Deterministic Sparse FFT Algorithms

by Katrin Ulrike Wannenwetsch
Doctoral thesis
Date of Examination:2016-08-09
Date of issue:2016-09-30
Advisor:Prof. Dr. Gerlind Plonka-Hoch
Referee:Prof. Dr. Gerlind Plonka-Hoch
Referee:Prof. Dr. Daniel Potts
Persistent Address: http://dx.doi.org/10.53846/goediss-5874

 

 

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Abstract

English

The discrete Fourier transform (DFT) is a well-known transform with many applications in various fields. By fast Fourier transform (FFT) algorithms, the DFT of a vector can be efficiently computed. Using these algorithms, one can reconstruct a complex vector x of length N from its discrete Fourier transform applying O(N log N) arithmetical operations. In order to improve the complexity of FFT algorithms, one needs additional a priori assumptions on the vector x. In this thesis, the focus is on vectors with small support or sparse vectors for which several new deterministic algorithms are proposed that have a lower complexity than regular FFT algorithms. We develop sublinear time algorithms for the reconstruction of complex vectors or matrices with small support from Fourier data as well as an algorithm for the reconstruction of real nonnegative vectors. The algorithms are analyzed and evaluated in numerical experiments. Furthermore, we generalize the algorithm for real nonnegative vectors with small support and propose an approach to the reconstruction of sparse vectors with real nonnegative entries.
Keywords: Discrete Fourier Transform; Fast Fourier Transform; sparse Fourier reconstruction; sparse FFT; vector reconstruction; small support
 
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