Manfred Tasche

Fast Fourier Transforms for Nonequispaced Data: A Tutorial

In this chapter we consider approximativemethods for the fast computation of multivariate discrete Fourier transforms for nonequispaced data (NDFT) in the time domain and in the frequency domain. In particularwe are interested in the approximation error as function of the arithmetic complexity of the algorithm. We discuss the robustness of NDFTiaalgorithms with respect to roundoff errors and applyNDFTalgorithms for the fast computation of Besseltransforms.

Fast algorithms for discrete polynomial transforms

Consider the Vandermonde-like matrix P:= (P k (cos jπ/N)) j,k=0 N , where the polynomials P k satisfy a three-term recurrence relation. If P k are the Chebyshev polynomials T k , then P coincides with C N+1 := (cos jkπ/N) j,k=0 N . This paper presents a new fast algorithm for the computation of the matrix-vector product Pa in O(N log 2 N) arithmetical operations. The algorithm divides into a fast transform which replaces Pa with C N+1 ā and a subsequent fast cosine transform. The first and central part of the algorithm is realized by a straightforward cascade summation based on properties of associated polynomials and by fast polynomial multiplications. Numerical tests demonstrate that our fast polynomial transform realizes Pa with almost the same precision as the Clenshaw algorithm, but is much faster for N ≥ 128.

Parameter estimation for exponential sums by approximate Prony method

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.

A Unified Approach to Periodic Wavelets

We sketch a new approach to p-periodic wavelets for general periodic scaling functions. Our method is based on properties of periodic shift-invariant spaces and related bracket products. A special way to construct periodic wavelets is the periodization of a known cardinal multiresolution. Efficient decomposition and reconstruction algorithms using FFT-algorithms are proposed.

Nonlinear Approximation by Sums of Exponentials and Translates

In this paper, we discuss the numerical solution of two nonlinear approximation problems. Many applications in electrical engineering, signal processing, and mathematical physics lead to the following problem. Let

FAST AND STABLE ALGORITHMS FOR DISCRETE SPHERICAL FOURIER TRANSFORMS

h

Fast polynomial multiplication and convolutions related to the discrete cosine transform

be a linear combination of exponentials with real frequencies. Determine all frequencies, all coefficients, and the number of summands if finitely many perturbed, uniformly sampled data of

On the construction of wavelets on a bounded interval

h

Multilevel Gauss-Newton methods for phase retrieval problems

are given. We solve this problem by an approximate Prony method (APM) and prove the stability of the solution in the square and uniform norm. Further, an APM for nonuniformly sampled data is proposed too. The second approximation problem is related to the first one and reads as follows: Let

Worst and Average Case Roundoff Error Analysis for FFT

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