High order generalized permutational fractional Fourier transforms
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 HermiteGaussian 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.
 Publication:

Chinese Physics
 Pub Date:
 February 2004
 DOI:
 10.1088/10091963/13/2/010
 Bibcode:
 2004ChPhy..13..178R
 Keywords:

 Fourier transform;
 fractional Fourier transform;
 permutational fractional Fourier transform