Abstract

The mutual intensity function for a partially coherent light is used to develop an expression for the output intensity distribution for a broadband optical information processor. The coherence requirement for smeared image deblurring and image subtraction is then determined using the intensity distribution. We also quantitatively show the dependence of coherence criteria on the spectral bandwidth, the source size, deblurring width, spatial frequency, and the separation of input object transparencies.

© 1982 Optical Society of America

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References

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  1. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  2. F. T. S. Yu, Introduction to Diffraction Information Processing and Holography (MIT Press, Cambridge, 1973).
  3. A. Vander Lugt, IEEE Proc. 62, 1300 (1974).
    [Crossref]
  4. G. L. Rogers, Opt. Laser Technol. 7, 153 (1975).
    [Crossref]
  5. M. A. Monahan, K. Bromley, R. P. Bocker, Proc. IEEE 65, 121 (1977).
    [Crossref]
  6. K. Bromley, Opt. Acta. 21, 35 (1974).
    [Crossref]
  7. G. L. Rogers, Noncoherent Optical Processing (Wiley, New York, 1977).
  8. S. Lowenthal, P. Chavel, in Proceedings, ICO Jerusalem 1976 Conference on Holography and Optical Processing, E. Marom, A. Friesem, E. Wiener-Avnear, Eds. (Pergamon, New York, 1977).
  9. A. Lohmann, Appl. Opt. 16, 261 (1977).
    [Crossref] [PubMed]
  10. E. N. Leith, J. Roth, Appl. Opt. 16, 2565 (1977).
    [Crossref] [PubMed]
  11. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970).
  12. F. T. S. Yu, Opt. Comm. 27, 23 (1978).
    [Crossref]
  13. F. T. S. Yu, Appl. Opt. 17, 3571 (1978).
    [Crossref] [PubMed]
  14. F. T. S. Yu, Proc. Soc. Photo-Opt. Instrum. Eng. 232, 9 (1980).
  15. S. L. Zhuang, T. H. Chao, F. T. S. Yu, Opt. Lett. 6, 102 (1981).
    [Crossref] [PubMed]
  16. K. Dutta, J. W. Goodman, J. Opt. Soc. Am. 67, 796 (1977).
    [Crossref]
  17. F. T. S. Yu, S. L. Zhuang, S. T. Wu, Appl. Phys. 827, 99 (1982).
  18. S. T. Wu, F. T. S. Yu, Appl. Opt. 20, 4082 (1981).
    [Crossref] [PubMed]

1982 (1)

F. T. S. Yu, S. L. Zhuang, S. T. Wu, Appl. Phys. 827, 99 (1982).

1981 (2)

1980 (1)

F. T. S. Yu, Proc. Soc. Photo-Opt. Instrum. Eng. 232, 9 (1980).

1978 (2)

1977 (4)

1975 (1)

G. L. Rogers, Opt. Laser Technol. 7, 153 (1975).
[Crossref]

1974 (2)

A. Vander Lugt, IEEE Proc. 62, 1300 (1974).
[Crossref]

K. Bromley, Opt. Acta. 21, 35 (1974).
[Crossref]

Bocker, R. P.

M. A. Monahan, K. Bromley, R. P. Bocker, Proc. IEEE 65, 121 (1977).
[Crossref]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970).

Bromley, K.

M. A. Monahan, K. Bromley, R. P. Bocker, Proc. IEEE 65, 121 (1977).
[Crossref]

K. Bromley, Opt. Acta. 21, 35 (1974).
[Crossref]

Chao, T. H.

Chavel, P.

S. Lowenthal, P. Chavel, in Proceedings, ICO Jerusalem 1976 Conference on Holography and Optical Processing, E. Marom, A. Friesem, E. Wiener-Avnear, Eds. (Pergamon, New York, 1977).

Dutta, K.

Goodman, J. W.

K. Dutta, J. W. Goodman, J. Opt. Soc. Am. 67, 796 (1977).
[Crossref]

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

Leith, E. N.

Lohmann, A.

Lowenthal, S.

S. Lowenthal, P. Chavel, in Proceedings, ICO Jerusalem 1976 Conference on Holography and Optical Processing, E. Marom, A. Friesem, E. Wiener-Avnear, Eds. (Pergamon, New York, 1977).

Monahan, M. A.

M. A. Monahan, K. Bromley, R. P. Bocker, Proc. IEEE 65, 121 (1977).
[Crossref]

Rogers, G. L.

G. L. Rogers, Opt. Laser Technol. 7, 153 (1975).
[Crossref]

G. L. Rogers, Noncoherent Optical Processing (Wiley, New York, 1977).

Roth, J.

Vander Lugt, A.

A. Vander Lugt, IEEE Proc. 62, 1300 (1974).
[Crossref]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970).

Wu, S. T.

F. T. S. Yu, S. L. Zhuang, S. T. Wu, Appl. Phys. 827, 99 (1982).

S. T. Wu, F. T. S. Yu, Appl. Opt. 20, 4082 (1981).
[Crossref] [PubMed]

Yu, F. T. S.

F. T. S. Yu, S. L. Zhuang, S. T. Wu, Appl. Phys. 827, 99 (1982).

S. T. Wu, F. T. S. Yu, Appl. Opt. 20, 4082 (1981).
[Crossref] [PubMed]

S. L. Zhuang, T. H. Chao, F. T. S. Yu, Opt. Lett. 6, 102 (1981).
[Crossref] [PubMed]

F. T. S. Yu, Proc. Soc. Photo-Opt. Instrum. Eng. 232, 9 (1980).

F. T. S. Yu, Appl. Opt. 17, 3571 (1978).
[Crossref] [PubMed]

F. T. S. Yu, Opt. Comm. 27, 23 (1978).
[Crossref]

F. T. S. Yu, Introduction to Diffraction Information Processing and Holography (MIT Press, Cambridge, 1973).

Zhuang, S. L.

F. T. S. Yu, S. L. Zhuang, S. T. Wu, Appl. Phys. 827, 99 (1982).

S. L. Zhuang, T. H. Chao, F. T. S. Yu, Opt. Lett. 6, 102 (1981).
[Crossref] [PubMed]

Appl. Opt. (4)

Appl. Phys. (1)

F. T. S. Yu, S. L. Zhuang, S. T. Wu, Appl. Phys. 827, 99 (1982).

IEEE Proc. (1)

A. Vander Lugt, IEEE Proc. 62, 1300 (1974).
[Crossref]

J. Opt. Soc. Am. (1)

Opt. Acta. (1)

K. Bromley, Opt. Acta. 21, 35 (1974).
[Crossref]

Opt. Comm. (1)

F. T. S. Yu, Opt. Comm. 27, 23 (1978).
[Crossref]

Opt. Laser Technol. (1)

G. L. Rogers, Opt. Laser Technol. 7, 153 (1975).
[Crossref]

Opt. Lett. (1)

Proc. IEEE (1)

M. A. Monahan, K. Bromley, R. P. Bocker, Proc. IEEE 65, 121 (1977).
[Crossref]

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

F. T. S. Yu, Proc. Soc. Photo-Opt. Instrum. Eng. 232, 9 (1980).

Other (5)

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

F. T. S. Yu, Introduction to Diffraction Information Processing and Holography (MIT Press, Cambridge, 1973).

G. L. Rogers, Noncoherent Optical Processing (Wiley, New York, 1977).

S. Lowenthal, P. Chavel, in Proceedings, ICO Jerusalem 1976 Conference on Holography and Optical Processing, E. Marom, A. Friesem, E. Wiener-Avnear, Eds. (Pergamon, New York, 1977).

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970).

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Figures (13)

Fig. 1
Fig. 1

Partially coherent optical processing system: P0, source plane; P1, input plane; P2, Fourier plane; P3, output plane; L, achromatic lenses.

Fig. 2
Fig. 2

Output intensity distribution of the deblurred image. Δλ, spectral bandwidth of the light source; W, smeared length.

Fig. 3
Fig. 3

Plots of the deblurring width ΔW as a function of the spectral bandwidth of the light source Δλ for various values of smeared light W.

Fig. 4
Fig. 4

Output intensity distribution of the deblurred image for various values of the source size Δs.

Fig. 5
Fig. 5

Plots of the deblurring width as a function of the source size for various values of the smear length W.

Fig. 6
Fig. 6

Partially coherent optical processing system for image subtraction: S1, light source; P0, encoded extended source plane; P1, input plane; P2, Fourier plane; P3, output plane; L, achromatic lenses.

Fig. 7
Fig. 7

Apparent modulation transfer function for a partially coherent image subtraction: (a) basic frequency; (b) second harmonic.

Fig. 8
Fig. 8

Relationship between the cutoff frequency ω0 and the spectral bandwidth of the light source Δλ for different minimum desirable contrasts Cm.

Fig. 9
Fig. 9

Relationship between the cutoff frequency and the spectral bandwidth of the light source Δλ for various values of separation H. 2H is the main separation between the input object transparencies.

Fig. 10
Fig. 10

Apparent modulation transfer function vs the separation H for different desirable contrasts Cm.

Fig. 11
Fig. 11

Relationship between MTF (ω), MTF (2ω), and the ratio of the slit width to spatial period d/D.

Tables (5)

Tables Icon

Table I Effect of Temporal Coherence Requirement

Tables Icon

Table II Effect of Spatial Coherence Requirement

Tables Icon

Table III Temporal Coherence Requirement for Different ωc and H.

Tables Icon

Table IV Source Size for Image Subtraction Under Different MTS and Separation H

Tables Icon

Table V Spatial Coherence Requirement for Various d/D

Equations (64)

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I ( u , υ ) = λ 0 Δ λ / 2 λ 0 + Δ λ / 2 γ ( x 0 , y 0 ) S ( λ ) C ( λ ) × | T ( x + x 0 , y + y 0 ; λ ) f ( x , y ) × exp [ j 2 π λ f ( x u + y υ ) ] dxdy | 2 d x 0 d y 0 d λ ,
I ( u , υ ) = λ l λ h | A ( u , υ ; λ ) | 2 d λ ,
A ( u , υ ; λ ) = T ( x , y ; λ ) f ( x , y ) exp [ j 2 π λ f ( x u + y υ ) ] dxdy
t ( u , υ ) = rect ( u / W ) ,
T ( x , y ; λ ) = [ t ( u , υ ) ] = sinc ( W λ f x ) ,
rect ( u / W ) = { 1 | u | W / 2 , 0 otherwise .
f ( x ; λ 0 ) = 1 T ( x ; λ 0 ) = 1 sinc ( W λ 0 f x ) ,
A ( u ; λ ) = T ( x ; λ ) f ( x ; λ 0 ) exp ( j 2 π λ f x u ) d x = rect ( u / W ) * 1 sinc ( W λ 0 f x ) exp ( j 2 π λ f x u ) d x ,
A ( u ; λ ) = { 4 λ 0 f π n = 1 2 m ( 1 ) n sin π n λ 0 λ sin 2 π n λ 0 λ W u sgn ( u ) , for | u | > W / 2 , 4 λ 0 f π n = 1 2 m ( 1 ) n cos π n λ 0 λ cos 2 π n λ 0 λ W u , for | u | W / 2 ,
I ( 1 ) ( u , Δ λ ) = 2 λ o 2 f 2 π 2 n n l = 1 8 ( 1 ) n + n + l ( 3 { λ a 3 cos [ a n n l ( u ) λ a ] λ b 3 cos [ a n n l ( u ) λ b ] } 6 [ a n n l ( u ) ] { λ a 2 sin [ a n n l ( u ) λ a ] λ b 2 sin [ a n n l ( u ) λ b ] } 6 [ a n n l ( u ) ] 2 { λ a 1 cos [ a n n l ( u ) λ a ] λ b 1 cos [ a n n l ( u ) λ b ] } 6 [ a n n l ( u ) ] 3 [ Si ( λ a ) Si ( λ b ) ] ) for | u | > W / 2 ,
I ( 2 ) ( u , Δ λ ) = 2 λ 0 2 f 2 π 2 n n l = 1 8 ( 1 ) n + n ( 6 [ a n n l ( u ) ] 3 [ Si ( λ a ) Si ( λ b ) ] + 6 [ a n n l ( u ) ] 2 { λ a 1 cos [ a n n l ( u ) λ a ] λ b 1 cos [ a n n l ( u ) λ b ] } + 6 [ a n n l ( u ) ] { λ a 2 sin [ a n n l ( u ) λ a ] λ b 1 sin [ a n n l ( u ) λ b ] } 3 { λ a 3 sin [ a n n l ( u ) λ a ] λ b 3 cos [ a n n l ( u ) λ b ] } ) for | u | W / 2 .
λ a = λ 0 / ( λ 0 Δ λ ) ,
λ b = λ 0 / ( λ 0 + Δ λ ) ,
Si ( x ) 0 x sin y y d y ,
a n n ( 1 ) ( u ) = π ( n n ) ( 1 2 u / W ) , a n n ( 3 ) ( u ) = π ( n n ) ( 1 + 2 u / W ) , a n n ( 5 ) ( u ) = π ( n + n ) ( 1 2 u / W ) , a n n ( 7 ) ( u ) = π ( n + n ) ( 1 + 2 u / W ) ,
a n n ( 2 ) ( u ) = π [ n ( 1 2 u / W ) n ( 1 + 2 u / W ) ] , a n n ( 4 ) ( u ) = π [ n ( 1 + 2 u / W ) n ( 1 2 u / W ) ] , a n n ( 6 ) ( u ) = π [ n ( 1 2 u / W ) + n ( 1 + 2 u / W ) ] , a n n ( 8 ) ( u ) = π [ n ( 1 + 2 u / W ) + n ( 1 2 u / W ) ] .
γ ( x 0 ) = rect ( x 0 / Δ s ) .
I ( u ) = rect ( x 0 / Δ s ) | A ( u , x 0 ; λ ) | 2 d x 0 ,
A ( u , x 0 ; λ ) = sinc [ W λ f ( x + x 0 ) ] exp ( j 2 π λ f x u ) d x * 1 sinc ( W λ f x ) exp ( j 2 π λ f x u ) d x ,
A ( u , x 0 ; λ ) = { 2 f λ 0 j n = 1 ( 1 ) n n exp ( j 2 π α u / W ) [ exp ( j 2 π n W u ) sinc ( α + n ) exp ( j 2 π n W u ) sinc ( α n ) ] , for | u | > W / 2 , 2 f λ 0 π n = 1 ( 1 ) n n exp ( j 2 π α u / W ) ( 1 α n { 1 exp [ j 2 π ( α n ) u / W ] cos π ( α n ) } ) + 1 α + n { 1 exp [ j 2 π ( α + n ) u / W ] cos π ( α + n ) } , for | u | W / 2 ,
I ( 1 ) ( u , Δ s ) = 16 f 3 λ 0 3 W π 2 n n { Φ n n ( 1 ) ( Δ s ) cos [ 2 π ( n n ) u / W ] + Φ n n ( 2 ) ( Δ s ) cos [ 2 π ( n + n ) u / W ] } .
S j ( m , Δ s ) = S i { 2 π [ α ̅ ( Δ s ) m ) ] } S i { 2 π [ α ̅ ( Δ s ) m ] } .
C j ( m , Δ s ) = C i [ 2 π | α ̅ ( Δ s ) + m ) | ] C i [ 2 π | α ̅ ( Δ s ) m | ] ,
Si ( x ) = 0 x sin β β d β , C i ( x ) = x cos β β d β , α ̅ ( Δ s ) = Δ S W / ( 2 λ f ) ,
Φ n n ( 1 ) ( Δ s ) = n n n n { | n [ n + α ̅ ( Δ s ) n α ̅ ( Δ s ) ] [ ( 1 ) n C j ( n , Δ s ) ( 1 ) n C j ( n , Δ s ) ] } , Φ n n ( 2 ) ( Δ s ) = n n n + n { | n [ n + α ̅ ( Δ s ) n α ̅ ( Δ s ) ] [ ( 1 ) n C j ( n , Δ s ) ( 1 ) n C j ( n , Δ s ) ] } .
I ( 2 ) ( u , Δ s ) = 16 λ 0 3 f 3 W π 2 n n { Φ n n ( 3 ) ( Δ s ) + Φ n n ( 4 ) ( Δ s ) sin [ 2 π ( n n ) u / W ] + Φ n n ( 5 ) ( Δ s ) cos [ 2 π ( n n ) u / W ] + Φ n n ( 6 ) ( Δ s ) cos [ 2 π ( n + n ) u / W ] } ,
Φ n n ( 3 ) ( Δ s ) = ( 1 ) n 2 n n 2 n 2 | n [ n α ̅ ( Δ s ) n + α ̅ ( Δ s ) ] + ( 1 ) n n n 2 n 2 [ C k + ( n , Δ s ) + C k ( n , Δ s ) ] , Φ n n ( 4 ) ( Δ s ) = n n 2 n 2 [ S k + ( n , Δ s ) + S k ( n , Δ s ) ] , Φ n n ( 5 ) ( Δ s ) = ( 1 ) n + 1 n n 2 n 2 [ C k + ( n , Δ s ) + C k ( n , Δ s ) ] + 2 n n { | n [ n + α ̅ ( Δ s ) n α ̅ ( Δ s ) ] + [ C j ( n , Δ s ) C j ( n , Δ s ) ] } , Φ n n ( 6 ) ( Δ s ) = 1 n + n { | n [ n α ̅ ( Δ s ) n + α ̅ ( Δ s ) ] [ C j ( n , Δ s ) + C j ( n , Δ s ) ] } ,
S k ± ( m , Δ s ) = S i { π ( 2 u / W ± 1 ) [ α ̅ ( Δ s ) + m ] } S i { π ( 2 u / W ± 1 ) [ α ̅ ( Δ s ) + m ] } , C k ± ( m , Δ s ) = C i [ π | 2 u / W ± 1 | | α ̅ ( Δ s ) + m | ] C i [ π | 2 u / W ± 1 | | α ̅ ( Δ s ) + m | ] .
γ ( x 0 , y 0 ) = rect ( x 0 / d ) * n = N N δ ( x 0 n D ) ,
J ( u 1 u 2 ; λ ) = γ ( x 0 , y 0 ) exp [ j 2 π λ f ( u 1 u 2 ) x 0 ] d x 0 ,
J ( u 1 u 2 ; λ ) = n = N N sinc d ( u 1 u 2 ) λ f exp [ j 2 π λ f ( u 1 u 2 ) n D ] .
t ( u ) = A ( u H ) + B ( u + H ) ,
J ( x 1 , x 2 ; λ ) = { J ( u 1 u 2 ; λ ) t ( u 1 ) t * ( u 2 ) } ( 1 + C sin 2 π ω 0 x 1 ) × ( 1 + C sin 2 π ω 0 x 2 ) .
I ( u ; λ ) = J ( x 1 , x 2 ; λ ) exp [ j 2 π λ f ( x 1 x 2 ) u ] d x 1 d x 2 ,
I ( u ; λ ) = N [ | A ( u H ) | 2 + | B ( u + H ) | 2 ] + N C 2 4 { 2 sin ( 2 d ω 0 ) × Re [ A ( u H + λ f ω 0 ) B ( u + H λ f ω 0 ) ] | A ( u H + λ f ω 0 ) | 2 | B ( u + H λ f ω 0 ) | 2 | A ( u H λ f ω 0 ) | 2 | B ( u + H + λ f ω 0 ) | 2 } ,
I ( 0 ) ( u ; λ ) = | A ( u H + λ f ω 0 ) | 2 2 sinc ( 2 d ω 0 ) × Re [ A ( u H + λ f ω 0 ) B ( u + H λ f ω 0 ) ] + | B ( u + H λ f ω 0 ) | 2 .
I ( 0 ) ( u ; λ ) | d = 0 = [ A ( u ) B ( u ) ] 2 .
I ( 0 ) ( u ) = Δ λ / 2 Δ λ / 2 { | A ( u + λ f ω 0 ) | 2 2 sinc ( 2 d ω 0 ) × Re [ A ( u + λ f ω 0 ) × B ( u λ f ω 0 ) ] + | B ( u λ f ω 0 ) | 2 } d λ ,
A ( u + λ f ω 0 ) A ( u ) + m = 1 1 m ! A ( m ) ( u ) ( λ f ω 0 ) m , B ( u λ f ω 0 ) B ( u ) + m = 1 1 m ! B ( m ) ( u ) ( λ f ω 0 ) m ,
A ( m ) ( u ) = d m A ( u ) / d u m , B ( m ) ( u ) = d m B ( u ) / d u m .
I ( 0 ) ( u ) = [ | A ( u ) | 2 2 sinc ( 2 d w 0 ) A ( u ) B ( u ) + | B ( u ) | 2 ] Δ λ + m = even m 0 1 2 m 1 ( m + 1 ) ! ( f w 0 ) m [ A ( m ) ( u ) sinc ( 2 d w 0 ) B ( m ) ( u ) ] × [ A ( u ) + B ( u ) ] ( Δ λ ) m + 1 + m m m m m + m = even 0 0 1 2 ( m + m 1 ) ( m + m + 1 ) m ! m ! × ( f w 0 ) m + m [ A ( m ) ( u ) A ( m ) ( u ) ( 1 ) m sinc ( 2 d w 0 ) A ( m ) ( u ) B ( m ) ( u ) + ( 1 ) m + m B ( m ) ( u ) B ( m ) ( u ) ] ( Δ λ ) m + m + 1 .
A ( u ) = 1 , B ( u ) = ½ [ 1 + C 0 cos ( 2 π ω u ) ] ,
Ī 0 ( u ) = [ A ( u ) B ( u ) ] 2 ,
Ī 0 ( u ) = 1 4 C 0 2 8 C 0 2 cos ( 2 π ω u ) C 0 2 8 cos 2 π ( 2 ω ) u .
Ī 0 ( u ) = [ 5 4 C 0 2 8 sinc ( 2 d ω 0 ) ] Δ λ + C 0 [ ½ sinc ( 2 d ω 0 ) ] × sinc ( f ω ω 0 Δ λ ) Δ λ cos ( 2 π ω u ) + C 0 2 8 sinc ( 2 f ω ω 0 Δ λ ) Δ λ cos 2 π ( 2 ω ) u .
MTF ( w ) = [ 1 2 sinc ( 2 d ω 0 ) ] ( 2 C 0 2 ) sinc ( f ω ω 0 Δ λ ) 10 C 0 2 8 sinc ( 2 d ω 0 ) ,
MTF ( 2 w ) = ( 2 C 0 2 ) sinc ( 2 f ω ω 0 Δ λ ) 10 C 0 2 8 sinc ( 2 d ω 0 ) .
MTF ( ω ) = sinc ( f ω ω 0 Δ λ ) ,
MTF ( 2 ω ) = sinc ( 2 f ω ω 0 Δ λ ) .
MTF 1 = ( 2 C 0 2 ) [ 1 2 sinc ( 2 d ω 0 ) ] 10 C 0 2 8 sinc ( 2 d ω 0 ) ,
MTF 2 = 2 C 0 2 10 C 0 2 8 sinc ( 2 d ω 0 ) .
ω 0 = H / ( λ f 0 ) .
Δ W W
1 sinc ( W x λ 0 f ) exp ( j 2 π λ f x u ) d x = { 4 f λ 0 W n = 1 ( 1 ) n × sin 2 π n λ 0 λ W u } sgn ( u ) ,
λ f l π f x sin ( π lfx ) exp ( j 2 π f x u ) dfx .
f x = n / l , n = 1 , 2 , . . . .
{ n l n l + + R } π f x sin ( π lfx ) exp ( j 2 π f x u ) dfx = 0 ,
z = R exp ( j θ ) , d z = j θ R exp ( j θ ) d θ .
R π f x sin ( π lfx ) exp ( j 2 π f x u ) dfx = π z sin ( π l z ) exp ( j 2 π z u ) d z = R 2 π R 2 exp ( j 2 π R u cos θ ) exp ( 2 π R sin θ u ) θ exp ( 2 j θ ) exp [ j π l R ( cos θ + j sin θ ) ] exp [ j π l R ( cos θ + j sin θ ) ] d θ ,
lim R R π f x sin ( π lfx ) exp ( j 2 π f x u ) dfx = 0 .
π f x sin ( π lfx ) exp ( j 2 π f x u ) dfx = 2 π j n = R n ,
R n = lim z l n ( z n l ) π z exp ( j 2 π z u ) sin ( π l z ) = ( 1 ) n n exp ( j 2 n π u / l ) π l 2 .
π f x sin ( π lfx ) exp ( j 2 π f x u ) dfx = 4 l 2 n = 1 ( 1 ) n n sin 2 π n l u , for u > 0 .
π f x sin ( π lfx ) exp ( j 2 π f x u ) dfx = 4 l 2 n = 1 ( 1 ) n n sin 2 π n l u , for u < 0 .

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