The Fractional Fourier Transform and Applications

This paper describes the \fractional Fourier transform", which admits computation by an algorithm that has complexity proportional to the fast Fourier transform algorithm. Whereas the discrete Fourier transform (DFT) is based on integral roots of unity e 2 , the fractional Fourier transform is based on fractional roots of unity e 2 i , where is arbitrary. The fractional Fourier transform and the corresponding fast algorithm are useful for such applications as computing DFTs of sequences with prime lengths, computing DFTs of sparse sequences, analyzing sequences with non-integer periodicities, performing high-resolution trigonometric interpolation, detecting lines in noisy images and detecting signals with linearly drifting frequencies. In many cases, the resulting algorithms are faster by arbitrarily large factors than conventional techniques. Bailey is with the Numerical Aerodynamic Simulation (NAS) Systems Division at NASA Ames Research Center, Mo ett Field, CA 94035. Swarztrauber is with the National Center for Atmospheric Research, Boulder, CO 80307, which is sponsored by National Science Foundation. This work was completed while Swarztrauber was visiting the Research Institute for Advanced Computer Science (RIACS) at NASA Ames. Swarztrauber's work was funded by the NAS Systems Division via Cooperative Agreement NCC 2-387 between NASA and the Universities Space Research Association.