Nonuniform fast Fourier transform

Abstract

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 ε)L + (∏ℓ=1L Mℓ) ∑l=1L log Mℓ) operations, where Mℓ is the number of Fourier components along dimension ℓ, N is the number of irregularly spaced samples, and ε is the required accuracy. This is a dramatic improvement over the O(N ∏l=1L Mℓ) operations required for the direct evaluation (NDFT). The performance of the NFFT depends on the low-pass 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. © 1999 Society of Exploration Geophysicists. All rights reserved.