Nonuniform fast Fourier transform

The nonuniform discrete Fourier transform (NDFT) can be computed with a fast algorithm, referred to as the nonuniform fast Fourier transform (NFFT). In L dimensions, the NFFT requires O(N(-ln epsilon)(L) + (IIl=1L Me) Sigma(l=1)(L) logM(l)) operations, where M-l is the number of Fourier components along dimension l, N is the number of irregularly spaced samples, and epsilon is the required accuracy. This is a dramatic improvement over the O (N IIl=1L M-l) operations required for the direct evaluation (NDFT). The performance of the NFFT depends on the lowpass filter used in the algorithm. A truncated Gauss pulse, proposed in the literature, is optimized. A newly proposed filter, a Gauss pulse tapered with a Hanning window, performs better than the truncated Gauss pulse and the B-spline, also proposed in the literature. For small filter length, a numerically optimized filter shows the best results. Numerical experiments for 1-D and 2-D implementations confirm the theoretically predicted accuracy and efficiency properties of the algorithm.