## Improved fast fractional-Fourier-transform algorithm

JOSA A, Vol. 21, Issue 9, pp. 1677-1681 (2004)

http://dx.doi.org/10.1364/JOSAA.21.001677

Enhanced HTML Acrobat PDF (179 KB)

### Abstract

Through the optimization of the main interval of the fractional order, an improved fast algorithm for numerical calculation of the fractional Fourier transforms is proposed. With this improved algorithm, the fractional Fourier transforms of a rectangular function and a Gaussian function are calculated. Its calculation errors are compared with those calculated with the previously published algorithm, and the results show that the calculation accuracy of the improved algorithm is much higher.

© 2004 Optical Society of America

**OCIS Codes**

(070.2590) Fourier optics and signal processing : ABCD transforms

(350.6980) Other areas of optics : Transforms

**History**

Original Manuscript: November 20, 2003

Revised Manuscript: March 15, 2004

Manuscript Accepted: March 15, 2004

Published: September 1, 2004

**Citation**

Xingpeng Yang, Qiaofeng Tan, Xiaofeng Wei, Yong Xiang, Yingbai Yan, and Guofan Jin, "Improved fast fractional-Fourier-transform algorithm," J. Opt. Soc. Am. A **21**, 1677-1681 (2004)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-21-9-1677

Sort: Year | Journal | Reset

### References

- V. Namias, “The fractional Fourier transform and its application in quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980). [CrossRef]
- B. Yurke, W. Schleich, D. F. Walls, “Quantum superposition generated by quantum nondemolition measurements,” Phys. Rev. A 42, 1703–1711 (1990). [CrossRef] [PubMed]
- W. Vogel, W. Schleich, “Phase distribution of a quantum state without using phase states,” Phys. Rev. A 44, 7642–7646 (1991). [CrossRef] [PubMed]
- M. Beck, M. G. Raymer, I. A. Walmsley, V. Kong, “Chronocyclic tomography for measuring the amplitude and phase structure of optical pulses,” Opt. Lett. 18, 2041–2043 (1993). [CrossRef] [PubMed]
- 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]
- 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]
- A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993). [CrossRef]
- D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Graded-index fibers, Wigner-distribution functions, and the fractional Fourier transform,” Appl. Opt. 33, 6188–6193 (1994). [CrossRef] [PubMed]
- H. M. Ozaktas, D. Mendlovic, “The fractional Fourier transform as a tool for analyzing beam propagation andspherical mirror resonators,” Opt. Lett. 19, 1678–1680 (1994). [CrossRef] [PubMed]
- P. Pellat-Finet, “Fresnel diffraction and the fractional-order Fourier transform,” Opt. Lett. 19, 1388–1390 (1994). [CrossRef] [PubMed]
- L. M. Bernardo, O. D. D. Soares, “Fractional Fourier transform and Fourier optical systems,” Opt. Commun. 110, 517–522 (1994). [CrossRef]
- D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transformations and their optical implementation: part I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993). [CrossRef]
- H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transformations and their optical implementation: part II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993). [CrossRef]
- H. M. Ozaktas, O. Arikan, M. A. Kutay, “Digital computation of the fractional Fourier transform,” Signal Process. 44, 2141–2150 (1996).
- X. Deng, Y. Li, D. Fan, Y. Qiu, “A fast algorithm for fractional Fourier transforms,” Opt. Commun. 138, 270–274 (1997). [CrossRef]
- D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999). [CrossRef]
- X. Deng, B. Bihari, J. Gan, F. Zhao, R. T. Chen, “Fast algorithm for chirp transforms with zooming-in ability and its applications,” J. Opt. Soc. Am. A 17, 762–771 (2000). [CrossRef]
- A. Bultheel, H. E. Martinez Sulbaran, “Computation of the fractional Fourier transform” (2003); http://www.cs.kuleuven.ac.be/cwis/research/nalag/papers/ade/frftcomp/ccomp.pdf .
- S.-C. Pei, M.-H. Yeh, C.-C. Tseng, “Discrete fractional Fourier transform based on orthogonal projections,” IEEE Trans. Signal Process. 47, 1335–1348 (1999). [CrossRef]
- F. J. Marinho, L. M. Bernardo, “Numerical calculation of fractional Fourier transforms with a single fast-Fourier-transform algorithm,” J. Opt. Soc. Am. A 15, 2111–2116 (1998). [CrossRef]
- J. Garcia, D. Mas, R. G. Dorsch, “Fractional-Fourier-transform calculation through the fast-Fourier-transform algorithm,” Appl. Opt. 35, 7013–7018 (1996). [CrossRef] [PubMed]
- H. M. Ozaktas, Z. Zalevski, M. A. Kutay, The Fractional Fourier Transform (Wiley, New York, 2001), Chap. 6, pp. 218–220.
- J. O’Neill, “Discrete TFDs (fracft.m)”: a collection of Matlab files for time-frequency analysis (1999); ftp.mathworks.com/pub/contrib/v5/signal/DiscreteTFDs/.
- M. A. Kutay, “FracF: Fast computation of the fractional Fourier transform (1996)”; http://www.ee.bilkent.edu.tr/∼haldun/fracF.m .
- V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980). [CrossRef]

## 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.