Fast Fourier Transforms for Nonequispaced Data: A Tutorial

Potts, Daniel; Steidl, Gabriele; Tasche, Manfred

doi:10.1007/978-1-4612-0143-4_12

Cited by 222 publications

(248 citation statements)

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“…Note that V is also a nonequispaced Fourier matrix (see [9,10,4]). We discuss the properties of V. Especially, we show that V is left-invertible and estimate the spectral norm of its left inverse.…”

Section: Approximate Prony Methods Based On Nfftmentioning

confidence: 99%

“…k=0, j=−M is realized in each iteration step by the nonequispaced fast Fourier transform (NFFT, see [9,10]. 7.…”

Section: Nmmentioning

confidence: 99%

“…For large P and N, use the CGNR method, where the multiplication of the Vandermonde-type matrix e ikω j 2N, P+1 k=0, j=−P−1 is realized in each iteration step by NFFT (see [9,10]). 7.…”

Section: Form the Perturbed Rectangular Hankel Matrix (32)mentioning

confidence: 99%

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Parameter estimation for exponential sums by approximate Prony method

2010

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The recovery of signal parameters from noisy sampled data is a fundamental problem in digital signal processing. In this paper, we consider the following spectral analysis problem: Let f be a real-valued sum of complex exponentials. Determine all parameters of f , i.e., all different frequencies, all coefficients, and the number of exponentials from finitely many equispaced sampled data of f . This is a nonlinear inverse problem. In this paper, we present new results on an approximate Prony method (APM) which is based on [1]. In contrast to [1], we apply matrix perturbation theory such that we can describe the properties and the numerical behavior of the APM in detail. The number of sampled data acts as regularization parameter. The first part of APM estimates the frequencies and the second part solves an overdetermined linear Vandermonde-type system in a stable way. We compare the first part of APM also with the known ESPRIT method. The second part is related to the nonequispaced fast Fourier transform (NFFT). Numerical experiments show the performance of our method.

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Section: Approximate Prony Methods Based On Nfftmentioning

confidence: 99%

“…k=0, j=−M is realized in each iteration step by the nonequispaced fast Fourier transform (NFFT, see [9,10]. 7.…”

Section: Nmmentioning

confidence: 99%

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Parameter estimation for exponential sums by approximate Prony method

2010

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“…The second objective of the proposed method is to present new applications of Ooura's methods (Ooura, 2001(Ooura, , 2005 for the Fourier transforms to obtain the solution of PIDEs. The high accuracy of the proposed method is due to the very fast convergence of Ooura's methods, and speeding up of computations by Ooura's methods is realized by combining them with the nonuniform FFT (Dutt & Rokhlin, 1993, 1995Greengard & Lee, 2004;Potts et al, 2001;Steidl, 1998) or the fractional FFT (Bailey & Swarztrauber, 1991;Chourdakis, 2005;Tanaka, 2014a). In addition, in order to treat a (seemingly) singular integral in (1.10), we propose an indefinite integration formula using the sinc-Gauss sampling formula (Tanaka et al, 2008), which is also accurate and combined with the FFT.…”

Section: R\{0}mentioning

confidence: 99%

A fast and accurate numerical method for symmetric Lévy processes based on the Fourier transform and sinc-Gauss sampling formula

Tanaka¹

2015

IMA J Numer Anal

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In this paper, we propose a fast and accurate numerical method based on Fourier transform to solve Kolmogorov forward equations of symmetric scalar Lévy processes. The method is based on the accurate numerical formulas for Fourier transform proposed by Ooura. These formulas are combined with nonuniform fast Fourier transform (FFT) and fractional FFT to speed up the numerical computations. Moreover, we propose a formula for numerical indefinite integration on equispaced grids as a component of the method. The proposed integration formula is based on the sinc-Gauss sampling formula, which is a function approximation formula. This integration formula is also combined with the FFT. Therefore, all steps of the proposed method are executed using the FFT and its variants. The proposed method allows us to be free from some special treatments for a non-smooth initial condition and numerical time integration. The numerical solutions obtained by the proposed method appeared to be exponentially convergent on the interval if the corresponding exact solutions do not have sharp cusps. Furthermore, the real computational times are approximately consistent with the theoretical estimates.

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“…for indices (k, n) ∈ I M , can also be carried out by the NFFT as described in [14]. The transposed algorithm for the intermediate steps, hence the multiplication with the matrix C T M , can be derived directly from the explicit representation in (2.19).…”

Section: Associated Legendre Functions and Polynomialsmentioning

confidence: 99%

Fast evaluation of quadrature formulae on the sphere

Keiner

Potts

2008

Math. Comp.

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Abstract. Recently, a fast approximate algorithm for the evaluation of expansions in terms of standard L 2 S 2 -orthonormal spherical harmonics at arbitrary nodes on the sphere S 2 has been proposed in [S. Kunis and D. Potts. Fast spherical Fourier algorithms. J. Comput. Appl. Math., 161:75-98, 2003]. The aim of this paper is to develop a new fast algorithm for the adjoint problem which can be used to compute expansion coefficients from sampled data by means of quadrature rules.We give a formulation in matrix-vector notation and an explicit factorisation of the spherical Fourier matrix based on the former algorithm. Starting from this, we obtain the corresponding factorisation of the adjoint spherical Fourier matrix and are able to describe the associated algorithm for the adjoint transformation which can be employed to evaluate quadrature rules for arbitrary weights and nodes on the sphere. We provide results of numerical tests showing the stability of the obtained algorithm using as examples classical Gauß-Legendre and Clenshaw-Curtis quadrature rules as well as the HEALPix pixelation scheme and an equidistribution.

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Fast Fourier Transforms for Nonequispaced Data: A Tutorial

Cited by 222 publications

References 0 publications

Parameter estimation for exponential sums by approximate Prony method

Parameter estimation for exponential sums by approximate Prony method

A fast and accurate numerical method for symmetric Lévy processes based on the Fourier transform and sinc-Gauss sampling formula

Fast evaluation of quadrature formulae on the sphere

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