Multiple-parameter real discrete fractional Fourier and Hartley transforms

In this paper, two new real fractional transforms with many parameters are constructed. They are the real discrete fractional Fourier transform (RDFRFT) and the real discrete fractional Hartley transform (RDFRHT). The eigenvectors of these two new transforms are all random, and they both have only two distinct eigenvalues: 1 or -1. Real eigenvectors of both two transforms are constructed from random DFT-commuting matrices. We also propose an alternative definition of RDFRHT based on a diagonal-like matrix. All of the proposed new transforms have required good properties to be fractional transforms. Finally, since outputs of proposed new transforms are random, they can be applied in image encryptions.