## Simplified fractional Fourier transforms

JOSA A, Vol. 17, Issue 12, pp. 2355-2367 (2000)

http://dx.doi.org/10.1364/JOSAA.17.002355

Acrobat PDF (231 KB)

### 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

**OCIS Codes**

(070.2580) Fourier optics and signal processing : Paraxial wave optics

(070.2590) Fourier optics and signal processing : ABCD transforms

(070.6020) Fourier optics and signal processing : Continuous optical signal processing

(070.6110) Fourier optics and signal processing : Spatial filtering

**Citation**

Soo-Chang Pei and Jian-Jiun Ding, "Simplified fractional Fourier transforms," J. Opt. Soc. Am. A **17**, 2355-2367 (2000)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-17-12-2355

Sort: Year | Journal | Reset

### References

- V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
- L. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42, 3084–3091 (1994).
- M. Moshinsky and C. Quesne, “Linear canonical transformations and their unitary representations,” J. Math. Phys. 12, 1772–1783 (1971).
- K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, New York, 1979), Chap. 9.
- S. Abe and J. T. Sheridan, “Optical operations on wave functions as the Abelian subgroups of the special affine Fourier transformation,” Opt. Lett. 19, 1801–1803 (1994).
- L. M. Bernardo, “ABCD matrix formalism of fractional Fourier optics,” Opt. Eng. 35, 732–740 (1996).
- H. M. Ozaktas and D. Mendlovic, “Every Fourier optical system is equivalent to consecutive fractional-Fourier-domain filtering,” Appl. Opt. 35, 3167–3170 (1996).
- S. C. Pei and J. J. Ding, “Closed form discrete fractional and affine Fourier transforms,” IEEE Trans. Signal Process. 48, 1338–1353 (2000).
- H. M. Ozaktas, O. Arikan, M. A. Kutay, and G. Bozdagi, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150 (1996).
- A. M. Almanasreh and M. G. Abushagur, “Fractional correlation based on the modified fractional order Fourier transform,” Opt. Eng. 37, 175–184 (1998).
- M. A. Kutay, H. M. Ozaktas, O. Arikan, and L. Onural, “Optimal filter in fractional Fourier domains,” IEEE Trans. Signal Process. 45, 1129–1143 (1997).
- A. W. Lohmann, D. Mendlovic, and Z. Zalevsky, “Fractional Hilbert transform,” Opt. Lett. 21, 281–283 (1996).
- H. M. Ozaktas, B. Barshan, D. Mendlovic, and 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).
- S. C. Pei and J. J. Ding, “Relations between the fractional operations and the Wigner distribution, ambiguity function,” submitted to IEEE Trans. Signal Process.
- L. B. Almeida, “Product and convolution theorems for the fractional Fourier transform,” IEEE Signal Process. Lett. 4, 15–17 (1997).
- A. I. Zayed, “A convolution and product theorem for the fractional Fourier transform,” IEEE Signal Process. Lett. 5, 101–103 (1998).
- B. Barshan, M. A. Kutay, and H. M. Ozaktas, “Optimal filtering with linear canonical transformations,” Opt. Commun. 135, 32–36 (1997).
- L. Yu, M. Huang, L. Wu, Y. Lu, W. Huang, M. Chen, and Z. Zhu, “Fractional Fourier transform and the elliptic gradient-index medium,” Opt. Commun. 152, 23–25 (1998).
- P. Pellat-Finet and G. Bonnet, “Fractional order Fourier transform and Fourier optics,” Opt. Commun. 111, 141–154 (1994).
- D. Mendlovic, H. M. Zalevsky, and A. W. Lohmann, “Fractional correlation,” Appl. Opt. 34, 303–309 (1995).
- A. W. Lohmann, Z. Zalevsky, and D. Mendlovic, “Synthesis of pattern recognition filters for fractional Fourier processing,” Opt. Commun. 128, 199–204 (1996).
- S. Granieri, R. Arizaga, and E. E. Sicre, “Optical correlation based on the fractional Fourier transform,” Appl. Opt. 36, 6636–6645 (1997).

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.