Corpus ID: 210919910

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

@article{Barnett2020AliasingEO,
  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},
  year={2020}
}
  • 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… Expand
2 Citations
How exponentially ill-conditioned are contiguous submatrices of the Fourier matrix?
TLDR
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. Expand
Continuous window functions for NFFT
TLDR
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. Expand

References

SHOWING 1-10 OF 41 REFERENCES
A parallel non-uniform fast Fourier transform library based on an "exponential of semicircle" kernel
TLDR
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. Expand
Accelerating the Nonuniform Fast Fourier Transform
TLDR
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. Expand
Nonuniform fast Fourier transforms using min-max interpolation
TLDR
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. Expand
Optimized Least-Square Nonuniform Fast Fourier Transform
  • M. Jacob
  • Mathematics, Computer Science
  • IEEE Transactions on Signal Processing
  • 2009
TLDR
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. Expand
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.Expand
PROBABILITY AGAINST CONDITION NUMBER AND SAMPLING OF MULTIVARIATE TRIGONOMETRIC RANDOM POLYNOMIALS
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 moreExpand
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 paperExpand
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.Expand
Automated parameter tuning based on RMS errors for nonequispaced FFTs
  • F. Nestler
  • Mathematics, Computer Science
  • Adv. Comput. Math.
  • 2016
TLDR
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. Expand
Graphics processing unit accelerated non-uniform fast Fourier transform for ultrahigh-speed, real-time Fourier-domain OCT
TLDR
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. Expand
...
1
2
3
4
5
...