Showing papers on "Non-uniform discrete Fourier transform published in 2018"

Fast Fourier Transforms for Nonequispaced Data

Abstract: In this chapter, we describe fast algorithms for the computation of the DFT for d-variate nonequispaced data, since in a variety of applications the restriction to equispaced data is a serious drawback. These algorithms are called nonequispaced fast Fourier transforms and abbreviated by NFFT.

The Discrete Fourier Transform

Abstract: First, the real exponential is used to explain the concept of transform in reducing the multiplication operation into much simpler addition operation Then, the complex exponential with a pure imaginary exponent is introduced and the relation between the exponential and sinusoidal signals is established It is pointed out that the DFT gives a complex exponential polynomial representation of a signal The orthogonality property of the complex exponentials is used to derive the definitions of the DFT and the IDFT The least squares error criterion of signal representation is established Several examples of computing the DFT and IDFT are presented using the matrix form of the DFT and the IDFT An image processing application concludes the chapter

An Iterative Interpolated DFT to Remove Spectral Leakage in Time-Domain Near-Field Scanning

Abstract: Time-domain near-field scanning is gaining more and more interest within EMC engineering to analyze electromagnetic near-fields of, eg, quasi-stationar devices When using a digital oscilloscope to scan the near-fields of an electronic device, the oscilloscope measures time-domain signals that comprise in most cases a large number of frequency components For many of these components, noncoherent sampling occurs, resulting in spectral leakage when calculating the frequency spectrum of the time-domain signals with the discrete Fourier transform This paper proposes an improved method to detect the presence of such noncoherently sampled signals as well as an iterative algorithm to obtain accurate approximations of all the frequency components with their accompanying amplitudes and phase angles The algorithm excels over existing algorithms in obtaining these values especially in situations where several sinusoidal components are close to each other in the spectrum This is achieved, thanks to an iterative process of removing the influence of the multiple sinusoidal components on each other This paper contains the mathematical description of the algorithm and a numerical example evaluating the accuracy of the algorithm The algorithm has a higher accuracy than the existing approaches, eg, multipoint Interpolated Discrete Fourier Transform (IpDFTs), with only a slight increase of the computational cost

Efficient type-4 and type-5 non-uniform FFT methods in the one-dimensional case

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.

The Fourier Transform

Abstract: Of all the numerical methods we have seen so far, the Fourier transform has arguably had the most significant impact over the course of the 20th century.