• Corpus ID: 210919910

Aliasing error of the exp$(\beta \sqrt{1-z^2})$ kernel in the nonuniform fast Fourier transform

  title={Aliasing error of the exp\$(\beta \sqrt\{1-z^2\})\$ kernel in the nonuniform fast Fourier transform},
  author={Alex H. Barnett},
  journal={arXiv: Numerical Analysis},
  • A. Barnett
  • Published 26 January 2020
  • Mathematics, Computer Science
  • arXiv: Numerical Analysis
The most popular algorithm for the nonuniform fast Fourier transform (NUFFT) uses the dilation of a kernel $\phi$ to spread (or interpolate) between given nonuniform points and a uniform upsampled grid, combined with an FFT and diagonal scaling (deconvolution) in frequency space. The high performance of the recent FINUFFT library is in part due to its use of a new ``exponential of semicircle'' kernel $\phi(z)=e^{\beta \sqrt{1-z^2}}$, for $z\in[-1,1]$, zero otherwise, whose Fourier transform… 
2 Citations
How exponentially ill-conditioned are contiguous submatrices of the Fourier matrix?
The proof uses the Kaiser-Bessel transform pair, and estimates on sums over distorted sinc functions, to construct a localized trial vector whose DFT is also localized, and proves a lower bound on the condition number of any cyclically contiguous submatrix of the discrete Fourier transform (DFT) matrix.
Continuous window functions for NFFT
This paper considers the continuous/discontinuous Kaiser--Bessel, continuous $\exp$- type, and continuous $\cosh$-type window functions and presents novel explicit error estimates for NFFT with such a window function and derive rules for the optimal choice from the parameters involved in N FFT.


A parallel non-uniform fast Fourier transform library based on an "exponential of semicircle" kernel
FINUFFT is presented, an efficient parallel library for type 1 (nonuiform to uniform), type 2 (uniform to nonuniform), or type 3 (non uniform toNonuniform) transforms, in dimensions 1, 2, or 3, which uses minimal RAM, requires no precomputation or plan steps, and has a simple interface to several languages.
Accelerating the Nonuniform Fast Fourier Transform
This paper observes that one of the standard interpolation or "gridding" schemes, based on Gaussians, can be accelerated by a significant factor without precomputation and storage of the interpolation weights, of particular value in two- and three- dimensional settings.
Nonuniform fast Fourier transforms using min-max interpolation
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 and indicates that the proposed method easily generalizes to multidimensional signals.
Optimized Least-Square Nonuniform Fast Fourier Transform
  • M. Jacob
  • Mathematics, Computer Science
    IEEE Transactions on Signal Processing
  • 2009
A memory efficient approximation to the nonuniform Fourier transform of a support limited sequence is derived based on the theory of shift-invariant representations and an exact expression for the worst-case mean square approximation error is derived.
Uniform asymptotic expansions for prolate spheriodal functions with large parameters
By application of the theory for second order linear differential equations with a turning point and a regular (double pole) singularity developed by Boyd and Dunster (this Journal, 17 (1986), pp.
The difficult factor in the condition number of a large linear system is the spectral norm of . To eliminate this factor, we here replace worst case analysis by a probabilistic argument. To be more
Fast Fourier Transforms for Nonequispaced Data
A group of algorithms is presented generalizing the fast Fourier transform to the case of noninteger frequencies and nonequispaced nodes on the interval $[ - \pi ,\pi ]$. The schemes of this paper
On the Fast Fourier Transform of Functions with Singularities
Abstract We consider a simple approach for the fast evaluation of the Fourier transform of functions with singularities based on projecting such functions on a subspace of Multiresolution Analysis.
Automated parameter tuning based on RMS errors for nonequispaced FFTs
  • F. Nestler
  • Mathematics, Computer Science
    Adv. Comput. Math.
  • 2016
This paper studies the error behavior of the well known fast Fourier transform for nonequispaced data (NFFT) with respect to the ℒ2$\mathcal {L}_{2}-norm and states an easy and efficient method to tune the involved parameters automatically.
Graphics processing unit accelerated non-uniform fast Fourier transform for ultrahigh-speed, real-time Fourier-domain OCT
GPU-NUFFT provides an accurate approximation to GPU-NUDFT in terms of image quality, but offers >10 times higher processing speed and improved sensitivity roll-off, higher local signal-to-noise ratio and immunity to side-lobe artifacts caused by the interpolation error.