Abstract

Signal-to-noise ratios for the amplitude of the object Fourier components are compared assuming either redundant or nonredundant beam recombination. A general condition is given for the object brightness below which redundant beam recombination is superior. A similar condition is found when the variance of the closure phases is considered.

© 1987 Optical Society of America

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References

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  1. A. Labeyrie, “Interference fringes obtained on Vega with two optical telescopes,” Astrophys. J. 196, L71–L75 (1975).
    [CrossRef]
  2. M. Shao, D. H. Staelin, “First fringe measurements with a phase-tracking stellar interferometer,” Appl. Opt. 19, 1519–1522 (1980).
    [CrossRef] [PubMed]
  3. J. Davis, W. J. Tango, “The Sydney University 11.4 m prototype stellar interferometer,” Proc. Astron. Soc. Austr. 6, 34–38 (1985).
  4. R. C. Jennison, “A phase sensitive interferometer technique for the measurement of the Fourier transforms of spatial brightness distribution is of small angular extent,” Mon. Not. R. Astr. Soc. 118, 276–284 (1958).
  5. A. E. E. Rogers, “Methods of using closure phases in radio aperture synthesis,” in 1980 International Computing Conference (Book I), W. T. Rhodes, ed., Proc. Soc. Photo-Opt. Instrum. Eng.231, 10–17 (1980).
    [CrossRef]
  6. D. H. Rogstad, “A technique for measuring visibility phase with an optical interferometer in the presence of atmospheric seeing,” Appl. Opt. 7, 585–588 (1968).
    [CrossRef] [PubMed]
  7. F. D. Russel, J. W. Goodman, “Non-redundant arrays and postdetection processing for aberration compensation in incoherent imaging,”J. Opt. Soc. Am. 61, 182–191 (1971).
    [CrossRef]
  8. W. T. Rhodes, J. W. Goodman, “Interferometric technique for recording and restoring images degraded by unknown aberrations,”J. Opt. Soc. Am. 63, 647–657 (1973).
    [CrossRef]
  9. J. E. Baldwin, C. A. Haniff, C. D. Mackay, P. J. Warner, “Closure phase in high-resolution optical imaging,” Nature 320, 595–597 (1986).
    [CrossRef]
  10. A. H. Greenaway, “Diffraction-limited pictures from single turbulence-degraded images in astronomy,” Opt. Commun. 42, 157–161 (1982).
    [CrossRef]
  11. N. R. Arnot, “A technique for obtaining diffraction-limited pictures from a single large aperture small-exposure image,” Opt. Commun. 45, 380–384 (1983).
    [CrossRef]
  12. A. H. Greenaway, “Terrestrial optical aperture synthesis technique (TOAST),” Opt. Commun. 58, 149–154 (1986).
    [CrossRef]
  13. G. Weigelt, B. Wirnitzer, “Image reconstruction by the speckle-masking method,” Opt. Lett. 8, 389–391 (1983).
    [CrossRef] [PubMed]
  14. A. W. Lohmann, G. Weigelt, B. Wirnitzer, “Speckle masking in astronomy: triple correlation theory and applications,” Appl. Opt. 22, 4028–4037 (1983).
    [CrossRef] [PubMed]
  15. F. Roddier, “Triple correlation as a phase closure technique,” Opt. Commun. 60, 145–148 (1986).
    [CrossRef]
  16. F. Sibille, A. Chelli, P. Lena, “Infrared speckle interferometry,” Astron. Astrophys. 79, 315–328 (1979).
  17. F. Roddier, P. Lena, “Long-baseline Michelson interferometry with large ground-based telescopes operating at optical wavelengths: general formalism. Interferometry at visible wavelengths,”J. Opt. (Paris) 15, 171–182 (1984).
    [CrossRef]

1986 (3)

J. E. Baldwin, C. A. Haniff, C. D. Mackay, P. J. Warner, “Closure phase in high-resolution optical imaging,” Nature 320, 595–597 (1986).
[CrossRef]

A. H. Greenaway, “Terrestrial optical aperture synthesis technique (TOAST),” Opt. Commun. 58, 149–154 (1986).
[CrossRef]

F. Roddier, “Triple correlation as a phase closure technique,” Opt. Commun. 60, 145–148 (1986).
[CrossRef]

1985 (1)

J. Davis, W. J. Tango, “The Sydney University 11.4 m prototype stellar interferometer,” Proc. Astron. Soc. Austr. 6, 34–38 (1985).

1984 (1)

F. Roddier, P. Lena, “Long-baseline Michelson interferometry with large ground-based telescopes operating at optical wavelengths: general formalism. Interferometry at visible wavelengths,”J. Opt. (Paris) 15, 171–182 (1984).
[CrossRef]

1983 (3)

G. Weigelt, B. Wirnitzer, “Image reconstruction by the speckle-masking method,” Opt. Lett. 8, 389–391 (1983).
[CrossRef] [PubMed]

A. W. Lohmann, G. Weigelt, B. Wirnitzer, “Speckle masking in astronomy: triple correlation theory and applications,” Appl. Opt. 22, 4028–4037 (1983).
[CrossRef] [PubMed]

N. R. Arnot, “A technique for obtaining diffraction-limited pictures from a single large aperture small-exposure image,” Opt. Commun. 45, 380–384 (1983).
[CrossRef]

1982 (1)

A. H. Greenaway, “Diffraction-limited pictures from single turbulence-degraded images in astronomy,” Opt. Commun. 42, 157–161 (1982).
[CrossRef]

1980 (1)

M. Shao, D. H. Staelin, “First fringe measurements with a phase-tracking stellar interferometer,” Appl. Opt. 19, 1519–1522 (1980).
[CrossRef] [PubMed]

1979 (1)

F. Sibille, A. Chelli, P. Lena, “Infrared speckle interferometry,” Astron. Astrophys. 79, 315–328 (1979).

1975 (1)

A. Labeyrie, “Interference fringes obtained on Vega with two optical telescopes,” Astrophys. J. 196, L71–L75 (1975).
[CrossRef]

1973 (1)

W. T. Rhodes, J. W. Goodman, “Interferometric technique for recording and restoring images degraded by unknown aberrations,”J. Opt. Soc. Am. 63, 647–657 (1973).
[CrossRef]

1971 (1)

F. D. Russel, J. W. Goodman, “Non-redundant arrays and postdetection processing for aberration compensation in incoherent imaging,”J. Opt. Soc. Am. 61, 182–191 (1971).
[CrossRef]

1968 (1)

D. H. Rogstad, “A technique for measuring visibility phase with an optical interferometer in the presence of atmospheric seeing,” Appl. Opt. 7, 585–588 (1968).
[CrossRef] [PubMed]

1958 (1)

R. C. Jennison, “A phase sensitive interferometer technique for the measurement of the Fourier transforms of spatial brightness distribution is of small angular extent,” Mon. Not. R. Astr. Soc. 118, 276–284 (1958).

Arnot, N. R.

N. R. Arnot, “A technique for obtaining diffraction-limited pictures from a single large aperture small-exposure image,” Opt. Commun. 45, 380–384 (1983).
[CrossRef]

Baldwin, J. E.

J. E. Baldwin, C. A. Haniff, C. D. Mackay, P. J. Warner, “Closure phase in high-resolution optical imaging,” Nature 320, 595–597 (1986).
[CrossRef]

Chelli, A.

F. Sibille, A. Chelli, P. Lena, “Infrared speckle interferometry,” Astron. Astrophys. 79, 315–328 (1979).

Davis, J.

J. Davis, W. J. Tango, “The Sydney University 11.4 m prototype stellar interferometer,” Proc. Astron. Soc. Austr. 6, 34–38 (1985).

Goodman, J. W.

W. T. Rhodes, J. W. Goodman, “Interferometric technique for recording and restoring images degraded by unknown aberrations,”J. Opt. Soc. Am. 63, 647–657 (1973).
[CrossRef]

F. D. Russel, J. W. Goodman, “Non-redundant arrays and postdetection processing for aberration compensation in incoherent imaging,”J. Opt. Soc. Am. 61, 182–191 (1971).
[CrossRef]

Greenaway, A. H.

A. H. Greenaway, “Terrestrial optical aperture synthesis technique (TOAST),” Opt. Commun. 58, 149–154 (1986).
[CrossRef]

A. H. Greenaway, “Diffraction-limited pictures from single turbulence-degraded images in astronomy,” Opt. Commun. 42, 157–161 (1982).
[CrossRef]

Haniff, C. A.

J. E. Baldwin, C. A. Haniff, C. D. Mackay, P. J. Warner, “Closure phase in high-resolution optical imaging,” Nature 320, 595–597 (1986).
[CrossRef]

Jennison, R. C.

R. C. Jennison, “A phase sensitive interferometer technique for the measurement of the Fourier transforms of spatial brightness distribution is of small angular extent,” Mon. Not. R. Astr. Soc. 118, 276–284 (1958).

Labeyrie, A.

A. Labeyrie, “Interference fringes obtained on Vega with two optical telescopes,” Astrophys. J. 196, L71–L75 (1975).
[CrossRef]

Lena, P.

F. Roddier, P. Lena, “Long-baseline Michelson interferometry with large ground-based telescopes operating at optical wavelengths: general formalism. Interferometry at visible wavelengths,”J. Opt. (Paris) 15, 171–182 (1984).
[CrossRef]

F. Sibille, A. Chelli, P. Lena, “Infrared speckle interferometry,” Astron. Astrophys. 79, 315–328 (1979).

Lohmann, A. W.

A. W. Lohmann, G. Weigelt, B. Wirnitzer, “Speckle masking in astronomy: triple correlation theory and applications,” Appl. Opt. 22, 4028–4037 (1983).
[CrossRef] [PubMed]

Mackay, C. D.

J. E. Baldwin, C. A. Haniff, C. D. Mackay, P. J. Warner, “Closure phase in high-resolution optical imaging,” Nature 320, 595–597 (1986).
[CrossRef]

Rhodes, W. T.

W. T. Rhodes, J. W. Goodman, “Interferometric technique for recording and restoring images degraded by unknown aberrations,”J. Opt. Soc. Am. 63, 647–657 (1973).
[CrossRef]

Roddier, F.

F. Roddier, “Triple correlation as a phase closure technique,” Opt. Commun. 60, 145–148 (1986).
[CrossRef]

F. Roddier, P. Lena, “Long-baseline Michelson interferometry with large ground-based telescopes operating at optical wavelengths: general formalism. Interferometry at visible wavelengths,”J. Opt. (Paris) 15, 171–182 (1984).
[CrossRef]

Rogers, A. E. E.

A. E. E. Rogers, “Methods of using closure phases in radio aperture synthesis,” in 1980 International Computing Conference (Book I), W. T. Rhodes, ed., Proc. Soc. Photo-Opt. Instrum. Eng.231, 10–17 (1980).
[CrossRef]

Rogstad, D. H.

D. H. Rogstad, “A technique for measuring visibility phase with an optical interferometer in the presence of atmospheric seeing,” Appl. Opt. 7, 585–588 (1968).
[CrossRef] [PubMed]

Russel, F. D.

F. D. Russel, J. W. Goodman, “Non-redundant arrays and postdetection processing for aberration compensation in incoherent imaging,”J. Opt. Soc. Am. 61, 182–191 (1971).
[CrossRef]

Shao, M.

M. Shao, D. H. Staelin, “First fringe measurements with a phase-tracking stellar interferometer,” Appl. Opt. 19, 1519–1522 (1980).
[CrossRef] [PubMed]

Sibille, F.

F. Sibille, A. Chelli, P. Lena, “Infrared speckle interferometry,” Astron. Astrophys. 79, 315–328 (1979).

Staelin, D. H.

M. Shao, D. H. Staelin, “First fringe measurements with a phase-tracking stellar interferometer,” Appl. Opt. 19, 1519–1522 (1980).
[CrossRef] [PubMed]

Tango, W. J.

J. Davis, W. J. Tango, “The Sydney University 11.4 m prototype stellar interferometer,” Proc. Astron. Soc. Austr. 6, 34–38 (1985).

Warner, P. J.

J. E. Baldwin, C. A. Haniff, C. D. Mackay, P. J. Warner, “Closure phase in high-resolution optical imaging,” Nature 320, 595–597 (1986).
[CrossRef]

Weigelt, G.

A. W. Lohmann, G. Weigelt, B. Wirnitzer, “Speckle masking in astronomy: triple correlation theory and applications,” Appl. Opt. 22, 4028–4037 (1983).
[CrossRef] [PubMed]

G. Weigelt, B. Wirnitzer, “Image reconstruction by the speckle-masking method,” Opt. Lett. 8, 389–391 (1983).
[CrossRef] [PubMed]

Wirnitzer, B.

G. Weigelt, B. Wirnitzer, “Image reconstruction by the speckle-masking method,” Opt. Lett. 8, 389–391 (1983).
[CrossRef] [PubMed]

A. W. Lohmann, G. Weigelt, B. Wirnitzer, “Speckle masking in astronomy: triple correlation theory and applications,” Appl. Opt. 22, 4028–4037 (1983).
[CrossRef] [PubMed]

Appl. Opt. (3)

D. H. Rogstad, “A technique for measuring visibility phase with an optical interferometer in the presence of atmospheric seeing,” Appl. Opt. 7, 585–588 (1968).
[CrossRef] [PubMed]

M. Shao, D. H. Staelin, “First fringe measurements with a phase-tracking stellar interferometer,” Appl. Opt. 19, 1519–1522 (1980).
[CrossRef] [PubMed]

A. W. Lohmann, G. Weigelt, B. Wirnitzer, “Speckle masking in astronomy: triple correlation theory and applications,” Appl. Opt. 22, 4028–4037 (1983).
[CrossRef] [PubMed]

Astron. Astrophys. (1)

F. Sibille, A. Chelli, P. Lena, “Infrared speckle interferometry,” Astron. Astrophys. 79, 315–328 (1979).

Astrophys. J. (1)

A. Labeyrie, “Interference fringes obtained on Vega with two optical telescopes,” Astrophys. J. 196, L71–L75 (1975).
[CrossRef]

J. Opt. (Paris) (1)

F. Roddier, P. Lena, “Long-baseline Michelson interferometry with large ground-based telescopes operating at optical wavelengths: general formalism. Interferometry at visible wavelengths,”J. Opt. (Paris) 15, 171–182 (1984).
[CrossRef]

J. Opt. Soc. Am. (2)

F. D. Russel, J. W. Goodman, “Non-redundant arrays and postdetection processing for aberration compensation in incoherent imaging,”J. Opt. Soc. Am. 61, 182–191 (1971).
[CrossRef]

W. T. Rhodes, J. W. Goodman, “Interferometric technique for recording and restoring images degraded by unknown aberrations,”J. Opt. Soc. Am. 63, 647–657 (1973).
[CrossRef]

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

R. C. Jennison, “A phase sensitive interferometer technique for the measurement of the Fourier transforms of spatial brightness distribution is of small angular extent,” Mon. Not. R. Astr. Soc. 118, 276–284 (1958).

Nature (1)

J. E. Baldwin, C. A. Haniff, C. D. Mackay, P. J. Warner, “Closure phase in high-resolution optical imaging,” Nature 320, 595–597 (1986).
[CrossRef]

Opt. Commun. (4)

A. H. Greenaway, “Diffraction-limited pictures from single turbulence-degraded images in astronomy,” Opt. Commun. 42, 157–161 (1982).
[CrossRef]

N. R. Arnot, “A technique for obtaining diffraction-limited pictures from a single large aperture small-exposure image,” Opt. Commun. 45, 380–384 (1983).
[CrossRef]

A. H. Greenaway, “Terrestrial optical aperture synthesis technique (TOAST),” Opt. Commun. 58, 149–154 (1986).
[CrossRef]

F. Roddier, “Triple correlation as a phase closure technique,” Opt. Commun. 60, 145–148 (1986).
[CrossRef]

Opt. Lett. (1)

G. Weigelt, B. Wirnitzer, “Image reconstruction by the speckle-masking method,” Opt. Lett. 8, 389–391 (1983).
[CrossRef] [PubMed]

Proc. Astron. Soc. Austr. (1)

J. Davis, W. J. Tango, “The Sydney University 11.4 m prototype stellar interferometer,” Proc. Astron. Soc. Austr. 6, 34–38 (1985).

Other (1)

A. E. E. Rogers, “Methods of using closure phases in radio aperture synthesis,” in 1980 International Computing Conference (Book I), W. T. Rhodes, ed., Proc. Soc. Photo-Opt. Instrum. Eng.231, 10–17 (1980).
[CrossRef]

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Figures (1)

Fig. 1
Fig. 1

Schematic example of a nonredundant beam recombination. I.P., Redundant input pupil made of three equally spaced telescopes; O.P., nonredundant output pupil.

Equations (65)

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j ( r ) = i ( r ) + b ( r ) .
J ( f ) = I ( f ) + B ( f ) ,
J ( f ) 2 = I ( f ) 2 + B ( f ) 2 .
σ 2 ( J ( f ) 2 ) = σ 2 ( I ( f ) 2 ) + σ 2 ( B ( f ) 2 ) + 2 I ( f ) 2 B ( f ) 2 ,
σ 2 ( J ( f ) 2 ) = σ 2 ( I ( f ) 2 ) + B ( f ) 2 2 + 2 I ( f ) 2 B ( f ) 2 .
S = k = 1 N [ J ( f k ) 2 - B ( f k ) 2 ] .
S = k = 1 N [ J ( f k ) 2 - B ( f k ) 2 ] = N I ( f k ) 2 .
σ 2 ( I ( f ) 2 ) = 0.
σ 2 ( S ) = k = 1 N σ 2 ( J ( f k ) 2 ) .
σ 2 ( J ( f k ) 2 ) = B ( f k ) 2 2 + 2 I ( f k ) 2 B ( f k ) 2 ;
σ 2 ( S ) = N [ B ( f k ) 2 2 + 2 I ( f k ) 2 B ( f k ) 2 ] .
S σ ( S ) = N I ( f k ) 2 [ N B ( f k ) 2 2 + 2 N I ( f k ) 2 B ( f k ) 2 ] 1 / 2 .
I ( f ) = k = 1 N I k ( f ) .
I ( f ) 2 = | k = 1 N I k ( f ) | 2 = k = 1 N I k ( f ) 2 = N I k ( f ) 2 .
S = J ( f ) 2 - B ( f ) 2 .
S = I ( f ) 2 .
S = N I k ( f ) 2 .
σ 2 ( I ( f ) 2 ) = I ( f ) 2 2 ,
σ 2 ( S ) = I ( f ) 2 2 + B ( f ) 2 2 + 2 I ( f ) 2 B ( f ) 2 = [ I ( f ) 2 + B ( f ) 2 ] 2 ,
σ ( S ) = N I k ( f ) 2 + B ( f ) 2 .
S σ ( S ) = N I k ( f ) 2 N I k ( f ) 2 + B ( f ) 2 .
I k ( f ) = I ( f k ) ,
B ( f ) 2 = B ( f k ) 2 .
S σ ( S ) = N I ( f k ) 2 N I ( f k ) 2 + B ( f k ) 2 ,
N B ( f k ) 2 2 + 2 N I ( f k ) 2 B ( f k ) 2 < N 2 I ( f k ) 2 2 + B ( f k ) 2 2 + 2 N I ( f k ) 2 B ( f k ) 2
I ( f k ) 2 B ( f k ) 2 > N - 1 N .
I ( f k ) 2 B ( f k ) 2 > N - 1 / 2 .
x ( r ) = i δ ( r - r i ) ,
Q ( f ) = X ( f ) 2 - X ( 0 )
Q ( f ) = W ( f ) n 2 ,
σ 2 ( Q ) = σ 1 2 + σ 2 2 .
σ 2 2 = 2 W ( f ) n 3 + n 2 .
S = k = 1 N Q ( f k ) ,
S = k = 1 N Q ( f k ) = N W k n 2 ,
σ 2 [ Q ( f k ) ] = 2 W k n 3 + n 2 .
σ 2 ( S ) = k = 1 N σ 2 [ Q ( f k ) ] = N ( 2 W k n 3 + n 2 ) ,
S σ S = N W k n [ N ( 2 W k n + 1 ) ] 1 / 2 .
S = Q ( f ) = W ( f ) n 2 = N W k n 2 .
σ 2 ( S ) = σ 2 ( Q ) = σ 1 2 + σ 2 2 ,
σ 1 2 = S 2 = N 2 W k 2 n 4 ,
σ 2 ( S ) = N 2 W k 2 n 4 + 2 N W k n 3 + n 2 .
S σ ( S ) = N W k n 1 + N W k n .
1 + N W k n > [ N ( 2 W k n + 1 ) ] 1 / 2 ,
W k n > N - 1 / N
W k n > N - 1 / 2 .
I ( f ) = A p = 1 N ( f ) exp ( i ϕ p ) ,
I ( 3 ) ( f 1 , f 2 ) = I ( f 1 ) I ( f 2 ) I * ( f 1 + f 2 ) .
I ( 3 ) ( f 1 , f 2 ) = A 3 p = 1 N ( f 1 ) q = 1 N ( f 2 ) r = 1 N ( f 1 + f 2 ) exp ( i ϕ p ) exp ( i ϕ q ) exp ( - i ϕ r ) ,
N r = N ( f 1 ) N ( f 2 ) N ( f 1 + f 2 ) - N l ( f 1 , f 2 ) .
I ( 3 ) ( f 1 , f 2 ) = A 3 N l ( f 1 , f 2 ) .
σ 0 2 = A 6 N r = A 6 N ( f 1 ) N ( f 2 ) N ( f 1 + f 2 ) - A 6 N l ( f 1 , f 2 ) .
var [ β ( f 1 , f 2 ) ] = 1 2 M σ 0 2 I ( 3 ) ( f 1 , f 2 ) 2 ,
var [ β ( f 1 , f 2 ) ] = N ( f 1 ) N ( f 2 ) N ( f 1 + f 2 ) - N l ( f 1 , f 2 ) 2 M N l 2 ( f 1 , f 2 ) .
[ I ( f 1 ) + B ( f 1 ) ] [ I ( f 2 ) + B ( f 2 ) ] [ I * ( f 1 + f 2 ) + B * ( f 1 + f 2 ) ] = I ( f 1 ) I ( f 2 ) I * ( f 1 + f 2 ) + B ( f 1 ) I ( f 2 ) I * ( f 1 + f 2 ) + I ( f 1 ) B ( f 2 ) I * ( f 1 + f 2 ) + I ( f 1 ) I ( f 2 ) B * ( f 1 + f 2 ) + B ( f 1 ) B ( f 2 ) I * ( f 1 + f 2 ) + B ( f 1 ) I ( f 2 ) B * ( f 1 + f 2 ) + I ( f 1 ) B ( f 2 ) B * ( f 1 + f 2 ) + B ( f 1 ) B ( f 2 ) B * ( f 1 + f 2 ) .
I ( f ) 2 = A 2 | p = 1 N ( f ) exp ( i ϕ p ) | 2 = A 2 N ( f ) .
B ( f ) 2 = σ n 2 .
σ 2 = σ 0 2 + A 4 σ n 2 [ N ( f 1 ) N ( f 2 ) + N ( f 1 ) N ( f 1 + f 2 ) + N ( f 2 ) N ( f 1 + f 2 ) ] + A 2 σ n 4 [ N ( f 1 ) + N ( f 2 ) + N ( f 1 + f 2 ) ] + σ n 6 ,
σ 2 = [ A 2 N ( f 1 ) + σ n 2 ] [ A 2 N ( f 2 ) + σ n 2 ] [ A 2 N ( f 1 + f 2 ) + σ n 2 ] - A 6 N l ( f 1 , f 2 ) .
var [ β ( f 1 , f 2 ) ] = [ N ( f 1 ) + σ n 2 / A 2 ] [ N ( f 2 ) + σ n 2 / A 2 ] [ N ( f 1 + f 2 ) + σ n 2 / A 2 ] - N l ( f 1 , f 2 ) 2 M N l 2 ( f 1 , f 2 ) .
var [ β ( f 1 , f 2 ) ] = ( 1 + σ n 2 / A 2 ) 3 - 1 2 M .
var [ β ( f 1 , f 2 ) ] = ( 1 + σ n 2 / A 2 ) 3 - 1 2 M N l ( f 1 , f 2 ) .
[ N ( f 1 ) + σ n 2 / A 2 ] [ N ( f 2 ) + σ n 2 / A 2 ] [ N ( f 1 + f 2 ) + σ n 2 / A 2 ] > N l ( f 1 , f 2 ) [ 1 + σ n 2 / A 2 ] 3 .
N ( f 1 ) N ( f 2 ) N ( f 1 + f 2 ) N l ( f 1 , f 2 ) N ,
A 2 σ n 2 > N 1 / 3 - 1 N - N 1 / 3 ,
A 2 σ n 2 > N - 2 / 3 ,

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