Abstract

We show that a fractional version of the finite Fourier transform may be defined by using prolate spheroidal wave functions of order zero. The transform is linear and additive in its index and asymptotically goes over to Namias’s definition of the fractional Fourier transform. As a special case of this definition, it is shown that the finite Fourier transform may be inverted by using information over a finite range of frequencies in Fourier space, the inversion being sensitive to noise. Numerical illustrations for both forward (fractional) and inverse finite transforms are provided.

© 2004 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. V. Namias, “The fractional order Fourier transform and its applications to quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
    [CrossRef]
  2. H. M. Ozaktas, D. Mendlovic, “Fourier transforms of fractional order and their optical interpretation,” Opt. Commun. 101, 163–169 (1993).
    [CrossRef]
  3. D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation: I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
    [CrossRef]
  4. H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transforms and their optical implementation: II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993).
    [CrossRef]
  5. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
    [CrossRef]
  6. P. Pellat-Finet, “Fresnel diffraction and the fractional order Fourier transform,” Opt. Lett. 19, 1388–1390 (1994).
    [CrossRef] [PubMed]
  7. H. M. Ozaktas, D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–751 (1995).
    [CrossRef]
  8. H. M. Ozaktas, B. Barshan, D. Mendlovic, L. Onural, “Convolution, filtering, and multiplexing in fractional Fourier domains and their relation to chirp and wavelet transforms,” J. Opt. Soc. Am. A 11, 547–559 (1994).
    [CrossRef]
  9. C. Candan, M. A. Kutay, H. M. Ozaktas, “The discretefractional Fourier transform,” IEEE Trans. Signal Process. 48, 1329–1337 (2000).
    [CrossRef]
  10. S. Pei, J. Ding, “Simplified fractional Fourier transforms,” J. Opt. Soc. Am. A 17, 2355–2367 (2000).
    [CrossRef]
  11. H. M. Ozaktas, Z. Zalevsky, M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, Chichester, UK, 2001; this book contains a comprehensive list of publications on this subject.)
  12. K. Khare, N. George, “Sampling theory approach to prolate spheroidal wave functions,” J. Phys. A 36, 10011–10021 (2003).
    [CrossRef]
  13. C. E. Shannon, “Communication in the presence of noise,” Proc. IRE 37, 10–21 (1949).
    [CrossRef]
  14. C. Flammer, Spheroidal Wave Functions (Stanford U. Press, Stanford, Calif., 1957).
  15. J. A. Stratton, P. M. Morse, L. J. Chu, J. D. C. Little, F. J. Corbato, Spheroidal Wave Functions (Wiley, New York, 1956).
  16. D. Slepian, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—I,” Bell Syst. Tech. J. 40, 43–63 (1961).
    [CrossRef]
  17. B. R. Frieden, “Evaluation, design and extrapolation methods for optical signals, based on based on the use of prolate functions,” in Progress in Optics, Vol. IX, E. Wolf ed. (Elsevier, New York, 1971), pp. 311–407.
  18. M. Bertero, C. De Mol, “Super-resolution by data inversion,” in Progress in Optics, Vol. XXXVIE. Wolf, ed. (Elsevier, New York, 1996), pp. 129–178.
  19. D. Slepian, “Some asymptotic expansions for prolate spheroidal wave functions,” J. Math. Phys. 44, 99–140 (1965).
  20. P. M. Morse, H. Feschbach, Methods of Theoretical Physics (McGraw-Hill, London, 1953), p. 781.

2003 (1)

K. Khare, N. George, “Sampling theory approach to prolate spheroidal wave functions,” J. Phys. A 36, 10011–10021 (2003).
[CrossRef]

2000 (2)

C. Candan, M. A. Kutay, H. M. Ozaktas, “The discretefractional Fourier transform,” IEEE Trans. Signal Process. 48, 1329–1337 (2000).
[CrossRef]

S. Pei, J. Ding, “Simplified fractional Fourier transforms,” J. Opt. Soc. Am. A 17, 2355–2367 (2000).
[CrossRef]

1995 (1)

1994 (2)

1993 (4)

1980 (1)

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

1965 (1)

D. Slepian, “Some asymptotic expansions for prolate spheroidal wave functions,” J. Math. Phys. 44, 99–140 (1965).

1961 (1)

D. Slepian, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—I,” Bell Syst. Tech. J. 40, 43–63 (1961).
[CrossRef]

1949 (1)

C. E. Shannon, “Communication in the presence of noise,” Proc. IRE 37, 10–21 (1949).
[CrossRef]

Barshan, B.

Bertero, M.

M. Bertero, C. De Mol, “Super-resolution by data inversion,” in Progress in Optics, Vol. XXXVIE. Wolf, ed. (Elsevier, New York, 1996), pp. 129–178.

Candan, C.

C. Candan, M. A. Kutay, H. M. Ozaktas, “The discretefractional Fourier transform,” IEEE Trans. Signal Process. 48, 1329–1337 (2000).
[CrossRef]

Chu, L. J.

J. A. Stratton, P. M. Morse, L. J. Chu, J. D. C. Little, F. J. Corbato, Spheroidal Wave Functions (Wiley, New York, 1956).

Corbato, F. J.

J. A. Stratton, P. M. Morse, L. J. Chu, J. D. C. Little, F. J. Corbato, Spheroidal Wave Functions (Wiley, New York, 1956).

De Mol, C.

M. Bertero, C. De Mol, “Super-resolution by data inversion,” in Progress in Optics, Vol. XXXVIE. Wolf, ed. (Elsevier, New York, 1996), pp. 129–178.

Ding, J.

Feschbach, H.

P. M. Morse, H. Feschbach, Methods of Theoretical Physics (McGraw-Hill, London, 1953), p. 781.

Flammer, C.

C. Flammer, Spheroidal Wave Functions (Stanford U. Press, Stanford, Calif., 1957).

Frieden, B. R.

B. R. Frieden, “Evaluation, design and extrapolation methods for optical signals, based on based on the use of prolate functions,” in Progress in Optics, Vol. IX, E. Wolf ed. (Elsevier, New York, 1971), pp. 311–407.

George, N.

K. Khare, N. George, “Sampling theory approach to prolate spheroidal wave functions,” J. Phys. A 36, 10011–10021 (2003).
[CrossRef]

Khare, K.

K. Khare, N. George, “Sampling theory approach to prolate spheroidal wave functions,” J. Phys. A 36, 10011–10021 (2003).
[CrossRef]

Kutay, M. A.

C. Candan, M. A. Kutay, H. M. Ozaktas, “The discretefractional Fourier transform,” IEEE Trans. Signal Process. 48, 1329–1337 (2000).
[CrossRef]

H. M. Ozaktas, Z. Zalevsky, M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, Chichester, UK, 2001; this book contains a comprehensive list of publications on this subject.)

Little, J. D. C.

J. A. Stratton, P. M. Morse, L. J. Chu, J. D. C. Little, F. J. Corbato, Spheroidal Wave Functions (Wiley, New York, 1956).

Lohmann, A. W.

Mendlovic, D.

Morse, P. M.

P. M. Morse, H. Feschbach, Methods of Theoretical Physics (McGraw-Hill, London, 1953), p. 781.

J. A. Stratton, P. M. Morse, L. J. Chu, J. D. C. Little, F. J. Corbato, Spheroidal Wave Functions (Wiley, New York, 1956).

Namias, V.

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

Onural, L.

Ozaktas, H. M.

C. Candan, M. A. Kutay, H. M. Ozaktas, “The discretefractional Fourier transform,” IEEE Trans. Signal Process. 48, 1329–1337 (2000).
[CrossRef]

H. M. Ozaktas, D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–751 (1995).
[CrossRef]

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

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 transforms and their optical implementation: II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993).
[CrossRef]

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

H. M. Ozaktas, Z. Zalevsky, M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, Chichester, UK, 2001; this book contains a comprehensive list of publications on this subject.)

Pei, S.

Pellat-Finet, P.

Pollak, H. O.

D. Slepian, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—I,” Bell Syst. Tech. J. 40, 43–63 (1961).
[CrossRef]

Shannon, C. E.

C. E. Shannon, “Communication in the presence of noise,” Proc. IRE 37, 10–21 (1949).
[CrossRef]

Slepian, D.

D. Slepian, “Some asymptotic expansions for prolate spheroidal wave functions,” J. Math. Phys. 44, 99–140 (1965).

D. Slepian, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—I,” Bell Syst. Tech. J. 40, 43–63 (1961).
[CrossRef]

Stratton, J. A.

J. A. Stratton, P. M. Morse, L. J. Chu, J. D. C. Little, F. J. Corbato, Spheroidal Wave Functions (Wiley, New York, 1956).

Zalevsky, Z.

H. M. Ozaktas, Z. Zalevsky, M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, Chichester, UK, 2001; this book contains a comprehensive list of publications on this subject.)

Bell Syst. Tech. J. (1)

D. Slepian, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—I,” Bell Syst. Tech. J. 40, 43–63 (1961).
[CrossRef]

IEEE Trans. Signal Process. (1)

C. Candan, M. A. Kutay, H. M. Ozaktas, “The discretefractional Fourier transform,” IEEE Trans. Signal Process. 48, 1329–1337 (2000).
[CrossRef]

J. Inst. Math. Appl. (1)

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

J. Math. Phys. (1)

D. Slepian, “Some asymptotic expansions for prolate spheroidal wave functions,” J. Math. Phys. 44, 99–140 (1965).

J. Opt. Soc. Am. A (6)

J. Phys. A (1)

K. Khare, N. George, “Sampling theory approach to prolate spheroidal wave functions,” J. Phys. A 36, 10011–10021 (2003).
[CrossRef]

Opt. Commun. (1)

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

Opt. Lett. (1)

Proc. IRE (1)

C. E. Shannon, “Communication in the presence of noise,” Proc. IRE 37, 10–21 (1949).
[CrossRef]

Other (6)

C. Flammer, Spheroidal Wave Functions (Stanford U. Press, Stanford, Calif., 1957).

J. A. Stratton, P. M. Morse, L. J. Chu, J. D. C. Little, F. J. Corbato, Spheroidal Wave Functions (Wiley, New York, 1956).

H. M. Ozaktas, Z. Zalevsky, M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, Chichester, UK, 2001; this book contains a comprehensive list of publications on this subject.)

P. M. Morse, H. Feschbach, Methods of Theoretical Physics (McGraw-Hill, London, 1953), p. 781.

B. R. Frieden, “Evaluation, design and extrapolation methods for optical signals, based on based on the use of prolate functions,” in Progress in Optics, Vol. IX, E. Wolf ed. (Elsevier, New York, 1971), pp. 311–407.

M. Bertero, C. De Mol, “Super-resolution by data inversion,” in Progress in Optics, Vol. XXXVIE. Wolf, ed. (Elsevier, New York, 1996), pp. 129–178.

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 (6)

Fig. 1
Fig. 1

First four prolate spheroidal functions of order zero calculated by using the sampling-theorem-based approach for the case L=1 and B=1.

Fig. 2
Fig. 2

Ten highest eigenvalues for the sinc kernel on a logarithmic scale. L=1 and B=1.

Fig. 3
Fig. 3

Fractional finite Fourier transform of Λ(u) corresponding to orders α=0.25, 0.5, 0.75, 1.0. The real and imaginary parts of the transform are shown by solid and dotted curves, respectively. The first ten prolate spheroids are used for computation.

Fig. 4
Fig. 4

Inverse finite Fourier transform of sinc2(u) with use of Fourier space information over u:(-1, 1). The real and imaginary parts of the transform are shown by solid and dotted curves, respectively. The first ten prolate spheroids are used for computation.

Fig. 5
Fig. 5

Inverse finite Fourier transform of sinc2(u) with use of Fourier space information over u:(-1, 1). Tolerance in computation of an in Eq. (12) is set to 0.1. The series in Eq. (21) is truncated to n=5. Only the real part of the recovery is shown.

Fig. 6
Fig. 6

Inverse finite Fourier transform of sinc2(u) with use of Fourier space information over u:(-1, 1). Tolerance in computation of an in Eq. (12) is set to 0.1. The series in Eq. (21) is truncated to n=6. Only the real part of the recovery is shown.

Tables (1)

Tables Icon

Table 1 Ten Highest Eigenvalues of the Sinc Kernel for L=1 and B=1 Computed with the Sampling-Theorem-Based Approach

Equations (47)

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

F{g(x)}=-LLdx exp(-i2πfx)g(x).
λnϕn(x)=-LLdxsinc[2B(x-x)]ϕn(x).
sinc(x)=sin πxπx.
-dxϕn(x)ϕl(x)=δn,l,
-dxϕn(x)ϕl(x)=2Bλnδn,l,
-LLdx exp(-i2πfx)ϕn(x)=i-n(2Lλn)1/2ϕn(Lf/B).
exp(-i2πfx)=LBn=0i-n(2Lλn)1/2 ϕn(x)ϕnLfB,
for|x|<L
δ(x-x)=12Bn=01λn ϕn(x)ϕn(x),
for|x|,|x|<L.
u=BL x,u=LB f.
-BLBLdu exp(-i2πuu)ϕnLB u
=i-n(2Bλn)1/2ϕnLB u.
gLB u=n=0anϕnLB u,
an=LB12Bλn-BLBLdugLB uϕnLB u.
FgLB u=n=0ani-n(2Bλn)1/2ϕnLB u,
FαgLB u=-BLBLduK(u, u;α)gLB u,
K(u, u;α)=LBn=0i-nα(2Bλn)α/2-1×ϕnLB uϕnLB u.
K(u, u;0)=LBn=012Bλn ϕnLB uϕnLB u=δ(u-u),
for|u|,|u|<BL.
-BLBLdugLB uK(u, u; 0)=gLB u,
for|u|<BL.
K(u, u; 1)=exp(-i2πuu),
K(u, u; α1, α2)=-BLBLduK(u, u; α1)K(u, u; α2)
K(u, u; α1, α2)=LBn=0i-n(α1+α2)×(2Bλn)(α1+α2)/2-1ϕnLB u×ϕnLB u=K(u, u; α1+α2).
FαgLB u=n=0ani-nα(2Bλn)α/2ϕnLB u.
K(u, u; -1)
=LBn=0in(2Bλn)-3/2ϕnLB uϕnLB u
K(u, u; α)=n=0i-nα(2Bλn)α/2-1ϕn(u)ϕn(u).
ϕn(u)=2BλnLNnDn4πBL u+j=1(2c)-jk=-2j2jAkjDn+2k4πBL u.
Dn(y)=2-n/2exp(-y2/4)Hn(y/2),
Nn=12BL n!1+127(4c2)×(n4+2n3+23n2+12).
2Bλn=1-4 πn! cn+1/2exp(-2c)×1-132c (6n2-2n+3).
K(u, u; α)=2exp[-π(u2+u2)]×n=0i-nα2nn! Hn(2πu)Hn(2πu).
K(u, u; α)=exp[-i(π4sgn(sin ϕ)-ϕ2)]|sin ϕ|1/2×exp[iπ(u2cot ϕ-2uucsc ϕ+u2cot ϕ)],
Amk=-LLdxsinc(2Bx-m)sinc(2Bx-k)
un=[ϕn(m/2B)]T,
ϕn(x)=m=-ϕn(m/2B)sinc(2Bx-m).
g(u)=Λ(u)=1-|u|,|u|10,otherwise 
Φn(f )=-LLdx exp(-i2πfx)ϕn(x).
2Bλnϕn(x)=-LLdxϕn(x)-BBdfexp[i2πf(x-x)]=-BBdfexp(i2πfx)Φn(f).
BL λnΦn(f )=-BBdfsinc[2L(f-f)]Φn(f).
Φn(f )=βnϕnLfB.
-BBdf exp(-i2πfx)Φn(f )
=2B-LLdx sinc[2B(x+x)]ϕn(x)=2Bλnϕn(-x).
βn-BBdf exp(-i2πfx)ϕnLfB=βn2BL ϕn(x).
βn=i-n(2Lλn)1/2.

Metrics