Abstract

It is well known that atmospheric turbulence severely degrades the performance of ground-based imaging systems. Techniques to overcome the effects of the atmosphere have been developing at a rapid pace over the past 10 years. These techniques can be grouped into two broad categories: predetection and postdetection techniques. A recent newcomer to the postdetection scene is deconvolution from wave-front sensing (DWFS). DWFS is a postdetection image-reconstruction technique that makes use of one feature of predetection techniques. A wave-front sensor (WFS) is used to record the wave-front phase distortion in the pupil of the telescope for each short-exposure image. The additional information provided by the WFS is used to estimate the system’s point-spread function (PSF). The PSF is then used in conjunction with the ensemble of short-exposure images to obtain an estimate of the object intensity distribution through deconvolution. With the addition of DWFS to the suite of possible postdetection image-reconstruction techniques, it is natural to ask “How does DWFS compare with both traditional linear and speckle image-reconstruction techniques?” In the results we make a direct comparison based on a frequency-domain signal-to-noise-ratio performance metric. This metric is applied to each technique’s image-reconstruction estimator. We find that DWFS nearly always results in improved performance over the estimators of traditional linear image reconstruction such as Wiener filtering. On the other hand, DWFS does not always outperform speckle-imaging techniques, and in cases that it does the improvement is small.

© 1995 Optical Society of America

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  1. J. H. Hardy, “Active optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
    [CrossRef]
  2. A. Labeyrie, “Attainment of diffraction-limited resolution in large telescopes by Fourier analysing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).
  3. J. C. Dainty, “Stellar speckle interferometry,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), pp. 255–280.
    [CrossRef]
  4. G. Weigelt, “Speckle imaging and speckle spectroscopy,” in New Technology for Astronomy, J. Swings, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1130, 148–151 (1989).
  5. T. J. Cornwell, “The applications of closure phase to astronomical imaging,” Science 245, 263–269 (1989).
    [CrossRef] [PubMed]
  6. 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]
  7. 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]
  8. V. Michau, T. Marais, J. Laurent, J. Primot, J. C. Fontanella, M. Tallon, J. Fuensalida, “High-resolution astronomical observations using deconvolution form wavefront sensing,” in Propagation Engineering: Fourth in a Series, L. R. Bissonnette, W. B. Miller, eds., 1487, 64–71 (1991).
  9. B. M. Welsh, R. N. V. 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, “Biased estimators and object-spectrum estimation in deconvolution from wave-front sensing,” Appl. Opt. 33, 5754–5763 (1994).
    [CrossRef] [PubMed]
  11. M. C. Roggemann, B. M. Welsh, “Signal-to-noise ratio for astronomical imaging using deconvolution from wave-front sensing,” Appl. Opt. 33, 5400–5414 (1994).
    [CrossRef] [PubMed]
  12. 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).
  13. A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1965), pp. 385–429.
  14. J. Biemond, R. L. Lagendijk, R. M. Mersereau, “Iterative methods for image deblurring,” Proc. IEEE 78, 856–883 (1990).
    [CrossRef]
  15. M. C. Roggemann, “Limited degree-of-freedom adaptive optics and image reconstruction,” Appl. Opt. 30, 4227–4233 (1991).
    [CrossRef] [PubMed]
  16. M. C. Roggemann, “Optical performance of fully and partially compensated adaptive optics systems using least-squares and minimum variance phase reconstruction,” Comput. Electr. Eng. 18, 451–466 (1992).
    [CrossRef]
  17. J. C. Dainty, A. H. Greenaway, “Estimation of spatial power spectra in speckle imaging,” J. Opt. Soc. Am. 69, 786–790 (1979).
    [CrossRef]
  18. D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for very long and very short exposures,” J. Opt. Soc. Am. 56, 1372–1379 (1966).
    [CrossRef]
  19. I. S. Reed, “On a moment theorem for complex Gaussian processes,” IRE Trans. Inf. Theory IT-8, 194–195 (1962).
    [CrossRef]
  20. 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]

1994 (2)

1993 (1)

1992 (2)

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

1991 (1)

1990 (3)

J. Biemond, R. L. Lagendijk, R. M. Mersereau, “Iterative methods for image deblurring,” Proc. IEEE 78, 856–883 (1990).
[CrossRef]

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]

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]

1989 (1)

T. J. Cornwell, “The applications of closure phase to astronomical imaging,” Science 245, 263–269 (1989).
[CrossRef] [PubMed]

1979 (1)

1978 (1)

J. H. Hardy, “Active optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
[CrossRef]

1970 (1)

A. Labeyrie, “Attainment of diffraction-limited resolution in large telescopes by Fourier analysing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).

1966 (1)

1962 (1)

I. S. Reed, “On a moment theorem for complex Gaussian processes,” IRE Trans. Inf. Theory IT-8, 194–195 (1962).
[CrossRef]

Biemond, J.

J. Biemond, R. L. Lagendijk, R. M. Mersereau, “Iterative methods for image deblurring,” Proc. IEEE 78, 856–883 (1990).
[CrossRef]

Cornwell, T. J.

T. J. Cornwell, “The applications of closure phase to astronomical imaging,” Science 245, 263–269 (1989).
[CrossRef] [PubMed]

Dainty, J. C.

J. C. Dainty, A. H. Greenaway, “Estimation of spatial power spectra in speckle imaging,” J. Opt. Soc. Am. 69, 786–790 (1979).
[CrossRef]

J. C. Dainty, “Stellar speckle interferometry,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), pp. 255–280.
[CrossRef]

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.

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]

V. Michau, T. Marais, J. Laurent, J. Primot, J. C. Fontanella, M. Tallon, J. Fuensalida, “High-resolution astronomical observations using deconvolution form wavefront sensing,” in Propagation Engineering: Fourth in a Series, L. R. Bissonnette, W. B. Miller, eds., 1487, 64–71 (1991).

Fried, D. L.

Fuensalida, J.

V. Michau, T. Marais, J. Laurent, J. Primot, J. C. Fontanella, M. Tallon, J. Fuensalida, “High-resolution astronomical observations using deconvolution form wavefront sensing,” in Propagation Engineering: Fourth in a Series, L. R. Bissonnette, W. B. Miller, eds., 1487, 64–71 (1991).

Gonglewski, J. D.

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.

Hardy, J. H.

J. H. Hardy, “Active optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
[CrossRef]

Labeyrie, A.

A. Labeyrie, “Attainment of diffraction-limited resolution in large telescopes by Fourier analysing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).

Lagendijk, R. L.

J. Biemond, R. L. Lagendijk, R. M. Mersereau, “Iterative methods for image deblurring,” Proc. IEEE 78, 856–883 (1990).
[CrossRef]

Laurent, J.

V. Michau, T. Marais, J. Laurent, J. Primot, J. C. Fontanella, M. Tallon, J. Fuensalida, “High-resolution astronomical observations using deconvolution form wavefront sensing,” in Propagation Engineering: Fourth in a Series, L. R. Bissonnette, W. B. Miller, eds., 1487, 64–71 (1991).

Marais, T.

V. Michau, T. Marais, J. Laurent, J. Primot, J. C. Fontanella, M. Tallon, J. Fuensalida, “High-resolution astronomical observations using deconvolution form wavefront sensing,” in Propagation Engineering: Fourth in a Series, L. R. Bissonnette, W. B. Miller, eds., 1487, 64–71 (1991).

Matson, C. L.

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]

Mersereau, R. M.

J. Biemond, R. L. Lagendijk, R. M. Mersereau, “Iterative methods for image deblurring,” Proc. IEEE 78, 856–883 (1990).
[CrossRef]

Michau, V.

V. Michau, T. Marais, J. Laurent, J. Primot, J. C. Fontanella, M. Tallon, J. Fuensalida, “High-resolution astronomical observations using deconvolution form wavefront sensing,” in Propagation Engineering: Fourth in a Series, L. R. Bissonnette, W. B. Miller, eds., 1487, 64–71 (1991).

Niederhausern, R. N. V.

Papoulis, A.

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1965), pp. 385–429.

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]

V. Michau, T. Marais, J. Laurent, J. Primot, J. C. Fontanella, M. Tallon, J. Fuensalida, “High-resolution astronomical observations using deconvolution form wavefront sensing,” in Propagation Engineering: Fourth in a Series, L. R. Bissonnette, W. B. Miller, eds., 1487, 64–71 (1991).

Reed, I. S.

I. S. Reed, “On a moment theorem for complex Gaussian processes,” IRE Trans. Inf. Theory IT-8, 194–195 (1962).
[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, 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.

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.

Tallon, M.

V. Michau, T. Marais, J. Laurent, J. Primot, J. C. Fontanella, M. Tallon, J. Fuensalida, “High-resolution astronomical observations using deconvolution form wavefront sensing,” in Propagation Engineering: Fourth in a Series, L. R. Bissonnette, W. B. Miller, eds., 1487, 64–71 (1991).

Voelz, D. G.

Weigelt, G.

G. Weigelt, “Speckle imaging and speckle spectroscopy,” in New Technology for Astronomy, J. Swings, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1130, 148–151 (1989).

Welsh, B. M.

Appl. Opt. (5)

Astron. Astrophys. (1)

A. Labeyrie, “Attainment of diffraction-limited resolution in large telescopes by Fourier analysing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).

Comput. Electr. Eng. (2)

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

IRE Trans. Inf. Theory (1)

I. S. Reed, “On a moment theorem for complex Gaussian processes,” IRE Trans. Inf. Theory IT-8, 194–195 (1962).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (1)

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]

Proc. IEEE (2)

J. Biemond, R. L. Lagendijk, R. M. Mersereau, “Iterative methods for image deblurring,” Proc. IEEE 78, 856–883 (1990).
[CrossRef]

J. H. Hardy, “Active optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
[CrossRef]

Science (1)

T. J. Cornwell, “The applications of closure phase to astronomical imaging,” Science 245, 263–269 (1989).
[CrossRef] [PubMed]

Other (5)

J. C. Dainty, “Stellar speckle interferometry,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), pp. 255–280.
[CrossRef]

G. Weigelt, “Speckle imaging and speckle spectroscopy,” in New Technology for Astronomy, J. Swings, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1130, 148–151 (1989).

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).

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1965), pp. 385–429.

V. Michau, T. Marais, J. Laurent, J. Primot, J. C. Fontanella, M. Tallon, J. Fuensalida, “High-resolution astronomical observations using deconvolution form wavefront sensing,” in Propagation Engineering: Fourth in a Series, L. R. Bissonnette, W. B. Miller, eds., 1487, 64–71 (1991).

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Figures (3)

Fig. 1
Fig. 1

Graph of SNRDWFS/|γ| SNRLinear versus GH, in which we have assumed that GH = GĤ

Fig. 2
Fig. 2

Graph of S N R DWFS/ S N R SI versus GH, in which we have assumed that γ = 1 and GH = GĤ: shot-noise-limited case.

Fig. 3
Fig. 3

Graph of S N R DWFS / S N R SI versus GH, in which we have assumed γ = 1 and GH = GĤ: OTF-variance-limited case.

Equations (32)

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SNR ( u ) = | O ˆ ( u ) | { Var [ O ˆ ( u ) ] } 1 / 2 ,
S N R ( u ) = | O ˆ ( u ) | 2 { Var [ | O ˆ ( u ) | 2 ] } 1 / 2 ,
i ( x ) = o ( x ) h ( x ) ,
I ( u ) = O ( u ) H ( u ) ,
SNR Linear = | I | ( Var [ I ] ) 1 / 2 , = K ¯ | O n | | H | ( K ¯ + K 2 ¯ | O n | 2 Var [ H ] ) 1 / 2 , = | O n | | H | ( K ¯ 1 + | O n | 2 Var [ H ] ) 1 / 2 ,
SNR Linear = K ¯ | O n | | H | . ( shot-noise-limited case )
SNR Linear = | H | { Var [ H ] } 1 / 2 . ( OTF-variance-limited case )
I ' = I H ˆ * = O H H ˆ * ,
SNR DWFS = I ( Var [ I ] ) 1 / 2 , = K ¯ | O n | | H H ˆ * | ( K ¯ | H ˆ | 2 + K ¯ 2 | O n | 2 Var [ H H ˆ * ] ) 1 / 2 , = | O n | | H H ˆ * | ( K ¯ 1 | H ˆ | 2 + | O n | 2 Var [ H H ˆ * ] ) 1 / 2 .
S N R SI = K ¯ 2 | O n | 2 | H | 2 ( K ¯ 2 + 2 K ¯ 3 | O n | 2 | H | 2 + K ¯ 4 | O n | 4 Var [ | H | 2 ] ) 1 / 2 = | O n | 2 | H | 2 ( K ¯ 2 + 2 K ¯ | O n | 2 | H | 2 + | O n | 4 Var [ | H | 2 ] ) 1 / 2 ,
S N R DWFS = K ¯ 2 | O n | 2 | H H ˆ * | 2 ( K ¯ 2 | H ˆ | 4 + 2 K ¯ 3 | O n | 2 | H | 2 | H ˆ | 4 + K ¯ 4 | O n | 4 Var [ | H H ˆ * | 2 ] ) 1 / 2 = | O n | 2 | H H ˆ * | 2 ( K ¯ 2 | H ˆ | 4 + 2 K ¯ 1 | O n | 2 | H | 2 | H ˆ | 4 + | O n | 4 Var [ | H H ˆ * | 2 ] ) 1 / 2 ,
G H = | H | 2 | H | 2 ,
γ H H ˆ = H H ˆ * ( | H | 2 | H ˆ | 2 ) 1 / 2 .
SNR DWFS = | γ H H ˆ | G H × | O n | | H | ( K ¯ 1 + | O n | 2 Var [ H ] [ Var [ H H ˆ * ] Var [ H ] | H ˆ | 2 ] ) 1 / 2 .
H = H + H ¯ ,
H ˆ = H ˆ + H ˆ ¯ ,
Var [ H H ˆ * ] Var [ H ] | H ˆ | 2 = G H 1 G H ˆ G H 1 ,
SNR DWFS = | γ H H ˆ | G H × | O n | | H | ( K ¯ 1 + | O n | 2 Var [ H ] [ G H 1 G H ˆ G H 1 ] ) 1 / 2 .
SNR DWFS = | γ H H ˆ | G H K ¯ | O n | | H | , = | γ H H ˆ | G H SNR Linear , ( shot-noise-limited case )
SNR DWFS = | γ H H ˆ | G H ( G H 1 G H 1 G H ˆ ) 1 / 2 | H | Var [ H ] = | γ H H ˆ | G H ( G H 1 G H 1 G H ˆ ) 1 / 2 SNR Linear , ( OTF-variance-limited case )
G H = | | H | 4 | H | 2 ,
Γ H H ˆ = | H H ˆ | 2 ( | H | 4 | H ˆ | 4 ) 1 / 2 .
S N R DWFS = Γ H H ˆ G H K ¯ 2 | O n | 2 | H | 2 ( K ¯ 2 + 2 K ¯ 3 | O n | 2 | H | 2 [ | H | 2 | H ˆ | 4 | H | 2 | H ˆ | 4 ] + K ¯ 4 | O n | 4 Var [ | H | 2 ] [ Var [ | H H ˆ | 2 ] Var [ | H | 2 ] | H ˆ | 4 ] ) 1 / 2 = Γ H H ˆ G H | O n | 2 | H | 2 ( K ¯ 2 + 2 K ¯ 1 | O n | 2 | H | 2 [ | H | 2 | H ˆ | 4 | H | 2 | H ˆ | 4 ] + | O n | 4 Var [ | H | 2 ] [ Var [ | H H ˆ | 2 ] Var [ | H | 2 ] | H ˆ | 4 ] ) 1 / 2 .
S N R DWFS = Γ H H ˆ G H K ¯ | O n | 2 | H | 2 = Γ H H ˆ G H S N R SI , ( shot-noise-limited case )
G H = ( 2 1 G H 2 ) 1 / 2 ,
Γ H H ˆ = 1 + | γ H H ˆ | 2 1 G H G H ˆ [ ( 2 1 G H ˆ 2 ) ( 2 1 G H 2 ) ] 1 / 2 .
S N R DWFS = 1 + | γ H H ˆ | 2 1 G H G H ˆ ( 2 1 G H ˆ 2 ) 1 / 2 S N R SI . ( shot-noise-limited case )
S N R DWFS = Γ H H ˆ G H ( Var [ | H H ˆ | 2 ] Var [ | H | 2 ] | H ˆ | 4 ) 1 / 2 | H | 2 ( Var [ | H | 2 ] ) 1 / 2 = Γ H H ˆ G H ( Var [ | H H ˆ | 2 ] Var [ | H | 2 ] | H ˆ | 4 ) 1 / 2 S N R SI , ( OTF-variance-limited case )
Var [ | H H ˆ | 2 ] Var [ | H | 2 ] | H ˆ | 4 = f 1 ( G H , G H ˆ , γ H H ˆ ) f 2 ( G H , G H ˆ ) ,
f 1 ( G H , G H ˆ , γ H H ˆ ) = ( 2 1 G H 2 ) ( 2 1 G H ˆ 2 ) 1 + | 2 γ H H ˆ 1 G H G H ˆ | 2 | γ H H ˆ | 4 + 4 [ ( | γ H H ˆ | 2 1 G H G H ˆ ) | γ H H ˆ 1 ( G H G H ˆ ) 1 / 2 | 2 ] × ( 2 1 G H ) ( 2 1 G H ˆ ) 2 ( | γ H H ˆ | 2 1 G H G H ˆ ) × ( 1 1 G H G H ˆ ) 8 | γ H H ˆ 1 ( G H G H ˆ ) 1 / 2 | 2 × [ ( 1 1 G H ) ( 1 1 G H ˆ ) ( 1 1 ( G H G H ˆ ) ) ] ,
f 2 ( G H , G H ˆ ) = ( 1 1 G H 2 ) ( 2 1 G H ˆ 2 ) .
S N R DWFS 1 + | γ H H ˆ | 2 ( 3 + 3 | γ H H ˆ | 4 2 | γ H H ˆ | 2 ) 1 / 2 S N R SI . ( OTF-variance-limited case )

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