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
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
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.
Received: 15 April 2003
Revised: 05 June 2003
Accepted manuscript online:
Fund: Project supported by the Multidiscipline Scientific Research Foundation of the Harbin Institute of Technology, China (Grant No HITMD200018).
Cite this article:
Ran Qi-Wen (冉启文), Yuan Lin (袁琳), Tan Li-Ying (谭立英), Ma Jing (马晶), Wang Qi (王骐) High order generalized permutational fractional Fourier transforms 2004 Chinese Physics 13 178
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