A sparse data fast Fourier transform (SDFFT)

A multilevel algorithm that efficiently Fourier transforms sparse spatial data to sparse spectral data with controllable error is presented. The algorithm termed "sparse data fast Fourier transform" (SDFFT) is particularly useful for signal processing applications where only part of the k-space is to be computed - regardless of whether it is a regular region like an angular section of the Ewald sphere or it consists of completely arbitrary points scattered in the k-space. In addition, like the various nonuniform fast Fourier transforms, the O(NlogN) algorithm can deal with a sparse, nonuniform spatial domain. In this paper, the parabolic reflector antenna problem is studied as an example to demonstrate its use in the computation of far-field patterns due to arbitrary aperture antennas and antenna arrays. The algorithm is also promising for various applications such as backprojection tomography, diffraction tomography, and synthetic aperture radar imaging.