Abstract

One method for improving the quality of astronomical images measured through a atmospheric turbulence uses simultaneous short-exposure measurements of both an image and the output of a wave-front sensor exposed to an image of the telescope pupil. The wave-front sensor measurements are used to reconstruct an estimate of the instantaneous generalized pupil function of the telescope, which is used to compute an estimate of the instantaneous optical transfer function, which is then used in a deconvolution procedure. This imaging method has been called both deconvolution from wave-front sensor (DWFS) measurements and self-referenced speckle holography. We analyze the signal-to-noise ratio (SNR) behavior of this imaging method in the spatial frequency domain. The analysis includes effects arising from differences in the correlation properties of the incident and the estimated pupil phases and the fact that the object-spectrum estimator is a randomly filtered doubly stochastic Poisson random process. SNR results obtained for the DWFS method are compared with the speckle-imaging power-spectrum SNR for equivalent seeing conditions and light levels. It is shown that for unresolved stars the power-spectrum SNR is superior to the DWFS SNR. However, for extended objects the power-spectrum SNR and the DWFS SNR are similar. Since speckle imaging uses a separate Fourier phase-reconstruction process not required by the DWFS method, the DWFS method provides an alternative to speckle imaging that uses simple postprocessing at the cost of a wave-front sensor measurement but with no loss of SNR performance for extended objects.

© 1994 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” Prog. Opt. 19, (1981).
    [Crossref]
  2. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, NewYork, 1968), pp. 101–136.
  3. J. W. Goodman, Statistical Optics (Wiley, New York, 1985), pp. 361–460.
  4. D. L. Fried, “Post detection wavefront compensation,” in Digital Image Recovery and Synthesis, P. S. Idell, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 828, 127–133 (1987).
  5. J. Primot, G. Rousset, J. C. Fontanella, “Deconvolution from wave-front sensing: a new technique for compensating turbulence-degraded images,” J. Opt. Soc. Am. A 7, 1589–1608 (1990).
    [Crossref]
  6. J. D. Gonglewski, D. G. Voelz, J. S. Fender, D. C. Dayton, B. K. Spielbusch, R. E. Pierson, “First astronomical application of postdetection turbulence compensation: images of α Aurigae, ν Ursae Majoris, and α Geminorum using self-referenced speckle holography,” Appl. Opt. 29, 4527–4529 (1990).
    [Crossref] [PubMed]
  7. F. Roddier, C. Roddier, S. V. Peursem, “Diffraction-limited imaging through aberrated optics using pupil-plane and/or image-plane information,” in Space Sensing, Communications, and Networking, M. Ross, R. J. Temkin, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1059, 173–179 (1989).
  8. B. M. Welsh, R. N. VonNiederhausern, “Performance analysis of self-referenced speckle holography,” in Atmospheric Propagation and Remote Sensing, A. Kohnle, W. B. Miller, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1688, 572–582 (1992).
  9. B. M. Welsh, R. N. Von Niederhausern, “Performance analysis of the self-referenced speckle holography image reconstruction technique,” Appl. Opt. 32, 5071–5078 (1993).
    [Crossref] [PubMed]
  10. M. C. Roggemann, B. M. Welsh, J. Devey, “Statistical properties of transfer function estimates obtaining using wave-front sensor measurements looking through atmospheric turbulence,” Appl. Opt. (to be published).
  11. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [Crossref]
  12. E. P. Wallner, “Optimal wave-front correction using slope measurements,” J. Opt. Soc. Am. 73, 1771–1776 (1983).
    [Crossref]
  13. M. C. Roggemann, “Optical performance of fully and partially compensated adaptive optics systems using least-squares and minimum variance phase reconstruction,” Comput. Electron. Eng. 18, 451–466 (1992).
    [Crossref]
  14. J. C. Dainty, A. H. Greenaway, “Estimation of spatial power spectra in speckle imaging,” J. Opt. Soc. Am. 69, 786–790 (1979).
    [Crossref]
  15. M. C. Roggemann, C. L. Matson, “Power spectrum and Fourier phase spectrum estimation by using fully and partially compensating adaptive optics and bispectrum postprocessing,” J. Opt. Soc. Am. A 9, 1525–1535 (1992).
    [Crossref]
  16. C. L. Matson, I. A. DeLarue, T. M. Gray, I. E. Drunzer, “Optimal Fourier spectrum estimation from the bispectrum,” Comput. Electr. Eng. 18, 485–497 (1992).
    [Crossref]
  17. B. M. Welsh, C. S. Gardner, “Performance analysis of adaptive optics systems using slope sensors,” J. Opt. Soc. Am. A 6, 1913–1923 (1989).
    [Crossref]
  18. P. S. Idell, A. Webster, “Resolution limits for coherent optical imaging: signal-to-signal analysis in the spatial-frequency domain,” J. Opt. Soc. Am. 9, 43–56 (1992).
    [Crossref]
  19. G. Cochran, “Phase screen generation,” Tech. Rep. 663 (Optical Sciences Company, Placentia, California, 1985).
  20. M. C. Roggemann, “Limited degree-of-freedom adpative optics and image reconstruction,” Appl. Opt. 30, 4227–4233 (1991).
    [Crossref] [PubMed]
  21. M. C. Roggemann, J. A. Meinhardt, “Image reconstruction by means of wave-front sensor measurements in closed loop adaptive optics systems,” J. Opt. Soc. Am. A 10, 1996–2007 (1993).
    [Crossref]

1993 (2)

1992 (4)

M. C. Roggemann, C. L. Matson, “Power spectrum and Fourier phase spectrum estimation by using fully and partially compensating adaptive optics and bispectrum postprocessing,” J. Opt. Soc. Am. A 9, 1525–1535 (1992).
[Crossref]

C. L. Matson, I. A. DeLarue, T. M. Gray, I. E. Drunzer, “Optimal Fourier spectrum estimation from the bispectrum,” Comput. Electr. Eng. 18, 485–497 (1992).
[Crossref]

P. S. Idell, A. Webster, “Resolution limits for coherent optical imaging: signal-to-signal analysis in the spatial-frequency domain,” J. Opt. Soc. Am. 9, 43–56 (1992).
[Crossref]

M. C. Roggemann, “Optical performance of fully and partially compensated adaptive optics systems using least-squares and minimum variance phase reconstruction,” Comput. Electron. Eng. 18, 451–466 (1992).
[Crossref]

1991 (1)

1990 (2)

1989 (1)

1983 (1)

1981 (1)

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” Prog. Opt. 19, (1981).
[Crossref]

1979 (1)

1976 (1)

Cochran, G.

G. Cochran, “Phase screen generation,” Tech. Rep. 663 (Optical Sciences Company, Placentia, California, 1985).

Dainty, J. C.

Dayton, D. C.

DeLarue, I. A.

C. L. Matson, I. A. DeLarue, T. M. Gray, I. E. Drunzer, “Optimal Fourier spectrum estimation from the bispectrum,” Comput. Electr. Eng. 18, 485–497 (1992).
[Crossref]

Devey, J.

M. C. Roggemann, B. M. Welsh, J. Devey, “Statistical properties of transfer function estimates obtaining using wave-front sensor measurements looking through atmospheric turbulence,” Appl. Opt. (to be published).

Drunzer, I. E.

C. L. Matson, I. A. DeLarue, T. M. Gray, I. E. Drunzer, “Optimal Fourier spectrum estimation from the bispectrum,” Comput. Electr. Eng. 18, 485–497 (1992).
[Crossref]

Fender, J. S.

Fontanella, J. C.

J. Primot, G. Rousset, J. C. Fontanella, “Deconvolution from wave-front sensing: a new technique for compensating turbulence-degraded images,” J. Opt. Soc. Am. A 7, 1589–1608 (1990).
[Crossref]

Fried, D. L.

D. L. Fried, “Post detection wavefront compensation,” in Digital Image Recovery and Synthesis, P. S. Idell, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 828, 127–133 (1987).

Gardner, C. S.

Gonglewski, J. D.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, NewYork, 1968), pp. 101–136.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), pp. 361–460.

Gray, T. M.

C. L. Matson, I. A. DeLarue, T. M. Gray, I. E. Drunzer, “Optimal Fourier spectrum estimation from the bispectrum,” Comput. Electr. Eng. 18, 485–497 (1992).
[Crossref]

Greenaway, A. H.

Idell, P. S.

Matson, C. L.

M. C. Roggemann, C. L. Matson, “Power spectrum and Fourier phase spectrum estimation by using fully and partially compensating adaptive optics and bispectrum postprocessing,” J. Opt. Soc. Am. A 9, 1525–1535 (1992).
[Crossref]

C. L. Matson, I. A. DeLarue, T. M. Gray, I. E. Drunzer, “Optimal Fourier spectrum estimation from the bispectrum,” Comput. Electr. Eng. 18, 485–497 (1992).
[Crossref]

Meinhardt, J. A.

Noll, R. J.

Peursem, S. V.

F. Roddier, C. Roddier, S. V. Peursem, “Diffraction-limited imaging through aberrated optics using pupil-plane and/or image-plane information,” in Space Sensing, Communications, and Networking, M. Ross, R. J. Temkin, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1059, 173–179 (1989).

Pierson, R. E.

Primot, J.

J. Primot, G. Rousset, J. C. Fontanella, “Deconvolution from wave-front sensing: a new technique for compensating turbulence-degraded images,” J. Opt. Soc. Am. A 7, 1589–1608 (1990).
[Crossref]

Roddier, C.

F. Roddier, C. Roddier, S. V. Peursem, “Diffraction-limited imaging through aberrated optics using pupil-plane and/or image-plane information,” in Space Sensing, Communications, and Networking, M. Ross, R. J. Temkin, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1059, 173–179 (1989).

Roddier, F.

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” Prog. Opt. 19, (1981).
[Crossref]

F. Roddier, C. Roddier, S. V. Peursem, “Diffraction-limited imaging through aberrated optics using pupil-plane and/or image-plane information,” in Space Sensing, Communications, and Networking, M. Ross, R. J. Temkin, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1059, 173–179 (1989).

Roggemann, M. C.

M. C. Roggemann, J. A. Meinhardt, “Image reconstruction by means of wave-front sensor measurements in closed loop adaptive optics systems,” J. Opt. Soc. Am. A 10, 1996–2007 (1993).
[Crossref]

M. C. Roggemann, “Optical performance of fully and partially compensated adaptive optics systems using least-squares and minimum variance phase reconstruction,” Comput. Electron. Eng. 18, 451–466 (1992).
[Crossref]

M. C. Roggemann, C. L. Matson, “Power spectrum and Fourier phase spectrum estimation by using fully and partially compensating adaptive optics and bispectrum postprocessing,” J. Opt. Soc. Am. A 9, 1525–1535 (1992).
[Crossref]

M. C. Roggemann, “Limited degree-of-freedom adpative optics and image reconstruction,” Appl. Opt. 30, 4227–4233 (1991).
[Crossref] [PubMed]

M. C. Roggemann, B. M. Welsh, J. Devey, “Statistical properties of transfer function estimates obtaining using wave-front sensor measurements looking through atmospheric turbulence,” Appl. Opt. (to be published).

Rousset, G.

J. Primot, G. Rousset, J. C. Fontanella, “Deconvolution from wave-front sensing: a new technique for compensating turbulence-degraded images,” J. Opt. Soc. Am. A 7, 1589–1608 (1990).
[Crossref]

Spielbusch, B. K.

Voelz, D. G.

Von Niederhausern, R. N.

VonNiederhausern, R. N.

B. M. Welsh, R. N. VonNiederhausern, “Performance analysis of self-referenced speckle holography,” in Atmospheric Propagation and Remote Sensing, A. Kohnle, W. B. Miller, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1688, 572–582 (1992).

Wallner, E. P.

Webster, A.

Welsh, B. M.

B. M. Welsh, R. N. Von Niederhausern, “Performance analysis of the self-referenced speckle holography image reconstruction technique,” Appl. Opt. 32, 5071–5078 (1993).
[Crossref] [PubMed]

B. M. Welsh, C. S. Gardner, “Performance analysis of adaptive optics systems using slope sensors,” J. Opt. Soc. Am. A 6, 1913–1923 (1989).
[Crossref]

M. C. Roggemann, B. M. Welsh, J. Devey, “Statistical properties of transfer function estimates obtaining using wave-front sensor measurements looking through atmospheric turbulence,” Appl. Opt. (to be published).

B. M. Welsh, R. N. VonNiederhausern, “Performance analysis of self-referenced speckle holography,” in Atmospheric Propagation and Remote Sensing, A. Kohnle, W. B. Miller, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1688, 572–582 (1992).

Appl. Opt. (3)

Comput. Electr. Eng. (1)

C. L. Matson, I. A. DeLarue, T. M. Gray, I. E. Drunzer, “Optimal Fourier spectrum estimation from the bispectrum,” Comput. Electr. Eng. 18, 485–497 (1992).
[Crossref]

Comput. Electron. Eng. (1)

M. C. Roggemann, “Optical performance of fully and partially compensated adaptive optics systems using least-squares and minimum variance phase reconstruction,” Comput. Electron. Eng. 18, 451–466 (1992).
[Crossref]

J. Opt. Soc. Am. (4)

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

Prog. Opt. (1)

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” Prog. Opt. 19, (1981).
[Crossref]

Other (7)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, NewYork, 1968), pp. 101–136.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), pp. 361–460.

D. L. Fried, “Post detection wavefront compensation,” in Digital Image Recovery and Synthesis, P. S. Idell, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 828, 127–133 (1987).

M. C. Roggemann, B. M. Welsh, J. Devey, “Statistical properties of transfer function estimates obtaining using wave-front sensor measurements looking through atmospheric turbulence,” Appl. Opt. (to be published).

F. Roddier, C. Roddier, S. V. Peursem, “Diffraction-limited imaging through aberrated optics using pupil-plane and/or image-plane information,” in Space Sensing, Communications, and Networking, M. Ross, R. J. Temkin, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1059, 173–179 (1989).

B. M. Welsh, R. N. VonNiederhausern, “Performance analysis of self-referenced speckle holography,” in Atmospheric Propagation and Remote Sensing, A. Kohnle, W. B. Miller, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1688, 572–582 (1992).

G. Cochran, “Phase screen generation,” Tech. Rep. 663 (Optical Sciences Company, Placentia, California, 1985).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1

Schematic diagram of hardware and computer system used in DWFS.

Fig. 2
Fig. 2

Block diagram of model used for deriving the DWFS SNR: (a) photon-limited case, (b) photon-noise and CCD read-noise case.

Fig. 3
Fig. 3

Expected single-frame DWFS SNR and speckle-imaging power-spectrum SNR for a single star for the following conditions: (a) DWFS SNR for r 0 = 50 cm, (b) power-spectrum SNR for r 0 = 50 cm, (c) DWFS SNR for r 0 = 10 cm, (d) power-spectrum SNR for r 0 = 10 cm, (e) DWFS SNR for r 0 = 7 cm, (f) power-spectrum SNR for r 0 = 7 cm. For the power-spectrum SNR’s, the image-plane average photoevent levels are consistent with the SNR W values shown.

Fig. 4
Fig. 4

Same as Fig. 3 but for an extended satellite object.

Fig. 5
Fig. 5

Satellite model used for extended-object calculations.

Equations (79)

Equations on this page are rendered with MathJax. Learn more.

i ( x , y ) = o ( x , y ) * h ( x , y ) ,
I ( u , v ) = O ( u , v ) H ( u , v ) ,
H ˜ i ( u , v ) = P ( x , y ) P ( x - u λ f , y - v λ f ) exp { j [ θ ˜ i ( x , y ) - θ ˜ i ( x - u λ f , y - v λ f ) ] } d x d y P ( x , y ) 2 d x d y ,
O ˜ ( u , v ) = I i ( u , v ) H ˜ i * ( u , v ) H ˜ i ( u , v ) 2 ,
O ˜ ( u , v ) = O ( u , v ) H i ( u , v ) H ˜ i * ( u , v ) H ˜ i ( u , v ) 2 ,
S ( u , v ) = H i ( u , v ) H ˜ i * ( u , v ) H ˜ i ( u , v ) 2 ,
S i ( u , v ) = H i ( u , v ) H ˜ i * ( u , v ) H ˜ i ( u , v ) 2 .
H i ( u , v ) H ˜ i * ( u , v ) - H ˜ i ( u , v ) 2 0
O ˜ 1 ( u , v ) = I i ( u , v ) [ H ˜ i obj ( u , v ) ] * H k ref ( u , v ) [ H ˜ i ref ( u , v ) ] * .
S 1 ( u , v ) = H i obj ( u , v ) [ H ˜ i obj ( u , v ) ] * H k ref ( u , v ) [ H ˜ k ref ( u , v ) ] * .
Num [ S i 1 ( u , v ) ] = H i obj ( u , v ) [ H ˜ i obj ( u , v ) ] * .
Num [ O ˜ 1 ( u , v ) ] = I i ( u , v ) [ H ˜ i obj ( u , v ) ] * ,
Num [ O ˜ i 1 ( u , v ) ] = I i ( u , v ) [ H ˜ i obj ( u , v ) ] * .
H i ( u , v ) = N - 1 P ( x , y ) P ( x - u λ f , y - v λ f ) × exp { j [ θ i ( x , y ) - θ i ( x - u λ f , y - v λ f ) ] } d x d y ,
N = P ( x , y ) 2 d x d y
H i ( u , v ) H ˜ i * ( u , v ) - H ˜ i ( u , v ) 2 0
θ ˜ i ( x , y ) = j c j r j ( x , y ) ,
c j = n M j n s n ,
Δ = ( s - H c ) ,
h n j = s n c j .
c opt = ( H T H ) - 1 H T s .
M j n = ( H T H ) - 1 H T .
SNR O ˜ 1 = Num [ O ˜ i 1 ( u , v ) ] ( var { Num [ O ˜ i 1 ( u , v ) ] } ) 1 / 2 ,
var ( Z ) = Z 2 - Z 2 .
SNR ( u , v ) = K ¯ O n ( u , v ) H i ( u , v ) H ˜ i * ( u , v ) { K ¯ H ˜ i ( u , v ) 2 + ( K ¯ ) 2 O n ( u , v ) 2 var [ H i ( u , v ) H ˜ i ( u , v ) ] } 1 / 2 ,
SNR ( u , v ) = K ¯ O n ( u , v ) H i ( u , v ) H ˜ i * ( u , v ) { K ¯ H ˜ i ( u , v ) 2 + ( K ¯ ) 2 O n ( u , v ) 2 var [ H i ( u , v ) H ˜ i ( u , v ) ] + P σ p 2 H ˜ i ( u , v ) 2 } 1 / 2 ,
SNR N ( u , v ) = N 1 / 2 SNR ( u , v ) .
α n α n = σ α 2 δ n n ,
σ α = 0.74 π η SNR W L ,
SNR W = ( K ¯ W ) 1 / 2 ,
SNR W = K ¯ W ( K ¯ W + P σ r 2 ) 1 / 2 .
D ( Δ x , Δ y ) = 6.88 [ ( Δ x 2 + Δ y 2 ) 1 / 2 r 0 ] 5 / 3 ,
SNR ( u ) = Num [ O ˜ i 1 ( u ) ] ( var { Num [ O ˜ i 1 ( u ) ] } ) 1 / 2 ,
var ( Z ) = Z 2 - Z 2 .
i ( x ) = n = 1 K δ ( x - x n ) ,
I ( u ) = - i ( x ) exp ( - j 2 π u x ) d x ,
I ( u ) = n = 1 K exp ( - j 2 π u x n ) .
i ( x ) = n = 1 K δ ( x - x n ) + p = 1 P n p δ ( x - x p ) ,
n p n p = σ p 2 ,             p = p ,
n p n p = 0 ,             p p ,
Num [ O ˜ 1 ( u ) ] = I i ( u ) H ˜ i * ( u ) .
I i ( u ) H ˜ i * ( u ) = E H , H ˜ ( E K H , H ˜ { E x n K ; H , H ˜ [ I i ( u ) H ˜ i * ( u ) ] } ) ,
E x n K ; H , H ˜ [ I i ( u ) H ˜ i * ( u ) ] = E x n K ; H , H ˜ [ n = 1 K exp ( - j 2 π u x n ) H ˜ i * ( u ) ] .
E x n K ; H , H ˜ [ I i ( u ) H ˜ i * ( u ) ] = E x n K ; H , H ˜ [ n = 1 K exp ( - j 2 π u x n ) ] H ˜ i * ( u ) .
E x n K ; H , H ˜ [ I i ( u ) H ˜ i * ( u ) ] = ( n = 1 K { E x n K ; H , H ˜ [ exp ( - j 2 π u x n ) ] } ) H ˜ i * ( u ) .
p x ( x n ) = λ ( x n ) - λ ( x n ) d x n
- λ ( x n ) d x n = K ¯ ,
F [ λ i ( x n ) ] = Λ i ( u ) .
Λ i ( u ) = O ( u ) H i ( u ) ,
o ( x ) = K ¯ o act ( x ) o act ( x ) d x ,
F [ o ( x ) ] = O ( u ) = K ¯ F [ o act ( x ) ] o act ( x ) d x .
O n ( u ) = F [ o act ( x ) ] o act ( x ) d x ,
O ( u ) = K ¯ O n ( u ) .
E x n K ; H , H ˜ [ I i ( u ) H ˜ i * ( u ) ] = K H ˜ i * ( u ) Λ i ( u ) K ¯ = K H ˜ i * ( u ) H i ( u ) O ( u ) K ¯ .
E x n K ; H , H ˜ [ I i ( u ) H ˜ i * ( u ) ] = K H ˜ i * ( u ) H i ( u ) O n ( u ) .
E K H , H ˜ [ K O n ( u ) H i ( u ) H ˜ i * ( u ) ] = K ¯ O n ( u ) H i ( u ) H ˜ i * ( u ) .
E H , H ˜ [ K ¯ O n ( u ) H i ( u ) H ˜ i * ( u ) ] = K ¯ O n ( u ) H i ( u ) H ˜ i * ( u ) .
Num [ O ˜ i 1 ( u ) ] 2 = I i ( u ) H ˜ i * ( u ) 2 .
Num [ O ˜ i 1 ( u ) ] 2 = [ n = 1 K exp ( - j 2 π u x n ) ] [ m = 1 K exp ( j 2 π u x n ) ] H ˜ i ( u ) 2 .
Num [ O ˜ i 1 ( u ) ] 2 = E H , H ˜ [ E K H , H ˜ ( E x n , x m K ; H , H ˜ { [ n = 1 K exp ( - j 2 π u x n ) ] × [ m = 1 K exp ( j 2 π u x m ) ] H ˜ i ( u ) 2 } ) ] ,
E x n , x m K ; H , H ˜ { [ n = 1 K exp ( - j 2 π u x n ) ] [ m = 1 K exp ( j 2 π u x m ) ] H ˜ i ( u ) 2 } = E x n , x m K ; H , H ˜ ( { n = 1 K m = 1 K exp [ - j 2 π u ( x n - x m ) ] } H ˜ i ( u ) 2 ) ,
E x n , x m K ; H , H ˜ ( { n = 1 K m = 1 K exp [ - j 2 π u ( x n - x m ) ] } H ˜ i ( u ) 2 ) = E x n , x m K ; H , H ˜ { n = 1 K m = 1 K exp [ - j 2 π u ( x n - x m ) ] } H ˜ i ( u ) 2 .
p x ( x n x m ) = p x ( x n ) p x ( x m ) ,
E x n , x m K ; H , H ˜ { exp [ - j 2 π u ( x n - x m ) ] } = Λ i ( u ) 2 ( K ¯ ) 2 .
E x n , x m K ; H , H ˜ ( { n = 1 K m = 1 K exp [ - j 2 π u ( x n - x m ) ] } H ˜ i ( u ) 2 ) = [ K + ( K 2 - K ) O n ( u ) H i ( u ) 2 ] H ˜ i ( u ) 2 .
E K H , H ˜ { [ K + ( K 2 - K ) O n ( u ) H i ( u ) 2 ] H ˜ i ( u ) 2 } = [ K ¯ + ( K ¯ ) 2 O n ( u ) H i ( u ) 2 ] H ˜ i ( u ) 2 ,
E H , H ˜ { [ K ¯ + ( K ¯ ) 2 O n ( u ) H i ( u ) 2 ] H ˜ i ( u ) 2 } = K ¯ H ˜ i ( u ) 2 + ( K ¯ ) 2 O n ( u ) 2 H i ( u ) H ˜ i ( u ) 2 .
var { Num [ O i 1 ( u ) ] } = K ¯ H ˜ i ( u ) 2 + ( K ¯ ) 2 O n ( u ) 2 var [ H i ( u ) H ˜ i ( u ) ] .
σ RN 2 ( u ) = F [ p = 1 P n p δ ( x - x p ) ] H ˜ i ( u ) 2 .
σ RN 2 ( u ) = | F [ p = 1 P n p δ ( x - x p ) ] | 2 H ˜ i ( u ) 2 .
F [ p = 1 P n p δ ( x - x p ) ] = p = 1 P n p exp ( - j 2 π u x p ) .
| F [ p = 1 P n p δ ( x - x p ) ] | 2 = p = 1 P p = 1 P n p n p exp [ - j 2 π u ( x p - x p ) ] .
p = 1 P p = 1 P n p n p exp [ - j 2 π u ( x p - x p ) ] = p = 1 P p ' = 1 P n p n p exp [ - j 2 π u ( x p - x p ) ] .
p = 1 P p = 1 P n p n p exp [ - j 2 π u ( x p - x p ) ] = P σ p 2 .
σ RN 2 ( u ) = P σ p 2 H ˜ i ( u ) 2 .
var { Num [ O ˜ i 1 ( u ) ] = K ¯ H ˜ i ( u ) 2 + ( K ¯ ) 2 O n ( u ) 2 × var [ H i ( u ) H ˜ i ( u ) ] + P σ p 2 H ˜ i ( u ) 2 .
SNR ( u ) = K ¯ O n ( u ) H i ( u ) H ˜ i * ( u ) { K ¯ H ˜ i ( u ) 2 + ( K ¯ ) 2 O n ( u ) 2 var [ H i ( u ) H ˜ i ( u ) ] } 1 / 2 .
SNR ( u ) = K ¯ O n ( u ) H i ( u ) H ˜ i * ( u ) { K ¯ H ˜ i ( u ) 2 + ( K ¯ ) 2 O n ( u ) 2 var [ H i ( u ) H ˜ i ( u ) ] + P σ p 2 H ˜ i ( u ) 2 } 1 / 2 .
SNR ( u ) = K ¯ O n ( u ) H i ( u ) { K ¯ + ( K ¯ ) 2 O n ( u ) 2 var [ H i ( u ) ] } 1 / 2 .

Metrics