Efficient type‐4 and type‐5 non‐uniform FFT methods in the one‐dimensional case. Issue 1 (1st February 2018)

• Record Type: Journal Article
• Authors: Selva, Jesus
• Is Part Of: IET signal processing. Volume 12:Issue 1(2018)
• Page Start: 74
• Page End: 81
• Publication Date: 2018-02-01
• Subjects: signal processing -- fast Fourier transforms -- conjugate gradient methods -- inverse transforms; type‐4 nonuniform FFT method -- type‐5 nonuniform FFT method -- type‐4 nonuniform fast Fourier transform method -- type‐5 nonuniform fast Fourier transform method -- finite‐length signals -- time grid -- frequency grid -- inversion problem -- fast multipole method -- FM method -- iterative conjugate gradient method -- iterative CG method -- Lagrange formula structure; Signal processing -- Periodicals; 621.3822
• DOI: 10.1049/iet-spr.2016.0509
• Languages: English
• ISSNs: 1751-9675

Abstract

Abstract : 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.