Non-iterative Type 4 and 5 Nonuniform FFT Methods in the One-Dimensional Case

The so-called non-uniform FFT (NFFT) is a family of algorithms for efficiently computing the Fourier transform of finite-length signals, whenever the time or frequency grid is nonuniformly spaced. Following the usual classification, there exist five NFFT types. Types 1 and 2 make it possible to pass from the time to the frequency domain with nonuniform input and output grids respectively. Type 3 allows for both input and output nonuniform grids. Finally, types 4 and 5 are the inverses of types 1 and 2 and are expensive computationally, given that they involve iterative methods. In this paper, we solve this last drawback in the one-dimensional case by presenting non-iterative type 4 and 5 NFFT methods that just involve three NFFTs of types 1 or 2 plus some additional FFTs. The methods are based on exploiting the structure of the Lagrange interpolation formula. The paper includes several numerical examples in which the proposed methods are compared with the Gaussian elimination (GE) and conjugate gradient (CG) methods, both in terms of round-off error and computational burden.
Comment: Submitted to the IEEE Trans. on Signal Processing