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 + (Pi L (sub l = 1) M l ) Sigma L (sub l = 1) log M 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(NPi L (sub l = 1) 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.