Abstract

We have generalized the Hilbert transform by defining the fractional Hilbert transform (FHT) operation. In the first stage, two different approaches for defining the FHT are suggested. One is based on modifying only the spatial filter, and the other proposes using the fractional Fourier plane for filtering. In the second stage, the two definitions are combined into a fractional Hilbert transform, which is characterized by two parameters. Computer simulations are presented.

© 1996 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. KastlerRev. Opt. 29, 308 (1950).
  2. H. WolterAnn. Phys. 7, 341 (1950).
    [Crossref]
  3. D. Hauk, A. LohmannOptik 15, 275 (1958).
  4. S. Lowenthal, Y. BelvauxAppl. Phys. Lett. 11, 49 (1967).
    [Crossref]
  5. J. K. T. Eu, A. W. LohmannOpt. Commun. 9, 257 (1973).
    [Crossref]
  6. B. R. Brown, A. W. LohmannAppl. Opt. 5, 967 (1966).
    [Crossref] [PubMed]
  7. A. W. Lohmann, D. P. ParisAppl. Opt. 6, 1739 (1967).
    [Crossref] [PubMed]
  8. D. Mendlovic, H. M. OzaktasJ. Opt. Soc. Am. A 10, 1875 (1993).
    [Crossref]
  9. D. Mendlovic, H. M. Ozaktas, A. W. LohmannAppl. Opt. 34, 303 (1995).
    [Crossref] [PubMed]
  10. M. Gedzioriwski, J. GarciaOpt. Commun. 119, 207 (1995).
    [Crossref]

1995 (2)

1993 (1)

1973 (1)

J. K. T. Eu, A. W. LohmannOpt. Commun. 9, 257 (1973).
[Crossref]

1967 (2)

A. W. Lohmann, D. P. ParisAppl. Opt. 6, 1739 (1967).
[Crossref] [PubMed]

S. Lowenthal, Y. BelvauxAppl. Phys. Lett. 11, 49 (1967).
[Crossref]

1966 (1)

1958 (1)

D. Hauk, A. LohmannOptik 15, 275 (1958).

1950 (2)

A. KastlerRev. Opt. 29, 308 (1950).

H. WolterAnn. Phys. 7, 341 (1950).
[Crossref]

Belvaux, Y.

S. Lowenthal, Y. BelvauxAppl. Phys. Lett. 11, 49 (1967).
[Crossref]

Brown, B. R.

Eu, J. K. T.

J. K. T. Eu, A. W. LohmannOpt. Commun. 9, 257 (1973).
[Crossref]

Garcia, J.

M. Gedzioriwski, J. GarciaOpt. Commun. 119, 207 (1995).
[Crossref]

Gedzioriwski, M.

M. Gedzioriwski, J. GarciaOpt. Commun. 119, 207 (1995).
[Crossref]

Hauk, D.

D. Hauk, A. LohmannOptik 15, 275 (1958).

Kastler, A.

A. KastlerRev. Opt. 29, 308 (1950).

Lohmann, A.

D. Hauk, A. LohmannOptik 15, 275 (1958).

Lohmann, A. W.

Lowenthal, S.

S. Lowenthal, Y. BelvauxAppl. Phys. Lett. 11, 49 (1967).
[Crossref]

Mendlovic, D.

Ozaktas, H. M.

Paris, D. P.

Wolter, H.

H. WolterAnn. Phys. 7, 341 (1950).
[Crossref]

Ann. Phys. (1)

H. WolterAnn. Phys. 7, 341 (1950).
[Crossref]

Appl. Opt. (3)

Appl. Phys. Lett. (1)

S. Lowenthal, Y. BelvauxAppl. Phys. Lett. 11, 49 (1967).
[Crossref]

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

Opt. Commun. (2)

J. K. T. Eu, A. W. LohmannOpt. Commun. 9, 257 (1973).
[Crossref]

M. Gedzioriwski, J. GarciaOpt. Commun. 119, 207 (1995).
[Crossref]

Optik (1)

D. Hauk, A. LohmannOptik 15, 275 (1958).

Rev. Opt. (1)

A. KastlerRev. Opt. 29, 308 (1950).

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

Optical setup for performing the conventional HT. Later, the same configuration is used for performing the FHT.

Fig. 2
Fig. 2

Block diagram for performing the FHT based on the FRT operation.

Fig. 3
Fig. 3

Block diagram that demonstrates a combination of the two fractional Hilbert definitions with the two free parameters P and Q.

Fig. 4
Fig. 4

Computer simulation of the case when Q = 1.

Fig. 5
Fig. 5

Computer simulation of the case when Q = 0.8.

Fig. 6
Fig. 6

Computer simulation of the case when Q = 0.5.

Equations (7)

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

H ˜ 1 ( v ) = exp ( + i π 2 ) S ( v ) + exp ( i π 2 ) S ( v ) ,
H ˜ P ( v ) = exp ( + i ϕ ) S ( v ) + exp ( i ϕ ) S ( v ) ,
ϕ = P ( π / 2 ) .
H ˜ P ( v ) = cos ϕ H ˜ 0 ( v ) + sin ϕ H ˜ 1 ( v ) ,
V P ( x ) = U ˜ ( v ) H ˜ P ( v ) exp ( 2 π i v x ) d v .
V P ( x ) = cos ϕ V 0 ( x ) + sin ϕ V 1 ( x ) .
V Q ( x ) = Q ( H ^ 1 Q [ U 0 ( x ) ] } ,

Metrics