Abstract

The fractional Fourier transform (FRFT) has been used for many years, and it is useful in many applications. Most applications of the FRFT are based on the design of fractional filters (such as removal of chirp noise and the fractional Hilbert transform) or on fractional correlation (such as scaled space-variant pattern recognition). In this study we introduce several types of simplified fractional Fourier transform (SFRFT). Such transforms are all special cases of a linear canonical transform (an affine Fourier transform or an ABCD transform). They have the same capabilities as the original FRFT for design of fractional filters or for fractional correlation. But they are simpler than the original FRFT in terms of digital computation, optical implementation, implementation of gradient-index media, and implementation of radar systems. Our goal is to search for the simplest transform that has the same capabilities as the original FRFT. Thus we discuss not only the formulas and properties of the SFRFT’s but also their implementation. Although these SFRFT’s usually have no additivity properties, they are useful for the practical applications. They have great potential for replacing the original FRFT’s in many applications.

© 2000 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
    [CrossRef]
  2. L. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42, 3084–3091 (1994).
    [CrossRef]
  3. M. Moshinsky, C. Quesne, “Linear canonical transformations and their unitary representations,” J. Math. Phys. 12, 1772–1783 (1971).
    [CrossRef]
  4. K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, New York, 1979), Chap. 9.
  5. S. Abe, J. T. Sheridan, “Optical operations on wave functions as the Abelian subgroups of the special affine Fourier transformation,” Opt. Lett. 19, 1801–1803 (1994).
    [CrossRef] [PubMed]
  6. L. M. Bernardo, “ABCD matrix formalism of fractional Fourier optics,” Opt. Eng. 35, 732–740 (1996).
    [CrossRef]
  7. H. M. Ozaktas, D. Mendlovic, “Every Fourier optical system is equivalent to consecutive fractional-Fourier-domain filtering,” Appl. Opt. 35, 3167–3170 (1996).
    [CrossRef] [PubMed]
  8. S. C. Pei, J. J. Ding, “Closed form discrete fractional and affine Fourier transforms,” IEEE Trans. Signal Process. 48, 1338–1353 (2000).
    [CrossRef]
  9. 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]
  10. A. M. Almanasreh, M. G. Abushagur, “Fractional correlation based on the modified fractional order Fourier transform,” Opt. Eng. 37, 175–184 (1998).
    [CrossRef]
  11. M. A. Kutay, H. M. Ozaktas, O. Arikan, L. Onural, “Optimal filter in fractional Fourier domains,” IEEE Trans. Signal Process. 45, 1129–1143 (1997).
    [CrossRef]
  12. A. W. Lohmann, D. Mendlovic, Z. Zalevsky, “Fractional Hilbert transform,” Opt. Lett. 21, 281–283 (1996).
    [CrossRef] [PubMed]
  13. H. M. Ozaktas, B. Barshan, D. Mendlovic, L. Onural, “Convolution, filtering, and multiplexing in fractional Fourier domains and their rotation to chirp and wavelet transform,” J. Opt. Soc. Am. A 11, 547–559 (1994).
    [CrossRef]
  14. S. C. Pei, J. J. Ding, “Relations between the fractional operations and the Wigner distribution, ambiguity function,” submitted to IEEE Trans. Signal Process.
  15. L. B. Almeida, “Product and convolution theorems for the fractional Fourier transform,” IEEE Signal Process. Lett. 4, 15–17 (1997).
    [CrossRef]
  16. A. I. Zayed, “A convolution and product theorem for the fractional Fourier transform,” IEEE Signal Process. Lett. 5, 101–103 (1998).
    [CrossRef]
  17. B. Barshan, M. A. Kutay, H. M. Ozaktas, “Optimal filtering with linear canonical transformations,” Opt. Commun. 135, 32–36 (1997).
    [CrossRef]
  18. L. Yu, M. Huang, L. Wu, Y. Lu, W. Huang, M. Chen, Z. Zhu, “Fractional Fourier transform and the elliptic gradient-index medium,” Opt. Commun. 152, 23–25 (1998).
    [CrossRef]
  19. P. Pellat-Finet, G. Bonnet, “Fractional order Fourier transform and Fourier optics,” Opt. Commun. 111, 141–154 (1994).
    [CrossRef]
  20. D. Mendlovic, H. M. Zalevsky, A. W. Lohmann, “Fractional correlation,” Appl. Opt. 34, 303–309 (1995).
    [CrossRef] [PubMed]
  21. A. W. Lohmann, Z. Zalevsky, D. Mendlovic, “Synthesis of pattern recognition filters for fractional Fourier processing,” Opt. Commun. 128, 199–204 (1996).
    [CrossRef]
  22. S. Granieri, R. Arizaga, E. E. Sicre, “Optical correlation based on the fractional Fourier transform,” Appl. Opt. 36, 6636–6645 (1997).
    [CrossRef]

2000 (1)

S. C. Pei, J. J. Ding, “Closed form discrete fractional and affine Fourier transforms,” IEEE Trans. Signal Process. 48, 1338–1353 (2000).
[CrossRef]

1998 (3)

A. M. Almanasreh, M. G. Abushagur, “Fractional correlation based on the modified fractional order Fourier transform,” Opt. Eng. 37, 175–184 (1998).
[CrossRef]

A. I. Zayed, “A convolution and product theorem for the fractional Fourier transform,” IEEE Signal Process. Lett. 5, 101–103 (1998).
[CrossRef]

L. Yu, M. Huang, L. Wu, Y. Lu, W. Huang, M. Chen, Z. Zhu, “Fractional Fourier transform and the elliptic gradient-index medium,” Opt. Commun. 152, 23–25 (1998).
[CrossRef]

1997 (4)

B. Barshan, M. A. Kutay, H. M. Ozaktas, “Optimal filtering with linear canonical transformations,” Opt. Commun. 135, 32–36 (1997).
[CrossRef]

M. A. Kutay, H. M. Ozaktas, O. Arikan, L. Onural, “Optimal filter in fractional Fourier domains,” IEEE Trans. Signal Process. 45, 1129–1143 (1997).
[CrossRef]

L. B. Almeida, “Product and convolution theorems for the fractional Fourier transform,” IEEE Signal Process. Lett. 4, 15–17 (1997).
[CrossRef]

S. Granieri, R. Arizaga, E. E. Sicre, “Optical correlation based on the fractional Fourier transform,” Appl. Opt. 36, 6636–6645 (1997).
[CrossRef]

1996 (5)

A. W. Lohmann, Z. Zalevsky, D. Mendlovic, “Synthesis of pattern recognition filters for fractional Fourier processing,” Opt. Commun. 128, 199–204 (1996).
[CrossRef]

A. W. Lohmann, D. Mendlovic, Z. Zalevsky, “Fractional Hilbert transform,” Opt. Lett. 21, 281–283 (1996).
[CrossRef] [PubMed]

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]

L. M. Bernardo, “ABCD matrix formalism of fractional Fourier optics,” Opt. Eng. 35, 732–740 (1996).
[CrossRef]

H. M. Ozaktas, D. Mendlovic, “Every Fourier optical system is equivalent to consecutive fractional-Fourier-domain filtering,” Appl. Opt. 35, 3167–3170 (1996).
[CrossRef] [PubMed]

1995 (1)

1994 (4)

1980 (1)

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

1971 (1)

M. Moshinsky, C. Quesne, “Linear canonical transformations and their unitary representations,” J. Math. Phys. 12, 1772–1783 (1971).
[CrossRef]

Abe, S.

Abushagur, M. G.

A. M. Almanasreh, M. G. Abushagur, “Fractional correlation based on the modified fractional order Fourier transform,” Opt. Eng. 37, 175–184 (1998).
[CrossRef]

Almanasreh, A. M.

A. M. Almanasreh, M. G. Abushagur, “Fractional correlation based on the modified fractional order Fourier transform,” Opt. Eng. 37, 175–184 (1998).
[CrossRef]

Almeida, L. B.

L. B. Almeida, “Product and convolution theorems for the fractional Fourier transform,” IEEE Signal Process. Lett. 4, 15–17 (1997).
[CrossRef]

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

Arikan, O.

M. A. Kutay, H. M. Ozaktas, O. Arikan, L. Onural, “Optimal filter in fractional Fourier domains,” IEEE Trans. Signal Process. 45, 1129–1143 (1997).
[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]

Arizaga, R.

Barshan, B.

Bernardo, L. M.

L. M. Bernardo, “ABCD matrix formalism of fractional Fourier optics,” Opt. Eng. 35, 732–740 (1996).
[CrossRef]

Bonnet, G.

P. Pellat-Finet, G. Bonnet, “Fractional order Fourier transform and Fourier optics,” Opt. Commun. 111, 141–154 (1994).
[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]

Chen, M.

L. Yu, M. Huang, L. Wu, Y. Lu, W. Huang, M. Chen, Z. Zhu, “Fractional Fourier transform and the elliptic gradient-index medium,” Opt. Commun. 152, 23–25 (1998).
[CrossRef]

Ding, J. J.

S. C. Pei, J. J. Ding, “Closed form discrete fractional and affine Fourier transforms,” IEEE Trans. Signal Process. 48, 1338–1353 (2000).
[CrossRef]

S. C. Pei, J. J. Ding, “Relations between the fractional operations and the Wigner distribution, ambiguity function,” submitted to IEEE Trans. Signal Process.

Granieri, S.

Huang, M.

L. Yu, M. Huang, L. Wu, Y. Lu, W. Huang, M. Chen, Z. Zhu, “Fractional Fourier transform and the elliptic gradient-index medium,” Opt. Commun. 152, 23–25 (1998).
[CrossRef]

Huang, W.

L. Yu, M. Huang, L. Wu, Y. Lu, W. Huang, M. Chen, Z. Zhu, “Fractional Fourier transform and the elliptic gradient-index medium,” Opt. Commun. 152, 23–25 (1998).
[CrossRef]

Kutay, M. A.

B. Barshan, M. A. Kutay, H. M. Ozaktas, “Optimal filtering with linear canonical transformations,” Opt. Commun. 135, 32–36 (1997).
[CrossRef]

M. A. Kutay, H. M. Ozaktas, O. Arikan, L. Onural, “Optimal filter in fractional Fourier domains,” IEEE Trans. Signal Process. 45, 1129–1143 (1997).
[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]

Lohmann, A. W.

Lu, Y.

L. Yu, M. Huang, L. Wu, Y. Lu, W. Huang, M. Chen, Z. Zhu, “Fractional Fourier transform and the elliptic gradient-index medium,” Opt. Commun. 152, 23–25 (1998).
[CrossRef]

Mendlovic, D.

Moshinsky, M.

M. Moshinsky, C. Quesne, “Linear canonical transformations and their unitary representations,” J. Math. Phys. 12, 1772–1783 (1971).
[CrossRef]

Namias, V.

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

Onural, L.

Ozaktas, H. M.

M. A. Kutay, H. M. Ozaktas, O. Arikan, L. Onural, “Optimal filter in fractional Fourier domains,” IEEE Trans. Signal Process. 45, 1129–1143 (1997).
[CrossRef]

B. Barshan, M. A. Kutay, H. M. Ozaktas, “Optimal filtering with linear canonical transformations,” Opt. Commun. 135, 32–36 (1997).
[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]

H. M. Ozaktas, D. Mendlovic, “Every Fourier optical system is equivalent to consecutive fractional-Fourier-domain filtering,” Appl. Opt. 35, 3167–3170 (1996).
[CrossRef] [PubMed]

H. M. Ozaktas, B. Barshan, D. Mendlovic, L. Onural, “Convolution, filtering, and multiplexing in fractional Fourier domains and their rotation to chirp and wavelet transform,” J. Opt. Soc. Am. A 11, 547–559 (1994).
[CrossRef]

Pei, S. C.

S. C. Pei, J. J. Ding, “Closed form discrete fractional and affine Fourier transforms,” IEEE Trans. Signal Process. 48, 1338–1353 (2000).
[CrossRef]

S. C. Pei, J. J. Ding, “Relations between the fractional operations and the Wigner distribution, ambiguity function,” submitted to IEEE Trans. Signal Process.

Pellat-Finet, P.

P. Pellat-Finet, G. Bonnet, “Fractional order Fourier transform and Fourier optics,” Opt. Commun. 111, 141–154 (1994).
[CrossRef]

Quesne, C.

M. Moshinsky, C. Quesne, “Linear canonical transformations and their unitary representations,” J. Math. Phys. 12, 1772–1783 (1971).
[CrossRef]

Sheridan, J. T.

Sicre, E. E.

Wolf, K. B.

K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, New York, 1979), Chap. 9.

Wu, L.

L. Yu, M. Huang, L. Wu, Y. Lu, W. Huang, M. Chen, Z. Zhu, “Fractional Fourier transform and the elliptic gradient-index medium,” Opt. Commun. 152, 23–25 (1998).
[CrossRef]

Yu, L.

L. Yu, M. Huang, L. Wu, Y. Lu, W. Huang, M. Chen, Z. Zhu, “Fractional Fourier transform and the elliptic gradient-index medium,” Opt. Commun. 152, 23–25 (1998).
[CrossRef]

Zalevsky, H. M.

Zalevsky, Z.

A. W. Lohmann, D. Mendlovic, Z. Zalevsky, “Fractional Hilbert transform,” Opt. Lett. 21, 281–283 (1996).
[CrossRef] [PubMed]

A. W. Lohmann, Z. Zalevsky, D. Mendlovic, “Synthesis of pattern recognition filters for fractional Fourier processing,” Opt. Commun. 128, 199–204 (1996).
[CrossRef]

Zayed, A. I.

A. I. Zayed, “A convolution and product theorem for the fractional Fourier transform,” IEEE Signal Process. Lett. 5, 101–103 (1998).
[CrossRef]

Zhu, Z.

L. Yu, M. Huang, L. Wu, Y. Lu, W. Huang, M. Chen, Z. Zhu, “Fractional Fourier transform and the elliptic gradient-index medium,” Opt. Commun. 152, 23–25 (1998).
[CrossRef]

Appl. Opt. (3)

IEEE Signal Process. Lett. (2)

L. B. Almeida, “Product and convolution theorems for the fractional Fourier transform,” IEEE Signal Process. Lett. 4, 15–17 (1997).
[CrossRef]

A. I. Zayed, “A convolution and product theorem for the fractional Fourier transform,” IEEE Signal Process. Lett. 5, 101–103 (1998).
[CrossRef]

IEEE Trans. Signal Process. (4)

M. A. Kutay, H. M. Ozaktas, O. Arikan, L. Onural, “Optimal filter in fractional Fourier domains,” IEEE Trans. Signal Process. 45, 1129–1143 (1997).
[CrossRef]

S. C. Pei, J. J. Ding, “Closed form discrete fractional and affine Fourier transforms,” IEEE Trans. Signal Process. 48, 1338–1353 (2000).
[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]

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

J. Inst. Math. Appl. (1)

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

J. Math. Phys. (1)

M. Moshinsky, C. Quesne, “Linear canonical transformations and their unitary representations,” J. Math. Phys. 12, 1772–1783 (1971).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Commun. (4)

B. Barshan, M. A. Kutay, H. M. Ozaktas, “Optimal filtering with linear canonical transformations,” Opt. Commun. 135, 32–36 (1997).
[CrossRef]

L. Yu, M. Huang, L. Wu, Y. Lu, W. Huang, M. Chen, Z. Zhu, “Fractional Fourier transform and the elliptic gradient-index medium,” Opt. Commun. 152, 23–25 (1998).
[CrossRef]

P. Pellat-Finet, G. Bonnet, “Fractional order Fourier transform and Fourier optics,” Opt. Commun. 111, 141–154 (1994).
[CrossRef]

A. W. Lohmann, Z. Zalevsky, D. Mendlovic, “Synthesis of pattern recognition filters for fractional Fourier processing,” Opt. Commun. 128, 199–204 (1996).
[CrossRef]

Opt. Eng. (2)

L. M. Bernardo, “ABCD matrix formalism of fractional Fourier optics,” Opt. Eng. 35, 732–740 (1996).
[CrossRef]

A. M. Almanasreh, M. G. Abushagur, “Fractional correlation based on the modified fractional order Fourier transform,” Opt. Eng. 37, 175–184 (1998).
[CrossRef]

Opt. Lett. (2)

Other (2)

S. C. Pei, J. J. Ding, “Relations between the fractional operations and the Wigner distribution, ambiguity function,” submitted to IEEE Trans. Signal Process.

K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, New York, 1979), Chap. 9.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (11)

Fig. 1
Fig. 1

Implementation of a LCT with two cylinder lenses and one free space.

Fig. 2
Fig. 2

Implementation of a LCT with one cylinder lens and two free spaces.

Fig. 3
Fig. 3

Optical implementation (OPI) for the filter designed by the LCT. For b>0: Left system, the OPI of the LCT with parameters {a, b, c, d}; right system, the OPI of the LCT with parameters {-d, b, c,-a}. For b<0: left system, the OPI of the LCT with parameters {-a,-b,-c,-d}; right system, the OPI of the LCT with parameters {d,-b,-c, a}.

Fig. 4
Fig. 4

Optical implementation of the fractional filter designed by the type 1 SFRFT.

Fig. 5
Fig. 5

Optical implementation of the type 2 SFRFT (tan α>0).

Fig. 6
Fig. 6

Filter designed by the type 2 SFRFT (tan α>0).

Fig. 7
Fig. 7

Filter designed by the type 2 SFRFT when tan α<0.

Fig. 8
Fig. 8

GRIN medium and directions of the x, y, and z axes.

Fig. 9
Fig. 9

GRIN medium implementation of the fractional filter designed by the type 3 SFRFT.

Fig. 10
Fig. 10

This system contains two spherical disks and one free space.

Fig. 11
Fig. 11

Radar system to implement the fractional filter (designed by type 4 SFRFT).

Tables (1)

Tables Icon

Table 1 Comparison of Implementation Complexity of Fourier-Transform Types Analyzed in This Study

Equations (146)

Equations on this page are rendered with MathJax. Learn more.

OFα[f(t)]=1-j cot α2π1/2 expj2u2 cot α×- exp-jut csc α+j2t2 cot αf(t)dt.
OFα{OFβ[f(t)]}=OFβ{OFα[f(t)]}=OFα+β[f(t)].
OF(a,b,c,d)[f(t)]=1j2πb1/2 expj2dbu2×- exp-jbut+j2abt2f(t)dt,
b0,
OF(a,0,c,d)[f(t)]=(d)1/2 expj2cdu2f(du)b=0,
ad-bc=1
OF(a2, b2, c2, d2){OF(a1, b1, c1, d1)[f(t)]}=OF(e, f, g, h)[f(t)],
efgh=a2b2c2d2a1b1c1d1.
Y(a,b,c,d)(m)=(j2πb)-1/2Δ expj2dbm2Δu2×n=-NN exp-jbnmΔuΔt+j2abn2Δt2y(n),
y(n)=f(nΔt),Y(a,b,c,d)(m)=F(a,b,c,d)(mΔu), n, m=-N,-N+1,, N.
ΔtΔu=2πb/(2N+1),
Y(a,b,c,d)(m)=(j2πb)-1/2Δt expj2dbm2Δu2×n=-NN exp-j2πmn2N+1+j2abn2Δt2×y(n).
Yα(m)=1-j cot α2π1/2Δt expj2m2 cot αΔu2×n=-NN exp-j2πmn2N+1+j2n2 cot αΔt2×y(n),
ΔtΔu=2π sin α/(2N+1).
Oconvα[x(t), y(t)]=OF-α{OFα[x(t)]OFα{y[(t)]}.
z(t)=OF-α{OFα[x(t)]H(u)},
Oconv(a,b,c,d)[x(t), y(t)]
=OF(d,-b,-c,a){OF(a,b,c,d)[x(t)]OF(a,b,c,d)[y(t)]},
z(t)=OF(d,-b,-c,a){OF(a,b,c,d)[x(t)]H(u)}.
b/a
z1(t)=OF(d1,-b1,-c1,a1){OF(a1, b1, c1, d1)[x(t)]H1(u)},
z2(t)=OF(d2,-b2,-c2,a2){OF(a2, b2, c2, d2)[x(t)]H2(u)}.
z1(t)=1j2πb1exp-j2a1b1t2- expjb1u(t-k)×H1(u)du- expj2a1b1k2x(k)dk,
z2(t)=1j2πb2exp-j2a2b2t2- expjb2u(t-k)×H2(u)du- expj2a2b2k2x(k)dk.
a1b1=a2b2,b1, b20,
H2(u)=H1(b1u/b2),
z2(t)=1j2πb2exp-j2a1b1t2- expjb2u(t-k)×H1b1b2udu- expj2a1b1k2x(k)dk=1j2πb1exp-j2a1b1t2- expjb1u(t-k)×H1(s)ds- expj2a1b1k2x(k)dk=z1(t).
a/b=cot α.
b=1(fixed value of ΔtΔu),
d=0(savingonechirp-multiplicationoperation),
a/b=cot α,
abcd=cot α1-10.
OF(1)α[f(t)]=(j2π)-1/2×- exp-jut+j2t2 cot αf(t)dt,
OIF(1)α[Fα(u)]=j2π1/2 exp-j2u2 cot α×- exp(jut)Fα(u)dt.
OIF(1)α{OF(1)α[f(t)]}=f(t)forallα.
Yα(m)=(j2π)-1/2Δtn=-NN exp-j2πmn2N+1+j2n2 cot αΔt2y(n),
 y(n)=f(nΔt),Yα(m)=Fα(mΔu), n, m=-N,-N+1,, N,ΔtΔu=2π/(2N+1),
y(n)=j2π1/21(2N+1)Δtexp-j2n2 cot αΔt2×m=-NN expj2πmn2N+1Yα(m).
z(t)=OIF(1)α{OF(1)α[x(t)]OF(1)α[y(t)]};
z(t)=j-1/2(2π)-3/2 exp-j2t2 cot α- exp(jut)×- exp-juk+j2k2 cot αx(k)dk×- exp-jus+j2s2 cot αy(s)dsdu=(j2π)-1/2 exp-j2t2 cot α--δ(t-k-s)×expj2(k2+s2)cot αx(k)y(s)dsdk=(j2π)-1/2 exp-j2t2 cot α×- expj2[k2+(t-k)2]cot α×x(k)y(t-k)dsdk=(j2π)-1/2- exp[jk(k-t)cot α]×x(k)y(t-k)dsdk,
z(t)=exp-j2cot αt2Convexpj2t2 cot αx(t),expj2t2 cot αy(t),
f1=kb(1-a),d0=kb,f2=kb(1-d),
f1=f2=k sin α(1-cos α),d0=k sin α.
b>0.
d1=k(d-1)c,f0=-kc,d2=k(a-1)c,
d1=d2=k(csc α-cot α),f0=k csc α.
a1,d1,b>0;
a=d=1,b=0;
a1,d1,ad<1,b>0;
a1,d1,ad>1,b<0.
for Fig.1f1=k(1-cot a)-1,d0=k,f2=k;
for Fig.2d1=k,f0=k,d2=k(1-cot α).
for Fig.1f1=k,d0=k,f2=k(1+cot α)-1;
for Fig.2d1=k(1+cot α),f0=k,d2=k.
-1<cot α<1.
Fβ(u)=[j2π(cot α-cot β)]-1/2×- expj(u-t)22(cot α-cot β)Fα(t)dt,
Fα(u)=OF(1)α[f(t)],Fβ(u)=OF(1)β[f(t)].
cot β1-10=1cot α-cot β01cot α1-10.
OF(1)α[f(t-τ)]=exp(-jτu)Fα(u-τ cot α).
OF(1)α[exp(jvt)f(t)]=Fα(u-v),
OF(1)α[f(t)]=cot α Fα(u)+juFα(u).
OF(1)α[-jtf(t)]=Fα(u),
OF(1)α[f(t)/t]=-j-uFα(p)dp,
a=1,d=-1,a/b=cot α,
OF(2)α[f(t)]=cot αj2π1/2 exp-j2u2 cot α×- exp-jut cot α+j2t2 cot α×f(t)dt.
OIF(2)α[Fα(u)]=j cot α2π1/2 exp-j2t2 cot α×- expjut cot α+j2u2 cot α×Fα(u)dt.
forFig.1f1,d0=k tan α,
f2=(k/2)tan α;
forFig.2d1=k tan α,f0=(k/2)tan α,
d2=0.
d=k tan α,f=(k/2)tan α.
f=(k/2)|tan α|,d=k|tan α|.
n2(x, y)=n02[1-(nx/n0)x2-(ny/n0)y2],
nx, nyn0.
fL(x, y)=exp-j2πn0LλOFx(ax,bx,cx,dx)×{OFy(ay,by,cy,dy)[f0(x, y)]}.
axbxcxdx=cos ϕwx sin ϕ-sin ϕwxcos ϕ,
aybycydy=cos φwy sin φ-sin φwycos φ,
wx=(1/k)(nxn0)-1/2,wy=(1/k)(nyn0)-1/2,
ϕ=L(nx/n0)1/2,φ=L(ny/n0)1/2.
OF(3)ϕ[f(t)]=OF(cos ϕ,wx sin ϕ,-sin ϕ/wx,cos ϕ)[f(t)],
ϕ0,i.e.,L0.
OF(3)ϕ[f(t)]=csc ϕj2πwx1/2 expj2wxu2 cot ϕ×- exp-jutcsc ϕwx+j2wxt2 cot ϕ×f(t)dt.
b/a=wx tan ϕ.
α=tan-1(wx tan ϕ)orϕ=tan-1(wx-1 tan α).
L=(n0/nx)1/2 tan-1(wx-1 tan α).
cos ϕwx sin ϕ-wx-1 sin ϕcos ϕ-1=cos ϕ-wx sin ϕwx-1 sin ϕcos ϕ=cos(-ϕ)wx sin(-ϕ)-wx-1 sin(-ϕ)cos(-ϕ),
OF(3)2π-ϕ{OF(3)ϕ[f(t)]}=f(t).
OF(3)ϕ{OF(3)φ[f(t)]}=OF(3)φ{OF(3)ϕ[f(t)]}=OF(3)ϕ+φ[f(t)],
OF(3)ϕ[f(t)]=OF(3)ϕ+2π[f(t)].
x0(t)=OF(3)2π-ϕ{OF(3)ϕ[xi(t)]H(u)},
x0(-t)=OF(3)π-ϕ{OF(3)ϕ[xi(t)]H(u)},ϕ<π,
L1=ϕ(n0/nx)1/2,L2=(π-ϕ)(n0/nx)1/2.
FB(k, h)=exp(-j2πDλ-1)OSx(RA,RB,D)×{OSy(RA,RB,D)[FA(x, y)]},
OSx(RA,RB,D)[f(x)]=jλD exp-jπλ(RB-1+D-1)s2- expj2πλDsx+jπλ(RA-1-D-1)s2f(x)dx,D0,
OSx(RA,RB,0)[f(x)]=expjπλ(RA-1-RB-1)s2f(s),D=0,
abcd=1-RA-1D-D/kk(RA-1-RB-1+RA-1RB-1D)1+RB-1D.
b0.
ba=k-1(RA-1-D-1)-1.
RA=D/2
RB.
abcd=-1-D/k2k/D1,
OF(4)(D)[f(x)]=jλD exp-jk2Ds2×- expjkDsx+jk2Dx2f(x)dx.
D=k tan α.
f(-x)=OF(4)(D){OF(4)(D)[f(x)]},
f0(-x)=OF(4)(D){OF(4)(D)(fi(x))H(s)}.
Ocorrp,q,r[x(t), y(t)]=OFrOFp[x(t)]OFq[y(t)]¯,
cot p+cot r=cot q.
Ocorr(a,b,c,d),(e,f,g,h)(m,n,s,v)[x(t), y(t)]  =OF(m,n,s,v)OF(a,b,c,d)[x(t)]OF(e,f,g,h)[y(t)]¯  .
z(t)=12πj2πbfn1/2 expjv2nt2×- exp-jnut+jm2nu2expja2bu2×- exp-jbut+ja2bu2x(τ)dτ exp-jk2 fu2×- expjfuη-je2 fη2y(n)¯dηdu=12πj2πbfn1/2 expjv2nt2×-- expja2bτ2-je2 fη2×-expju-tn-τb+ηf×expj2nm+db-hfu2dux(τ)y(η)¯dτ dη.
db+mn=hf.
z(t)=j2πbfn1/2 expjv2nt2×-- expja2bτ2-je2 fη2×δ-tn-τb+ηfx(τ)y(η)¯dτdη=jf2πbn1/2 expjv2nt2×- expja2bτ2-jef2tn+τb2×x(τ)y*ftn+fτbdτ=jf2πbn1/2 expj(vn-ef)2n2t2×- expj(ab-ef)2b2τ2-jefnbtτ×x(τ)y*ftn+fτbdτ,
|z(t)|=|f|2π|bn|1/2- expj(ab-ef)2b2τ2-jefnbtτx(τ)y*ftn+fτbdτ.
y(t)=xbf(t-t0).
|z(t)|=|f|2π|bn|1/2- expj(ab-ef)2b2τ2-jefnbtτx(τ)x*τ+bnt-bft0dτ,
znft0=|f|2π|bn|1/2×- expj(ab-ef)2b2τ2-jebt0τ×|x(τ)|2dτ.
znft0=|f|2π|bn|1/2- exp-jebt0τ|x(τ)|2dτ;
znft0=|f|2π|bn|1/2- expje22(ab-ef)×t0-ab-efbeτ2|x(τ)|2dτ.
ab=ef,
-πBbe<t0<πBbe,
x(t)0,|t|>B,
abef,
ae-fbB-2πbeae-fb1/2<t0<2πbeae-fb1/2-ae-fbB,
Ocorrb,f[x(t), y(t)]=FTOF(1,b,-b-1,0)[x(t)]×OF(1,f,-f-1,0)[y(t)]¯.
|z(f-1t0)|=|f|2π|b|1/2- expj(b-f)2b2τ2-jbt0τ×|x(τ)|2dτ.
b=f,
y(t)=x(t-t0),
-bπB<t0<bπB.
bf,
y(t)=x[b(t-t0)/f],
1-fbB-2πb1-fb1/2<t0
<2πb1-fb1/2-1-fbB,
b=TB/2π,
b=12π(1-σ)T2+|1-σ|B2.
f=bσ.
Fb(u)=OF(5)b[f(t)]=(j2πb)-1/2- exp-jbut+j2bt2f(t)dt,
f(t)=OIF(5)b[Fb(u)]=j2πb1/2 exp-j2bt2- expjbtuFb(u)du.
Ocorrb,f[x(t), y(t)]=FTOF(5)b[x(t)]OF(5)f[y(t)]¯,
OF(5)b(OF(5)b{OF(5)b[h(t)]})=h(-t)anyb,
OIF(5)b[h(u)]=OF(5)b{OF(5)b[h(-u)]}anyb.
OF(5)b[exp(jvt)f(t)]=Fb(u-bv).
OF(5)b[tf(t)]=jbFb(u).
Fa(u)=(j2π)-1/2- exp-jut+j2cot αt2f(t)dt.
Fa(u)=cot αj2π1/2 exp-j2u2 cot α×- exp-jut cot α+j2t2 cot αf(t)dt.
Fϕ(u)=csc ϕj2πwx1/2 expj2wxu2 cot ϕ×- exp-jutcsc ϕwx+j2wxt2 cot ϕ×f(t)dt,ϕ=tan-1(wx-1 tan α).
FD(u)=jλD exp-jk2Du2- expjkDut+jk2Dt2×f(t)dt,D=2π tan α/λ.
Fb(u)=OF(5)b[f(t)]=(j2πb)-1/2- exp-jbut+j2bt2f(t)dt.

Metrics