Abstract

The importance of the amplitude and phase in the fractional Fourier transform (FT) domain is analyzed on the basis of the rectangular signal and the real-world image. The quality of signal restoration from only the amplitude or from only the phase of its fractional FT by applying the inverse fractional FT is considered. It is shown that the signal reconstructed from the amplitude of the fractional FT usually reveals the main features of the original signal only for relatively low fractional orders. On the basis of phase information in the fractional FT domains, significant details of the signal can be obtained for nearly all fractional orders.

© 2003 Optical Society of America

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References

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    [CrossRef] [PubMed]
  5. D. Mendlovic, Y. Bitran, R. G. Dorsch, A. Lohmann, “Optical fractional correlation: experimental results,” J. Opt. Soc. Am. A 12, 1665–1670 (1995).
    [CrossRef]
  6. J. Garcı́a, D. Mendlovic, Z. Zalevsky, A. Lohmann, “Space variant simultaneous detection of several objects using multiple anamorphic fractional Fourier transform filters,” Appl. Opt. 35, 3945–3952 (1996).
    [CrossRef]
  7. J. Garcia, R. G. Dorsch, A. W. Lohmann, C. Ferreira, Z. Zalevsky, “Flexible optical implementation of fractional Fourier processors: application to correlation and filtering,” Opt. Commun. 133, 393–400 (1997).
    [CrossRef]
  8. O. Akay, G. F. Boudreaux-Bartels, “Fractional convolution and correlation via operator methods and an application to detection of linear FM signals,” IEEE Trans. Signal Process. 49, 979–993 (2001).
    [CrossRef]
  9. T. Alieva, M. L. Calvo, “Generalized fractional convolution,” in Perspectives in Modern Optics and Optical Instrumentation (Anita Publications, New Delhi, 2002), pp. 282–292.
  10. H. M. Ozaktas, Z. Zalevsky, M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, New York, 2001).
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    [CrossRef]
  13. L. B. Lesem, P. M. Hirsch, J. A. Jordan, “The kinoform: a new wavefront reconstruction device,” IBM J. Res. Dev. 13, 150–155 (1969).
    [CrossRef]
  14. A. W. Lohmann, D. Mendlovic, G. Shabtay, “Significance of phase and amplitude in the Fourier domain,” J. Opt. Soc. Am. A 14, 2901–2904 (1997).
    [CrossRef]
  15. D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation: I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
    [CrossRef]
  16. H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transformations and their optical implementation: II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993).
    [CrossRef]
  17. L. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42, 3084–3091 (1994).
    [CrossRef]
  18. T. Alieva, M. L. Calvo, M. J. Bastiaans, “Power filtering of n-order in the fractional Fourier domain,” J. Phys. A 35, 7779–7785 (2002).
    [CrossRef]
  19. M. Abramovich, I. A. Segun, Handbook of Mathematical Functions (Dover, New York, 1965).
  20. H. M. Ozaktas, O. Arikan, M. A. Kutay, G. Bozdagi, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150 (1996).
    [CrossRef]

2002 (1)

T. Alieva, M. L. Calvo, M. J. Bastiaans, “Power filtering of n-order in the fractional Fourier domain,” J. Phys. A 35, 7779–7785 (2002).
[CrossRef]

2001 (1)

O. Akay, G. F. Boudreaux-Bartels, “Fractional convolution and correlation via operator methods and an application to detection of linear FM signals,” IEEE Trans. Signal Process. 49, 979–993 (2001).
[CrossRef]

1997 (2)

A. W. Lohmann, D. Mendlovic, G. Shabtay, “Significance of phase and amplitude in the Fourier domain,” J. Opt. Soc. Am. A 14, 2901–2904 (1997).
[CrossRef]

J. Garcia, R. G. Dorsch, A. W. Lohmann, C. Ferreira, Z. Zalevsky, “Flexible optical implementation of fractional Fourier processors: application to correlation and filtering,” Opt. Commun. 133, 393–400 (1997).
[CrossRef]

1996 (2)

H. M. Ozaktas, O. Arikan, M. A. Kutay, G. Bozdagi, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150 (1996).
[CrossRef]

J. Garcı́a, D. Mendlovic, Z. Zalevsky, A. Lohmann, “Space variant simultaneous detection of several objects using multiple anamorphic fractional Fourier transform filters,” Appl. Opt. 35, 3945–3952 (1996).
[CrossRef]

1995 (2)

1994 (1)

L. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42, 3084–3091 (1994).
[CrossRef]

1993 (2)

1984 (1)

1981 (1)

A. V. Oppenheim, J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69, 529–541 (1981).
[CrossRef]

1969 (1)

L. B. Lesem, P. M. Hirsch, J. A. Jordan, “The kinoform: a new wavefront reconstruction device,” IBM J. Res. Dev. 13, 150–155 (1969).
[CrossRef]

1965 (1)

Abramovich, M.

M. Abramovich, I. A. Segun, Handbook of Mathematical Functions (Dover, New York, 1965).

Akay, O.

O. Akay, G. F. Boudreaux-Bartels, “Fractional convolution and correlation via operator methods and an application to detection of linear FM signals,” IEEE Trans. Signal Process. 49, 979–993 (2001).
[CrossRef]

Alieva, T.

T. Alieva, M. L. Calvo, M. J. Bastiaans, “Power filtering of n-order in the fractional Fourier domain,” J. Phys. A 35, 7779–7785 (2002).
[CrossRef]

T. Alieva, M. L. Calvo, “Generalized fractional convolution,” in Perspectives in Modern Optics and Optical Instrumentation (Anita Publications, New Delhi, 2002), pp. 282–292.

Almeida, L. B.

L. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42, 3084–3091 (1994).
[CrossRef]

Arikan, O.

H. M. Ozaktas, O. Arikan, M. A. Kutay, G. Bozdagi, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150 (1996).
[CrossRef]

Bastiaans, M. J.

T. Alieva, M. L. Calvo, M. J. Bastiaans, “Power filtering of n-order in the fractional Fourier domain,” J. Phys. A 35, 7779–7785 (2002).
[CrossRef]

Bitran, Y.

Boudreaux-Bartels, G. F.

O. Akay, G. F. Boudreaux-Bartels, “Fractional convolution and correlation via operator methods and an application to detection of linear FM signals,” IEEE Trans. Signal Process. 49, 979–993 (2001).
[CrossRef]

Bozdagi, G.

H. M. Ozaktas, O. Arikan, M. A. Kutay, G. Bozdagi, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150 (1996).
[CrossRef]

Calvo, M. L.

T. Alieva, M. L. Calvo, M. J. Bastiaans, “Power filtering of n-order in the fractional Fourier domain,” J. Phys. A 35, 7779–7785 (2002).
[CrossRef]

T. Alieva, M. L. Calvo, “Generalized fractional convolution,” in Perspectives in Modern Optics and Optical Instrumentation (Anita Publications, New Delhi, 2002), pp. 282–292.

DeVelis, J. B.

G. O. Reynolds, J. B. DeVelis, G. B. Parrent, B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE Press, Bellingham, Wash., 1989).

Dorsch, R. G.

J. Garcia, R. G. Dorsch, A. W. Lohmann, C. Ferreira, Z. Zalevsky, “Flexible optical implementation of fractional Fourier processors: application to correlation and filtering,” Opt. Commun. 133, 393–400 (1997).
[CrossRef]

D. Mendlovic, Y. Bitran, R. G. Dorsch, A. Lohmann, “Optical fractional correlation: experimental results,” J. Opt. Soc. Am. A 12, 1665–1670 (1995).
[CrossRef]

Ferreira, C.

J. Garcia, R. G. Dorsch, A. W. Lohmann, C. Ferreira, Z. Zalevsky, “Flexible optical implementation of fractional Fourier processors: application to correlation and filtering,” Opt. Commun. 133, 393–400 (1997).
[CrossRef]

Garci´a, J.

Garcia, J.

J. Garcia, R. G. Dorsch, A. W. Lohmann, C. Ferreira, Z. Zalevsky, “Flexible optical implementation of fractional Fourier processors: application to correlation and filtering,” Opt. Commun. 133, 393–400 (1997).
[CrossRef]

Gianino, P. D.

Hirsch, P. M.

L. B. Lesem, P. M. Hirsch, J. A. Jordan, “The kinoform: a new wavefront reconstruction device,” IBM J. Res. Dev. 13, 150–155 (1969).
[CrossRef]

Horner, J. L.

Jordan, J. A.

L. B. Lesem, P. M. Hirsch, J. A. Jordan, “The kinoform: a new wavefront reconstruction device,” IBM J. Res. Dev. 13, 150–155 (1969).
[CrossRef]

Kelly, D.

Kozma, A.

Kutay, M. A.

H. M. Ozaktas, O. Arikan, M. A. Kutay, G. Bozdagi, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150 (1996).
[CrossRef]

H. M. Ozaktas, Z. Zalevsky, M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, New York, 2001).

Lesem, L. B.

L. B. Lesem, P. M. Hirsch, J. A. Jordan, “The kinoform: a new wavefront reconstruction device,” IBM J. Res. Dev. 13, 150–155 (1969).
[CrossRef]

Lim, J. S.

A. V. Oppenheim, J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69, 529–541 (1981).
[CrossRef]

Lohmann, A.

Lohmann, A. W.

A. W. Lohmann, D. Mendlovic, G. Shabtay, “Significance of phase and amplitude in the Fourier domain,” J. Opt. Soc. Am. A 14, 2901–2904 (1997).
[CrossRef]

J. Garcia, R. G. Dorsch, A. W. Lohmann, C. Ferreira, Z. Zalevsky, “Flexible optical implementation of fractional Fourier processors: application to correlation and filtering,” Opt. Commun. 133, 393–400 (1997).
[CrossRef]

D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Fractional correlation,” Appl. Opt. 34, 303–309 (1995).
[CrossRef] [PubMed]

Mendlovic, D.

Oppenheim, A. V.

A. V. Oppenheim, J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69, 529–541 (1981).
[CrossRef]

Ozaktas, H. M.

H. M. Ozaktas, O. Arikan, M. A. Kutay, G. Bozdagi, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150 (1996).
[CrossRef]

D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Fractional correlation,” Appl. Opt. 34, 303–309 (1995).
[CrossRef] [PubMed]

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

H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transformations and their optical implementation: II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993).
[CrossRef]

H. M. Ozaktas, Z. Zalevsky, M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, New York, 2001).

Parrent, G. B.

G. O. Reynolds, J. B. DeVelis, G. B. Parrent, B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE Press, Bellingham, Wash., 1989).

Reynolds, G. O.

G. O. Reynolds, J. B. DeVelis, G. B. Parrent, B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE Press, Bellingham, Wash., 1989).

Segun, I. A.

M. Abramovich, I. A. Segun, Handbook of Mathematical Functions (Dover, New York, 1965).

Shabtay, G.

Thompson, B. J.

G. O. Reynolds, J. B. DeVelis, G. B. Parrent, B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE Press, Bellingham, Wash., 1989).

Vanderlugt, A.

A. Vanderlugt, Optical Signal Processing (Wiley, New York, 1992).

Zalevsky, Z.

J. Garcia, R. G. Dorsch, A. W. Lohmann, C. Ferreira, Z. Zalevsky, “Flexible optical implementation of fractional Fourier processors: application to correlation and filtering,” Opt. Commun. 133, 393–400 (1997).
[CrossRef]

J. Garcı́a, D. Mendlovic, Z. Zalevsky, A. Lohmann, “Space variant simultaneous detection of several objects using multiple anamorphic fractional Fourier transform filters,” Appl. Opt. 35, 3945–3952 (1996).
[CrossRef]

H. M. Ozaktas, Z. Zalevsky, M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, New York, 2001).

Appl. Opt. (4)

IBM J. Res. Dev. (1)

L. B. Lesem, P. M. Hirsch, J. A. Jordan, “The kinoform: a new wavefront reconstruction device,” IBM J. Res. Dev. 13, 150–155 (1969).
[CrossRef]

IEEE Trans. Signal Process. (3)

L. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42, 3084–3091 (1994).
[CrossRef]

O. Akay, G. F. Boudreaux-Bartels, “Fractional convolution and correlation via operator methods and an application to detection of linear FM signals,” IEEE Trans. Signal Process. 49, 979–993 (2001).
[CrossRef]

H. M. Ozaktas, O. Arikan, M. A. Kutay, G. Bozdagi, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150 (1996).
[CrossRef]

J. Opt. Soc. Am. A (4)

J. Phys. A (1)

T. Alieva, M. L. Calvo, M. J. Bastiaans, “Power filtering of n-order in the fractional Fourier domain,” J. Phys. A 35, 7779–7785 (2002).
[CrossRef]

Opt. Commun. (1)

J. Garcia, R. G. Dorsch, A. W. Lohmann, C. Ferreira, Z. Zalevsky, “Flexible optical implementation of fractional Fourier processors: application to correlation and filtering,” Opt. Commun. 133, 393–400 (1997).
[CrossRef]

Proc. IEEE (1)

A. V. Oppenheim, J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69, 529–541 (1981).
[CrossRef]

Other (5)

M. Abramovich, I. A. Segun, Handbook of Mathematical Functions (Dover, New York, 1965).

T. Alieva, M. L. Calvo, “Generalized fractional convolution,” in Perspectives in Modern Optics and Optical Instrumentation (Anita Publications, New Delhi, 2002), pp. 282–292.

H. M. Ozaktas, Z. Zalevsky, M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, New York, 2001).

A. Vanderlugt, Optical Signal Processing (Wiley, New York, 1992).

G. O. Reynolds, J. B. DeVelis, G. B. Parrent, B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE Press, Bellingham, Wash., 1989).

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Figures (7)

Fig. 1
Fig. 1

Amplitude of the fractional FT of the rectangular signal for nine different values of angle α: a, α=0, original signal; b, α=π/20; c, α=π/10; d, α=3π/20; e, α=π/5; f, α=π/4; g, α=3π/10; h, α=2π/5; i, α=π/2, amplitude of the Fourier transform.

Fig. 2
Fig. 2

Amplitude of the signal reconstructed from the amplitude of the fractional Fourier transform for the same nine values of angle α as in Fig. 1.

Fig. 3
Fig. 3

Phase profile in the Fourier domain for the rectangular signal, simulating the effect of the signal reconstruction from the phase of its fractional FT for the same nine values of angle α as in Fig. 1.

Fig. 4
Fig. 4

Amplitude of the signal reconstructed from the phase of the fractional FT for the same nine values of angle α as in Fig. 1.

Fig. 5
Fig. 5

Amplitude of the signal reconstructed from the phase of the fractional FT, taken in the region where the amplitude of the fractional FT is relatively large, for the same nine values of angle α as in Fig. 1.

Fig. 6
Fig. 6

Amplitude of the fractional FT of the cameraman image for different values of angle α: a, α=0, original signal; b, α=π/20; c, α=π/4; d, α=π/2, amplitude of the FT.

Fig. 7
Fig. 7

Amplitude of the image reconstructed from (a, c, e) the amplitude-only data and (b, d, f) the phase-only data of the fractional FT of the cameraman image for different values of angle α: (a, b) α=π/20, (c, d) α=π/4, (e, f) α=π/2, FT domain.

Equations (38)

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Rα[f(x)](u)=Fα(u)=f(x)K(α, x, u)dx,
K(α, x, u)=exp(iα/2)(i sinα)1/2expiπ(x2+u2)cosα-2xusinα.
Fα(u)|Fα(u)|exp[i2πφα(u)],1sinαexp(iπu2cotα+iα/2-iπ/4)×f(x)expiπx2 cosα-2xusinαdx,1sinαexp(iπu2cotα+iα/2-iπ/4)×Aα(u)exp[i2πϕα(u)],
tan[2πϕα(u)]=-A0(y)sin[πy2 cotα-2πuy/sinα+2πφ0(y)]dy-A0(y)cos[πy2 cotα-2πuy/sinα+2πφ0(y)]dy.
fA,α(x)=1(sinα)1/2R-α[Aα(u)](x),
fφ,α(x)=1(sinα)1/2R-αexp[i2πφα(u)]rectuΔu(x),
Rα[exp(icπu2)](x)=exp(iα/2)(cosα+c sinα)1/2expiπu2c-tanα1+c tanα.
|fφ,α(x)|=1sinαexp(-iπx2 cotα)rectuΔu×expi2πxusinα+i2πϕα(u)du=|R-π/2˜{exp[i2πϕα(u sinα)]}(x)|.
Rα[f(x-s)](u)=Fα(u-s cosα)exp[iπ sinα×(s2 cosα-2su)],
Rα[f(x)exp(i2πνx)](u)=Fα(u-ν sinα)exp[-iπ cosα(ν2 sinα-2νu)],
|fA,α,s(x)|=|R-α[|Fα(u-s cosα)|](x)|=|fA,α,0(x-s cos2 α)|.
|fφ,α,s(x)|=|sin-1/2αR-α(exp{i2πφα[u-s cosα)]}×exp(-2iπsu sinα))(x)|,  =|sin-1/2αR-α{exp[i2πφα(u-s cosα)]}×(x-s sin2 α)|,  =|sin-1/2αR-α{exp[i2πφα(u)]}×(x-s sin2 α-s cos2 α)|,  =|fφ,α,0(x-s)|.
Fπ/2(u)=sin(πud)πu.
g(u)=sgnsin(πud)πu,
fφ,π/2(x)=-g(u)exp(i2πxu)du.
fφ,π/2(x)=0g(u)[exp(i2πux)-exp(-i2πux)]du=2i0g(u)sin(2πux)du=-iπxn=0cos(2πux)2n/d(2n+1)/d-cos(2πux)(2n+1)/d2(n+1)/d=-2iπx1-cos2πxd×n=0cos2π(2n+1)xd.
Fα(u)=exp(iα/2)(i sinα)1/2exp(iπu2 cotα)×-d/2d/2expiπx2 cosα-2xusinαdx,  =exp(iα/2)(i sinα)1/2exp(iπu2 cotα)Aα(u)exp[i2πϕα(u)],  =exp(iα/2)(i sinα)1/2exp(iπu2 cotα)hα(u),
hα(u)=-d/2d/2expiπx2 cosα-2xusinαdx=exp-i2πu2sin(2α)-d/2d/2expiπ(x-u/cosα)2tanαdx.
C(z)=0zcosπ2t2dt,
S(z)=0zsinπ2t2dt,
hα(u)=[(tanα)/2]1/2 exp[-iβ(u)](ΔC+iΔS),
ΔC=C(z2)-C(z1),
ΔS=S(z2)-S(z1),
Aα(u)=tanα2[(ΔC)2+(ΔS)2]1/2,
2πϕα(u)=arctan(ΔS/ΔC)-β(u).
2πϕα(u sinα)=(πd2 cotα)/4+η(d, α, u),
tan[η(d, α, u)]=cosγ[f(y1)-f(y2)]+sinγ[g(y1)+g(y2)]cosγ[g(y1)-g(y2)]-sinγ[f(y1)+f(y2)],
|fφ,α(x)|=rectuΔuexp[iη(d, α, u)+i2πxu]du,
C(z)=12+f(z)sinπ2z2-g(z)cosπ2z2,
S(z)=12-f(z)cosπ2z2-g(z)sinπ2z2,
ΔC+iΔS=f(z)sinπ2z2-g(z)cosπ2z2-if(z)cosπ2z2+g(z)sinπ2z2z1z2=-expiπ2z2[g(z)+if(z)]z1z2=expiπ4(z12+z22)expiπ4(z12-z22)[g(z1)+if(z1)]-exp-iπ4(z12-z22)[g(z2)+if(z2)],
π4(z12+z22)=π cotαd24+2πu2sin (2α)=π cotαd24+β(u),
π4(z12-z22)=πudsinα,
hα(u)=[(tanα)/2]1/2expiπd2 cotα4expiπudsinα×[g(z1)+if(z1)]-exp-iπudsinα×[g(z2)+if(z2)].
f(z)=1+0.926z2+1.792z+3.104z2+(z),
g(z)=12+4.142z+3.492z2+6.670z3+(z),
f(z)+f(-z)=cosπ2z2-sinπ2z2,
g(z)+g(-z)=cosπ2z2+sinπ2z2,

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