Abstract

The ambiguity function and Cohen’s class of bilinear phase-space distributions are represented in a quasi-polar coordinate system instead of in a Cartesian system. Relationships between these distributions and the fractional Fourier transform are derived; in particular, derivatives of the ambiguity function are related to moments of the fractional power spectra. A simplification is achieved for the description of underspread signals, for optical beam characterization, and for the generation of signal-adaptive phase-space distributions.

© 2000 Optical Society of America

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  1. E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
    [CrossRef]
  2. A. Papoulis, “Ambiguity function in Fourier optics,” J. Opt. Soc. Am. 64, 779–788 (1974).
    [CrossRef]
  3. M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
    [CrossRef]
  4. L. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42, 3084–3091 (1994).
    [CrossRef]
  5. M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
    [CrossRef] [PubMed]
  6. J. Tu, S. Tamura, “Analytic relation for recovering the mutual intensity by means of intensity information,” J. Opt. Soc. Am. A 15, 202–206 (1998).
    [CrossRef]
  7. A. W. Lohmann, D. Mendlovic, Z. Zalevsky, “Fractional transformation in optics,” in Progress in Optics, XXXVIII, E. Wolf, ed. (Elsevier, Amsterdam, 1998), pp. 263–342.
  8. H. M. Ozaktas, M. A. Kutay, D. Mendlovic, “Introduction to the fractional Fourier transform and its applications,” in Advances in Imaging and Electron Physics, P. W. Hawkes, ed. (Academic, San Diego, Calif., 1991), Vol. 106, pp. 239–291.
  9. L. Cohen, “Time-frequency distributions—a review,” Proc. IEEE 77, 941–981 (1989).
    [CrossRef]
  10. L. J. Stankovic, I. Djurovic, “Relationship between the ambiguity function coordinate transformations and the fractional Fourier transform,” Ann. Telecommun. 53, 316–319 (1998).
  11. J. C. Wood, D. T. Barry, “Tomographic time-frequency analysis and its applications toward time-varying filtering and adaptive kernel design for multi-component linear-FM signals,” IEEE Trans. Signal Process. 42, 2094–2104 (1994).
    [CrossRef]
  12. F. Hlawatsch, G. Matz, “Time-frequency signal processing: a statistical perspective,” Proceedings of CSSP-98, ProRISC/IEEE Workshop on Circuits, Systems and Signal Processing, J. P. Veen, ed. (STW, Technology Foundation, Utrecht, The Netherlands, 1998), pp. 207–219.
  13. C. J. R. Sheppard, K. G. Larkin, “Focal shift, optical transfer function, and phase-space representations,” J. Opt. Soc. Am. A 17, 772–779 (2000).
    [CrossRef]
  14. F. Boudreaux-Bartels, “Mixed time-frequency signal transformations,” in The Transforms and Applications Handbook, A. D. Poularikas, ed. (CRC, Boca Raton, Fla., 1996), pp. 887–962.
  15. M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. A 73, 1434–1441 (1983).
    [CrossRef]
  16. N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6–10 (1984).
    [CrossRef]
  17. K. Ichikawa, A. W. Lohmann, M. Takeda, “Phase retrieval based on the irradiance transport equation and the Fourier transform method: experiments,” Appl. Opt. 27, 3433–3436 (1988).
    [CrossRef] [PubMed]
  18. T. Alieva, V. Lopez, F. Agullo-Lopez, L. B. Almeida, “The fractional Fourier transform in the optical propagation problems,” J. Mod. Opt. 41, 1037–1044 (1994).
    [CrossRef]
  19. R. G. Baraniuk, D. L. Jones, “A radially Gaussian, signal dependent time-frequency representation,” Proceedings of ICASSP-91, IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1991), pp. 3181–3184.
  20. A. W. Lohmann, B. H. Soffer, “Relationships between the Radon–Wigner and fractional Fourier transforms,” J. Opt. Soc. Am. A 11, 1798–1801 (1994).
    [CrossRef]

2000

1998

J. Tu, S. Tamura, “Analytic relation for recovering the mutual intensity by means of intensity information,” J. Opt. Soc. Am. A 15, 202–206 (1998).
[CrossRef]

L. J. Stankovic, I. Djurovic, “Relationship between the ambiguity function coordinate transformations and the fractional Fourier transform,” Ann. Telecommun. 53, 316–319 (1998).

1994

J. C. Wood, D. T. Barry, “Tomographic time-frequency analysis and its applications toward time-varying filtering and adaptive kernel design for multi-component linear-FM signals,” IEEE Trans. Signal Process. 42, 2094–2104 (1994).
[CrossRef]

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

M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

T. Alieva, V. Lopez, F. Agullo-Lopez, L. B. Almeida, “The fractional Fourier transform in the optical propagation problems,” J. Mod. Opt. 41, 1037–1044 (1994).
[CrossRef]

A. W. Lohmann, B. H. Soffer, “Relationships between the Radon–Wigner and fractional Fourier transforms,” J. Opt. Soc. Am. A 11, 1798–1801 (1994).
[CrossRef]

1989

L. Cohen, “Time-frequency distributions—a review,” Proc. IEEE 77, 941–981 (1989).
[CrossRef]

1988

1984

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6–10 (1984).
[CrossRef]

1983

M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. A 73, 1434–1441 (1983).
[CrossRef]

1978

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
[CrossRef]

1974

1932

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Agullo-Lopez, F.

T. Alieva, V. Lopez, F. Agullo-Lopez, L. B. Almeida, “The fractional Fourier transform in the optical propagation problems,” J. Mod. Opt. 41, 1037–1044 (1994).
[CrossRef]

Alieva, T.

T. Alieva, V. Lopez, F. Agullo-Lopez, L. B. Almeida, “The fractional Fourier transform in the optical propagation problems,” J. Mod. Opt. 41, 1037–1044 (1994).
[CrossRef]

Almeida, L. B.

T. Alieva, V. Lopez, F. Agullo-Lopez, L. B. Almeida, “The fractional Fourier transform in the optical propagation problems,” J. Mod. Opt. 41, 1037–1044 (1994).
[CrossRef]

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

Baraniuk, R. G.

R. G. Baraniuk, D. L. Jones, “A radially Gaussian, signal dependent time-frequency representation,” Proceedings of ICASSP-91, IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1991), pp. 3181–3184.

Barry, D. T.

J. C. Wood, D. T. Barry, “Tomographic time-frequency analysis and its applications toward time-varying filtering and adaptive kernel design for multi-component linear-FM signals,” IEEE Trans. Signal Process. 42, 2094–2104 (1994).
[CrossRef]

Bastiaans, M. J.

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
[CrossRef]

Beck, M.

M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

Boudreaux-Bartels, F.

F. Boudreaux-Bartels, “Mixed time-frequency signal transformations,” in The Transforms and Applications Handbook, A. D. Poularikas, ed. (CRC, Boca Raton, Fla., 1996), pp. 887–962.

Cohen, L.

L. Cohen, “Time-frequency distributions—a review,” Proc. IEEE 77, 941–981 (1989).
[CrossRef]

Djurovic, I.

L. J. Stankovic, I. Djurovic, “Relationship between the ambiguity function coordinate transformations and the fractional Fourier transform,” Ann. Telecommun. 53, 316–319 (1998).

Hlawatsch, F.

F. Hlawatsch, G. Matz, “Time-frequency signal processing: a statistical perspective,” Proceedings of CSSP-98, ProRISC/IEEE Workshop on Circuits, Systems and Signal Processing, J. P. Veen, ed. (STW, Technology Foundation, Utrecht, The Netherlands, 1998), pp. 207–219.

Ichikawa, K.

Jones, D. L.

R. G. Baraniuk, D. L. Jones, “A radially Gaussian, signal dependent time-frequency representation,” Proceedings of ICASSP-91, IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1991), pp. 3181–3184.

Kutay, M. A.

H. M. Ozaktas, M. A. Kutay, D. Mendlovic, “Introduction to the fractional Fourier transform and its applications,” in Advances in Imaging and Electron Physics, P. W. Hawkes, ed. (Academic, San Diego, Calif., 1991), Vol. 106, pp. 239–291.

Larkin, K. G.

Lohmann, A. W.

Lopez, V.

T. Alieva, V. Lopez, F. Agullo-Lopez, L. B. Almeida, “The fractional Fourier transform in the optical propagation problems,” J. Mod. Opt. 41, 1037–1044 (1994).
[CrossRef]

Matz, G.

F. Hlawatsch, G. Matz, “Time-frequency signal processing: a statistical perspective,” Proceedings of CSSP-98, ProRISC/IEEE Workshop on Circuits, Systems and Signal Processing, J. P. Veen, ed. (STW, Technology Foundation, Utrecht, The Netherlands, 1998), pp. 207–219.

McAlister, D. F.

M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

Mendlovic, D.

A. W. Lohmann, D. Mendlovic, Z. Zalevsky, “Fractional transformation in optics,” in Progress in Optics, XXXVIII, E. Wolf, ed. (Elsevier, Amsterdam, 1998), pp. 263–342.

H. M. Ozaktas, M. A. Kutay, D. Mendlovic, “Introduction to the fractional Fourier transform and its applications,” in Advances in Imaging and Electron Physics, P. W. Hawkes, ed. (Academic, San Diego, Calif., 1991), Vol. 106, pp. 239–291.

Ozaktas, H. M.

H. M. Ozaktas, M. A. Kutay, D. Mendlovic, “Introduction to the fractional Fourier transform and its applications,” in Advances in Imaging and Electron Physics, P. W. Hawkes, ed. (Academic, San Diego, Calif., 1991), Vol. 106, pp. 239–291.

Papoulis, A.

Raymer, M. G.

M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

Sheppard, C. J. R.

Soffer, B. H.

Stankovic, L. J.

L. J. Stankovic, I. Djurovic, “Relationship between the ambiguity function coordinate transformations and the fractional Fourier transform,” Ann. Telecommun. 53, 316–319 (1998).

Streibl, N.

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6–10 (1984).
[CrossRef]

Takeda, M.

Tamura, S.

Teague, M. R.

M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. A 73, 1434–1441 (1983).
[CrossRef]

Tu, J.

Wigner, E.

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Wood, J. C.

J. C. Wood, D. T. Barry, “Tomographic time-frequency analysis and its applications toward time-varying filtering and adaptive kernel design for multi-component linear-FM signals,” IEEE Trans. Signal Process. 42, 2094–2104 (1994).
[CrossRef]

Zalevsky, Z.

A. W. Lohmann, D. Mendlovic, Z. Zalevsky, “Fractional transformation in optics,” in Progress in Optics, XXXVIII, E. Wolf, ed. (Elsevier, Amsterdam, 1998), pp. 263–342.

Ann. Telecommun.

L. J. Stankovic, I. Djurovic, “Relationship between the ambiguity function coordinate transformations and the fractional Fourier transform,” Ann. Telecommun. 53, 316–319 (1998).

Appl. Opt.

IEEE Trans. Signal Process.

J. C. Wood, D. T. Barry, “Tomographic time-frequency analysis and its applications toward time-varying filtering and adaptive kernel design for multi-component linear-FM signals,” IEEE Trans. Signal Process. 42, 2094–2104 (1994).
[CrossRef]

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

J. Mod. Opt.

T. Alieva, V. Lopez, F. Agullo-Lopez, L. B. Almeida, “The fractional Fourier transform in the optical propagation problems,” J. Mod. Opt. 41, 1037–1044 (1994).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6–10 (1984).
[CrossRef]

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
[CrossRef]

Phys. Rev.

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Phys. Rev. Lett.

M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

Proc. IEEE

L. Cohen, “Time-frequency distributions—a review,” Proc. IEEE 77, 941–981 (1989).
[CrossRef]

Other

A. W. Lohmann, D. Mendlovic, Z. Zalevsky, “Fractional transformation in optics,” in Progress in Optics, XXXVIII, E. Wolf, ed. (Elsevier, Amsterdam, 1998), pp. 263–342.

H. M. Ozaktas, M. A. Kutay, D. Mendlovic, “Introduction to the fractional Fourier transform and its applications,” in Advances in Imaging and Electron Physics, P. W. Hawkes, ed. (Academic, San Diego, Calif., 1991), Vol. 106, pp. 239–291.

R. G. Baraniuk, D. L. Jones, “A radially Gaussian, signal dependent time-frequency representation,” Proceedings of ICASSP-91, IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1991), pp. 3181–3184.

F. Boudreaux-Bartels, “Mixed time-frequency signal transformations,” in The Transforms and Applications Handbook, A. D. Poularikas, ed. (CRC, Boca Raton, Fla., 1996), pp. 887–962.

F. Hlawatsch, G. Matz, “Time-frequency signal processing: a statistical perspective,” Proceedings of CSSP-98, ProRISC/IEEE Workshop on Circuits, Systems and Signal Processing, J. P. Veen, ed. (STW, Technology Foundation, Utrecht, The Netherlands, 1998), pp. 207–219.

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Equations (56)

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Afh(x, u)=- f(x+12x)h*(x-12x)×exp(-i2πux)dx
Afh(x, u)=- Fπ/2 (u+12u)Hπ/2* (u-12u)×exp(i2πxu)du.
Afh(0, u)=- f(x)h*(x)exp(-i2πux)dx.
Afh(x, 0)=- Fπ/2(u)Hπ/2*(u)exp(i2πxu)du.
Rα[f(x)](u)=Fα(u)=-K(α, x, u)f(x)dx,
K(α, x, u)=exp[i(a/2)](i sin α)1/2×expiπ(x2+u2)cos α-2uxsin α.
x=R cos β,
u=R sin β,
f(x), h(x)cross-ambiguityA¯fh(R, β)
 fractionalFT  rotationofAF
Rα[f], Rα[h]cross-ambiguityA˜Rα[f],Rα[h](R, β)
=A˜fh(R, β+α),
A˜fh(R, α)=A˜Fα Hα(R, 0)=AFαHα(R, 0).
A˜fh(R, α)=-Fα+π/2(x)Hα+π/2*(x)exp(i2πRx)dx.
A˜f (R, α-12π)=-|Fα(x)|2 exp(i2πRx)dx,
|A˜f (R, α)|0forRR0,
E=-|Fα(x)|2dx=A˜f (R, α-12π)|R=0=Af (0, 0).
mα=1E-|Fα(x)|2xdx=1E12πiA˜f (R, α-π/2)RR=0.
A˜f (R, α-π/2)RR=0,α=π/2=Af (x, u)xx=0,u=0=2πi-|Fπ/2(u)|2udu,
A˜f (R, α-π/2)RR=0,α=π=Af (x, u)ux=0,u=0=2πi-|f(-x)|2xdx.
A˜f (R,α-π/2)R=Af (x, u)xxR+Af (x, u)uuR=Af (x, u)xsin α-Af (x, u)ucos α,
mα=m0 cos α+mπ/2 sin α.
wα=1E-|Fα(x)|2x2dx=1E12πi2 2A˜f (R, α-π/2)R2R=0,
2Af (x, u)xux=0,u=0=πi-Fπ/2(u)uFπ/2*(u)-Fπ/2(u)Fπ/2*(u)uudu=πi-f(x)zf*(x)-f(x)f*(x)x×(-x)dx,
2A˜f (R, α-π/2)RRR=0=πi-Fα(x)xFα*(x)-Fα(x)Fα*(x)xxdx,
μα=πiE12πi2-Fα(x)xFα*(x)-Fα(x)Fα*(x)xxdx.
2A˜f (R, α-π/2)R2=2Af (x, u)x2sin2 α+2Af (x, u)u2×cos2 α-2Af (x, u)xusin 2α
wα=wo cos2 α+wπ/2 sin2 α-μ0 sin 2α.
wαμαμαwα+π/2=cos α-sin αsin αcos α×w0μ0μ0wπ/2cos αsin α-sin αcos α,
μα=12(w0-wπ/2)sin 2α+μ0 cos 2α.
U0(r)=12πi1|f(r)|2- Af (x, u)xx=0 exp(i2πru)du,
Af (x, u)xx=0=Af (R, α-π/2)RRxx=0+A˜f (R, α-π/2)ααxx=0=-1uA˜f (-u, α-π/2)αα=0,
Af (x, u)xx=0=-1u- |Fα(x)|2αα=0 exp(-i2πux)dx,
U0(r)=12|F0(r)|2- |Fα(x)|2αα=0×-1πi-1uexp[i2πu(r-x)]dudx.
-1/(πi)-(1/u)exp[i2πu(r-x)]du=sgn(x-r),
U0(r)=12|F0(r)|2- |Fα(x)|2αα=0 sgn(x-r)dx,
Uβ(r)=12|Fβ(r)|2- |Fα(x)|2αα=β sgn(x-r)dx.
Uβ(r)=12πdφβ(r)dr.
Pf (y, v)=--Φ(x, u)Af (x, u)×exp[i2π(yu-vx)]dxdu,
y=ρ cos φ,x=R cos α,
v=ρ sin φ,u=R sin α,
P˜f (ρ, φ)=-RdR0πdαΦ˜(R, α)A˜f (R, α)×exp[i2πRρ sin(α-φ)].
Wfh(x, u)=-f (x+12x)h*(x-12x)×exp(-i2πux)dx,
Wfh(x, u)=-Fπ/2(u+12u)Hπ/2*(u-12u)×exp(i2πux)du,
Wfh(x, u)=--Afh(y, v)exp[i2π(xv-uy)]dydv.
W˜fh(R, α)=-Fα+π/2(12x)Hα+π/2*(-12x)×exp(i2πxR)dx.
W˜f (ρ, α)=±-|Fα+π/2(12x)|2 exp(i2πxρ)dx±20|Fα+π/2(12x)|2 cos(2πxρ)dx.
W˜f (ρ, φ)=-RdR0πdαA˜f (R, α)×exp[i2πRρ sin(α-φ)].
Gfh(x, u)=-f(x)h*(x-x)exp(-i2πux)dx
Gfh(x, u)=exp(-iπux)Afh(x, u).
G˜fh(R, α)=exp(-i12πR2 sin 2α)A˜fh(R, α)
=exp(-i12πR2 sin 2α)-Fα+π/2(x)×Hα+π/2*(x)exp(i2πRx)dx.
|G˜fh(R, α)|2
=|A˜fh(R, α)|2
=-Fα+π/2(x)Hα+π/2*(x)exp(i2πRx)dx2.
G˜fh(R, α)=exp(-i12πR2 sin 2α)-Fα+π/2(x)×exp(-πx2)exp(i2πRx)dx.

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