A Deterministic Sparse FFT for Functions with Structured Fourier Sparsity

Bittens, Sina; Zhang, Ruochuan; Iwen, Mark A.

doi:10.48550/arxiv.1705.05256

Cited by 1 publication

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“…F f =: f 4 Of course deterministic algorithms with error guarantees of the type of (1) do exist for more restricted classes of periodic functions f . See, e.g., [2,3,23] for some examples. These include USSFT methods developed for periodic functions with structured Fourier support [2] which are of use for, among other things, the fast approximation of functions which exhibit sparsity with respect to other bounded orthonormal basis functions [13].…”

Section: Notation and Setupmentioning

confidence: 99%

“…Furthermore, g will always denote the periodic Gaussian as defined above in (3). Finally, we will also make use of the Dirichlet kernel D M : R → C, defined by…”

Section: 1mentioning

confidence: 99%

“…2 The following DSFT theorem is proven below. 3 Theorem 1. Let N ∈ N, s ∈ [2, N ] ∩ N, 1 ≤ r ≤ N 36 , and f ∈ C N .…”

Section: Introductionmentioning

confidence: 99%

“…In addition, the method used to develop this new deterministic DSFT algorithm is general enough that it can be applied to any fast and noise robust USSFT method of the type mentioned above (be it deterministic, or randomized) in order to yield a new fast and robust DSFT algorithm. As a result, we are also able to use the fastest of the currently existing USSFT methods [15,17,19,25,21,5,3] in order to create new publicly available DSFT implementations herein which are both faster and more robust to noise than currently existing noise robust DSFT methods for large N .…”

Section: Introductionmentioning

confidence: 99%

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A New Class of Fully Discrete Sparse Fourier Transforms: Faster Stable Implementations with Guarantees

Merhi

Zhang

Iwen

et al. 2018

J Fourier Anal Appl

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In this paper we consider Sparse Fourier Transform (SFT) algorithms for approximately computing the best s-term approximation of the Discrete Fourier Transform (DFT)f ∈ C N of any given input vector f ∈ C N in just (s log N ) O(1) -time using only a similarly small number of entries of f . In particular, we present a deterministic SFT algorithm which is guaranteed to always recover a near best s-term approximation of the DFT of any given input vector f ∈ C N in O s 2 log 11 2 (N )time. Unlike previous deterministic results of this kind, our deterministic result holds for both arbitrary vectors f ∈ C N and vector lengths N . In addition to these deterministic SFT results, we also develop several new publicly available randomized SFT implementations for approximately computingf from f using the same general techniques. The best of these new implementations is shown to outperform existing discrete sparse Fourier transform methods with respect to both runtime and noise robustness for large vector lengths N . 2 ∩Z, by sampling its associated trigonometric polynomial f (x) = ω∈Bf ω e iωx

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Section: Notation and Setupmentioning

confidence: 99%

“…Furthermore, g will always denote the periodic Gaussian as defined above in (3). Finally, we will also make use of the Dirichlet kernel D M : R → C, defined by…”

Section: 1mentioning

confidence: 99%

“…2 The following DSFT theorem is proven below. 3 Theorem 1. Let N ∈ N, s ∈ [2, N ] ∩ N, 1 ≤ r ≤ N 36 , and f ∈ C N .…”

Section: Introductionmentioning

confidence: 99%