中国物理B ›› 2004, Vol. 13 ›› Issue (2): 178-186.doi: 10.1088/1009-1963/13/2/010

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High order generalized permutational fractional Fourier transforms

袁琳1, 冉启文2, 谭立英3, 马晶3, 王骐3   

  1. (1)Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China; (2)Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China; State Key Laboratory of Tunable Laser Technology, Research Institute of Optic-Electronics, Harbin Institute of Technology, Harbin 150001, China; (3)State Key Laboratory of Tunable Laser Technology, Research Institute of Optic-Electronics, Harbin Institute of Technology, Harbin 150001, China
  • 收稿日期:2003-04-15 修回日期:2003-06-05 出版日期:2004-02-06 发布日期:2005-07-06
  • 基金资助:
    Project supported by the Multidiscipline Scientific Research Foundation of the Harbin Institute of Technology, China (Grant No HITMD200018).

High order generalized permutational fractional Fourier transforms

Ran Qi-Wen (冉启文)ab, Yuan Lin (袁琳)a, Tan Li-Ying (谭立英)b, Ma Jing (马晶)b, Wang Qi (王骐)b   

  1. a Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China; b State Key Laboratory of Tunable Laser Technology, Research Institute of Optic-Electronics, Harbin Institute of Technology, Harbin 150001, China
  • Received:2003-04-15 Revised:2003-06-05 Online:2004-02-06 Published:2005-07-06
  • Supported by:
    Project supported by the Multidiscipline Scientific Research Foundation of the Harbin Institute of Technology, China (Grant No HITMD200018).

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摘要： We generalize the definition of the fractional Fourier transform (FRFT) by extending the new definition proposed by Shih. The generalized FRFT, called the high order generalized permutational fractional Fourier transform (HGPFRFT), is a generalized permutational transform. It is shown to have arbitrary natural number M periodic eigenvalues not only with respect to the order of Hermite－Gaussian functions but also to the order of the transform. This HGPFRFT will be reduced to the original FRFT proposed by Namias and Liu's generalized FRFT and Shih's FRFT at the three limits with M=+∞, M=4k(k is a natural number), and M=4, respectively. Therefore the HGPFRFT introduces a comprehensive approach to Shih's FRFT and the original definition. Some important properties of HGPFRFT are discussed. Lastly the results of computer simulations and symbolic representations of the transform are provided.

关键词: Fourier transform, fractional Fourier transform, permutational fractional Fourier transform

Abstract: We generalize the definition of the fractional Fourier transform (FRFT) by extending the new definition proposed by Shih. The generalized FRFT, called the high order generalized permutational fractional Fourier transform (HGPFRFT), is a generalized permutational transform. It is shown to have arbitrary natural number M periodic eigenvalues not only with respect to the order of Hermite－Gaussian functions but also to the order of the transform. This HGPFRFT will be reduced to the original FRFT proposed by Namias and Liu's generalized FRFT and Shih's FRFT at the three limits with $M=+\infty, M=4k$(k is a natural number), and $M=4$, respectively. Therefore the HGPFRFT introduces a comprehensive approach to Shih's FRFT and the original definition. Some important properties of HGPFRFT are discussed. Lastly the results of computer simulations and symbolic representations of the transform are provided.

Key words: Fourier transform, fractional Fourier transform, permutational fractional Fourier transform

中图分类号:  (Fourier optics)

  • 42.30.Kq
02.10.Ud (Linear algebra)