Abstract

Calculations of the signal-to-noise ratio (SNR) of estimates of the power spectra of spatially varying random processes, such as stellar speckle patterns, usually include realizations that contain less than two detected photons. It is shown in this paper that if these cases are excluded from the analysis procedure either implicitly or explicitly, then under the usual definition of SNR, the overall SNR of an estimate can increase by up to a factor of N¯1/2, where N¯1 is the average number of detected photons per realization.

© 1979 Optical Society of America

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References

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  1. F. Roddier, “Signal-to-noise ratio in speckle interferometry,” in Imaging in Astronomy, AAS/SAO/OSA/SPIE Topical Meeting. Preprints paper Th C6, Boston1975 (unpublished).
  2. J. W. Goodman and J. F. Belsher, “Fundamental limitations in linear invariant restoration of atmospherically degraded images,” SPIE Seminar Proc.,  75, 141–154 (1976).
    [Crossref]
  3. J. W. Goodman and J. F. Belsher, “Photon limited images and their restoration,” RADC-TR-76-50, March, 1976;“Precompensation and postcompensation of photon limited degraded images,” RADC-TR-76-382, Dec, 1976;“Photon limitations in imaging and image restoration”, RADC-TR-77-175, May1977.All ARPA Order No.2646 (Rome Air Development Center, Griffiss AFB, NY 13441).
  4. M. G. Miller, “Noise considerations in stellar speckle interferometry,” J. Opt. Soc. Am.,  67, 1176–1184 (1977).
    [Crossref]
  5. J. C. Dainty and A. H. Greenaway, “The signal-to-noise ratio in speckle interferometry,” Proc. IAU Colloquium No. 50, “High Angular Resolution Stellar Interferometry,” Maryland Aug/Sept.1978 (available from Chatterton Astronomy Department, University of Sydney, NSW, Australia 2006, $20 (Australian) + postage).
  6. A. H. Greenaway, “The signal-to-noise ratio in long baseline stellar interferometry,” Optica Acta (in press).
  7. J. J. Burke and J. B. Breckinridge, “Passive imaging through the turbulent atmosphere: Fundamental limits on the spatial frequency resolution of a rotational shearing interferometer,” J. Opt. Soc. Am.,  68, 67–77 (1978).
    [Crossref]
  8. A. Labeyrie, “High resolution techniques in optical astronomy,” in Progress in Optics; Vol. 14, edited by E. Wolf, (North-Holland, Amsterdam, 1976) p 47–87.
    [Crossref]
  9. J. C. Dainty, “The transfer function, signal-to-noise ratio and limiting magnitude in stellar speckle interferometry,” Mon. Not. R. Astr. Soc.,  169, 631–641 (1974).
  10. J. C. Dainty, “Computer simulations of speckle interferometry of binary stars in the photon counting mode,” Mon. Not. R. Astr. Soc.,  183, 223–236 (1978).
  11. J. G. Walker, “Optimum exposure time and filter bandwidth in speckle interferometry,” (see Ref 5).
  12. D. Korff, “Analysis of a method for obtaining near diffraction-limited information in the presence of atmospheric turbulence,” J. Opt. Soc. Am.,  63, 971–980 (1973).
    [Crossref]
  13. D. L. Fried, “Optical resolution through a randomly inhomogeneous medium,” J. Opt. Soc. Am.,  56, 1372–1379 (1966).
    [Crossref]
  14. The factor 0.435 on the bottom line of Eq. (21) arises from the normalization of the overall transfer function to 1.0 at the origin. Defining a normalized seeing transfer function,Ts(ν)≡Cz(ν)Cz(0)=exp{−3.44(|ν|r0)5/3},Eq. (7) of Ref. 5 can be written, for r0≪ D,〈|T(ν)|2〉=|Ts(ν)|2|T0(ν)|2+∫−∞∞|Ts(ν)|2dν·1P·TD(ν),where T0(ν) is the telescope OTF and P is the area of the (unshaded) pupil. With P = πD)2/4, performing the integration yields〈|T(ν)|2〉=|Ts(ν)|2|T0(ν)|2+(r0D)20.435TD(ν).

1978 (2)

J. C. Dainty, “Computer simulations of speckle interferometry of binary stars in the photon counting mode,” Mon. Not. R. Astr. Soc.,  183, 223–236 (1978).

J. J. Burke and J. B. Breckinridge, “Passive imaging through the turbulent atmosphere: Fundamental limits on the spatial frequency resolution of a rotational shearing interferometer,” J. Opt. Soc. Am.,  68, 67–77 (1978).
[Crossref]

1977 (1)

1976 (2)

J. W. Goodman and J. F. Belsher, “Fundamental limitations in linear invariant restoration of atmospherically degraded images,” SPIE Seminar Proc.,  75, 141–154 (1976).
[Crossref]

J. W. Goodman and J. F. Belsher, “Photon limited images and their restoration,” RADC-TR-76-50, March, 1976;“Precompensation and postcompensation of photon limited degraded images,” RADC-TR-76-382, Dec, 1976;“Photon limitations in imaging and image restoration”, RADC-TR-77-175, May1977.All ARPA Order No.2646 (Rome Air Development Center, Griffiss AFB, NY 13441).

1974 (1)

J. C. Dainty, “The transfer function, signal-to-noise ratio and limiting magnitude in stellar speckle interferometry,” Mon. Not. R. Astr. Soc.,  169, 631–641 (1974).

1973 (1)

1966 (1)

Belsher, J. F.

J. W. Goodman and J. F. Belsher, “Fundamental limitations in linear invariant restoration of atmospherically degraded images,” SPIE Seminar Proc.,  75, 141–154 (1976).
[Crossref]

J. W. Goodman and J. F. Belsher, “Photon limited images and their restoration,” RADC-TR-76-50, March, 1976;“Precompensation and postcompensation of photon limited degraded images,” RADC-TR-76-382, Dec, 1976;“Photon limitations in imaging and image restoration”, RADC-TR-77-175, May1977.All ARPA Order No.2646 (Rome Air Development Center, Griffiss AFB, NY 13441).

Breckinridge, J. B.

Burke, J. J.

Dainty, J. C.

J. C. Dainty, “Computer simulations of speckle interferometry of binary stars in the photon counting mode,” Mon. Not. R. Astr. Soc.,  183, 223–236 (1978).

J. C. Dainty, “The transfer function, signal-to-noise ratio and limiting magnitude in stellar speckle interferometry,” Mon. Not. R. Astr. Soc.,  169, 631–641 (1974).

J. C. Dainty and A. H. Greenaway, “The signal-to-noise ratio in speckle interferometry,” Proc. IAU Colloquium No. 50, “High Angular Resolution Stellar Interferometry,” Maryland Aug/Sept.1978 (available from Chatterton Astronomy Department, University of Sydney, NSW, Australia 2006, $20 (Australian) + postage).

Fried, D. L.

Goodman, J. W.

J. W. Goodman and J. F. Belsher, “Photon limited images and their restoration,” RADC-TR-76-50, March, 1976;“Precompensation and postcompensation of photon limited degraded images,” RADC-TR-76-382, Dec, 1976;“Photon limitations in imaging and image restoration”, RADC-TR-77-175, May1977.All ARPA Order No.2646 (Rome Air Development Center, Griffiss AFB, NY 13441).

J. W. Goodman and J. F. Belsher, “Fundamental limitations in linear invariant restoration of atmospherically degraded images,” SPIE Seminar Proc.,  75, 141–154 (1976).
[Crossref]

Greenaway, A. H.

J. C. Dainty and A. H. Greenaway, “The signal-to-noise ratio in speckle interferometry,” Proc. IAU Colloquium No. 50, “High Angular Resolution Stellar Interferometry,” Maryland Aug/Sept.1978 (available from Chatterton Astronomy Department, University of Sydney, NSW, Australia 2006, $20 (Australian) + postage).

A. H. Greenaway, “The signal-to-noise ratio in long baseline stellar interferometry,” Optica Acta (in press).

Korff, D.

Labeyrie, A.

A. Labeyrie, “High resolution techniques in optical astronomy,” in Progress in Optics; Vol. 14, edited by E. Wolf, (North-Holland, Amsterdam, 1976) p 47–87.
[Crossref]

Miller, M. G.

Roddier, F.

F. Roddier, “Signal-to-noise ratio in speckle interferometry,” in Imaging in Astronomy, AAS/SAO/OSA/SPIE Topical Meeting. Preprints paper Th C6, Boston1975 (unpublished).

Walker, J. G.

J. G. Walker, “Optimum exposure time and filter bandwidth in speckle interferometry,” (see Ref 5).

J. Opt. Soc. Am. (4)

Mon. Not. R. Astr. Soc. (2)

J. C. Dainty, “The transfer function, signal-to-noise ratio and limiting magnitude in stellar speckle interferometry,” Mon. Not. R. Astr. Soc.,  169, 631–641 (1974).

J. C. Dainty, “Computer simulations of speckle interferometry of binary stars in the photon counting mode,” Mon. Not. R. Astr. Soc.,  183, 223–236 (1978).

RADC-TR-76-50 (1)

J. W. Goodman and J. F. Belsher, “Photon limited images and their restoration,” RADC-TR-76-50, March, 1976;“Precompensation and postcompensation of photon limited degraded images,” RADC-TR-76-382, Dec, 1976;“Photon limitations in imaging and image restoration”, RADC-TR-77-175, May1977.All ARPA Order No.2646 (Rome Air Development Center, Griffiss AFB, NY 13441).

SPIE Seminar Proc. (1)

J. W. Goodman and J. F. Belsher, “Fundamental limitations in linear invariant restoration of atmospherically degraded images,” SPIE Seminar Proc.,  75, 141–154 (1976).
[Crossref]

Other (6)

F. Roddier, “Signal-to-noise ratio in speckle interferometry,” in Imaging in Astronomy, AAS/SAO/OSA/SPIE Topical Meeting. Preprints paper Th C6, Boston1975 (unpublished).

A. Labeyrie, “High resolution techniques in optical astronomy,” in Progress in Optics; Vol. 14, edited by E. Wolf, (North-Holland, Amsterdam, 1976) p 47–87.
[Crossref]

J. G. Walker, “Optimum exposure time and filter bandwidth in speckle interferometry,” (see Ref 5).

J. C. Dainty and A. H. Greenaway, “The signal-to-noise ratio in speckle interferometry,” Proc. IAU Colloquium No. 50, “High Angular Resolution Stellar Interferometry,” Maryland Aug/Sept.1978 (available from Chatterton Astronomy Department, University of Sydney, NSW, Australia 2006, $20 (Australian) + postage).

A. H. Greenaway, “The signal-to-noise ratio in long baseline stellar interferometry,” Optica Acta (in press).

The factor 0.435 on the bottom line of Eq. (21) arises from the normalization of the overall transfer function to 1.0 at the origin. Defining a normalized seeing transfer function,Ts(ν)≡Cz(ν)Cz(0)=exp{−3.44(|ν|r0)5/3},Eq. (7) of Ref. 5 can be written, for r0≪ D,〈|T(ν)|2〉=|Ts(ν)|2|T0(ν)|2+∫−∞∞|Ts(ν)|2dν·1P·TD(ν),where T0(ν) is the telescope OTF and P is the area of the (unshaded) pupil. With P = πD)2/4, performing the integration yields〈|T(ν)|2〉=|Ts(ν)|2|T0(ν)|2+(r0D)20.435TD(ν).

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Figures (1)

FIG. 1
FIG. 1

SNR per frame (or realization) as a function of N ¯, the average number of detected photons per frame, for values of q from 0 to 5.

Equations (64)

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ϕ ( ν ) = lim X | Λ ( ν ) | 2 ,
Λ ( ν ) = 1 2 X X X λ ( X ) exp ( 2 π i ν x ) d x
( SNR ) 1 1 ;
( SNR ) 1 n ¯ T ¯ D ( ν ) ϕ ̂ 0 ( ν ) ;
( SNR ) 1 n ¯ 3 / 2 n s 1 / 2 T D ( ν ) ϕ ̂ 0 ( ν ) ,
( SNR ) 1 n ¯ .
d j ( x ) = k = 1 N j δ ( x x j k ) ,
Q = | D j ( ν ) | 2 N j ,
SNR = E { Q } / σ Q .
| D j ( ν ) | 2 = k = 1 N j l = 1 N j exp [ 2 π i ν ( x k x l ) ]
E k l { | D j ( ν ) | 2 } = k = 1 N j l = 1 N j E k l { exp [ 2 π i ν ( x k x l ) ] } .
( λ j ( x k ) / λ j ( x ) d x ) ( λ j ( x l ) / λ j ( x ) d x ) ,
| λ j ( x ) exp ( 2 π i ν x ) d x λ j ( x ) d x | 2 = | Λ j ( ν ) Λ j ( 0 ) | 2 .
E k l { | D j ( ν ) | 2 } = N j + N j ( N j 1 ) | Λ j ( ν ) Λ j ( 0 ) | 2 .
E { N j ( N j 1 ) ( N j p + 1 ) } = N ¯ j p ,
E k l , N j { | D j ( ν ) | 2 } = N ¯ j + | Λ j ( ν ) | 2 .
E { Q } = E { | D j ( ν ) | 2 N j } = ϕ λ ( ν ) ,
ϕ ̂ i ( ν ) = ϕ i ( ν ) / ϕ i ( 0 ) = ϕ ̂ λ ( ν ) = ϕ λ ( ν ) / ϕ λ ( 0 ) ,
E { N ¯ j 2 } = [ E { N j } ] 2 = N ¯ 2 ,
E { Q } = N ¯ 2 ϕ ̂ i ( ν ) .
σ Q 2 = E { Q 2 } [ E { Q } ] 2 = E { | D j ( ν ) | 4 } 2 E { N j | D j ( ν ) | 2 } + E { N j 2 } N ¯ 4 ϕ ̂ t 2 ( ν ) .
σ Q 2 = N ¯ 2 [ ( 1 + N ¯ ϕ ̂ i ( ν ) ) 2 + ϕ ̂ i ( 2 ν ) ] ,
( SNR ) 1 = N ¯ ϕ ̂ i ( ν ) { [ ( 1 + N ¯ ϕ ̂ i ( ν ) ] 2 + ϕ ̂ i ( 2 ν ) } 1 / 2 .
( SNR ) 1 = N ¯ ϕ ̂ i ( ν ) / [ 1 + N ¯ ϕ ̂ i ( ν ) ] , ν > ν max / 2 ,
ϕ ̂ i ( ν ) = [ T D ( ν ) / n s ] ϕ ̂ 0 ( ν ) , ν r 0 / λ f , n s 1 ,
n s = ( D / r 0 ) 2 1 0.435 .
( SNR ) 1 = n ¯ T 0 ( ν ) ϕ ̂ 0 ( ν ) 1 + n ¯ T D ( ν ) ϕ ̂ 0 ( ν ) , ν > D / 2 λ f , n s 1
( SNR ) 1 1
( SNR ) 1 n ¯ T D ( ν ) ϕ ̂ 0 ( ν )
Q 1 = | D j ( ν ) | 2 N ¯ c ,
P q ( i ) = m k / f q k ! k q , = 0 k < q ,
f q = q m k k ! = e m 0 q 1 m k k !
F q = 1 e m 0 q 1 m k k ! , q 1 = 1 , q = 0
M p , q = E { k ( k 1 ) ( k p + 1 ) } , p = 1 , 2 , 3
M p , q = m p q p m k k ! / f q = m p ( e m 0 q p 1 m k k ! ) / f q , q > p = m p e m f q , q p .
M p , q = H p , q m p ,
E { Q 1 } = H 2 , q N ¯ 2 ϕ ̂ i ( ν ) ,
σ Q 1 2 = H 1 , q N ¯ + [ 2 H 2 , q ( H 1 , q ) 2 ] N ¯ 2 + 4 H 2 , q N ¯ 2 ϕ ̂ i ( ν ) + H 2 , q N ¯ 2 ϕ ̂ i ( 2 ν ) + [ 4 H 3 , q 2 H 1 , q · H 2 , q ] N ¯ 3 ϕ ̂ i ( ν )
( SNR ) 1 = H 2 , q F q 1 / 2 n ¯ T D ( ν ) ϕ ̂ 0 ( ν ) / { 2 H 2 , q ( H 1 , q ) 2 + ( 2 H 3 , q H 1 , q H 2 , q ) × 2 n ¯ T D ( ν ) ϕ ̂ 0 ( ν ) + [ 2 H 4 , q ( H 2 , q ) 2 ] n ¯ 2 T D 2 ( ν ) ϕ ̂ 0 2 ( ν ) + 1 N ¯ [ H 1 , q + 4 H 2 , q n ¯ T D ( ν ) ϕ ̂ 0 ( ν ) ] } 1 / 2 .
( SNR ) 1 n ¯ 3 / 2 n s 1 / 2 T D ( ν ) ϕ 0 ( ν ) , N ¯ 1 .
( SNR ) 1 n ¯ T D ( ν ) ϕ 0 ( ν ) ( 2 3 ) 1 / 2 , N ¯ 1 .
R 1 N ¯ 1 / 2 ( 2 / 3 ) 1 / 2 , N ¯ 1 .
( SNR ) 1 n ¯ T D ( ν ) ϕ ̂ 0 ( ν ) , N ¯ 1
R 2 N ¯ 1 / 2 , N ¯ 1 .
σ Q 2 = E [ | D j ( ν ) | 4 ] 2 E [ N j | D j ( ν ) | 2 ] + E [ N j 2 ] N ¯ 4 ϕ ̂ i 2 ( ν ) ,
E { | D j ( ν ) | 4 } = N ¯ + 2 N ¯ 2 + 4 ( 1 + N ¯ ) N ¯ 2 ϕ ̂ i ( ν ) + N ¯ 2 ϕ ̂ i ( 2 ν ) + 2 N ¯ 4 ϕ ̂ i 2 ( ν ) .
E { N j | D j ( ν ) | 2 } = E { N j 2 + ( N j 3 N j 2 ) | Λ j ( ν ) Λ j ( 0 ) | 2 } .
E { N j | D j ( ν ) | 2 } = E [ N ¯ j + N ¯ j 2 + ( N ¯ j 3 + 3 N ¯ j 2 + N ¯ j N ¯ j 2 N ¯ j ) | Λ j ( ν ) | 2 N ¯ j 2 ] = E [ N ¯ j + N ¯ j 2 + ( N ¯ j + 2 ) ϕ λ ( ν ) ] .
E { N j | D j ( ν ) | 2 } = N ¯ + N ¯ 2 + N ¯ 2 ( N ¯ + 2 ) ϕ ¯ i ( ν ) ,
E { N j 2 } = N ¯ + N ¯ 2 .
σ Q 2 = N ¯ 2 + 2 N ¯ 3 ϕ ̂ i ( ν ) + N ¯ 4 ϕ ̂ i ( ν ) + N ¯ 2 ϕ ̂ i ( 2 ν ) .
Q 1 = ( | D j ( ν ) | 2 N ¯ c ) .
E k l { | D j ( ν ) | 2 } = N j + N j ( N j 1 ) | Λ j ( ν ) Λ j ( 0 ) | 2 .
E k l , j { | D j ( ν ) | 2 } = N ¯ j , c + H 2 , q ( H 1 , q ) 2 · | Λ i ( ν ) | 2 .
E { Q 1 } = H 2 , q N ¯ 2 ϕ ̂ i ( ν )
σ Q 1 2 = E { | D j ( ν ) | 4 } 2 N ¯ c E { | D j ( ν ) | 2 } + N ¯ c 2 ( H 2 , q ) 2 N ¯ 4 ϕ ̂ i 2 ( ν ) .
E { | D j ( ν ) | 4 } = H 1 , q N ¯ j + 2 H 2 , q N ¯ j 2 + 4 [ H 2 , q / ( H 1 , q ) 2 + [ H 3 , q / ( H 1 , q ) 2 ] N ¯ j ] | Λ j ( ν ) | 2 + [ H 2 , q / ( H 1 , q ) 2 ] | | Λ j ( 2 ν ) | 2 + [ H 4 , q / ( H 1 , q ) 4 ] | Λ j ( ν ) | 4 + [ H 3 , q / ( H 1 , q ) 3 ] { Λ j ( 2 ν ) [ Λ j * ( ν ) ] 2 + Λ j * ( 2 ν ) [ Λ j ( ν ) ] 2 } .
E λ { | Λ j ( ν ) | 2 } = ϕ λ ( ν ) , E λ ( { Λ j ( 2 ν ) [ Λ j * ( ν ) ] 2 + Λ j * ( 2 ν ) [ Λ j ( ν ) ] 2 } ) = 0 , E λ { | Λ j ( ν ) | 4 } = 2 ϕ λ 2 ( ν ) ,
E { | D j ( ν ) | 4 } = H 1 , q N ¯ + 2 H 2 , q N ¯ 2 + 4 [ H 2 , q / ( H 1 , q ) 2 + [ H 3 , q / ( H 1 , q ) 2 ] N ¯ ] ϕ λ ( ν ) + [ H 2 , q / ( H 1 , q ) 2 ] ϕ λ ( 2 ν ) + [ 2 H 4 , q / ( H 1 , q ) 4 ] ϕ λ 2 ( ν ) .
ϕ λ ( ν ) = ϕ λ ( 0 ) ϕ ̂ i ( ν ) = ( H 1 , q ) 2 N ¯ 2 ϕ ̂ i ( ν ) ,
σ Q 1 2 = H 1 , q N ¯ + [ 2 H 2 , q ( H 1 , q ) 2 ] N ¯ 2 + 4 H 2 , q N ¯ 2 ϕ ̂ i ( ν ) + H 2 , q N ¯ 2 ϕ ̂ i ( 2 ν ) + [ 4 H 3 , q 2 H 1 , q H 2 , q ] N ¯ 3 ϕ ̂ i ( ν ) + [ 2 H 4 , q ( H 2 , q ) 2 ] N ¯ 4 ϕ ̂ i 2 ( ν ) ,
Ts(ν)Cz(ν)Cz(0)=exp{3.44(|ν|r0)5/3},
|T(ν)|2=|Ts(ν)|2|T0(ν)|2+|Ts(ν)|2dν·1P·TD(ν),
|T(ν)|2=|Ts(ν)|2|T0(ν)|2+(r0D)20.435TD(ν).