Abstract

Random laser-writer position errors in raster-scanned monochrome halftone images are examined by using Fourier analysis techniques. For a high-contrast, narrow-exposure latitude recording material (typically used in halftone reproduction) with medium-sized halftone dots (25–85%), a one-dimensional halftone model is developed to derive the signal power spectrum of a halftone image containing position errors in the slow-scan (page-scan) direction. The spectrum of such an image is shown to consist of a periodic component and a random component, which is a function of position error but independent of dot size. The term signal power spectrum, in the context of this paper, is the average squared modulus of the Fourier transform of an image containing these errors. A comparison is made between the percent position-error specification quoted for continuous-tone laser writing and the corresponding specification made in writing digital halftones. For a given absolute position error, the ratio of the resulting percent position error of a single pixel in continuous-tone writing to that for a halftone cell is shown to be roughly 2(X/L), where X is the halftone cell period and L is the pixel size used in continuous-tone reconstruction. Thus, whereas the position-error tolerance for halftone writing is almost an order of magnitude greater than its continuous-tone counterpart on a percent basis, on an absolute scale they are approximately equal. The results can be generalized to include any digitally generated halftone image based on a center-growing dot configuration and containing dot-size/shape distortions.

© 1988 Optical Society of America

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References

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  1. F. Bestenreiner, U. Geis, J. Helmberger, K. Stadler, “Visibility and correction of periodic interference structures in line-byline recorded images,”J. Appl. Photogr. Eng. 2, 86–92 (1976).
  2. D. Haas, “Contrast modulation in halftone images produced by variation in scan line spacing,” in Proceedings of the SPSE Third International Congress: Advances in Non-Impact Printing Technologies (Society of Photographic Scientists and Engineers, Springfield, Va., 1986), p. 183.
  3. K. Takiguchi, T. Miyagi, A. Okamura, H. Ishoshi, F. Shibata, “Effect of photoreceptor drum rotational speed variation on laser beam printer halftone reproduction,” in Proceedings of the SPSE Third International Congress: Advances in Non-Impact Printing Technologies (Society of Photographic Scientists and Engineers, Springfield, Va., 1986), pp. 168–172.
  4. P. D. Burns, M. Rabbani, L. A. Ray, “Analysis of image noise due to position errors in laser writers,” Appl. Opt. 25, 2158–2168 (1986).
    [CrossRef] [PubMed]
  5. R. Näsänen, “Visibility of halftone dot textures,”IEEE Trans. Syst. Man Cybern. SMC-14, 1447–1450 (1984).
    [CrossRef]
  6. D. Kermisch, P. G. Roetling, “Fourier spectrum of halftone images,”J. Opt. Soc. Am. 65, 1177 (A) (1975).
    [CrossRef]
  7. J. P. Allebach, B. Liu, “Random quasiperiodic halftone process,”J. Opt. Soc. Am. 66, 909–917 (1976).
    [CrossRef]
  8. S. Bloomberg, P. Engeldrum, “Color error due to pixel placement errors in a dot matrix printer,” in Proceedings of the SPSE Third International Congress: Advances in Non-Impact Printing Technologies (Society of Photographic Scientists and Engineers, Springfield, Va., 1986), pp. 257–260.
  9. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1986).
  10. J. Lawson, G. Uhlenbeck, Threshold Signals (McGraw-Hill, New York, 1950), p. 44.
  11. L. E. Franks, “A model for the random video process,” Bell Syst. Tech. J. 45, 609–630 (1966).
  12. J. G. Proakis, Digital Communications (McGraw-Hill, New York, 1983), Chap. 3.
  13. W. A. Gardner, L. E. Franks, “Characterization of cyclostationary random signal processes,”IEEE Trans. Inf. Theory IT-21, 4–14 (1975).
    [CrossRef]
  14. W. A. Gardner, “Common pitfalls in the application of stationary process theory to time-sampled and modulated signals,”IEEE Trans. Commun. COM-35, 529–534 (1987).
    [CrossRef]
  15. A. Papoulis, The Fourier Integral and Its Applications (McGraw-Hill, New York, 1962), Chap. 12.
  16. P. D. Burns, “aMeasurement of random and periodic image noise in raster-written images,” in Proceedings of the SPSE Second International Congress: Advances in Non-Impact Printing Technologies (Society of Photographic Scientists and Engineers, Springfield, Va., 1984), pp. 139–142.
  17. Y. W. Lee, Statistical Theory of Communication (Wiley, New York, 1960), pp. 60–66.
  18. A. Papoulis, The Fourier Integral and Its Applications (McGraw-Hill, New York, 1962), p. 44.

1987 (1)

W. A. Gardner, “Common pitfalls in the application of stationary process theory to time-sampled and modulated signals,”IEEE Trans. Commun. COM-35, 529–534 (1987).
[CrossRef]

1986 (1)

1984 (1)

R. Näsänen, “Visibility of halftone dot textures,”IEEE Trans. Syst. Man Cybern. SMC-14, 1447–1450 (1984).
[CrossRef]

1976 (2)

F. Bestenreiner, U. Geis, J. Helmberger, K. Stadler, “Visibility and correction of periodic interference structures in line-byline recorded images,”J. Appl. Photogr. Eng. 2, 86–92 (1976).

J. P. Allebach, B. Liu, “Random quasiperiodic halftone process,”J. Opt. Soc. Am. 66, 909–917 (1976).
[CrossRef]

1975 (2)

W. A. Gardner, L. E. Franks, “Characterization of cyclostationary random signal processes,”IEEE Trans. Inf. Theory IT-21, 4–14 (1975).
[CrossRef]

D. Kermisch, P. G. Roetling, “Fourier spectrum of halftone images,”J. Opt. Soc. Am. 65, 1177 (A) (1975).
[CrossRef]

1966 (1)

L. E. Franks, “A model for the random video process,” Bell Syst. Tech. J. 45, 609–630 (1966).

Allebach, J. P.

Bestenreiner, F.

F. Bestenreiner, U. Geis, J. Helmberger, K. Stadler, “Visibility and correction of periodic interference structures in line-byline recorded images,”J. Appl. Photogr. Eng. 2, 86–92 (1976).

Bloomberg, S.

S. Bloomberg, P. Engeldrum, “Color error due to pixel placement errors in a dot matrix printer,” in Proceedings of the SPSE Third International Congress: Advances in Non-Impact Printing Technologies (Society of Photographic Scientists and Engineers, Springfield, Va., 1986), pp. 257–260.

Burns, P. D.

P. D. Burns, M. Rabbani, L. A. Ray, “Analysis of image noise due to position errors in laser writers,” Appl. Opt. 25, 2158–2168 (1986).
[CrossRef] [PubMed]

P. D. Burns, “aMeasurement of random and periodic image noise in raster-written images,” in Proceedings of the SPSE Second International Congress: Advances in Non-Impact Printing Technologies (Society of Photographic Scientists and Engineers, Springfield, Va., 1984), pp. 139–142.

Engeldrum, P.

S. Bloomberg, P. Engeldrum, “Color error due to pixel placement errors in a dot matrix printer,” in Proceedings of the SPSE Third International Congress: Advances in Non-Impact Printing Technologies (Society of Photographic Scientists and Engineers, Springfield, Va., 1986), pp. 257–260.

Franks, L. E.

W. A. Gardner, L. E. Franks, “Characterization of cyclostationary random signal processes,”IEEE Trans. Inf. Theory IT-21, 4–14 (1975).
[CrossRef]

L. E. Franks, “A model for the random video process,” Bell Syst. Tech. J. 45, 609–630 (1966).

Gardner, W. A.

W. A. Gardner, “Common pitfalls in the application of stationary process theory to time-sampled and modulated signals,”IEEE Trans. Commun. COM-35, 529–534 (1987).
[CrossRef]

W. A. Gardner, L. E. Franks, “Characterization of cyclostationary random signal processes,”IEEE Trans. Inf. Theory IT-21, 4–14 (1975).
[CrossRef]

Geis, U.

F. Bestenreiner, U. Geis, J. Helmberger, K. Stadler, “Visibility and correction of periodic interference structures in line-byline recorded images,”J. Appl. Photogr. Eng. 2, 86–92 (1976).

Haas, D.

D. Haas, “Contrast modulation in halftone images produced by variation in scan line spacing,” in Proceedings of the SPSE Third International Congress: Advances in Non-Impact Printing Technologies (Society of Photographic Scientists and Engineers, Springfield, Va., 1986), p. 183.

Helmberger, J.

F. Bestenreiner, U. Geis, J. Helmberger, K. Stadler, “Visibility and correction of periodic interference structures in line-byline recorded images,”J. Appl. Photogr. Eng. 2, 86–92 (1976).

Ishoshi, H.

K. Takiguchi, T. Miyagi, A. Okamura, H. Ishoshi, F. Shibata, “Effect of photoreceptor drum rotational speed variation on laser beam printer halftone reproduction,” in Proceedings of the SPSE Third International Congress: Advances in Non-Impact Printing Technologies (Society of Photographic Scientists and Engineers, Springfield, Va., 1986), pp. 168–172.

Kermisch, D.

D. Kermisch, P. G. Roetling, “Fourier spectrum of halftone images,”J. Opt. Soc. Am. 65, 1177 (A) (1975).
[CrossRef]

Lawson, J.

J. Lawson, G. Uhlenbeck, Threshold Signals (McGraw-Hill, New York, 1950), p. 44.

Lee, Y. W.

Y. W. Lee, Statistical Theory of Communication (Wiley, New York, 1960), pp. 60–66.

Liu, B.

Miyagi, T.

K. Takiguchi, T. Miyagi, A. Okamura, H. Ishoshi, F. Shibata, “Effect of photoreceptor drum rotational speed variation on laser beam printer halftone reproduction,” in Proceedings of the SPSE Third International Congress: Advances in Non-Impact Printing Technologies (Society of Photographic Scientists and Engineers, Springfield, Va., 1986), pp. 168–172.

Näsänen, R.

R. Näsänen, “Visibility of halftone dot textures,”IEEE Trans. Syst. Man Cybern. SMC-14, 1447–1450 (1984).
[CrossRef]

Okamura, A.

K. Takiguchi, T. Miyagi, A. Okamura, H. Ishoshi, F. Shibata, “Effect of photoreceptor drum rotational speed variation on laser beam printer halftone reproduction,” in Proceedings of the SPSE Third International Congress: Advances in Non-Impact Printing Technologies (Society of Photographic Scientists and Engineers, Springfield, Va., 1986), pp. 168–172.

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1986).

A. Papoulis, The Fourier Integral and Its Applications (McGraw-Hill, New York, 1962), Chap. 12.

A. Papoulis, The Fourier Integral and Its Applications (McGraw-Hill, New York, 1962), p. 44.

Proakis, J. G.

J. G. Proakis, Digital Communications (McGraw-Hill, New York, 1983), Chap. 3.

Rabbani, M.

Ray, L. A.

Roetling, P. G.

D. Kermisch, P. G. Roetling, “Fourier spectrum of halftone images,”J. Opt. Soc. Am. 65, 1177 (A) (1975).
[CrossRef]

Shibata, F.

K. Takiguchi, T. Miyagi, A. Okamura, H. Ishoshi, F. Shibata, “Effect of photoreceptor drum rotational speed variation on laser beam printer halftone reproduction,” in Proceedings of the SPSE Third International Congress: Advances in Non-Impact Printing Technologies (Society of Photographic Scientists and Engineers, Springfield, Va., 1986), pp. 168–172.

Stadler, K.

F. Bestenreiner, U. Geis, J. Helmberger, K. Stadler, “Visibility and correction of periodic interference structures in line-byline recorded images,”J. Appl. Photogr. Eng. 2, 86–92 (1976).

Takiguchi, K.

K. Takiguchi, T. Miyagi, A. Okamura, H. Ishoshi, F. Shibata, “Effect of photoreceptor drum rotational speed variation on laser beam printer halftone reproduction,” in Proceedings of the SPSE Third International Congress: Advances in Non-Impact Printing Technologies (Society of Photographic Scientists and Engineers, Springfield, Va., 1986), pp. 168–172.

Uhlenbeck, G.

J. Lawson, G. Uhlenbeck, Threshold Signals (McGraw-Hill, New York, 1950), p. 44.

Appl. Opt. (1)

Bell Syst. Tech. J. (1)

L. E. Franks, “A model for the random video process,” Bell Syst. Tech. J. 45, 609–630 (1966).

IEEE Trans. Commun. (1)

W. A. Gardner, “Common pitfalls in the application of stationary process theory to time-sampled and modulated signals,”IEEE Trans. Commun. COM-35, 529–534 (1987).
[CrossRef]

IEEE Trans. Inf. Theory (1)

W. A. Gardner, L. E. Franks, “Characterization of cyclostationary random signal processes,”IEEE Trans. Inf. Theory IT-21, 4–14 (1975).
[CrossRef]

IEEE Trans. Syst. Man Cybern. (1)

R. Näsänen, “Visibility of halftone dot textures,”IEEE Trans. Syst. Man Cybern. SMC-14, 1447–1450 (1984).
[CrossRef]

J. Appl. Photogr. Eng. (1)

F. Bestenreiner, U. Geis, J. Helmberger, K. Stadler, “Visibility and correction of periodic interference structures in line-byline recorded images,”J. Appl. Photogr. Eng. 2, 86–92 (1976).

J. Opt. Soc. Am. (2)

D. Kermisch, P. G. Roetling, “Fourier spectrum of halftone images,”J. Opt. Soc. Am. 65, 1177 (A) (1975).
[CrossRef]

J. P. Allebach, B. Liu, “Random quasiperiodic halftone process,”J. Opt. Soc. Am. 66, 909–917 (1976).
[CrossRef]

Other (10)

D. Haas, “Contrast modulation in halftone images produced by variation in scan line spacing,” in Proceedings of the SPSE Third International Congress: Advances in Non-Impact Printing Technologies (Society of Photographic Scientists and Engineers, Springfield, Va., 1986), p. 183.

K. Takiguchi, T. Miyagi, A. Okamura, H. Ishoshi, F. Shibata, “Effect of photoreceptor drum rotational speed variation on laser beam printer halftone reproduction,” in Proceedings of the SPSE Third International Congress: Advances in Non-Impact Printing Technologies (Society of Photographic Scientists and Engineers, Springfield, Va., 1986), pp. 168–172.

A. Papoulis, The Fourier Integral and Its Applications (McGraw-Hill, New York, 1962), Chap. 12.

P. D. Burns, “aMeasurement of random and periodic image noise in raster-written images,” in Proceedings of the SPSE Second International Congress: Advances in Non-Impact Printing Technologies (Society of Photographic Scientists and Engineers, Springfield, Va., 1984), pp. 139–142.

Y. W. Lee, Statistical Theory of Communication (Wiley, New York, 1960), pp. 60–66.

A. Papoulis, The Fourier Integral and Its Applications (McGraw-Hill, New York, 1962), p. 44.

J. G. Proakis, Digital Communications (McGraw-Hill, New York, 1983), Chap. 3.

S. Bloomberg, P. Engeldrum, “Color error due to pixel placement errors in a dot matrix printer,” in Proceedings of the SPSE Third International Congress: Advances in Non-Impact Printing Technologies (Society of Photographic Scientists and Engineers, Springfield, Va., 1986), pp. 257–260.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1986).

J. Lawson, G. Uhlenbeck, Threshold Signals (McGraw-Hill, New York, 1950), p. 44.

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Figures (9)

Fig. 1
Fig. 1

Halftone banding produced by random laser writer position errors in the slow-scan direction. (Photo courtesy of B. Narayan, Eastman Kodak Company.)

Fig. 2
Fig. 2

Two-dimensional construction of a halftone cell for K = 8.

Fig. 3
Fig. 3

Construction of a halftone dot along the slow-scan (y) direction.

Fig. 4
Fig. 4

Realization of a single halftone dot-intensity profile resulting from random laser-writer position errors.

Fig. 5
Fig. 5

Halftone dot-reflectance profile resulting from the hard-limiter threshold η.

Fig. 6
Fig. 6

Series of 2N + 1 halftone dot traces along the slow-scan direction, illustrating random pulse-width and pulse-position modulation as a consequence of σξ.

Fig. 7
Fig. 7

Contribution of outermost writing spots to the overall intensity profile of the halftone dot.

Fig. 8
Fig. 8

Random spectral component of the halftone reflectance signal power spectrum, shown with varying degrees of position error.

Fig. 9
Fig. 9

Periodic spectral component of the halftone reflectance signal power spectrum for b = 0.50, which also illustrates the spectrum of the ideal writing case with no position errors.

Equations (75)

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f ( x , y , ξ ) = k = 1 K l = 1 K m ( x ) t k , l I 0 s ( x l L , y k L ξ k ) ,
g ( x , y , ξ ) = [ f ( x , y , ξ ) ] γ ( E ) ,
f ( y , ξ ) = k = 1 K t k I 0 s ( y k L ξ k ) = k = 1 K t k I 0 exp [ ( y k L ξ k ) 2 2 σ s 2 ] ,
g ( y ; a n , b n ) = r max R rect ( y + a n b n ) ,
g ( y ; a , b ) = r max R n = N N rect ( y n X + a n b n ) ,
g ( y ) ¯ ϕ g g ( y 1 , y 2 ) E { g ( y ) } = g ( y + X ) ¯ , E { g ( y 1 ) g * ( y 2 ) } = ϕ g g ( y 1 + X , y 2 + X ) y ( , ) , y ( , ) .
| G ( f ; a , b ) | 2 = ( R π f ) 2 n = N N m = N N sin ( π b n f ) sin ( π b m f ) × cos [ 2 π f ( a n a m ) ] cos [ 2 π f X ( n m ) ]
| G ( f ; b ) | 2 E { | G ( f ; a , b ) | 2 } = | G ( f ; a , b ) | 2 p a b ( a , b ) d a d b ,
p a b ( a , b ) = n = N N p a b ( a n , b n ) ,
| G ( f ; b ) | 2 = | G ( f ; a , b ) | 2 n = p a b ( a n , b n ) d a n d b n = ( R π f ) 2 n = N N m = N N sin ( π b n f ) × sin ( π b m f ) cos [ 2 π ( a n a m ) ] cos [ 2 π X ( n m ) ] × n = N N p a b ( a n , b n ) d a n d b n × m = N N p a b ( a m , b m ) d a m d b m .
a n = y l + y t / 2 ,
b n = y l y t .
f ( y l , ξ ) = k = 1 K t k I 0 exp [ ( y l k L ξ k ) 2 2 σ s 2 ] = η ,
f ( y t , ξ ) = k = 1 K t k I 0 exp [ ( y t k L ξ k ) 2 2 σ s 2 ] = η .
( a n ) = ( ξ l ξ t ) / 2 , b n = 2 { [ 2 σ s 2 ln ( I 0 / η ) ] 1 / 2 + k L } + ξ l + ξ t .
p a b ( a n , b n ) = 1 2 π σ ξ 2 exp ( a n 2 / σ ξ 2 ) × exp [ ( b n 2 { [ 2 σ s 2 ln ( I 0 η ) ] 1 / 2 + k L } ) 2 / 4 σ ξ 2 ] ,
p a ( a n ) = 1 π σ ξ exp ( a n 2 / σ ξ 2 ) ,
p b ( b n ) = 1 2 π σ ξ exp [ ( b n 2 { [ 2 σ s 2 ln ( I 0 η ) ] 1 / 2 + k L } ) 2 / 4 σ ξ 2 ] .
| G ( f ; b ) | 2 = ( R π f ) 2 ( sin 2 ( π b f ) exp [ ( 2 π σ ξ f ) 2 ] × n = N N m = N N cos [ 2 π X f ( n m ) ] + ( 2 N + 1 2 ) { 1 exp [ ( 2 π σ ξ f ) 2 } ) .
S ( f ) = lim N 1 ( 2 N + 1 ) X | G ( f ; b ) | 2 .
S ( f ) = ( R π f X ) 2 sin 2 ( π b f ) exp [ ( 2 π σ ξ f ) 2 ] × n = δ ( f n X ) + 1 2 X ( R π f ) 2 × { 1 exp [ ( 2 π σ ξ f ) 2 ] } .
S ( f ) + S p ( f ) + S r ( f ) ,
SNR = [ Σ n = S p ( f ) S r ( f ) d f ] 1 / 2 ,
S 0 ( f ) = S p ( f ) = ( R π f X ) 2 sin 2 ( π b f ) n = δ ( f n X ) ,
| G ( f ; a , b ) | 2 = ( R π f ) 2 n = N N m = N N sin ( π f b n ) sin ( π f b m ) × cos [ 2 π f ( a n a m ) ] cos [ 2 π f X ( n m ) ] .
g ( y ; a , b ) = r max R n = N N rect ( y n X + a n b n ) ,
rect ( y y 0 b ) { 0 | y y 0 b | > 1 2 , 1 2 , | y y 0 b | = 1 2 , 1 | y y 0 b | < 1 2 .
F { g ( y ; a , b ) } G ( f ; a , b ) = r max δ ( f ) R n = N N b n sinc ( b n f ) exp [ j 2 π f ( a n n X ) ] ,
sinc ( b f ) sin ( π b f ) / π b f .
| G ( f ; a , b ) | 2 G ( f ; a , b ) G * ( f ; a , b ) = ( R π f ) 2 ( { n = N N sin ( π b n f ) cos [ 2 π f ( a n n X ) ] } 2 + { n = N N sin ( π b n f ) sin [ 2 π f ( a n n X ) ] } 2 ) .
| G ( f ; a , b ) | 2 = ( R π f ) 2 n = N N m = N N sin ( π b n f ) sin ( π b m f ) × { cos [ 2 π f ( a n n X ) ] cos [ 2 π f ( a m m X ) ] + sin [ 2 π f ( a n n X ) ] sin [ 2 π f ( a m m X ) ] } ,
y l = { [ 2 σ s 2 ln ( I 0 / η ) ] 1 / 2 + k L + ξ l } ,
y t = { [ 2 σ s 2 ln ( I 0 / η ) ] 1 / 2 + k L + ξ t } ,
( y l k L ξ l ) 2 = 2 σ s 2 ln ( I 0 / η ) ,
( y t + k L + ξ t ) 2 = 2 σ s 2 ln ( I 0 / η ) .
a n = ξ l ξ t 2 ,
b n = 2 [ [ 2 σ s 2 ln ( I 0 / η ) ] 1 / 2 + k L ] + ξ l + ξ t .
a n = ( y l + y t ) / 2 , b n = y l y t .
p ξ ( ξ l ) = p ξ ( ξ t = p ξ ( ξ k ) = 1 2 π σ ξ exp ( ξ k 2 / 2 σ s 2 ) ,
p a b ( a n , b n ) = p a ( a n ) p b ( b n ) = 1 2 π σ ξ 2 exp ( a n 2 / σ ξ 2 ) × exp [ ( b n 2 { [ 2 σ s 2 ln ( I 0 / η ) ] 1 / 2 + k L } ) 2 / 4 σ ξ 2 ] .
p a b ( a n , b n ) = p ξ ( ξ l ) p ξ ( ξ t ) | J ( ξ l , ξ t ) | ,
ξ l = a n + b n 2 [ 2 σ s 2 ln ( I 0 / η ) ] 1 / 2 k L
ξ t = a n + b n 2 [ 2 σ s 2 ln ( I 0 / η ) ] 1 / 2 k L ,
| J ( ξ l , ξ t ) | = [ b n ξ l b n ξ t a n ξ l a n ξ t ] = [ 1 1 1 2 1 2 ] = 1
p a b ( a n , b n ) = p ξ { a n + b n 2 [ 2 σ s 2 ln ( I 0 / η ) ] 1 / 2 k L } × p ξ { a n + b n 2 [ 2 σ s 2 ln ( I 0 / η ) ] 1 / 2 k L } ,
| G ( f ; b ) | 2 E { | G ( f ; a , b ) | 2 } = ( R π f ) 2 ( sin 2 ( π b f ) exp [ ( 2 π σ ξ f ) 2 ] × n = N N m = N N cos [ 2 π X f ( n m ) ] + ( 2 N + 1 2 ) { 1 exp [ ( 2 π σ ξ f ) 2 ] } ) .
| G ( f ; b ) | 2 = ( R π f ) 2 n = N N sin 2 ( π b n f ) n = p b ( b n ) d b n + ( R π f ) 2 n = N N m = N N sin ( π f b n ) × sin ( π f b m ) cos [ 2 π f ( a n a m ) ] cos [ 2 π X f ( n m ) ] × n = N N p a ( a n ) d a n m = N N p a ( a m ) d a m n = N N p b ( b n ) d b n × n = N N p b ( b m ) d b m , n m .
( R / π f ) 2 ( G 1 + G 2 ) .
G 1 = n = N N n = p b ( b n ) d b n × 1 2 [ 1 cos ( 2 π b j ) ] p b ( b j ) d b j ,
G 1 = ( N + 1 2 ) n = N N 1 4 π σ ξ × cos ( 2 π f b j ) exp [ ( b j b ) 2 / 4 σ ξ 2 ] d b j .
1 4 π σ ξ cos [ 2 π f ( z + b ) ] exp [ z 2 / 4 σ ξ 2 ] d z .
sin ( 2 π f z ) exp [ z 2 / 4 σ ξ 2 ] d z = 0 ,
G 1 = ( N + 1 2 ) n = N N cos ( 2 π b f ) 1 2 π σ ξ × 0 cos ( 2 π f z ) exp [ z 2 / 4 σ ξ 2 ] d z ,
G 1 = ( N + 1 2 ) { 1 cos ( 2 π b f ) exp [ 2 π σ ξ f ) 2 ] } .
G 2 = n = N N m = N N cos [ 2 π f X ( n m ) ] × cos [ 2 π f ( a n a m ) ] × n = N N p a ( a n ) d a n m = N N p a ( a m ) d a m × n = N N p b ( b n ) d b n m = N n m N p b ( b m ) d b m × sin ( π f b j ) p b ( b j ) d b j sin ( π f b k ) p b ( b k ) d b k .
sin ( π b k f ) p b ( b k ) d b k = 1 2 π σ ξ sin ( π f b k ) exp [ ( b k b ) 2 / 4 σ ξ 2 ] d b k .
1 2 π σ ξ sin [ π f ( z + b ) ] exp [ z 2 / 4 σ ξ 2 ] d z = sin ( π b f ) 1 π σ ξ 0 cos ( π f z ) exp [ z 2 / 4 σ ξ 2 ] d z = sin ( π b f ) exp [ ( π σ ξ f ) 2 ] .
G 2 = sin 2 ( π b f ) exp [ 2 ( π σ ξ f ) 2 ] n = N N m = N N cos [ 2 π f X ( n m ) ] × n = N N p a ( a n ) d a n m = N n m N p a ( a m ) d a m × cos ( 2 π f a j ) p a ( a j ) d a j cos ( 2 π f a k ) p a ( a k ) d a k ,
G 2 = sin 2 ( π b f ) exp [ 2 ( π σ ξ f ) 2 ] n = N n m N m = N N cos [ 2 π f X ( n m ) ] × cos ( 2 π a j f ) p a ( a j ) d a j cos ( 2 π a k f ) p a ( a k ) d a k .
cos ( 2 π a k f ) p a ( a k ) d a k = 1 π σ ξ cos ( 2 π f a k ) exp [ a k 2 / σ ξ 2 ] d a k = exp [ ( π σ ξ f ) 2 ]
G 2 = sin 2 ( π b f ) exp [ ( 2 π σ ξ f ) 2 ] n = N n n N m = N N cos [ 2 π f X ( n m ) ] .
| G ( f ; b ) | 2 = ( R π f ) 2 ( sin 2 ( π b f ) exp [ ( 2 π σ ξ f ) 2 ] × n = N n m N m = N N cos [ 2 π f X ( n m ) ] + ( 2 N + 1 2 ) { 1 cos ( 2 π b f ) exp [ ( 2 π σ ξ f ) 2 ] } ) .
( 2 N + 1 ) sin 2 ( π b f ) exp [ ( 2 π σ ξ f ) 2 ]
| G ( f ; b ) | 2 = ( R π f ) 2 ( sin 2 ( π b f ) exp [ ( 2 π σ ξ f ) 2 ] × n = N N m = N n , m N cos [ 2 π f X ( n m ) ] + ( 2 N + 1 2 ) × { 1 + exp [ ( 2 π σ ξ f ) 2 ] × [ 2 sin 2 ( π b f ) cos ( 2 π b f ) ] } ) ,
lim N n = N N m = N N cos [ 2 π f X ( n m ) ] = 1 X 2 n = δ ( f n X ) .
n = N N m = N N cos [ 2 π f X ( n m ) ]
n = 0 N m = 0 N exp [ j 2 π X f ( n m ) ] + n = 1 N m = 1 N exp [ j 2 π X f ( n m ) ] = n = 0 N exp ( j 2 π n X f ) m = 0 N exp ( j 2 π m X f ) + n = 0 N exp ( j 2 π n X f ) m = 1 N exp ( j 2 π m X f ) ,
2 × 1 exp ( j 2 π N X f ) 1 exp ( i 2 π X f ) × 1 exp ( j 2 π N X f ) 1 exp ( j 2 π X f ) 1 n = 1 N exp ( j 2 π n X f ) m = 1 N exp ( j 2 π m X f )
1 cos ( 2 π N X f ) 1 cos ( 2 π X f ) = sin 2 ( π N X f ) sin 2 ( π X f ) .
f 2 sin 2 ( π X f ) × sin 2 ( π N X f ) f 2 k N ( f ) .
sin 2 ( π N X f ) f 2
lim N k N ( f ) = n = δ ( f n X ) .
lim n sin 2 ( π N X f ) f 2 = π 2 δ ( f ) ,
f 2 sin 2 ( π X f )
lim N k N ( f ) = ( π f ) 2 sin 2 ( π X f ) | f = 0 δ ( f ) = 1 X 2 δ ( f ) .

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