Rational-ordered discrete fractional Fourier transform

The discrete fractional Fourier transform (DFRFT) whose order parameter is a rational number has special interesting properties that ordinary DFRFT does not possess. In this paper, periodicity and eigendecomposition properties of the rational-ordered DFRFT (RODFRFT) are investigated. We find that RODFRFT must be periodic and periods of RODFRFT are derived for all possible orders. As to the eigendecomposition of RODFRFT, we first derive eigenvalue multiplicities of the RODFRFT of order 4/p, where p is its period. The results are then generalized to RODFRFT of any rational orders.