The discrete fractional Fourier transform

Abstract

In this work, the discrete counterpart of the continuous Fractional Fourier Transform (FrFT) is proposed, discussed and consolidated. The discrete transform generalizes the Discrete Fourier Transform (DFT) to arbitrary orders, in the same sense that the continuous FrFT generalizes the continuous time Fourier Transform. The definition proposed satisfies the requirements of unitarity, additivity of the orders and reduction to DFT. The definition proposed tends to the continuous transform as the dimension of the discrete transform matrix increases and provides a good approximation to the continuous FrFT for the finite dimensional matrices. Simulation results and some properties of the discrete FrFT are also discussed.