We design a sublinear Fourier sampling algorithm for a case of sparse off-grid frequency recovery. These are signals with the form \(f(t) = \sum_{j=1}^k a_j {\rm e}^{i\omega_j t} + \hat\nu\), \(t \in{\mathbb Z\!}\); i.e., exponential polynomials with a noise term. The frequencies {ω j } satisfy ω j  ∈ [η,2π − η] and min i ≠ j |ω i  − ω j | ≥ η for some η > 0. We design a sublinear time randomized algorithm, which takes O(klogklog(1/η)(logk + log( ∥ a ∥ 1/ ∥ ν ∥ 1)) samples of f(t) and runs in time proportional to number of samples, recovering {ω j } and {a j } such that, with probability Ω(1), the approximation error satisfies |ω j ′ − ω j | ≤ η/k and |a j  − a j ′| ≤ ∥ ν ∥ 1/k for all j with |a j | ≥ ∥ ν ∥ 1/k.


Discrete Fourier Transform Failure Probability Pass Region Recovery Algorithm Hash Family 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Petros Boufounos
    • 1
  • Volkan Cevher
    • 2
  • Anna C. Gilbert
    • 3
  • Yi Li
    • 4
  • Martin J. Strauss
    • 5
  1. 1.Mitsubishi Electric Research LabsCambridgeUSA
  2. 2.Laboratory for Information and Inference SystemsEPFLLausanneSwitzerland
  3. 3.Department of MathematicsUniversity of MichiganAnn ArborUSA
  4. 4.Department of EECSUniversity of MichiganAnn ArborUSA
  5. 5.Departments of Mathematics and EECSUniversity of MichiganAnn ArborUSA

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