Nonuniform fast fourier transforms using min-max interpolation

TLDR

The FFT is widely used for efficient computation of the Fourier transform at uniformly spaced frequencies, but many applications require nonuniform frequency sampling, prompting fast approximations based on oversampled FFTs. This paper introduces an interpolation method for the nonuniform Fourier transform that is optimal in the min‑max sense, minimizing the worst‑case approximation error over all unit‑norm signals. The method readily generalizes to multidimensional signals. Numerical experiments demonstrate that the min‑max approach achieves substantially lower approximation errors than conventional interpolation techniques and facilitates optimization of interpolation kernel parameters such as the Kaiser‑Bessel function.

Abstract

The fast Fourier transform (FFT) is used widely in signal processing for efficient computation of the FT of finite-length signals over a set of uniformly spaced frequency locations. However, in many applications, one requires nonuniform sampling in the frequency domain, i.e., a nonuniform FT. Several papers have described fast approximations for the nonuniform FT based on interpolating an oversampled FFT. This paper presents an interpolation method for the nonuniform FT that is optimal in the min-max sense of minimizing the worst-case approximation error over all signals of unit norm. The proposed method easily generalizes to multidimensional signals. Numerical results show that the min-max approach provides substantially lower approximation errors than conventional interpolation methods. The min-max criterion is also useful for optimizing the parameters of interpolation kernels such as the Kaiser-Bessel function.