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14
Simple and Practical Algorithm for Sparse Fourier Transform
"... We consider the sparse Fourier transform problem: given a complex vector x of length n, and a parameter k, estimate the k largest (in magnitude) coefficients of the Fourier transform of x. The problem is of key interest in several areas, including signal processing, audio/image/video compression, an ..."
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Cited by 13 (5 self)
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We consider the sparse Fourier transform problem: given a complex vector x of length n, and a parameter k, estimate the k largest (in magnitude) coefficients of the Fourier transform of x. The problem is of key interest in several areas, including signal processing, audio/image/video compression, and learning theory. We propose a new algorithm for this problem. The algorithm leverages techniques from digital signal processing, notably Gaussian and DolphChebyshev filters. Unlike the typical approach to this problem, our algorithm is not iterative. That is, instead of estimating “large ” coefficients, subtracting them and recursing on the reminder, it identifies and estimates the k largest coefficients in “one shot”, in a manner akin to sketching/streaming algorithms. The resulting algorithm is structurally simpler than its predecessors. As a consequence, we are able to extend considerably the range of sparsity, k, for which the algorithm is faster than FFT, both in theory and practice. 1
Nearly Optimal Sparse Fourier Transform
"... We consider the problem of computing the ksparse approximation to the discrete Fourier transform of an ndimensional signal. We show: • An O(k log n)time algorithm for the case where the input signal has at most k nonzero Fourier coefficients, and • An O(k log n log(n/k))time algorithm for gener ..."
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Cited by 10 (4 self)
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We consider the problem of computing the ksparse approximation to the discrete Fourier transform of an ndimensional signal. We show: • An O(k log n)time algorithm for the case where the input signal has at most k nonzero Fourier coefficients, and • An O(k log n log(n/k))time algorithm for general input signals. Both algorithms achieve o(n log n) time, and thus improve over the Fast Fourier Transform, for any k = o(n). Further, they are the first known algorithms that satisfy this property. Also, if one assumes that the Fast Fourier Transform is optimal, the algorithm for the exactly ksparse case is optimal for any k = n Ω(1). We complement our algorithmic results by showing that any algorithm for computing the sparse Fourier transform of a general signal must use at least Ω(k log(n/k) / log log n) signal samples, even if it is allowed The discrete Fourier transform (DFT) is one of the most important and widely used computational tasks. Its applications are broad and include signal processing, communications, and audio/image/video compression. Hence, fast algorithms for DFT are highly valuable. Currently, the fastest such algorithm is the Fast Fourier
Improved Approximation Guarantees for SublinearTime Fourier Algorithms
, 2010
"... ABSTRACT. In this paper modified variants of the sparse Fourier transform algorithms from [14] are presented which improve on the approximation error bounds of the original algorithms. In addition, simple methods for extending the improved sparse Fourier transforms to higher dimensional settings are ..."
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Cited by 5 (3 self)
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ABSTRACT. In this paper modified variants of the sparse Fourier transform algorithms from [14] are presented which improve on the approximation error bounds of the original algorithms. In addition, simple methods for extending the improved sparse Fourier transforms to higher dimensional settings are developed. As a consequence, approximate Fourier transforms are obtained which will identify a nearoptimal kterm Fourier series for any given input function, f: [0, 2π] D → C, in O ( k 2 · D 4) time (neglecting logarithmic factors). Faster randomized Fourier algorithm variants with runtime complexities that scale linearly in the sparsity parameter k are also presented. 1.
Simple deterministically constructible rip matrices with sublinear fourier sampling requirements
 in in Proc. of Proceedings of CISS 2008
, 2008
"... Abstract—We present a deterministic number theoretic construction for matrices with the Restricted Isometry Property (RIP). Furthermore, we show that the number theoretic properties of our RIP matrices allow their products with Discrete Fourier Transform (DFT) matrices to be well approximated via a ..."
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Cited by 4 (1 self)
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Abstract—We present a deterministic number theoretic construction for matrices with the Restricted Isometry Property (RIP). Furthermore, we show that the number theoretic properties of our RIP matrices allow their products with Discrete Fourier Transform (DFT) matrices to be well approximated via a few highly sparse matrix multiplications. Hence, our RIP matrices may be approximately multiplied by the DFT of any input vector in sublineartime by reading only a small fraction of its entries. As a consequence, we obtain small deterministic sample sets which are guaranteed to allow the recovery of nearoptimal sparse Fourier representations for all periodic functions having an integrable second derivative over a single period. Explicit bounds are provided for the sizes of our RIP matrices, the sizes of their associated sublinear Fourier sampling sets, and the errors incurred by quickly approximating their products with DFT matrices. The Fourier sampling requirements obtained herein improve on previous deterministic Fourier sampling results in [1], [2]. Index Terms—Fourier transforms, Discrete Fourier transforms,
What’s the frequency, Kenneth?: Sublinear Fourier sampling off the grid. RANDOM/APPROX
, 2012
"... Abstract. We design a sublinear Fourier sampling algorithm for a case of sparse offgrid frequency recovery. These are signals with the form f(t) = ∑ k j=1 ajeiω j t +ˆν, t ∈ ZZ; i.e., exponential polynomials with a noise term. The frequencies {ωj} satisfy ωj ∈ [η, 2π −η] and mini=j ωi −ωj  ≥ η ..."
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Cited by 4 (0 self)
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Abstract. We design a sublinear Fourier sampling algorithm for a case of sparse offgrid frequency recovery. These are signals with the form f(t) = ∑ k j=1 ajeiω j t +ˆν, t ∈ ZZ; i.e., exponential polynomials with a noise term. The frequencies {ωj} satisfy ωj ∈ [η, 2π −η] and mini=j ωi −ωj  ≥ η for some η> 0. We design a sublinear time randomized algorithm, which takes O(k log k log(1/η)(log k +log(‖a‖1/‖ν‖1)) samples of f(t) and runs in time proportional to number of samples, recovering {ωj} and {aj} such that, with probability Ω(1), the approximation error satisfies ω ′ j − ωj  ≤η/k and aj − a ′ j≤‖ν‖1/k for all j with aj  ≥‖ν‖1/k. 1
Sampleoptimal averagecase sparse fourier transform in two dimensions. Unpublished manuscript
, 2012
"... We present the first sampleoptimal sublinear time algorithms for the sparse Discrete Fourier Transform over a twodimensional √ n × √ n grid. Our algorithms are analyzed for average case signals. For signals whose spectrum is exactly sparse, our algorithms use O(k) samples and run in O(klogk) time ..."
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Cited by 2 (1 self)
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We present the first sampleoptimal sublinear time algorithms for the sparse Discrete Fourier Transform over a twodimensional √ n × √ n grid. Our algorithms are analyzed for average case signals. For signals whose spectrum is exactly sparse, our algorithms use O(k) samples and run in O(klogk) time, wherek is the expected sparsity of the signal. For signals whose spectrum is approximately sparse, our algorithm usesO(klogn) samples and runs in O(klog 2 n) time; the latter algorithm works for k = Θ ( √ n). The number of samples used by our algorithms matches the known lower bounds for the respective signal models. By a known reduction, our algorithms give similar results for the onedimensional sparse Discrete Fourier Transform whennis a power of a small composite number (e.g.,n = 6 t). 1
EMPIRICAL EVALUATION OF TWO DETERMINISTIC SPARSE FOURIER TRANSFORMS
"... This paper empirically evaluates a recently proposed Deterministic Sparse Fourier Transform algorithm (hereafter called DSFT) for the first time. Our experiments indicate that DSFT is capable of guaranteed general frequencysparse signal recovery using subNyquist sampling for realistic bandwidth val ..."
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This paper empirically evaluates a recently proposed Deterministic Sparse Fourier Transform algorithm (hereafter called DSFT) for the first time. Our experiments indicate that DSFT is capable of guaranteed general frequencysparse signal recovery using subNyquist sampling for realistic bandwidth values. Furthermore, we show that both variants of DSFT have fast reconstruction runtimes. In fact, the sublineartime DSFT variant is shown to be faster than a traditional Fast Fourier Transform (FFT) for highlysparse wideband signals.
Localization in Noisy Environments ∗
, 2010
"... This paper studies the problem of recovering a signal with a sparse representation in a given orthonormal basis using as few noisy observations as possible. As opposed to previous studies, this paper models observations which are subject to the type of ‘clutter noise ’ encountered in radar applicati ..."
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This paper studies the problem of recovering a signal with a sparse representation in a given orthonormal basis using as few noisy observations as possible. As opposed to previous studies, this paper models observations which are subject to the type of ‘clutter noise ’ encountered in radar applications (i.e., the measurements used influence the observed noise). Given this model, the paper develops bounds on the number of measurements required to reconstruct the support of the signal and the signal itself up to any given accuracy level when the measurement noise is Gaussian using nonadaptive and adaptive measurement strategies. Further, the paper demonstrates that group testing measurement constructions may be combined with statistical binary detection and estimation methods to produce practical and computationally efficient adaptive algorithms for sparse signal approximation and support recovery. In particular, the paper proves that a wide class of sparse signals can be recovered by adaptive methods using fewer noisy linear measurements than required by any recovery method based on nonadaptive Gaussian measurement ensembles. This result demonstrates an improvement over previous nonadaptive methods in the compressed sensing literature for sparse support pattern recovery in the sublinearsparse support regime under the measurement model considered herein. 1
ON THE DESIGN OF DETERMINISTIC MATRICES FOR FAST RECOVERY OF FOURIER COMPRESSIBLE FUNCTIONS
"... Abstract. We present a general class of compressed sensing matrices which are then demonstrated to have associated sublineartime sparse approximation algorithms. We then develop methods for constructing specialized matrices from this class which are sparse when multiplied with a discrete Fourier tr ..."
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Abstract. We present a general class of compressed sensing matrices which are then demonstrated to have associated sublineartime sparse approximation algorithms. We then develop methods for constructing specialized matrices from this class which are sparse when multiplied with a discrete Fourier transform matrix. Ultimately, these considerations improve previous sampling requirements for deterministic sparse Fourier transform methods.