Abstract

We study the distribution of noise in optical images produced by the aperture synthesis technique, in which the principal source of noise is the intrinsic shot noise of photoelectric detection. The results of our analysis are directly applicable to any space-based optical interferometer. We show that the signal-to-noise ratio of images synthesized by such an ideal interferometric array is essentially independent of the details of the beam-combination geometry, the degree of array redundancy, and whether zero-spatial-frequency components are included in image synthesis. However, the distribution of noise does depend on the beam-combination geometry. A highly desirable distribution, one of uniform noise across the entire image, is obtained only when the beams from the n primary apertures are subdivided and combined pairwise on n(n − 1)/2 detectors.

© 1989 Optical Society of America

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  1. See, e.g., D. Mozurkewich, D. J. Hutter, K. J. Johnston, R. S. Simon, M. Shao, M. M. Colavita, D. H. Staelin, B. E. Hines, J. L. Hershey, J. A. Hughes, G. H. Kaplan, “Preliminary measurements of star positions with Mark III stellar interferometer,” Astron. Astrophys. 193, 1269–1277 (1988).
  2. C. A. Haniff, C. D. Mackay, D. J. Titterington, D. Sivia, P. J. Warner, “The first images from optical aperture synthesis,” Nature 328, 694–696 (1987).
    [CrossRef]
  3. T. Nakajima, S. R. Kulkarni, P. W. Gorham, A. M. Ghez, G. Neugebauer, B. J. Oke, T. A. Prince, A. C. S. Readhead, “Diffraction-limited imaging II: optical aperture synthesis imaging of two binary stars,” Astron. J. (to be published).
  4. See, e.g., J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 5.
  5. See, e.g., A. R. Thompson, J. M. Moran, G. W. Swenson, Interferometry and Aperture Synthesis (Wiley, New York, 1986).
  6. J. F. Walkup, J. W. Goodman, “Limitations of fringe-parameter estimation at low light levels,”J. Opt. Soc. Am. 63, 399–407 (1973); see also Ref. 4, Chap. 9.
    [CrossRef]
  7. R. A. Perley, F. R. Schwab, A. H. Bridle, Synthesis Imaging (National Radio Astronomy Observatory, Socorro, N. Mex., 1985).
  8. S. R. Kulkarni, “Self-noise in interferometers: radio and infrared,” submitted to Astron. J.
  9. K. Shimoda, H. Takahashi, C. H. Townes, “Fluctuations in amplification of quanta with application to maser amplifiers,” J. Phys. Soc. Jpn. 12, 686–700 (1957).
    [CrossRef]
  10. H. Steinberg, “The use of a laser amplifier in a laser communication system,” Proc. IEEE 51, 943 (1963).
    [CrossRef]
  11. M. Shao, Jet Propulsion Laboratory, 4800 Oak Grove Drive, Pasadena, California 91109 (personal communication, 1989). The National Aeronautics and Space Administration is working currently to improve the readout noise of CCD’s with the aim of making them more attractive for space interferometry than they are now.

1988 (1)

See, e.g., D. Mozurkewich, D. J. Hutter, K. J. Johnston, R. S. Simon, M. Shao, M. M. Colavita, D. H. Staelin, B. E. Hines, J. L. Hershey, J. A. Hughes, G. H. Kaplan, “Preliminary measurements of star positions with Mark III stellar interferometer,” Astron. Astrophys. 193, 1269–1277 (1988).

1987 (1)

C. A. Haniff, C. D. Mackay, D. J. Titterington, D. Sivia, P. J. Warner, “The first images from optical aperture synthesis,” Nature 328, 694–696 (1987).
[CrossRef]

1973 (1)

1963 (1)

H. Steinberg, “The use of a laser amplifier in a laser communication system,” Proc. IEEE 51, 943 (1963).
[CrossRef]

1957 (1)

K. Shimoda, H. Takahashi, C. H. Townes, “Fluctuations in amplification of quanta with application to maser amplifiers,” J. Phys. Soc. Jpn. 12, 686–700 (1957).
[CrossRef]

Bridle, A. H.

R. A. Perley, F. R. Schwab, A. H. Bridle, Synthesis Imaging (National Radio Astronomy Observatory, Socorro, N. Mex., 1985).

Colavita, M. M.

See, e.g., D. Mozurkewich, D. J. Hutter, K. J. Johnston, R. S. Simon, M. Shao, M. M. Colavita, D. H. Staelin, B. E. Hines, J. L. Hershey, J. A. Hughes, G. H. Kaplan, “Preliminary measurements of star positions with Mark III stellar interferometer,” Astron. Astrophys. 193, 1269–1277 (1988).

Ghez, A. M.

T. Nakajima, S. R. Kulkarni, P. W. Gorham, A. M. Ghez, G. Neugebauer, B. J. Oke, T. A. Prince, A. C. S. Readhead, “Diffraction-limited imaging II: optical aperture synthesis imaging of two binary stars,” Astron. J. (to be published).

Goodman, J. W.

Gorham, P. W.

T. Nakajima, S. R. Kulkarni, P. W. Gorham, A. M. Ghez, G. Neugebauer, B. J. Oke, T. A. Prince, A. C. S. Readhead, “Diffraction-limited imaging II: optical aperture synthesis imaging of two binary stars,” Astron. J. (to be published).

Haniff, C. A.

C. A. Haniff, C. D. Mackay, D. J. Titterington, D. Sivia, P. J. Warner, “The first images from optical aperture synthesis,” Nature 328, 694–696 (1987).
[CrossRef]

Hershey, J. L.

See, e.g., D. Mozurkewich, D. J. Hutter, K. J. Johnston, R. S. Simon, M. Shao, M. M. Colavita, D. H. Staelin, B. E. Hines, J. L. Hershey, J. A. Hughes, G. H. Kaplan, “Preliminary measurements of star positions with Mark III stellar interferometer,” Astron. Astrophys. 193, 1269–1277 (1988).

Hines, B. E.

See, e.g., D. Mozurkewich, D. J. Hutter, K. J. Johnston, R. S. Simon, M. Shao, M. M. Colavita, D. H. Staelin, B. E. Hines, J. L. Hershey, J. A. Hughes, G. H. Kaplan, “Preliminary measurements of star positions with Mark III stellar interferometer,” Astron. Astrophys. 193, 1269–1277 (1988).

Hughes, J. A.

See, e.g., D. Mozurkewich, D. J. Hutter, K. J. Johnston, R. S. Simon, M. Shao, M. M. Colavita, D. H. Staelin, B. E. Hines, J. L. Hershey, J. A. Hughes, G. H. Kaplan, “Preliminary measurements of star positions with Mark III stellar interferometer,” Astron. Astrophys. 193, 1269–1277 (1988).

Hutter, D. J.

See, e.g., D. Mozurkewich, D. J. Hutter, K. J. Johnston, R. S. Simon, M. Shao, M. M. Colavita, D. H. Staelin, B. E. Hines, J. L. Hershey, J. A. Hughes, G. H. Kaplan, “Preliminary measurements of star positions with Mark III stellar interferometer,” Astron. Astrophys. 193, 1269–1277 (1988).

Johnston, K. J.

See, e.g., D. Mozurkewich, D. J. Hutter, K. J. Johnston, R. S. Simon, M. Shao, M. M. Colavita, D. H. Staelin, B. E. Hines, J. L. Hershey, J. A. Hughes, G. H. Kaplan, “Preliminary measurements of star positions with Mark III stellar interferometer,” Astron. Astrophys. 193, 1269–1277 (1988).

Kaplan, G. H.

See, e.g., D. Mozurkewich, D. J. Hutter, K. J. Johnston, R. S. Simon, M. Shao, M. M. Colavita, D. H. Staelin, B. E. Hines, J. L. Hershey, J. A. Hughes, G. H. Kaplan, “Preliminary measurements of star positions with Mark III stellar interferometer,” Astron. Astrophys. 193, 1269–1277 (1988).

Kulkarni, S. R.

T. Nakajima, S. R. Kulkarni, P. W. Gorham, A. M. Ghez, G. Neugebauer, B. J. Oke, T. A. Prince, A. C. S. Readhead, “Diffraction-limited imaging II: optical aperture synthesis imaging of two binary stars,” Astron. J. (to be published).

S. R. Kulkarni, “Self-noise in interferometers: radio and infrared,” submitted to Astron. J.

Mackay, C. D.

C. A. Haniff, C. D. Mackay, D. J. Titterington, D. Sivia, P. J. Warner, “The first images from optical aperture synthesis,” Nature 328, 694–696 (1987).
[CrossRef]

Moran, J. M.

See, e.g., A. R. Thompson, J. M. Moran, G. W. Swenson, Interferometry and Aperture Synthesis (Wiley, New York, 1986).

Mozurkewich, D.

See, e.g., D. Mozurkewich, D. J. Hutter, K. J. Johnston, R. S. Simon, M. Shao, M. M. Colavita, D. H. Staelin, B. E. Hines, J. L. Hershey, J. A. Hughes, G. H. Kaplan, “Preliminary measurements of star positions with Mark III stellar interferometer,” Astron. Astrophys. 193, 1269–1277 (1988).

Nakajima, T.

T. Nakajima, S. R. Kulkarni, P. W. Gorham, A. M. Ghez, G. Neugebauer, B. J. Oke, T. A. Prince, A. C. S. Readhead, “Diffraction-limited imaging II: optical aperture synthesis imaging of two binary stars,” Astron. J. (to be published).

Neugebauer, G.

T. Nakajima, S. R. Kulkarni, P. W. Gorham, A. M. Ghez, G. Neugebauer, B. J. Oke, T. A. Prince, A. C. S. Readhead, “Diffraction-limited imaging II: optical aperture synthesis imaging of two binary stars,” Astron. J. (to be published).

Oke, B. J.

T. Nakajima, S. R. Kulkarni, P. W. Gorham, A. M. Ghez, G. Neugebauer, B. J. Oke, T. A. Prince, A. C. S. Readhead, “Diffraction-limited imaging II: optical aperture synthesis imaging of two binary stars,” Astron. J. (to be published).

Perley, R. A.

R. A. Perley, F. R. Schwab, A. H. Bridle, Synthesis Imaging (National Radio Astronomy Observatory, Socorro, N. Mex., 1985).

Prince, T. A.

T. Nakajima, S. R. Kulkarni, P. W. Gorham, A. M. Ghez, G. Neugebauer, B. J. Oke, T. A. Prince, A. C. S. Readhead, “Diffraction-limited imaging II: optical aperture synthesis imaging of two binary stars,” Astron. J. (to be published).

Readhead, A. C. S.

T. Nakajima, S. R. Kulkarni, P. W. Gorham, A. M. Ghez, G. Neugebauer, B. J. Oke, T. A. Prince, A. C. S. Readhead, “Diffraction-limited imaging II: optical aperture synthesis imaging of two binary stars,” Astron. J. (to be published).

Schwab, F. R.

R. A. Perley, F. R. Schwab, A. H. Bridle, Synthesis Imaging (National Radio Astronomy Observatory, Socorro, N. Mex., 1985).

Shao, M.

See, e.g., D. Mozurkewich, D. J. Hutter, K. J. Johnston, R. S. Simon, M. Shao, M. M. Colavita, D. H. Staelin, B. E. Hines, J. L. Hershey, J. A. Hughes, G. H. Kaplan, “Preliminary measurements of star positions with Mark III stellar interferometer,” Astron. Astrophys. 193, 1269–1277 (1988).

M. Shao, Jet Propulsion Laboratory, 4800 Oak Grove Drive, Pasadena, California 91109 (personal communication, 1989). The National Aeronautics and Space Administration is working currently to improve the readout noise of CCD’s with the aim of making them more attractive for space interferometry than they are now.

Shimoda, K.

K. Shimoda, H. Takahashi, C. H. Townes, “Fluctuations in amplification of quanta with application to maser amplifiers,” J. Phys. Soc. Jpn. 12, 686–700 (1957).
[CrossRef]

Simon, R. S.

See, e.g., D. Mozurkewich, D. J. Hutter, K. J. Johnston, R. S. Simon, M. Shao, M. M. Colavita, D. H. Staelin, B. E. Hines, J. L. Hershey, J. A. Hughes, G. H. Kaplan, “Preliminary measurements of star positions with Mark III stellar interferometer,” Astron. Astrophys. 193, 1269–1277 (1988).

Sivia, D.

C. A. Haniff, C. D. Mackay, D. J. Titterington, D. Sivia, P. J. Warner, “The first images from optical aperture synthesis,” Nature 328, 694–696 (1987).
[CrossRef]

Staelin, D. H.

See, e.g., D. Mozurkewich, D. J. Hutter, K. J. Johnston, R. S. Simon, M. Shao, M. M. Colavita, D. H. Staelin, B. E. Hines, J. L. Hershey, J. A. Hughes, G. H. Kaplan, “Preliminary measurements of star positions with Mark III stellar interferometer,” Astron. Astrophys. 193, 1269–1277 (1988).

Steinberg, H.

H. Steinberg, “The use of a laser amplifier in a laser communication system,” Proc. IEEE 51, 943 (1963).
[CrossRef]

Swenson, G. W.

See, e.g., A. R. Thompson, J. M. Moran, G. W. Swenson, Interferometry and Aperture Synthesis (Wiley, New York, 1986).

Takahashi, H.

K. Shimoda, H. Takahashi, C. H. Townes, “Fluctuations in amplification of quanta with application to maser amplifiers,” J. Phys. Soc. Jpn. 12, 686–700 (1957).
[CrossRef]

Thompson, A. R.

See, e.g., A. R. Thompson, J. M. Moran, G. W. Swenson, Interferometry and Aperture Synthesis (Wiley, New York, 1986).

Titterington, D. J.

C. A. Haniff, C. D. Mackay, D. J. Titterington, D. Sivia, P. J. Warner, “The first images from optical aperture synthesis,” Nature 328, 694–696 (1987).
[CrossRef]

Townes, C. H.

K. Shimoda, H. Takahashi, C. H. Townes, “Fluctuations in amplification of quanta with application to maser amplifiers,” J. Phys. Soc. Jpn. 12, 686–700 (1957).
[CrossRef]

Walkup, J. F.

Warner, P. J.

C. A. Haniff, C. D. Mackay, D. J. Titterington, D. Sivia, P. J. Warner, “The first images from optical aperture synthesis,” Nature 328, 694–696 (1987).
[CrossRef]

Astron. Astrophys. (1)

See, e.g., D. Mozurkewich, D. J. Hutter, K. J. Johnston, R. S. Simon, M. Shao, M. M. Colavita, D. H. Staelin, B. E. Hines, J. L. Hershey, J. A. Hughes, G. H. Kaplan, “Preliminary measurements of star positions with Mark III stellar interferometer,” Astron. Astrophys. 193, 1269–1277 (1988).

J. Opt. Soc. Am. (1)

J. Phys. Soc. Jpn. (1)

K. Shimoda, H. Takahashi, C. H. Townes, “Fluctuations in amplification of quanta with application to maser amplifiers,” J. Phys. Soc. Jpn. 12, 686–700 (1957).
[CrossRef]

Nature (1)

C. A. Haniff, C. D. Mackay, D. J. Titterington, D. Sivia, P. J. Warner, “The first images from optical aperture synthesis,” Nature 328, 694–696 (1987).
[CrossRef]

Proc. IEEE (1)

H. Steinberg, “The use of a laser amplifier in a laser communication system,” Proc. IEEE 51, 943 (1963).
[CrossRef]

Other (6)

M. Shao, Jet Propulsion Laboratory, 4800 Oak Grove Drive, Pasadena, California 91109 (personal communication, 1989). The National Aeronautics and Space Administration is working currently to improve the readout noise of CCD’s with the aim of making them more attractive for space interferometry than they are now.

T. Nakajima, S. R. Kulkarni, P. W. Gorham, A. M. Ghez, G. Neugebauer, B. J. Oke, T. A. Prince, A. C. S. Readhead, “Diffraction-limited imaging II: optical aperture synthesis imaging of two binary stars,” Astron. J. (to be published).

See, e.g., J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 5.

See, e.g., A. R. Thompson, J. M. Moran, G. W. Swenson, Interferometry and Aperture Synthesis (Wiley, New York, 1986).

R. A. Perley, F. R. Schwab, A. H. Bridle, Synthesis Imaging (National Radio Astronomy Observatory, Socorro, N. Mex., 1985).

S. R. Kulkarni, “Self-noise in interferometers: radio and infrared,” submitted to Astron. J.

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Figures (1)

Fig. 1
Fig. 1

Enhancement factor F of SNR versus the number n of apertures in the array. F1 and F2 refer to the nC2 array without and with the zero frequency, F3 and F4 refer to the nonredundant nCn array, and F5 and F6 refer to the maximally redundant nCn array.

Equations (89)

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I g h ( x ) = 2 I 0 [ 1 + γ g h cos ( κ x · B g h / d + ϕ g h ) ] ,
k g h ( p ) = 2 K 0 [ 1 + γ g h cos ( p ω g h + ϕ g h ) ] .
z g h = p = 1 P k g h ( p ) exp ( - i p ω g h ) .
Z g h = z g h = γ g h N exp ( i ϕ g h ) .
z g h 0 = p = 1 P k g h ( p ) ,
Z g h 0 = 2 N .
i 1 ( q ) Re [ r = 1 n b z r exp ( + i q ω r ) ] = r [ Re ( z r ) cos ( ω r q ) - Im ( z r ) sin ( ω r q ) ] .
I 1 ( q ) = Re [ r Z r exp ( + i q ω r ) ] ,
I 1 ( q ) = N r γ r cos ( ω r q + ϕ r ) .
V [ i 1 ( q ) ] i 1 ( q ) 2 - i 1 ( q ) 2 = r = 1 n b s = 1 n b [ Re ( z r ) Re ( z s ) - Re ( z r ) Re ( z s ) ] × cos ( ω r q ) cos ( ω s q ) - [ Re ( z r ) Im ( z s ) - Re ( z r ) Im ( z s ) ] cos ( ω r q ) sin ( ω s q ) - [ Im ( z r ) Re ( z s ) - Im ( z r ) Re ( z s ) ] sin ( ω r q ) × cos ( ω s q ) + [ Im ( z r ) Im ( z s ) - Im ( z r ) × Im ( z s ) ] sin ( ω r q ) sin ( ω s q ) .
cov [ Re ( z r ) , Re ( z s ) ] Re ( z r ) Re ( z s ) - Re ( z r ) Re ( z s ) = p = 1 P p = 1 P [ k r ( p ) k s ( p ) - k r ( p ) k s ( p ) ] cos ( ω r p ) cos ( ω s p ) .
k r ( p ) k s ( p ) - k r ( p ) k s ( p ) = δ r s δ p p k r ( p ) .
cov [ Re ( z r ) , Re ( z s ) ] = V [ Re ( z r ) ] δ r s = δ r s p k r ( p ) cos ( ω r p ) 2 = N δ r s .
cov [ Im ( z r ) , Im ( z s ) ] = N δ r s
cov [ Re ( z r ) , Im ( z s ) ] = 0.
V [ i 1 ( q ) ] = r { V [ Re ( z r ) ] cos 2 ( ω r q ) + V [ Im ( z r ) ] sin 2 ( ω r q ) } = r N = C 2 .
I 1 ( q ) = C 2 δ q 0
I 1 ( 0 ) { V [ i 1 ( 0 ) ] } 1 / 2 = ( C 2 ) 1 / 2 .
i 2 ( q ) r [ Re ( z r ) cos ( ω r q ) - Im ( z r ) sin ( ω r q ) + ( 1 / 2 ) z r 0 ] ,
I 2 ( q ) = N r [ γ r cos ( ϕ r + ω r q ) + 1 ] = I 1 ( q ) + C 2 .
V [ i 2 ( q ) ] = V [ i 1 ( q ) ] + r = 1 n b s = 1 n b { ( 1 / 2 ) cov [ Re ( z r ) , z s 0 ] cos ( ω r q ) - ( 1 / 2 ) cov [ Im ( z r ) , z s 0 ] sin ( ω r q ) + ( 1 / 2 ) cov [ z r 0 , Re ( z s ) ] cos ( ω s q ) - ( 1 / 2 ) cov [ z r 0 , Im ( z s ) ] sin ( ω s q ) + ( 1 / 4 ) cov [ z r 0 , z s 0 ] } .
cov [ Re ( z r ) , z r 0 ] = p k r ( p ) cos ( p ω r ) p k r ( p ) - p k r ( p ) cos ( p ω r ) p k r ( p ) = p k r ( p ) cos ( ω r p ) .
cov [ Im ( z r ) , z r 0 ] = - p k p ( p ) sin ( ω r p ) , cov ( z r 0 , z r 0 ) = p k r ( p ) = 2 N ,
V [ i 2 ( q ) ] = r [ N + ( 1 / 2 ) N + p k r ( p ) × cos ( ω r p - ω r q ) ] = N r [ ( 3 / 2 ) + γ r cos ( ω r q + ϕ r ) ] .
V [ i 2 ( q ) ] = ( 3 / 4 ) C + I 1 ( q ) = C 4 + I 2 ( q ) .
I 2 ( q ) = C ,
I 2 ( 0 ) { V [ i 2 ( 0 ) ] } 1 / 2 = ( 8 5 ) 1 / 2 ( C 2 ) 1 / 2 .
I ( x ) = I 0 [ n + 2 g < h γ g h cos ( κ x · B g h / d + ϕ g h ) ] ,
k ( p ) = Q 0 [ n + 2 g < h γ g h cos ( p ω g h + ϕ g h ) ] .
Z i j = Q 0 p exp ( - i p ω i j ) [ n + 2 g < h γ g h cos ( p ω g h + ϕ g h ) ] = Q 0 p g < h γ g h { exp ( i ϕ g h ) exp [ i p ( ω g h - ω i j ) ] + exp ( - i ϕ g h ) exp [ - i p ( ω g h + ω i j ) ] } .
Z i j = M γ i j exp ( i ϕ i j ) ,
Z 0 = Q 0 P n = C .
cov [ Re ( z i j ) , Re ( z k l ) ] = p k p cos ( p ω i j ) cos ( p ω k l ) = Q 0 p [ n + 2 g < h γ g h cos ( p ω g h + ϕ g h ) ] × cos ( p ω i j ) cos ( p ω k l ) .
ω g h ± ω i j ± ω k l 0 ,
cov [ Re ( z i j ) , Re ( z k l ) ] = M ( n 2 δ i k δ j l + γ cos ϕ 2 Δ i j , k l ) ,
cov [ Im ( z i j ) , Im ( z k l ) ] = M ( n 2 δ i k δ j l γ cos ϕ 2 Δ i j , k l ) .
cov [ Re ( z i j ) , Im ( z k l ) ] = - Q 0 p k p cos ( p ω i j ) sin ( p ω k l ) = - 2 Q 0 p g < h γ g h cos ( p ω g h + ϕ g h ) cos ( p ω i j ) sin ( p ω k l ) = M γ 2 ( sin ϕ ) Δ i j , k l × { + 1 sgn ( l - j ) for i = k , sgn ( i - k ) for j = l ,
cov [ Re ( z i j ) , Im ( z k l ) ] = ± cov [ Im ( z i j ) , Re ( z k l ) ] .
V ( z 0 ) = p k p = Z 0 = C ,
cov [ z 0 , Re ( z i j ) ] = Q 0 p cos ( p ω i j ) × [ n + 2 g < h γ g h cos ( p ω g h + ϕ g h ) ] = M γ i j cos ϕ i j
cov [ z 0 , Im ( z i j ) ] = M γ i j sin ϕ i j .
I 3 ( q ) = M i < j γ i j cos ( q ω i j + ϕ i j ) .
V [ i 3 ( q ) ] = i < j k < l { cov [ Re ( z i j ) , Re ( z k l ) ] cos ( q ω i j ) cos ( q ω k l ) + cov [ Im ( z i j ) , Im ( z k l ) ] sin ( q ω i j ) sin ( q ω k l ) - cov [ Re ( z i j ) , Im ( z k l ) ] cos ( q ω i j ) sin ( q ω k l ) - cov [ Im ( z i j ) , Re ( z k l ) ] sin ( q ω i j ) cos ( q ω k l ) } .
V [ i 3 ( q ) ] = M 2 { n i < j [ cos 2 ( q ω i j ) + sin 2 ( q ω i j ) ] + i < j k < l Δ i j , k l cos [ q ( ω i j ± ω k l ) ] γ cos ϕ - i < j = k < l sin [ q ( ω i j + ω j l ) ] γ i l sin ϕ i l - k < l = i < j sin [ q ( ω l j + ω k l ) γ k j sin ϕ k j - j < l ( n - l + j - 1 ) sin [ q ( ω i l - ω i j ) γ j l sin ϕ j l - l < j ( n - j + l - 1 ) sin [ q ( - ω i l + ω i j ) ] γ l j sin ϕ l j } .
V [ i 3 ( q ) ] = M 2 { n n b + 2 i < j = k < l cos ( q ω i l ) γ i l cos ϕ i l + 2 j < l ( n - 1 + j - 1 ) cos ( q ω j l ) γ j l cos ϕ j l - 2 i < l ( l - i - 1 ) sin ( q ω i l ) γ i l sin ϕ i l - 2 j < l ( n - l + j - 1 ) sin ( q ω j l γ j l sin ϕ j l } .
V [ i 3 ( q ) ] = M 2 × [ n n b + 2 ( n - 2 ) i < j γ i j cos ( q ω i j + ϕ i j ) ] .
I 3 ( 0 ) { V [ i 3 ( 0 ) ] } 1 / 2 = ( C 2 ) 1 / 2 ( 2 n - 2 3 n - 4 ) 1 / 2 .
I 4 ( q ) = M [ n 2 + i < j γ i j cos ( q ω i j + ϕ i j ) ]
V [ i 4 ( q ) ] = V [ i 3 ( q ) ] + 1 4 V ( z 0 ) + i < j cov [ z 0 , Re ( z i j ) ] cos ( q ω i j ) - i < j cov [ z 0 , Im ( z i j ) ] sin ( q ω i j ) = M 2 [ n ( 1 2 + n b ) + 2 ( n - 1 ) i < j 2 ( j - 1 ) γ i j × cos ( q ω i j + ϕ i j ) ] .
I 4 ( 0 ) { V [ i 4 ( 0 ) ] } 1 / 2 = ( C 2 ) 1 / 2 ( 2 n 2 3 n 2 - 5 n + 3 ) 1 / 2 ,
k ( p ) = Q 0 [ n + 2 r = 1 n - 1 ( n - r ) cos ( p r ω 0 ) ] .
z r = p = 1 P k ( p ) exp ( - i p r ω 0 )             ( 0 r n - 1 ) .
z r = M ( n - r ) ,
p exp ( i p m ω 0 ) P δ m , 0 .
cov [ Re ( z r ) , Re ( z s ) ] = p p cov [ k ( p ) , k ( p ) ] cos ( p r ω 0 ) × cos ( p s ω 0 ) = p k ( p ) cos ( p r ω 0 ) cos ( p s ω 0 ) ,
cov [ Re ( z r ) , Re ( z s ) ] = M 2 [ n δ r s + ( n - r - s ) ( 1 - δ r s ) + ( n - r - s ) Θ ( n - r - s ) ] ,
cov [ Im ( z r ) , Im ( z s ) ] = M 2 [ n δ r s + ( n - r - s ) ( 1 - δ r s ) - ( n - r - s ) Θ ( n - r - s ) ] .
I 5 ( q ) = M { n 2 cos ( n x 2 ) sin [ ( n - 1 ) x 2 ] sin ( x 2 ) - sin ( n x 2 ) d d x sin [ ( n - 1 ) x 2 ] sin ( x 2 ) } x = q ω 0 ,
V [ i 5 ( q ) ] = M 2 { n sin 2 ( n - 1 2 x ) sin 2 ( x 2 ) + n ( n - 2 ) 2 × cos ( n x 2 ) sin ( n - 1 2 x ) sin ( x 2 ) + n 2 sin ( n x 2 ) d d x sin ( n - 1 2 x ) sin ( x 2 ) + d d x cos ( n - 1 2 x ) sin ( n - 1 2 x ) sin 2 ( x 2 ) - ( n - 1 ) d d x cot ( x 2 ) - n - 2 2 d d x sin ( n x 2 ) sin ( n - 1 2 x ) sin ( x 2 ) + cos ( n x 2 ) d 2 d x 2 sin ( n - 1 2 x ) sin ( x 2 ) } x = q ω 0 .
I 5 ( 0 ) = C ( n - 1 ) 2 ,
V [ i 5 ( 0 ) ] = C 12 ( 5 n 2 - 9 n + 4 ) ,
I 5 ( 0 ) { V [ i 5 ( 0 ) ] } 1 / 2 = F ( C 2 ) 1 / 2 ,
F = [ 6 ( n - 1 ) 5 n - 4 ] 1 / 2
I 6 ( q ) = N ( n 2 + n 2 cos ( n x 2 ) sin ( n - 1 ) x 2 sin ( x 2 ) - sin n x 2 d d x { sin ( n - 1 ) x 2 sin ( x 2 ) } ) x = q ω 0
V [ i 6 ( q ) ] = V [ i 5 ( q ) ] - 3 4 M n + M [ n cos ( n - 1 2 ) x × sin ( n x 2 ) sin ( x 2 ) - d d x sin ( n - 1 2 ) x sin ( n x 2 ) sin ( x 2 ) ] x = q ω 0 ,
I 6 ( 0 ) = C n 2
V [ i 6 ( 0 ) ] = C 12 ( 5 n 2 - 3 n + 1 ) .
I 6 ( 0 ) { V ( i 6 ( 0 ) ] } 1 / 2 = F ( C 2 ) 1 / 2 ,
F = ( 6 n 2 5 n 2 - 3 n + 1 ) 1 / 2 .
i 5 ( R ) = Re [ r = 1 n - 1 z r exp ( i q r ω 0 ) ]
I 5 ( q ) = M Re [ r = 1 n - 1 ( n - r ) exp ( i q r ω 0 ) ] ,
V [ i 5 ( q ) ] = r = 1 n - 1 s = 1 n - 1 { cov [ Re ( z r ) , Re ( z s ) ] cos ( q r ω 0 ) cos ( q s ω 0 ) + cov [ Im ( z r ) , Im ( z s ) ] sin ( q r ω 0 ) sin ( q s ω 0 ) } ,
V [ i 5 ( q ) ] = r = 1 [ ( n - 1 ) / 2 ] { V [ Re ( z r ) ] cos 2 ( q r ω 0 ) + V [ Im ( z r ) ] sin 2 ( q r ω 0 ) } + r = [ ( n + 1 ) / 2 ] n - 1 { V [ Re ( z r ) ] cos 2 ( q r ω 0 ) + V [ Im ( z r ) ] sin 2 ( q r ω 0 ) } + 2 r > s = 1 n - 1 { cov [ Re ( z r ) , Re ( z s ) ] cos ( q r ω 0 ) cos ( q s ω 0 ) + cov [ Im ( z r ) , Im ( z s ) ] × sin ( q r ω 0 ) sin ( q s ω 0 ) } ,
S S = M 2 [ r = 1 n - 1 n + r = 1 [ ( n - 1 ) / 2 ] ( n - 2 r ) cos ( 2 q r ω 0 ) ] = M 2 { n ( n - 1 ) + [ n Re - ( q ω 0 ) Im ] × r = 1 [ ( n - 1 ) / 2 ] exp ( 2 i q ω 0 ) } .
S 1 = r > s = 1 n - 1 exp [ i q ( r - s ) ω 0 ]
S 2 = r > s = 1 n - 1 exp [ i q ( r + s ) ω 0 ] Θ ( n - r - s ) .
S S 1 + S 2 ,
V [ i 5 ( q ) ] = S S + M [ n Re - ( q ω 0 ) Im ] S .
s = 1 r - 1 exp ( - i s q ω 0 ) = exp ( - i q ω 0 ) - exp ( - i r q ω 0 ) 1 - exp ( - i q ω 0 ) ,
S 1 = 1 1 - exp ( - i q ω 0 ) r = 2 n - 1 { exp [ i ( r - 1 ) q ω 0 ] - 1 } .
S 1 = - i exp [ i ( n - 1 ) q ω 0 / 2 ] sin [ q ( n - 1 ) ω 0 / 2 ] 2 sin 2 ( q ω 0 / 2 ) + i ( n - 1 ) exp ( i q ω 0 / 2 ) 2 sin ( q ω 0 / 2 ) .
S 2 = 1 2 r = 1 n - 1 s = 1 n - 1 exp [ i q ( r + s ) ω 0 ] Θ ( n - r - s ) - 1 2 r = 1 [ ( n - 1 ) / 2 ] exp ( 2 q i r ω 0 ) .
S 2 = 1 2 R = 2 n - 1 exp ( i q R ω 0 ) ( R - 1 ) - 1 2 r = 1 [ ( n - 1 ) / 2 ] exp ( 2 i q r ω 0 ) .
S 2 = 1 2 exp ( i n q ω 0 / 2 ) [ n - 2 2 - i ( q w 0 ) ] × sin [ q ( n - 1 ) ω 0 / 2 ] sin ( q ω 0 / 2 ) - 1 2 r = 1 [ ( n - 1 ) / 2 ] exp ( 2 i q r ω 0 ) .
V [ i 5 ( q ) ] = M 2 n ( n - 1 ) + M [ n Re - ( q ω 0 ) Im ] × { - i exp [ i ( n - 1 ) q ω 0 / 2 ] sin [ q ( n - 1 ) ω 0 / 2 ] 2 sin 2 ( q ω 0 / 2 ) + i ( n - 1 ) exp ( i q ω 0 / 2 ) 2 sin ( q ω 0 / 2 ) + 1 2 exp ( i n q ω 0 / 2 ) × [ n - 2 2 - i ( q ω 0 ) ] sin [ q ( n - 1 ) ω 0 / 2 ] sin ( q ω 0 / 2 ) } .
V [ i 5 ( q ) ] = M 2 { n sin 2 ( n - 1 2 x ) sin 2 x 2 + n ( n - 2 ) 2 cos ( n x 2 ) × sin ( n - 1 2 x ) sin ( x 2 ) + n 2 sin ( n x 2 ) d d x sin ( n - 1 2 x ) sin ( x 2 ) + d d x cos ( n - 1 2 x ) sin ( n - 1 2 x ) sin 2 ( x 2 ) - ( n - 1 ) d d x cot ( x 2 ) - n - 2 2 d d x sin ( n x 2 ) sin ( n - 1 2 x ) sin ( x 2 ) + cos ( n x 2 ) d 2 d x 2 sin ( n - 1 2 x ) sin ( x 2 ) } x = q ω 0 .
i 6 ( q ) = Re [ r = 1 n - 1 z r exp ( i q r ω 0 ) ] + 1 2 z 0 = i 5 ( q ) + 1 2 z 0 .
I 6 ( q ) = I 5 ( q ) + 1 2 C .
V [ i 6 ( q ) ] = V [ i 5 ( q ) ] + 1 4 V ( z 0 ) + r = 1 n - 1 cov ( z 0 , Re z r ) cos ( q r ω 0 ) = V [ i 5 ( q ) ] + 1 4 M n + M r = 1 n - 1 ( n - r ) cos ( q r ω 0 ) = V [ i 5 ( q ) ] - 3 4 M n + M [ n Re - ( q ω 0 ) Im ] × r = 0 n - 1 exp ( i q r ω 0 ) = V [ i 5 ( q ) ] - 3 4 M n + M [ n cos ( n - 1 2 x ) × sin ( n x 2 ) sin ( x 2 ) - d d x sin ( n - 1 2 x ) sin ( n x 2 ) sin ( x 2 ) ] x = q ω 0 ,

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