Papers › A nonuniform fast Fourier transform based on low rank approximation
A nonuniform fast Fourier transform based on low rank approximation
Diego Ruiz-Antolin, Alex Townsend
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By viewing the nonuniform discrete Fourier transform (NUDFT) as a perturbed version of a uniform discrete Fourier transform, we propose a fast, stable, and simple algorithm for computing the NUDFT that costs 𝒪(NlogNlog(1/ϵ)/loglog(1/ϵ)) operations based on the fast Fourier transform, where N is the size of the transform and 0<ϵ<1 is a working precision. Our key observation is that a NUDFT and DFT matrix divided entry-by-entry is often well-approximated by a low rank matrix, allowing us to express a NUDFT matrix as a sum of diagonally-scaled DFT matrices. Our algorithm is simple to implement, automatically adapts to any working precision, and is competitive with state-of-the-art algorithms. In the fully uniform case, our algorithm is essentially the FFT. We also describe quasi-optimal algorithms for the inverse NUDFT and two-dimensional NUDFTs.
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