Abstract

A concise introduction to the concept of fractional Fourier transforms is followed by a discussion of their relation to chirp and wavelet transforms. The notion of fractional Fourier domains is developed in conjunction with the Wigner distribution of a signal. Convolution, filtering, and multiplexing of signals in fractional domains are discussed, revealing that under certain conditions one can improve on the special cases of these operations in the conventional space and frequency domains. Because of the ease of performing the fractional Fourier transform optically, these operations are relevant for optical information processing.

© 1994 Optical Society of America

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References

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  1. H. M. Ozaktas, D. Mendlovic, “Fourier transforms of fractional order and their optical interpretation,” Opt. Commun. 101, 163–169 (1993).
    [CrossRef]
  2. D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transformations and their optical implementation: I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
    [CrossRef]
  3. H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transformations and their optical implementation: II,” J. Opt. Soc. Am. A (to be published).
  4. A. C. McBride, F. H. Kerr, “On Namias’s fractional Fourier transform,” IMA J. Appl. Math. 39, 159–175 (1987).
    [CrossRef]
  5. V. Namias, “The fractional Fourier transform and its application in quantum mechanics,”J. Inst. Math. Its Appl. 25, 241–265 (1980).
    [CrossRef]
  6. D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Fourier transforms of fractional order and their optical interpretation,” in Optical Computing, Vol. 7 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 127–130.
  7. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,”J. Opt. Soc. Am. 10, 2181–2186 (1993).
    [CrossRef]
  8. D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “The effect of propagation in graded index media on the Wigner distribution function and the equivalence of two definitions of the fractional Fourier transform,” Appl. Opt. (to be published).
  9. N. Wiener, The Fourier Integral and Certain of Its Applications (Cambridge U. Press, Cambridge, 1933).
  10. G. Cincotti, F. Gori, M. Santarsiero, “Generalized self-Fourier functions,”J. Phys. A 25, 1191–1194 (1992).
    [CrossRef]
  11. H. O. Bartelt, K.-H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
    [CrossRef]
  12. T. A. C. M. Claasen, W. F. G. Mecklenbraucker, “The Wigner distribution—a tool for time–frequency signal analysis; part 1: continuous-time signals,” Philips J. Res. 35, 217–250 (1980).
  13. T. A. C. M. Claasen, W. F. G. Mecklenbraucker, “The Wigner distribution—a tool for time–frequency signal analysis; part 2: discrete-time signals,” Philips J. Res. 35, 276–300 (1980).
  14. A. W. Lohmann, B. H. Soffer, “Relationship between two transforms: Radon–Wigner and fractional Fourier,” in Annual Meeting, Vol. 16 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 109.
  15. A. Rosenfeld, A. C. Kak, Digital Picture Processing, 2nd ed. (Academic, San Diego, Calif., 1982), Vol. 1.
  16. B. W. Dickinson, K. Steiglitz, “Eigenvectors and functions of the discrete Fourier transform,”IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 25–31 (1982).
    [CrossRef]
  17. L. Onural, “Diffraction from a wavelet point of view,” Opt. Lett. 18, 846–848 (1993).
    [CrossRef] [PubMed]
  18. A. V. Oppenheim, R. W. Shafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).
  19. Our somewhat artificial examples have been chosen to illustrate the essential concepts in the simplest possible terms. In particular, it should be noted that real functions always have Wigner distributions exhibiting even symmetry with respect to ν.
  20. D. Mendlovic, H. M. Ozaktas, “Optical-coordinate transformation methods and optical interconnection architectures,” Appl. Opt. 32, 5119–5124 (1993).
    [CrossRef] [PubMed]
  21. M. J. Caola, “Self-Fourier functions,”J. Phys. A 24, 1143–1144 (1991).
    [CrossRef]
  22. A. W. Lohmann, D. Mendlovic, “Self-Fourier objects and other self-transform objects,” J. Opt. Soc. Am. A 9, 2009–2012 (1992).
    [CrossRef]
  23. A. W. Lohmann, University of Erlangen-Nürnberg, Staudtstrasse 7, Erlangen, Germany (personal communication).
  24. D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Self Fourier functions and fractional Fourier transforms,” Opt. Commun. (to be published).
  25. A. Yariv, Optical Electronics, 3rd ed. (Holt, New York, 1985).

1993 (5)

1992 (2)

A. W. Lohmann, D. Mendlovic, “Self-Fourier objects and other self-transform objects,” J. Opt. Soc. Am. A 9, 2009–2012 (1992).
[CrossRef]

G. Cincotti, F. Gori, M. Santarsiero, “Generalized self-Fourier functions,”J. Phys. A 25, 1191–1194 (1992).
[CrossRef]

1991 (1)

M. J. Caola, “Self-Fourier functions,”J. Phys. A 24, 1143–1144 (1991).
[CrossRef]

1987 (1)

A. C. McBride, F. H. Kerr, “On Namias’s fractional Fourier transform,” IMA J. Appl. Math. 39, 159–175 (1987).
[CrossRef]

1982 (1)

B. W. Dickinson, K. Steiglitz, “Eigenvectors and functions of the discrete Fourier transform,”IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 25–31 (1982).
[CrossRef]

1980 (4)

V. Namias, “The fractional Fourier transform and its application in quantum mechanics,”J. Inst. Math. Its Appl. 25, 241–265 (1980).
[CrossRef]

H. O. Bartelt, K.-H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

T. A. C. M. Claasen, W. F. G. Mecklenbraucker, “The Wigner distribution—a tool for time–frequency signal analysis; part 1: continuous-time signals,” Philips J. Res. 35, 217–250 (1980).

T. A. C. M. Claasen, W. F. G. Mecklenbraucker, “The Wigner distribution—a tool for time–frequency signal analysis; part 2: discrete-time signals,” Philips J. Res. 35, 276–300 (1980).

Bartelt, H. O.

H. O. Bartelt, K.-H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

Brenner, K.-H.

H. O. Bartelt, K.-H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

Caola, M. J.

M. J. Caola, “Self-Fourier functions,”J. Phys. A 24, 1143–1144 (1991).
[CrossRef]

Cincotti, G.

G. Cincotti, F. Gori, M. Santarsiero, “Generalized self-Fourier functions,”J. Phys. A 25, 1191–1194 (1992).
[CrossRef]

Claasen, T. A. C. M.

T. A. C. M. Claasen, W. F. G. Mecklenbraucker, “The Wigner distribution—a tool for time–frequency signal analysis; part 2: discrete-time signals,” Philips J. Res. 35, 276–300 (1980).

T. A. C. M. Claasen, W. F. G. Mecklenbraucker, “The Wigner distribution—a tool for time–frequency signal analysis; part 1: continuous-time signals,” Philips J. Res. 35, 217–250 (1980).

Dickinson, B. W.

B. W. Dickinson, K. Steiglitz, “Eigenvectors and functions of the discrete Fourier transform,”IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 25–31 (1982).
[CrossRef]

Gori, F.

G. Cincotti, F. Gori, M. Santarsiero, “Generalized self-Fourier functions,”J. Phys. A 25, 1191–1194 (1992).
[CrossRef]

Kak, A. C.

A. Rosenfeld, A. C. Kak, Digital Picture Processing, 2nd ed. (Academic, San Diego, Calif., 1982), Vol. 1.

Kerr, F. H.

A. C. McBride, F. H. Kerr, “On Namias’s fractional Fourier transform,” IMA J. Appl. Math. 39, 159–175 (1987).
[CrossRef]

Lohmann, A. W.

A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,”J. Opt. Soc. Am. 10, 2181–2186 (1993).
[CrossRef]

A. W. Lohmann, D. Mendlovic, “Self-Fourier objects and other self-transform objects,” J. Opt. Soc. Am. A 9, 2009–2012 (1992).
[CrossRef]

H. O. Bartelt, K.-H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

A. W. Lohmann, B. H. Soffer, “Relationship between two transforms: Radon–Wigner and fractional Fourier,” in Annual Meeting, Vol. 16 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 109.

D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Fourier transforms of fractional order and their optical interpretation,” in Optical Computing, Vol. 7 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 127–130.

D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Self Fourier functions and fractional Fourier transforms,” Opt. Commun. (to be published).

A. W. Lohmann, University of Erlangen-Nürnberg, Staudtstrasse 7, Erlangen, Germany (personal communication).

D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “The effect of propagation in graded index media on the Wigner distribution function and the equivalence of two definitions of the fractional Fourier transform,” Appl. Opt. (to be published).

McBride, A. C.

A. C. McBride, F. H. Kerr, “On Namias’s fractional Fourier transform,” IMA J. Appl. Math. 39, 159–175 (1987).
[CrossRef]

Mecklenbraucker, W. F. G.

T. A. C. M. Claasen, W. F. G. Mecklenbraucker, “The Wigner distribution—a tool for time–frequency signal analysis; part 2: discrete-time signals,” Philips J. Res. 35, 276–300 (1980).

T. A. C. M. Claasen, W. F. G. Mecklenbraucker, “The Wigner distribution—a tool for time–frequency signal analysis; part 1: continuous-time signals,” Philips J. Res. 35, 217–250 (1980).

Mendlovic, D.

H. M. Ozaktas, D. Mendlovic, “Fourier transforms of fractional order and their optical interpretation,” Opt. Commun. 101, 163–169 (1993).
[CrossRef]

D. Mendlovic, H. M. Ozaktas, “Optical-coordinate transformation methods and optical interconnection architectures,” Appl. Opt. 32, 5119–5124 (1993).
[CrossRef] [PubMed]

D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transformations and their optical implementation: I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
[CrossRef]

A. W. Lohmann, D. Mendlovic, “Self-Fourier objects and other self-transform objects,” J. Opt. Soc. Am. A 9, 2009–2012 (1992).
[CrossRef]

D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Self Fourier functions and fractional Fourier transforms,” Opt. Commun. (to be published).

H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transformations and their optical implementation: II,” J. Opt. Soc. Am. A (to be published).

D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “The effect of propagation in graded index media on the Wigner distribution function and the equivalence of two definitions of the fractional Fourier transform,” Appl. Opt. (to be published).

D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Fourier transforms of fractional order and their optical interpretation,” in Optical Computing, Vol. 7 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 127–130.

Namias, V.

V. Namias, “The fractional Fourier transform and its application in quantum mechanics,”J. Inst. Math. Its Appl. 25, 241–265 (1980).
[CrossRef]

Onural, L.

Oppenheim, A. V.

A. V. Oppenheim, R. W. Shafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Ozaktas, H. M.

H. M. Ozaktas, D. Mendlovic, “Fourier transforms of fractional order and their optical interpretation,” Opt. Commun. 101, 163–169 (1993).
[CrossRef]

D. Mendlovic, H. M. Ozaktas, “Optical-coordinate transformation methods and optical interconnection architectures,” Appl. Opt. 32, 5119–5124 (1993).
[CrossRef] [PubMed]

D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transformations and their optical implementation: I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
[CrossRef]

D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Fourier transforms of fractional order and their optical interpretation,” in Optical Computing, Vol. 7 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 127–130.

H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transformations and their optical implementation: II,” J. Opt. Soc. Am. A (to be published).

D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Self Fourier functions and fractional Fourier transforms,” Opt. Commun. (to be published).

D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “The effect of propagation in graded index media on the Wigner distribution function and the equivalence of two definitions of the fractional Fourier transform,” Appl. Opt. (to be published).

Rosenfeld, A.

A. Rosenfeld, A. C. Kak, Digital Picture Processing, 2nd ed. (Academic, San Diego, Calif., 1982), Vol. 1.

Santarsiero, M.

G. Cincotti, F. Gori, M. Santarsiero, “Generalized self-Fourier functions,”J. Phys. A 25, 1191–1194 (1992).
[CrossRef]

Shafer, R. W.

A. V. Oppenheim, R. W. Shafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Soffer, B. H.

A. W. Lohmann, B. H. Soffer, “Relationship between two transforms: Radon–Wigner and fractional Fourier,” in Annual Meeting, Vol. 16 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 109.

Steiglitz, K.

B. W. Dickinson, K. Steiglitz, “Eigenvectors and functions of the discrete Fourier transform,”IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 25–31 (1982).
[CrossRef]

Wiener, N.

N. Wiener, The Fourier Integral and Certain of Its Applications (Cambridge U. Press, Cambridge, 1933).

Yariv, A.

A. Yariv, Optical Electronics, 3rd ed. (Holt, New York, 1985).

Appl. Opt. (1)

IEEE Trans. Acoust. Speech Signal Process. (1)

B. W. Dickinson, K. Steiglitz, “Eigenvectors and functions of the discrete Fourier transform,”IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 25–31 (1982).
[CrossRef]

IMA J. Appl. Math. (1)

A. C. McBride, F. H. Kerr, “On Namias’s fractional Fourier transform,” IMA J. Appl. Math. 39, 159–175 (1987).
[CrossRef]

J. Inst. Math. Its Appl. (1)

V. Namias, “The fractional Fourier transform and its application in quantum mechanics,”J. Inst. Math. Its Appl. 25, 241–265 (1980).
[CrossRef]

J. Opt. Soc. Am. (1)

A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,”J. Opt. Soc. Am. 10, 2181–2186 (1993).
[CrossRef]

J. Opt. Soc. Am. A (2)

J. Phys. A (2)

G. Cincotti, F. Gori, M. Santarsiero, “Generalized self-Fourier functions,”J. Phys. A 25, 1191–1194 (1992).
[CrossRef]

M. J. Caola, “Self-Fourier functions,”J. Phys. A 24, 1143–1144 (1991).
[CrossRef]

Opt. Commun. (2)

H. M. Ozaktas, D. Mendlovic, “Fourier transforms of fractional order and their optical interpretation,” Opt. Commun. 101, 163–169 (1993).
[CrossRef]

H. O. Bartelt, K.-H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

Opt. Lett. (1)

Philips J. Res. (2)

T. A. C. M. Claasen, W. F. G. Mecklenbraucker, “The Wigner distribution—a tool for time–frequency signal analysis; part 1: continuous-time signals,” Philips J. Res. 35, 217–250 (1980).

T. A. C. M. Claasen, W. F. G. Mecklenbraucker, “The Wigner distribution—a tool for time–frequency signal analysis; part 2: discrete-time signals,” Philips J. Res. 35, 276–300 (1980).

Other (11)

A. W. Lohmann, B. H. Soffer, “Relationship between two transforms: Radon–Wigner and fractional Fourier,” in Annual Meeting, Vol. 16 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 109.

A. Rosenfeld, A. C. Kak, Digital Picture Processing, 2nd ed. (Academic, San Diego, Calif., 1982), Vol. 1.

D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “The effect of propagation in graded index media on the Wigner distribution function and the equivalence of two definitions of the fractional Fourier transform,” Appl. Opt. (to be published).

N. Wiener, The Fourier Integral and Certain of Its Applications (Cambridge U. Press, Cambridge, 1933).

D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Fourier transforms of fractional order and their optical interpretation,” in Optical Computing, Vol. 7 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 127–130.

A. V. Oppenheim, R. W. Shafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Our somewhat artificial examples have been chosen to illustrate the essential concepts in the simplest possible terms. In particular, it should be noted that real functions always have Wigner distributions exhibiting even symmetry with respect to ν.

A. W. Lohmann, University of Erlangen-Nürnberg, Staudtstrasse 7, Erlangen, Germany (personal communication).

D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Self Fourier functions and fractional Fourier transforms,” Opt. Commun. (to be published).

A. Yariv, Optical Electronics, 3rd ed. (Holt, New York, 1985).

H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transformations and their optical implementation: II,” J. Opt. Soc. Am. A (to be published).

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Figures (10)

Fig. 1
Fig. 1

Wigner distribution of a chirp function.

Fig. 2
Fig. 2

(a) Wigner distribution of a signal, (b) compaction in the ath domain.

Fig. 3
Fig. 3

Noise separation in the ath domain.

Fig. 4
Fig. 4

Noise separation by repeated filtering in several fractional domains.

Fig. 5
Fig. 5

Limits to noise separation imposed by the uncertainty relation.

Fig. 6
Fig. 6

Multiplexing in the frequency domain.

Fig. 7
Fig. 7

Multiplexing in the space domain.

Fig. 8
Fig. 8

Multiplexing in both space and frequency.

Fig. 9
Fig. 9

Inefficient multiplexing of a signal with an oblique Wigner distribution.

Fig. 10
Fig. 10

Efficient multiplexing of a signal with an oblique Wigner distribution.

Equations (89)

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F ( ν ) = - f ( x ) exp ( - 2 π i ν x ) d x ,
f ( x ) = - F ( ν ) exp ( 2 π i ν x ) d ν .
f ( x ) + 4 π 2 [ ( 2 n + 1 ) / 2 π - x 2 ] f ( x ) = 0.
F ( ν ) + 4 π 2 [ ( 2 n + 1 ) / 2 π - ν 2 ] F ( ν ) = 0 ,
Ψ ˜ n ( x ) = 2 1 / 4 2 n n ! H n ( 2 π x ) exp ( - π x 2 )
F [ Ψ ˜ n ( x ) ] = λ n Ψ ˜ n ( x ) ,
f ( x ) = n = 0 A ˜ n Ψ ˜ n ( x ) ,
A ˜ n = - Ψ ˜ n ( x ) f ( x ) d x ,
F [ f ( x ) ] = n = 0 A ˜ n i - n Ψ ˜ n ( x ) .
F a [ Ψ ˜ n ( x ) ] = λ n a Ψ ˜ n ( x ) = i - a n Ψ ˜ n ( x ) ;
{ F a [ f ( x ) ] } ( x ) = n = 0 A ˜ n i - a n Ψ ˜ n ( x ) .
{ F a [ f ( x ) ] } ( x ) = - B a ( x , x ) f ( x ) d x ,
B a ( x , x ) = n = 0 λ n a Ψ ˜ n ( x ) Ψ ˜ n ( x ) = 2 1 / 2 exp [ - π ( x 2 + x 2 ) ] × n = 0 i - a n 2 n n ! H n ( 2 π x ) H n ( 2 π x ) .
B a ( x , x ) = exp [ - i ( π ϕ ^ / 4 - ϕ / 2 ) ] sin ϕ 1 / 2 × exp [ i π ( x 2 cot ϕ - 2 x x csc ϕ + x 2 cot ϕ ) ] ,
W [ f ( x ) ] = W ( x , ν ) = f ( x + x / 2 ) f * ( x - x / 2 ) exp ( - 2 π i ν x ) d x .
W [ f a ] = R - ϕ W [ f 0 ] .
W [ f a 2 ] = R ( - ϕ 2 + ϕ 1 ) W [ f a 1 ] .
R ϕ [ W [ f ] ] = F a [ f ] 2 ,
R ϕ [ W ( x 0 , x 1 ) ] = R - ϕ [ W ( x 0 , x 1 ) ] d x 1 .
F R ϕ W [ f ] = S ϕ F 2 D W [ f ] ,
A ( μ , y ) = f ( x + y / 2 ) f * ( x - y / 2 ) exp ( - 2 π i μ x ) d x
F F a [ f ] 2 = S ϕ [ A ( μ , - y ) ] ,
G l = k = 0 N - 1 g k w k l ,
g k = l = 0 N - 1 G l w - k l ,
w k l = 1 N exp ( - 2 π i k l / N ) .
G = Fg ,
g = F - 1 G ,
l = 0 N - 1 w i j w - l k = δ k j ,
F a g = λ g .
λ n = exp ( - i n π / 2 ) = i - n ,
F g n = λ n g n ,
F j g n = λ n j g n .
F a = j = 0 3 F j α j ( a ) ,
α j ( a ) = 1 4 k = 1 4 exp [ i k ( a - j ) π / 2 ]
F a g n = λ n a g n
g = n = 1 N A n g n ,
A n = g n T g ,
F a g = n = 1 N A n λ n a g n
f ( x ) = exp ( 2 π i ν c x ) ,             W ( x , ν ) = δ ( ν - ν c ) ,
f ( x ) = δ ( x - x c ) ,             W ( x , ν ) = δ ( x - x c ) ,
f ( x ) = exp [ 2 π i ( b 2 x 2 / 2 + b 1 x + b 0 ) ] ,             W ( x , ν ) = δ ( b 2 x + b 1 - ν ) .
f a ( x a ) = exp [ - i ( π ϕ ^ / 4 - ϕ / 2 ) ] sin ϕ 1 / 2 × exp [ i π ( x a 2 cot ϕ - 2 x a x 0 c csc ϕ + x 0 c 2 cot ϕ ) ] .
W ( x 0 , x 1 ) = sin ϕ - 1 δ ( x 0 cot ϕ - x 0 c csc ϕ - x 1 ) = δ ( x 0 cos ϕ - x 0 c - x 1 sin ϕ ) ,
f ( x ) = f 0 ( x 0 ) = f 0 ( x ) δ ( x 0 - x ) d x .
f a ( x a ) = f 0 ( x ) F a [ δ ( x 0 - x ) ] d x .
f a ( x a ) = f a ( x ) δ ( x a - x ) d x ,
f a ( x a ) = F a ( ν a ) exp ( 2 π i ν a x a ) d ν a
f a ( x a ) = f a ( x a ) B a - a ( x a , x a ) d x a .
g ( y ) = f a ( y sec ϕ ) = C ( ϕ ) exp ( - i π y 2 sin 2 ϕ ) × exp [ i π ( y - x tan 1 / 2 ϕ ) 2 ] f ( x ) d x .
g a ( x a ) = F a [ g ] = F a [ f ] * F a [ h ] = f a ( x a ) * h a ( x a ) ,
W g ( x , ν ) = W f ( x , ν ) W h ( x - x , ν ) d x ;
W g ( x , ν ) = W f ( x , ν ) W h ( x , ν - ν ) d ν .
g a ( x a ) = F a [ g ] = F a [ f ] × F a [ h ] = f a ( x a ) h a ( x a ) ,
F a + 1 [ f × a h ] = F a + 1 [ f ] * F a + 1 [ h ] = F a + 1 [ f * a + 1 h ] ,
F a - 1 [ f × a h ] = F a - 1 [ f ] * F a - 1 [ h ] = F a - 1 [ f * a - 1 h ] .
F a + 1 [ f * a h ] = F a + 1 [ f ] × F a + 1 [ h ] = F a + 1 [ f × a + 1 h ] ,
F a - 1 [ f * a h ] = F a - 1 [ f ] × F a - 1 [ h ] = F a - 1 [ f × a - 1 h ] .
f × a h = f * a + 1 h = f * a - 1 h ,
f * a h = f × a + 1 h = f × a - 1 h .
F a [ f * a p h ] = F a [ f ] × F a [ h ] .
rect ( x a - x a c Δ x a )
g a ( x a ) = rect ( x a - x a c Δ x a ) f a ( x a ) .
W rect ( x a , ν a ) = rect ( x a + x / 2 - x a c Δ x a ) × rect ( x a - x / 2 - x a c Δ x a ) exp ( - 2 π i x ν a ) d x , = rect { x 2 Δ x a [ 1 - 2 ( x a - x a c ) / Δ x a ] } × exp ( - 2 π i x ν a ) d x , = 2 Δ x a [ 1 - | 2 ( x a - x a c ) Δ x a | ] × sinc { 2 Δ x a [ 1 - | 2 ( x a - x a c ) Δ x a | ] ν a }
F a g = F a h * F a f ,
F a + 1 g = Λ [ F a + 1 h ] F a + 1 f ,
g = F - ( a + 1 ) Λ [ F a + 1 h ] F a + 1 f .
T = F - ( a + 1 ) Λ [ F a + 1 h ] F a + 1 = F - ( a + 1 ) Λ [ Fh a ] F a + 1 ,
g = F 3 Λ [ h 3 ] F 1 / 2 Λ [ h 2 ] F 1 / 2 Λ [ h 1 ] f ,
g = Tf ,
T = F a M Λ [ h M ] Λ [ h 3 ] F a 2 Λ [ h 2 ] F a 1 Λ [ h 1 ] F a 0 .
T = [ h M ] Λ [ h 3 ] [ h 2 ] [ h 1 ] F .
[ f 0 ( x 0 ) * δ ( x 0 - x 0 c ) ] exp ( 2 π i ν 0 c x 0 ) = f ( x 0 - x 0 c ) exp ( 2 π i ν 0 c x 0 ) .
f a ( x a ) * δ ( x a - x a c ) = f a ( x a - x a c ) ,
4 f j ( x ) = f ( x ) + i j F ( x ) + i 2 j f ( - x ) + i 3 j F ( - x )
f ( x ) = j = 0 3 f j ( x ) ,
n 2 ( r ) = n 1 2 [ 1 - ( n 2 / n 1 ) r 2 ] ,
g ( x ) = n = 0 A ˜ n λ n ( Δ z ) Ψ ˜ n ( x ) ,
β n k - ( n 2 / n 1 ) 1 / 2 ( n + 1 / 2 )
λ n ( a L ) = exp ( i k a L - i a π / 4 ) exp ( - i a n π / 2 ) = C i - a n = C ( i - n ) a = C λ n a ,
exp [ i ( k a L - a π / 4 ) ] f a ( y / s ) ,
B a ( y , y ) = exp [ i ( k a L - a π / 4 ) ] s - 1 B a ( y / s , y / s ) .
f a ( x ) = ( F a [ f ] ) ( x ) = B a ( x , v ) f ( v ) d v .
exp { - 2 π i [ ν - x cot ϕ + ( v + v ) / 2 sin ϕ ] x } d x
exp [ - 4 π i ( ν sin ϕ - x cos ϕ ) x / sin ϕ ] × exp [ - 4 π i ( ν sin ϕ - x cos ϕ ) 2 ] cot ϕ × exp [ - 4 π i v ( x sin ϕ + ν cos ϕ ) ] × f ( v ) f * [ - v - 2 ( ν sin ϕ - x cos ϕ ) ] d v .
x = x cos ϕ - ν sin ϕ ,
ν = x sin ϕ + ν cos ϕ ,
F a g n = j = 0 3 k = 1 4 1 4 exp [ i k ( a - j ) π / 2 ] F j g n .
F a g n = k = 1 4 exp ( i k a π / 2 ) 1 4 j = 0 3 exp [ i j ( - n - k ) π / 2 ] g n .
F a g n = k = 1 4 exp ( i k a π / 2 ) δ - n , k g n = exp ( - i n a π ) / 2 g n = λ n a g n ,

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