telescope,” Astron. J. 98, 1783–1799 (1989).
    [CrossRef]
  14. T. J. Cornwell, P. N. Wilkinson, “A new method for making maps with unstable radio interferometers,” Mon. Not. R. Astron. Soc. 196, 1067–1086 (1981).
  15. F. R. Schwab, “Adaptive calibration of radio interferometer data,” in 1980 International Optical Computing Conference (Book I), W. T. Rhodes, ed., Proc. Soc. Photo-Opt. Instrum. Eng.231, 18–25 (1980).
    [CrossRef]
  16. C. A. Haniff, C. D. Mackay, D. J. Titterington, D. Sivia, P. J. Warner, “The first images from optical aperture synthesis,” Nature (London) 328, 694–696 (1987).
    [CrossRef]
  17. T. Nakajima, S. R. Kulkarni, P. W. Gorham, A. M. Ghez, G. Neugebauer, J. B. Oke, T. A. Prince, A. C. S. Readhead, “Diffraction-limited imaging. II. Optical aperture-synthesis imaging of two binary stars,” Astron. J. 97, 1510–1521 (1989).
    [CrossRef]
  18. J. W. Goodman, J. F. Belsher, “Photon limited images and their restoration,” (Rome Air Development Center, New York, 1976); “Precompensation and postcompensation of photon limited degraded images,” (Rome Air Development Center, New York, 1977).

1989 (3)

P. W. Gorham, A. M. Ghez, S. R. Kulkarni, T. Nakajima, G. Neugebauer, J. B. Oke, T. A. Prince, “Diffraction-limited imaging. III. 30 mas closure phase imaging of six binary stars with the Hale 5 m telescope,” Astron. J. 98, 1783–1799 (1989).
[CrossRef]

T. Nakajima, S. R. Kulkarni, P. W. Gorham, A. M. Ghez, G. Neugebauer, J. B. Oke, T. A. Prince, A. C. S. Readhead, “Diffraction-limited imaging. II. Optical aperture-synthesis imaging of two binary stars,” Astron. J. 97, 1510–1521 (1989).
[CrossRef]

S. Prasad, S. R. Kulkarni, “Noise in optical synthesis images. I. Ideal Michelson interferometer,” J. Opt. Soc. Am. A 6, 1702–1714 (1989).
[CrossRef]

1988 (2)

1987 (2)

T. J. Cornwell, “Radio-interferometric imaging of weak objects in conditions of poor phase stability: the relationship between speckle masking and phase closure methods,” Astron. Astrophys. 180, 269–274 (1987).

C. A. Haniff, C. D. Mackay, D. J. Titterington, D. Sivia, P. J. Warner, “The first images from optical aperture synthesis,” Nature (London) 328, 694–696 (1987).
[CrossRef]

1985 (1)

1984 (1)

T. J. Pearson, A. C. S. Readhead, “Image formation by self-calibration in radio astronomy,” Annu. Rev. Astron. Astrophys. 22, 97–130 (1984).
[CrossRef]

1981 (1)

T. J. Cornwell, P. N. Wilkinson, “A new method for making maps with unstable radio interferometers,” Mon. Not. R. Astron. Soc. 196, 1067–1086 (1981).

1980 (1)

A. R. Thompson, B. G. Clark, C. M. Wade, P. J. Napier, “The very large array,” Astrophys. J. Suppl. Ser. 44, 151–167 (1980).
[CrossRef]

1979 (1)

Ayers, G. R.

Belsher, J. F.

J. W. Goodman, J. F. Belsher, “Photon limited images and their restoration,” (Rome Air Development Center, New York, 1976); “Precompensation and postcompensation of photon limited degraded images,” (Rome Air Development Center, New York, 1977).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1970), p. 510.

Clark, B. G.

A. R. Thompson, B. G. Clark, C. M. Wade, P. J. Napier, “The very large array,” Astrophys. J. Suppl. Ser. 44, 151–167 (1980).
[CrossRef]

Cornwell, T. J.

T. J. Cornwell, “Radio-interferometric imaging of weak objects in conditions of poor phase stability: the relationship between speckle masking and phase closure methods,” Astron. Astrophys. 180, 269–274 (1987).

T. J. Cornwell, P. N. Wilkinson, “A new method for making maps with unstable radio interferometers,” Mon. Not. R. Astron. Soc. 196, 1067–1086 (1981).

Dainty, J. C.

Ghez, A. M.

T. Nakajima, S. R. Kulkarni, P. W. Gorham, A. M. Ghez, G. Neugebauer, J. B. Oke, T. A. Prince, A. C. S. Readhead, “Diffraction-limited imaging. II. Optical aperture-synthesis imaging of two binary stars,” Astron. J. 97, 1510–1521 (1989).
[CrossRef]

P. W. Gorham, A. M. Ghez, S. R. Kulkarni, T. Nakajima, G. Neugebauer, J. B. Oke, T. A. Prince, “Diffraction-limited imaging. III. 30 mas closure phase imaging of six binary stars with the Hale 5 m telescope,” Astron. J. 98, 1783–1799 (1989).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 5.

J. W. Goodman, J. F. Belsher, “Photon limited images and their restoration,” (Rome Air Development Center, New York, 1976); “Precompensation and postcompensation of photon limited degraded images,” (Rome Air Development Center, New York, 1977).

Gorham, P. W.

P. W. Gorham, A. M. Ghez, S. R. Kulkarni, T. Nakajima, G. Neugebauer, J. B. Oke, T. A. Prince, “Diffraction-limited imaging. III. 30 mas closure phase imaging of six binary stars with the Hale 5 m telescope,” Astron. J. 98, 1783–1799 (1989).
[CrossRef]

T. Nakajima, S. R. Kulkarni, P. W. Gorham, A. M. Ghez, G. Neugebauer, J. B. Oke, T. A. Prince, A. C. S. Readhead, “Diffraction-limited imaging. II. Optical aperture-synthesis imaging of two binary stars,” Astron. J. 97, 1510–1521 (1989).
[CrossRef]

Greenaway, A. H.

Haniff, C. A.

C. A. Haniff, C. D. Mackay, D. J. Titterington, D. Sivia, P. J. Warner, “The first images from optical aperture synthesis,” Nature (London) 328, 694–696 (1987).
[CrossRef]

Kulkarni, S. R.

S. Prasad, S. R. Kulkarni, “Noise in optical synthesis images. I. Ideal Michelson interferometer,” J. Opt. Soc. Am. A 6, 1702–1714 (1989).
[CrossRef]

T. Nakajima, S. R. Kulkarni, P. W. Gorham, A. M. Ghez, G. Neugebauer, J. B. Oke, T. A. Prince, A. C. S. Readhead, “Diffraction-limited imaging. II. Optical aperture-synthesis imaging of two binary stars,” Astron. J. 97, 1510–1521 (1989).
[CrossRef]

P. W. Gorham, A. M. Ghez, S. R. Kulkarni, T. Nakajima, G. Neugebauer, J. B. Oke, T. A. Prince, “Diffraction-limited imaging. III. 30 mas closure phase imaging of six binary stars with the Hale 5 m telescope,” Astron. J. 98, 1783–1799 (1989).
[CrossRef]

S. R. Kulkarni, in JPL Workshop on Space Interferometry, M. Shao, S. R. Kulkarni, eds. (NASA, Washington, D.C., to be published).

T. Nakajima, S. R. Kulkarni, “Noise in optical synthesis images. IV. Effects of atmospheric disturbances,” submitted to J. Opt. Soc. Am. A.

Mackay, C. D.

C. A. Haniff, C. D. Mackay, D. J. Titterington, D. Sivia, P. J. Warner, “The first images from optical aperture synthesis,” Nature (London) 328, 694–696 (1987).
[CrossRef]

Nakajima, T.

P. W. Gorham, A. M. Ghez, S. R. Kulkarni, T. Nakajima, G. Neugebauer, J. B. Oke, T. A. Prince, “Diffraction-limited imaging. III. 30 mas closure phase imaging of six binary stars with the Hale 5 m telescope,” Astron. J. 98, 1783–1799 (1989).
[CrossRef]

T. Nakajima, S. R. Kulkarni, P. W. Gorham, A. M. Ghez, G. Neugebauer, J. B. Oke, T. A. Prince, A. C. S. Readhead, “Diffraction-limited imaging. II. Optical aperture-synthesis imaging of two binary stars,” Astron. J. 97, 1510–1521 (1989).
[CrossRef]

T. Nakajima, “Signal-to-noise ratio of the bispectral analysis of speckle interferometry,” J. Opt. Soc. Am. A 5, 1477–1491 (1988).
[CrossRef]

T. Nakajima, S. R. Kulkarni, “Noise in optical synthesis images. IV. Effects of atmospheric disturbances,” submitted to J. Opt. Soc. Am. A.

Napier, P. J.

A. R. Thompson, B. G. Clark, C. M. Wade, P. J. Napier, “The very large array,” Astrophys. J. Suppl. Ser. 44, 151–167 (1980).
[CrossRef]

Neugebauer, G.

T. Nakajima, S. R. Kulkarni, P. W. Gorham, A. M. Ghez, G. Neugebauer, J. B. Oke, T. A. Prince, A. C. S. Readhead, “Diffraction-limited imaging. II. Optical aperture-synthesis imaging of two binary stars,” Astron. J. 97, 1510–1521 (1989).
[CrossRef]

P. W. Gorham, A. M. Ghez, S. R. Kulkarni, T. Nakajima, G. Neugebauer, J. B. Oke, T. A. Prince, “Diffraction-limited imaging. III. 30 mas closure phase imaging of six binary stars with the Hale 5 m telescope,” Astron. J. 98, 1783–1799 (1989).
[CrossRef]

Northcott, M. J.

Oke, J. B.

P. W. Gorham, A. M. Ghez, S. R. Kulkarni, T. Nakajima, G. Neugebauer, J. B. Oke, T. A. Prince, “Diffraction-limited imaging. III. 30 mas closure phase imaging of six binary stars with the Hale 5 m telescope,” Astron. J. 98, 1783–1799 (1989).
[CrossRef]

T. Nakajima, S. R. Kulkarni, P. W. Gorham, A. M. Ghez, G. Neugebauer, J. B. Oke, T. A. Prince, A. C. S. Readhead, “Diffraction-limited imaging. II. Optical aperture-synthesis imaging of two binary stars,” Astron. J. 97, 1510–1521 (1989).
[CrossRef]

Pearson, T. J.

T. J. Pearson, A. C. S. Readhead, “Image formation by self-calibration in radio astronomy,” Annu. Rev. Astron. Astrophys. 22, 97–130 (1984).
[CrossRef]

Prasad, S.

Prince, T. A.

T. Nakajima, S. R. Kulkarni, P. W. Gorham, A. M. Ghez, G. Neugebauer, J. B. Oke, T. A. Prince, A. C. S. Readhead, “Diffraction-limited imaging. II. Optical aperture-synthesis imaging of two binary stars,” Astron. J. 97, 1510–1521 (1989).
[CrossRef]

P. W. Gorham, A. M. Ghez, S. R. Kulkarni, T. Nakajima, G. Neugebauer, J. B. Oke, T. A. Prince, “Diffraction-limited imaging. III. 30 mas closure phase imaging of six binary stars with the Hale 5 m telescope,” Astron. J. 98, 1783–1799 (1989).
[CrossRef]

Readhead, A. C. S.

T. Nakajima, S. R. Kulkarni, P. W. Gorham, A. M. Ghez, G. Neugebauer, J. B. Oke, T. A. Prince, A. C. S. Readhead, “Diffraction-limited imaging. II. Optical aperture-synthesis imaging of two binary stars,” Astron. J. 97, 1510–1521 (1989).
[CrossRef]

T. J. Pearson, A. C. S. Readhead, “Image formation by self-calibration in radio astronomy,” Annu. Rev. Astron. Astrophys. 22, 97–130 (1984).
[CrossRef]

Schwab, F. R.

F. R. Schwab, “Adaptive calibration of radio interferometer data,” in 1980 International Optical Computing Conference (Book I), W. T. Rhodes, ed., Proc. Soc. Photo-Opt. Instrum. Eng.231, 18–25 (1980).
[CrossRef]

Sivia, D.

C. A. Haniff, C. D. Mackay, D. J. Titterington, D. Sivia, P. J. Warner, “The first images from optical aperture synthesis,” Nature (London) 328, 694–696 (1987).
[CrossRef]

Thompson, A. R.

A. R. Thompson, B. G. Clark, C. M. Wade, P. J. Napier, “The very large array,” Astrophys. J. Suppl. Ser. 44, 151–167 (1980).
[CrossRef]

Titterington, D. J.

C. A. Haniff, C. D. Mackay, D. J. Titterington, D. Sivia, P. J. Warner, “The first images from optical aperture synthesis,” Nature (London) 328, 694–696 (1987).
[CrossRef]

Wade, C. M.

A. R. Thompson, B. G. Clark, C. M. Wade, P. J. Napier, “The very large array,” Astrophys. J. Suppl. Ser. 44, 151–167 (1980).
[CrossRef]

Warner, P. J.

C. A. Haniff, C. D. Mackay, D. J. Titterington, D. Sivia, P. J. Warner, “The first images from optical aperture synthesis,” Nature (London) 328, 694–696 (1987).
[CrossRef]

Wilkinson, P. N.

T. J. Cornwell, P. N. Wilkinson, “A new method for making maps with unstable radio interferometers,” Mon. Not. R. Astron. Soc. 196, 1067–1086 (1981).

Wirnitzer, B.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1970), p. 510.

Annu. Rev. Astron. Astrophys. (1)

T. J. Pearson, A. C. S. Readhead, “Image formation by self-calibration in radio astronomy,” Annu. Rev. Astron. Astrophys. 22, 97–130 (1984).
[CrossRef]

Astron. Astrophys. (1)

T. J. Cornwell, “Radio-interferometric imaging of weak objects in conditions of poor phase stability: the relationship between speckle masking and phase closure methods,” Astron. Astrophys. 180, 269–274 (1987).

Astron. J. (2)

P. W. Gorham, A. M. Ghez, S. R. Kulkarni, T. Nakajima, G. Neugebauer, J. B. Oke, T. A. Prince, “Diffraction-limited imaging. III. 30 mas closure phase imaging of six binary stars with the Hale 5 m telescope,” Astron. J. 98, 1783–1799 (1989).
[CrossRef]

T. Nakajima, S. R. Kulkarni, P. W. Gorham, A. M. Ghez, G. Neugebauer, J. B. Oke, T. A. Prince, A. C. S. Readhead, “Diffraction-limited imaging. II. Optical aperture-synthesis imaging of two binary stars,” Astron. J. 97, 1510–1521 (1989).
[CrossRef]

Astrophys. J. Suppl. Ser. (1)

A. R. Thompson, B. G. Clark, C. M. Wade, P. J. Napier, “The very large array,” Astrophys. J. Suppl. Ser. 44, 151–167 (1980).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

Mon. Not. R. Astron. Soc. (1)

T. J. Cornwell, P. N. Wilkinson, “A new method for making maps with unstable radio interferometers,” Mon. Not. R. Astron. Soc. 196, 1067–1086 (1981).

Nature (London) (1)

C. A. Haniff, C. D. Mackay, D. J. Titterington, D. Sivia, P. J. Warner, “The first images from optical aperture synthesis,” Nature (London) 328, 694–696 (1987).
[CrossRef]

Other (6)

J. W. Goodman, J. F. Belsher, “Photon limited images and their restoration,” (Rome Air Development Center, New York, 1976); “Precompensation and postcompensation of photon limited degraded images,” (Rome Air Development Center, New York, 1977).

F. R. Schwab, “Adaptive calibration of radio interferometer data,” in 1980 International Optical Computing Conference (Book I), W. T. Rhodes, ed., Proc. Soc. Photo-Opt. Instrum. Eng.231, 18–25 (1980).
[CrossRef]

S. R. Kulkarni, in JPL Workshop on Space Interferometry, M. Shao, S. R. Kulkarni, eds. (NASA, Washington, D.C., to be published).

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1970), p. 510.

T. Nakajima, S. R. Kulkarni, “Noise in optical synthesis images. IV. Effects of atmospheric disturbances,” submitted to J. Opt. Soc. Am. A.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 5.

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Figures (5)

Fig. 1
Fig. 1

Baseline-based indexing scheme for the bispectrum phasor for a six-element array. The baseline connecting apertures A and B is assigned the index 1, that connecting B and C the index 2, etc. Not all the baselines are shown.

Fig. 2
Fig. 2

Ratio of the image SNR for the bispectrum case to that for the ideal case as a function of the photon number 〈N〉 per subbeam per frame for various values of the number n of apertures. The dashed line represents the ideal case.

Fig. 3
Fig. 3

Image SNR for the bispectrum case as a function of the number n of apertures for a fixed source intensity 〈M〉 = 0.1. 〈M〉 is the number of photoelectrons per primary aperture per integration time. The dashed curve represents the ideal case.

Fig. 4
Fig. 4

Same as Fig. 3 but with 〈M〉 = 1.0.

Fig. 5
Fig. 5

Same as Fig. 3 but with 〈M〉 = 10.0.

Equations (44)

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I j ( x ) = 2 I 0 [ 1 + γ j cos ( 2 π B i λ x d + ϕ j + θ j ) ] ,
k j ( p ) = 2 K [ 1 + cos ( p ω j + ϕ j + θ j ) ] .
z j = p = 0 P - 1 k j ( p ) exp ( - i ω j p ) .
p j = z j z j *
P j = p = 0 P - 1 exp ( - i ω j p ) k j ( p ) p = 0 P - 1 exp ( i ω j p ) k j ( p ) .
P j = Z j Z j * + p = 0 P - 1 k j ( p ) = γ j 2 N 2 + 2 N .
q j = p j - p = 0 P - 1 k j ( p ) ,
Q j q j = γ j 2 N 2 .
V ( q j ) = 4 N 2 ( 1 + γ j 2 N ) .
S Q j = γ j 2 N 2 ( 1 + γ j 2 N ) 1 / 2 .
C ( q j , q k * ) = q j q k * - q j q k * = p p r r [ k j ( p ) k j ( p ) - δ p p k j ( p ) ] [ k k ( r ) k k ( r ) - δ r r k k ( r ) ] exp [ - i ω j ( p - p ) ] exp [ - i ω k ( r - r ) ] - Q j Q k * = 0.
b 123 = z 1 z 2 z 3 .
B j k l = b j k l = p = 0 P - 1 k j ( p ) exp ( - i ω j p ) q = 0 P - 1 k k ( q ) × exp ( - i ω k q ) r = 0 P - 1 k l ( r ) exp ( - i ω l r ) = p q r exp ( - i ω j p - i ω k q - i ω l r ) k j ( p ) k k ( q ) k l ( r ) .
k j ( p ) k k ( q ) k l ( r ) = k j ( p ) k k ( q ) k l ( r ) .
B j k l = Z j Z k Z l = γ j γ k γ l N 3 exp ( - i ψ j k l ) .
V ( b 123 ) b 123 b 123 * - b 123 b 123 * = z 1 z 2 z 3 z 1 * z 2 * z 3 * - z 1 z 2 z 3 z 1 * z 2 * z 3 * .
V ( b 123 ) = z 1 z 1 * z 2 z 2 * z 3 z 3 * - γ 1 2 γ 2 2 γ 3 2 N 6 .
V ( b 123 ) = 2 N 5 ( γ 1 2 γ 2 2 + γ 2 2 γ 3 2 + γ 3 2 γ 1 2 ) + 4 N 4 ( γ 1 2 + γ 2 2 + γ 3 2 ) + 8 N 3 .
S B 123 = γ 1 γ 2 γ 3 N 3 / 2 [ 2 N 2 ( γ 1 2 γ 2 2 + γ 2 2 γ 3 2 + γ 3 2 γ 1 2 ) + 4 N ( γ 1 2 + γ 2 2 + γ 3 2 ) + 8 ] 1 / 2 .
C ( b 123 , b j k l * ) = b 123 b j k l * - b 123 b j k l * .
C ( b 123 , b 145 * ) = z 1 z 1 * z 2 z 3 z 4 * z 5 * - z 1 z 2 z 3 z 1 * z 4 * z 5 * = [ z 1 z 1 * - z 1 z 1 * ] z 2 z 3 z 4 * z 5 * = 2 γ 2 γ 3 γ 4 γ 5 N 5 exp ( i ψ 123 - i ψ 145 ) .
C ( b 123 , b 145 ) = z 1 z 2 z 3 z 1 z 4 z 5 - z 1 z 2 z 3 z 1 z 4 z 5 = ( z 1 z 1 - z 1 z 1 ) z 2 z 3 z 4 z 5 = p k ( p ) exp ( - i 2 ω 1 p ) γ 2 γ 3 γ 4 γ 5 × exp ( i ψ 123 + i ψ 145 ) = 0.
μ 123 , 145 C ( b 123 , b 145 ) + C ( b 123 , b 145 * ) [ V ( b 123 ) V ( b 145 ) ] 1 / 2 .
B B j k l = γ 3 N 3 , σ b 2 V ( b ) V ( b 123 ) = N 3 ( 6 γ 4 N 2 + 12 γ 2 N + 8 ) , S B B V ( b ) = ( γ 2 N ) 3 / 2 ( 6 γ 4 N 2 + 12 γ 2 N + 8 ) 1 / 2 , μ b μ 123 , 145 = γ 4 N 2 3 γ 4 N 2 + 6 γ 2 N + 4 .
F = f = s = 1 n t B s .
V [ Re ( f ) ] = 1 4 s = 1 n t t = 1 n t C ( B s , B t ) + C ( B s , B t * ) + C ( B s * , B t ) + C ( B s * , B t * ) = 1 2 s t Re [ μ b ( s , t ) ] σ b ( s ) σ b ( t ) .
S σ S = 3 Re ( F ) { V [ Re ( f ) ] } 1 / 2 = 3 S F .
V [ Re ( f ) ] = 1 2 n t σ b 2 + 3 2 ( n - 3 ) n t μ b σ b 2 .
S F = [ 2 n t σ b 2 + 3 ( n - 3 ) μ b σ b 2 ] 1 / 2 γ 3 N 3 .
S σ S = 3 n t γ 3 N 3 / 2 [ 3 ( n - 2 ) γ 4 N 2 + 6 γ 2 N + 4 ] 1 / 2 .
S σ S = γ ( L / 2 ) { 1 + [ ( 6 γ 2 N + 4 ) / 3 ( n - 2 ) γ 4 N 2 ] } 1 / 2
S F = [ n ( n - 1 ) 18 ] 1 / 2 γ N .
S F = γ ( n / 18 ) 1 / 2 M 1 / 2 = γ ( L / 18 ) 1 / 2 ,
S σ S = γ ( L / 2 ) 1 / 2 .
S F [ n ( n - 1 ) ( n - 2 ) 24 ] 1 / 2 γ 3 N 3 / 2 , = [ n ( n - 2 ) ( n - 1 ) ( n - 1 ) 24 ] 1 / 2 γ 3 M 3 / 2 .
S σ S = 3 / 8 γ 3 M 3 / 2 ,
( S σ S ) S C = γ [ ( n b - n ) N ] 1 / 2 = α [ 2 ( n b - n ) ] 1 / 2 .
V ( q ) = q 2 - q 2 = { p q exp [ i ω ( p - q ) ] k ( p ) k ( q ) - r k ( r ) } × { s t exp [ - i ω ( s - t ) ] k ( s ) k ( r ) - u k ( u ) } - p q exp [ i ω ( p - q ) ] k ( p ) k ( q ) - r k ( r ) 2 = p q s t exp [ i ω ( p - q - s + t ) ] × k ( p ) k ( q ) k ( s ) k ( t ) - 2 p q u exp [ i ω ( p - q ) ] × k ( p ) k ( q ) k ( u ) + r u k ( r ) k ( u ) - p q exp [ i ω ( p - q ) ] k ( p ) k ( q ) - r k ( r ) 2 ,
k ( k - 1 ) ( k - m + 1 ) = k m ,
k ( p ) k ( q ) = k ( p ) k ( q ) + δ p q k ( p ) ,
k ( p ) k ( q ) k ( r ) = k ( p ) k ( q ) k ( r ) + δ p q k ( p ) k ( r ) + δ q r k ( p ) k ( q ) + δ r p k ( q ) k ( r ) + δ p q r k ( p ) ,
k ( p ) k ( q ) k ( r ) k ( s ) = k ( p ) k ( q ) k ( r ) k ( s ) + δ p q k ( p ) k ( r ) k ( s ) + δ p r k ( p ) k ( q ) k ( s ) + δ p s k ( p ) k ( q ) k ( r ) + δ q r k ( p ) k ( q ) k ( s ) + δ q s k ( p ) k ( q ) k ( r ) + δ r s k ( p ) k ( q ) k ( r ) + δ p q r k ( p ) k ( s ) + δ p q s k ( p ) k ( r ) + δ p r s k ( p ) k ( q ) + δ q r s k ( p ) k ( q ) + δ p q δ r s k ( p ) k ( r ) + δ p r δ q s k ( p ) k ( q ) + δ p s δ q r k ( p ) k ( q ) + δ p q r s k ( p ) ,
V ( q ) = 4 N 2 + 4 N Z ( ω ) 2 + [ Z ( ω ) 2 Z * ( 2 ω ) + Z * ( ω ) 2 Z ( 2 ω ) ] + Z ( 2 ω ) 2 ,
V ( q ) = 4 N 2 ( 1 + γ j 2 N ) .

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