Abstract

The effectiveness of optical preprocessing is examined for image-estimation applications. The problem is to design the pupil of the imaging system so that the detected image is the minimum-mean-square-error estimate of the desired image. The designs of several optimal pupil screens are presented. These designs demonstrate that both the optimum pupil screen and the effectiveness of optical preprocessing are functions of the statistics of the signal and the noise and of the coherence of the object illumination. The results indicate that preprocessing is more effective for coherent systems than for incoherent systems.

© 1987 Optical Society of America

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References

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  1. P. Jacquinot, B. Roizen-Dossier, “Apodisation,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1964), Vol. 3, pp. 31–186.
    [CrossRef]
  2. R. Barakat, “Application of apodization to increase two-point resolution by the Sparrow criterion. I. Coherent illumination,”J. Opt. Soc. Am. 52, 276–283 (1962).
    [CrossRef]
  3. R. Barakat, “Application of apodization to increase two-point resolution by the Sparrow criterion. II. Incoherent illumination,”J. Opt. Soc. Am. 53, 274–282 (1963).
    [CrossRef]
  4. V. P. Nayvar, N. K. Verma, “Two-point resolution of Gaussian aperture operating in partially coherent light using various resolution criteria,” Appl. Opt. 17, 2176–2180 (1978).
    [CrossRef]
  5. T. Araki, T. Asakura, “Coherent apodisation problems,” Opt. Commun. 20, 373–377 (1977).
    [CrossRef]
  6. E. C. Kintner, R. M. Sillitto, “Edge-ringing in partially coherent imaging,” Opt. Acta 24, 591–605 (1977).
    [CrossRef]
  7. F. G. Leaver, R. W. Smith, “Apodisation to produce a monotonically-decreasing radially symmetric, point-spread function,” Opt. Commun. 15, 374–378 (1975).
    [CrossRef]
  8. K. P. Rao, P. K. Mondal, T. Seshagiri Rao, “Coherent imagery of straight edges with Straubel apodisation filters,” Optik (Stuttgart) 50, 73–81 (1978).
  9. J. Tsujiuchi, “Correction of optical images by compensation of aberrations and by spatial frequency filtering,” in Progress in Optics, E. Wolfe, ed. (North-Holland, Amsterdam, 1963), Vol. 2, pp. 131–180.
    [CrossRef]
  10. S. C. Biswas, A. Boivin, “Influence of spherical aberration on the performance of optimum apodizers,” Opt. Acta 23, 569–588 (1976).
    [CrossRef]
  11. J. P. Mills, B. J. Thompson, “Effect of aberrations and apodization on the performance of coherent optical systems. II. Imaging,” J. Opt. Soc. Am. A 3, 704–716 (1986).
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  13. P. S. Idell, J. W. Goodman, “Design of optimal imaging concentrators for partially coherent sources: absolute encircled energy criterion,” J. Opt. Soc. Am. A 3, 943–953 (1986).
    [CrossRef]
  14. D. Slepian, “Analytic solution to two apodization problems,”J. Opt. Soc. Am. 55, 1110–1115 (1965).
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  15. W. E. Smith, H. H. Barrett, “Hotelling trace criterion as a figure of merit for the optimization of imaging systems,” J. Opt. Soc. Am. A 3, 717–725 (1986).
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  16. B. E. A. Saleh, “Optical bilinear transformations: general properties,” Opt. Acta 26, 777–799 (1979).
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  17. B. E. A. Saleh, W. C. Goeke, “Linear restoration of bilinearly distorted images,”J. Opt. Soc. Am. 70, 506–515 (1980).
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  18. B. E. A. Saleh, S. I. Sayegh, “Restoration of partially coherent images by use of a second-degree nonlinear filter,” Appl. Opt. 20, 4089–4093 (1981).
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  19. M. Rabbani, B. E. A. Saleh, “Bayesian restoration of partially coherent imagery in conventional and scanning microscopy,” Opt. Acta 31, 63–79 (1984).
    [CrossRef]
  20. B. E. A. Saleh, M. Rabbani, “Restoration of bilinearly distorted images: I. Finite impulse response linear digital filtering,”J. Opt. Soc. Am. 73, 66–70 (1983).
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  25. N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, E. Teller, “Equation of state calculations by fast computing machines,”J. Chem. Phys. 21, 1087–1092 (1953).
    [CrossRef]
  26. S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
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  29. B. P. Hildebrand, “Bounds on the modulation transfer function of optical systems in incoherent illumination,”J. Opt. Soc. Am. 56, 12–13 (1966).
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  30. J. A. MacDonald, “Apodization and frequency response with incoherent light,” Proc. Phys. Soc. London 72, 749–756 (1958).
    [CrossRef]
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1986 (3)

1984 (1)

M. Rabbani, B. E. A. Saleh, “Bayesian restoration of partially coherent imagery in conventional and scanning microscopy,” Opt. Acta 31, 63–79 (1984).
[CrossRef]

1983 (2)

1982 (1)

1981 (1)

1980 (1)

1979 (1)

B. E. A. Saleh, “Optical bilinear transformations: general properties,” Opt. Acta 26, 777–799 (1979).
[CrossRef]

1978 (2)

V. P. Nayvar, N. K. Verma, “Two-point resolution of Gaussian aperture operating in partially coherent light using various resolution criteria,” Appl. Opt. 17, 2176–2180 (1978).
[CrossRef]

K. P. Rao, P. K. Mondal, T. Seshagiri Rao, “Coherent imagery of straight edges with Straubel apodisation filters,” Optik (Stuttgart) 50, 73–81 (1978).

1977 (2)

T. Araki, T. Asakura, “Coherent apodisation problems,” Opt. Commun. 20, 373–377 (1977).
[CrossRef]

E. C. Kintner, R. M. Sillitto, “Edge-ringing in partially coherent imaging,” Opt. Acta 24, 591–605 (1977).
[CrossRef]

1976 (1)

S. C. Biswas, A. Boivin, “Influence of spherical aberration on the performance of optimum apodizers,” Opt. Acta 23, 569–588 (1976).
[CrossRef]

1975 (1)

F. G. Leaver, R. W. Smith, “Apodisation to produce a monotonically-decreasing radially symmetric, point-spread function,” Opt. Commun. 15, 374–378 (1975).
[CrossRef]

1972 (1)

1966 (1)

1965 (1)

1963 (1)

1962 (3)

1958 (1)

J. A. MacDonald, “Apodization and frequency response with incoherent light,” Proc. Phys. Soc. London 72, 749–756 (1958).
[CrossRef]

1953 (1)

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, E. Teller, “Equation of state calculations by fast computing machines,”J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Araki, T.

T. Araki, T. Asakura, “Coherent apodisation problems,” Opt. Commun. 20, 373–377 (1977).
[CrossRef]

Asakura, T.

T. Araki, T. Asakura, “Coherent apodisation problems,” Opt. Commun. 20, 373–377 (1977).
[CrossRef]

Barakat, R.

Barrett, H. H.

Biswas, S. C.

S. C. Biswas, A. Boivin, “Influence of spherical aberration on the performance of optimum apodizers,” Opt. Acta 23, 569–588 (1976).
[CrossRef]

Boivin, A.

S. C. Biswas, A. Boivin, “Influence of spherical aberration on the performance of optimum apodizers,” Opt. Acta 23, 569–588 (1976).
[CrossRef]

Gelatt, C. D.

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Gill, P. E.

P. E. Gill, W. Murray, M. H. Wright, Practical Optimization (Academic, London, 1981).

P. E. Gill, W. Murray, M. A. Saunders, M. H. Wright, User’s Guide for NPSOL (Version 4.0): a fortran Package for Nonlinear Programming (Systems Optimization Laboratory, Department of Operations Research, Stanford University, Stanford, Calif., 1986).

Goeke, W. C.

Goodman, J. W.

P. S. Idell, J. W. Goodman, “Design of optimal imaging concentrators for partially coherent sources: absolute encircled energy criterion,” J. Opt. Soc. Am. A 3, 943–953 (1986).
[CrossRef]

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

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Hildebrand, B. P.

Idell, P. S.

Jacquinot, P.

P. Jacquinot, B. Roizen-Dossier, “Apodisation,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1964), Vol. 3, pp. 31–186.
[CrossRef]

Kintner, E. C.

E. C. Kintner, R. M. Sillitto, “Edge-ringing in partially coherent imaging,” Opt. Acta 24, 591–605 (1977).
[CrossRef]

Kirkpatrick, S.

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Leaver, F. G.

F. G. Leaver, R. W. Smith, “Apodisation to produce a monotonically-decreasing radially symmetric, point-spread function,” Opt. Commun. 15, 374–378 (1975).
[CrossRef]

Lukosz, W.

MacDonald, J. A.

J. A. MacDonald, “Apodization and frequency response with incoherent light,” Proc. Phys. Soc. London 72, 749–756 (1958).
[CrossRef]

Metropolis, N.

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, E. Teller, “Equation of state calculations by fast computing machines,”J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Mills, J. P.

Mondal, P. K.

K. P. Rao, P. K. Mondal, T. Seshagiri Rao, “Coherent imagery of straight edges with Straubel apodisation filters,” Optik (Stuttgart) 50, 73–81 (1978).

Murray, W.

P. E. Gill, W. Murray, M. A. Saunders, M. H. Wright, User’s Guide for NPSOL (Version 4.0): a fortran Package for Nonlinear Programming (Systems Optimization Laboratory, Department of Operations Research, Stanford University, Stanford, Calif., 1986).

P. E. Gill, W. Murray, M. H. Wright, Practical Optimization (Academic, London, 1981).

Nayvar, V. P.

Rabbani, M.

Rao, K. P.

K. P. Rao, P. K. Mondal, T. Seshagiri Rao, “Coherent imagery of straight edges with Straubel apodisation filters,” Optik (Stuttgart) 50, 73–81 (1978).

Roizen-Dossier, B.

P. Jacquinot, B. Roizen-Dossier, “Apodisation,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1964), Vol. 3, pp. 31–186.
[CrossRef]

Rosenbluth, A.

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, E. Teller, “Equation of state calculations by fast computing machines,”J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Rosenbluth, M.

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, E. Teller, “Equation of state calculations by fast computing machines,”J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Saleh, B. E. A.

Saunders, M. A.

P. E. Gill, W. Murray, M. A. Saunders, M. H. Wright, User’s Guide for NPSOL (Version 4.0): a fortran Package for Nonlinear Programming (Systems Optimization Laboratory, Department of Operations Research, Stanford University, Stanford, Calif., 1986).

Sayegh, S. I.

Seshagiri Rao, T.

K. P. Rao, P. K. Mondal, T. Seshagiri Rao, “Coherent imagery of straight edges with Straubel apodisation filters,” Optik (Stuttgart) 50, 73–81 (1978).

Sillitto, R. M.

E. C. Kintner, R. M. Sillitto, “Edge-ringing in partially coherent imaging,” Opt. Acta 24, 591–605 (1977).
[CrossRef]

Slepian, D.

Smith, R. W.

F. G. Leaver, R. W. Smith, “Apodisation to produce a monotonically-decreasing radially symmetric, point-spread function,” Opt. Commun. 15, 374–378 (1975).
[CrossRef]

Smith, W. E.

Teller, A.

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, E. Teller, “Equation of state calculations by fast computing machines,”J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Teller, E.

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, E. Teller, “Equation of state calculations by fast computing machines,”J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Thompson, B. J.

Tichenor, D. A.

Tsujiuchi, J.

J. Tsujiuchi, “Correction of optical images by compensation of aberrations and by spatial frequency filtering,” in Progress in Optics, E. Wolfe, ed. (North-Holland, Amsterdam, 1963), Vol. 2, pp. 131–180.
[CrossRef]

Vecchi, M. P.

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Verma, N. K.

Wright, M. H.

P. E. Gill, W. Murray, M. H. Wright, Practical Optimization (Academic, London, 1981).

P. E. Gill, W. Murray, M. A. Saunders, M. H. Wright, User’s Guide for NPSOL (Version 4.0): a fortran Package for Nonlinear Programming (Systems Optimization Laboratory, Department of Operations Research, Stanford University, Stanford, Calif., 1986).

Appl. Opt. (3)

J. Chem. Phys. (1)

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, E. Teller, “Equation of state calculations by fast computing machines,”J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

J. Opt. Soc. Am. (9)

J. Opt. Soc. Am. A (3)

Opt. Acta (4)

S. C. Biswas, A. Boivin, “Influence of spherical aberration on the performance of optimum apodizers,” Opt. Acta 23, 569–588 (1976).
[CrossRef]

B. E. A. Saleh, “Optical bilinear transformations: general properties,” Opt. Acta 26, 777–799 (1979).
[CrossRef]

E. C. Kintner, R. M. Sillitto, “Edge-ringing in partially coherent imaging,” Opt. Acta 24, 591–605 (1977).
[CrossRef]

M. Rabbani, B. E. A. Saleh, “Bayesian restoration of partially coherent imagery in conventional and scanning microscopy,” Opt. Acta 31, 63–79 (1984).
[CrossRef]

Opt. Commun. (2)

F. G. Leaver, R. W. Smith, “Apodisation to produce a monotonically-decreasing radially symmetric, point-spread function,” Opt. Commun. 15, 374–378 (1975).
[CrossRef]

T. Araki, T. Asakura, “Coherent apodisation problems,” Opt. Commun. 20, 373–377 (1977).
[CrossRef]

Optik (Stuttgart) (1)

K. P. Rao, P. K. Mondal, T. Seshagiri Rao, “Coherent imagery of straight edges with Straubel apodisation filters,” Optik (Stuttgart) 50, 73–81 (1978).

Proc. Phys. Soc. London (1)

J. A. MacDonald, “Apodization and frequency response with incoherent light,” Proc. Phys. Soc. London 72, 749–756 (1958).
[CrossRef]

Science (1)

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Other (6)

P. E. Gill, W. Murray, M. H. Wright, Practical Optimization (Academic, London, 1981).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

P. E. Gill, W. Murray, M. A. Saunders, M. H. Wright, User’s Guide for NPSOL (Version 4.0): a fortran Package for Nonlinear Programming (Systems Optimization Laboratory, Department of Operations Research, Stanford University, Stanford, Calif., 1986).

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

J. Tsujiuchi, “Correction of optical images by compensation of aberrations and by spatial frequency filtering,” in Progress in Optics, E. Wolfe, ed. (North-Holland, Amsterdam, 1963), Vol. 2, pp. 131–180.
[CrossRef]

P. Jacquinot, B. Roizen-Dossier, “Apodisation,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1964), Vol. 3, pp. 31–186.
[CrossRef]

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Figures (22)

Fig. 1
Fig. 1

Problem formulation. s(x) is the signal amplitude transmittance, n(x) is the noise amplitude transmittance, o(x) is the object amplitude transmittance, q(x1, x2) is the imaging-system kernel, qideal(x1, x2) is the kernel of the ideal imaging system, i(x) is the ideal image of s(x), î(x) is the detected image, and (x) = i(x) − î(x) is the error term.

Fig. 2
Fig. 2

Parameter-optimization approach. Each Pk is a discrete sample of the exit pupil and is a parameter to optimize.

Fig. 3
Fig. 3

Test problem. Lenses L2 and L3 form a telecentric imaging system. The amplitude point-spread function of this imaging system is determined by the mask P(u). up indicates the finite extent of the pupil mask. The object o(x0) is illuminated by a Kohler illumination system formed by lens L1 and incoherent source I(xs). f′ is the focal length of the lenses.

Fig. 4
Fig. 4

Optimum pupil screen as a function of the spatial bandwidths of the signal and the noise: low-pass cases. The illumination is coherent, and Es/En = 10.0.

Fig. 5
Fig. 5

Optimum pupil screen as a function of the spatial bandwidths of the signal and the noise: high-pass cases. The illumination is coherent, and Es/En = 10.0.

Fig. 6
Fig. 6

Optimum pupil screen as a function of the input signal-to-noise ratio. The illumination is coherent and vs/vn = 0.5.

Fig. 7
Fig. 7

Optimum pupil screen as a function of the input signal-to-noise ratio. The illumination is coherent and vs/vn = 5.0.

Fig. 8
Fig. 8

Improvement factor KMSE as a function of the spatial band- widths of the signal and the noise and as a function of the input signal-to-noise ratio. The illuminationis coherent.

Fig. 9
Fig. 9

Optimum pupil screen and OTF as functions of the spatial bandwidths of the signal and the noise: low-pass cases. The illumination is incoherent, and Es/En = 10.0. (a) Pupil screens and (b) corresponding OTF’s compared with the OTF of the diffraction-limited system.

Fig. 10
Fig. 10

Optimum pupil screen and OTF as functions of the spatial bandwidths of the signal and the noise: high-pass cases. The illumination is incoherent and Es/En = 10.0. (a) Pupil screens and (b) corresponding OTF’s compared with the OTF of the diffraction-limited system.

Fig. 11
Fig. 11

Optimum pupil screen and OTF as functions of the input signal-to-noise ratio. The illumination is incoherent, and vs/vn = 0.2. (a) Pupil screens and (b) corresponding OTF’s.

Fig. 12
Fig. 12

Optimum pupil screen and OTF as a function of the input signal-to-noise ratio. The illumination is incoherent and vs/vn = 5.0. (a) Pupil screens and (b) corresponding OTF’s.

Fig. 13
Fig. 13

Improvement factor KMSE as a function of the spatial bandwidths of the signal and noise and as a function of the input signal-to-noise ratio. The illumination is incoherent.

Fig. 14
Fig. 14

Optimum pupil screens as functions of illumination coherence. vs/vn = 0.2 and Es/En = 10.0.

Fig. 15
Fig. 15

Optimum pupil screens as functions of illumination coherence. vs/vn = 5.0 and Es/En = 10.0.

Fig. 16
Fig. 16

Improvement factor KMSE as a function of the illumination coherence. Es/En = 10.0.

Fig. 17
Fig. 17

Magnitude and phase of optimum pupil screen: complex pupil. The illumination is incoherent, vs/vn = 0.2, and Es/En = 10.0. (a) Magnitude transmittance of pupil screen. (b) Phase transmittance of pupil screen.

Fig. 18
Fig. 18

Comparison of OTF’s. The complex pupil screen is shown in Fig. 17. The real pupil screen is shown in Fig. 9. vs/vn = 0.2, Es/En = 10.0, and the illumination is incoherent.

Fig. 19
Fig. 19

Comparison of improvement factors for complex versus real pupil screens as functions of vs/vn. The illumination is incoherent, and Es/En = 10.0.

Fig. 20
Fig. 20

Simulated detected image intensities: coherent illumination. (a) Diffraction-limited image of object. (b) Image obtained by using optimized imaging system. Dashed lines represent the ideal image (diffraction-limited image of the signal without noise).

Fig. 21
Fig. 21

Simulated detected image intensities: incoherent illumination. (a) Diffraction-limited image of object. (b) Image obtained by using optimized imaging system. Dashed lines represent the ideal image (diffraction-limited image of the signal without noise).

Fig. 22
Fig. 22

Simulated detected image intensities: partially coherent illumination (vc = 0.1). (a) Diffraction-limited image of object. (b) Image obtained by using optimized imaging system. Dashed lines represent the ideal image (diffraction-limited image of the signal without noise).

Equations (22)

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

i ^ ( x ) = - d x 1 d x 2 q ( x - x 1 , x - x 2 ) o * ( x 1 ) o ( x 2 ) .
q ( x 1 , x 2 ) = J 0 ( x 1 - x 2 ) p * ( x 1 ) p ( x 2 ) ,
p ( x ) = 1 λ d i - d u P ( u ) exp ( - j 2 π λ d i x u ) ,
q ideal ( x 1 , x 2 ) = J 0 ( x 1 - x 2 ) b * ( x 1 ) b ( x 2 ) .
I ^ ( v ) = - d v 1 Q ( v 1 , v - v 1 ) O * ( - v 1 ) O ( v - v 1 ) ,
Q ( v 1 , v 2 ) = - d v Γ ( v ) P * ( v - v 1 ) P ( v + v 2 ) ,
P ( u ) = k = - ( N - 1 ) / 2 ( N - 1 ) / 2 P k rect ( u - k Δ u Δ u ) ,
rect ( x ) = { 1.0 if x 1 / 2 0.0 otherwise .
p ( x ) = 1 λ f - d u P ( u ) exp ( - j 2 π λ f x u ) ,
J 0 ( Δ x 0 ) = 2 a I s λ f sinc ( 2 a Δ x 0 λ f ) ,
K MSE = MSE [ B ( u ) ] MSE [ P ( u ) ] ,
1 N j = 1 N [ i ( j ) - i d ( j ) ] 2 ,
E [ 2 ( x ) ] = E [ { i ( x ) - i ^ ( x ) } 2 ] = E [ i 2 ( x ) ] - 2 E [ i ( x ) i ^ ( x ) ] + E [ i ^ 2 ( x ) ] .
i ( x ) = d x 1 d x 2 q ideal ( x - x 1 , x - x 2 ) s * ( x 1 ) s ( x 2 ) .
E [ i 2 ( x ) ] = d x 1 d x 2 d x 3 d x 4 q ideal ( x - x 1 , x - x 2 ) × q ideal ( x - x 3 , x - x 4 ) E [ s * ( x 1 s ) ( x 2 ) s * ( x 3 ) s ( x 4 ) ] .
E [ s * ( x 1 ) s ( x 2 ) s * ( x 3 ) s ( x 4 ) ] = R s s ( x 2 - x 1 ) R s s ( x 4 - x 3 ) + R s s ( x 2 - x 3 ) R s s ( x 4 - x 1 ) ,
E [ i 2 ( x ) ] = d α 1 d α 2 d α 3 d α 4 q ideal ( α 1 , α 2 ) q ideal ( α 3 , α 4 ) × [ R s s ( α 1 - α 2 ) R s s ( α 3 - α 4 ) + R s s ( α 1 - α 4 ) R s s ( α 3 - α 2 ) ] .
q ideal ( α 1 , α 2 ) = d v 1 d v 2 Q ideal ( v 1 , v 2 ) exp [ j 2 π ( α 1 v 1 + α 2 v 2 ) ] , q ideal ( α 3 , α 4 ) = d v 3 d v 4 Q ideal ( v 3 , v 4 ) exp [ j 2 π ( α 3 v 3 + α 4 v 4 ) ] .
E [ i 2 ( x ) ] = d v 2 d v 4 Q ideal ( - v 4 , v 2 ) Q ideal ( - v 2 , v 4 ) Φ s s ( v 2 ) × Φ s s ( v 4 ) + [ d v 2 Q ideal ( - v 2 , v 2 ) Φ s s ( v 2 ) ] 2 ,
E [ i ( x ) i ^ ( x ) ] = d v 2 d v 4 Q ideal ( - v 4 , v 2 ) Q ( - v 2 , v 4 ) × Φ s s ( v 2 ) Φ s s ( v 4 ) + d v 4 Q ( - v 4 , v 4 ) × [ Φ s s ( v 4 ) + Φ n n ( v 4 ) ] × d v 2 Q ideal ( - v 2 , v 2 ) Φ s s ( v 2 ) ,
E [ i ^ 2 ( x ) ] = { d v 2 Q ( - v 2 , v 2 ) [ Φ s s ( v 2 ) + Φ n n ( v 2 ) ] } 2 + d v 2 d v 4 [ Φ s s ( v 2 ) + Φ n n ( v 2 ) ] × [ Φ s s ( v 4 ) + Φ n n ( v 4 ) ] Q ( - v 2 , v 4 ) Q ( - v 4 , v 2 ) ,
E [ 2 ( x ) ] = { d v 2 Q ideal ( - v 2 , v 2 ) Φ s s ( v 2 ) } 2 + d v 2 d v 4 Q ideal ( - v 4 , v 2 ) Q ideal ( - v 2 , v 4 ) × Φ s s ( v 2 ) Φ s s ( v 4 ) - 2 d v d v 4 Γ ( v ) P ( v + v 4 ) 2 ] × [ Φ s s ( v 4 ) + Φ n n ( v 4 ) ] d v 2 Q ideal ( - v 2 , v 2 ) Φ s s ( v 2 ) - 2 d v d v 2 d v 4 Q ideal ( - v 4 , v 2 ) × Γ ( v ) P * ( v + v 2 ) P ( v + v 4 ) Φ s s ( v 2 ) Φ s s ( v 4 ) + { d v d v 2 Γ ( v ) P ( v + v 2 ) 2 [ Φ s s ( v 2 ) + Φ n n ( v 2 ) ] } 2 + d v 2 d v 4 [ Φ s s ( v 2 ) + Φ n n ( v 2 ) ] [ Φ s s ( v 4 ) + Φ n n ( v 4 ) ] × | d v Γ ( v ) P * ( v + v 2 ) P ( v + v 4 ) | 2 .

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