Open AccessEfficient type‐4 and type‐5 non‐uniform FFT methods in the one‐dimensional case
Author(s) -
Selva Jesus
Publication year - 2018
Publication title -
iet signal processing
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.384
H-Index - 42
ISSN - 1751-9683
DOI - 10.1049/iet-spr.2016.0509
Subject(s) - fast fourier transform , algorithm , conjugate gradient method , split radix fft algorithm , fourier transform , computer science , discrete fourier transform (general) , type (biology) , prime factor fft algorithm , mathematics , fast multipole method , grid , gaussian , short time fourier transform , fourier analysis , multipole expansion , mathematical analysis , geometry , ecology , physics , quantum mechanics , biology
The so‐called non‐uniform fast Fourier transform (NFFT) is a family of algorithms for efficiently computing the Fourier transform of finite‐length signals, whenever the time or frequency grid is non‐uniformly spaced. Among the five usual NFFT types, types 4 and 5 involve an inversion problem, and this makes them the most intensive computationally. The usual efficient methods for these last types are either based on a fast multipole (FM) or on an iterative conjugate gradient (CG) method. The purpose of this study is to present efficient methods for these type‐4 and type‐5 NFFTs in the one‐dimensional case that just require three NFFTs of types 1 or 2 plus some additional fast Fourier transforms (FFTs). Fundamentally, they are based on exploiting the Lagrange formula structure. The proposed methods roughly provide a factor‐ten improvement on the FM and CG alternatives in computational burden. The study includes several numerical examples in double precision, in which the proposed and the Gaussian elimination, CG and FM methods are compared, both in terms of round‐off error and computational burden.