What’s the Frequency, Kenneth?: Sublinear Fourier Sampling Off the Grid

  • Petros Boufounos
  • Volkan Cevher
  • Anna C. Gilbert
  • Yi Li
  • Martin J. Strauss
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7408)

Abstract

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 {ωj} satisfy ωj ∈ [η,2π − η] and min i ≠ j |ωi − ωj| ≥ 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 {ωj} and {aj} such that, with probability Ω(1), the approximation error satisfies |ωj′ − ωj| ≤ η/k and |aj − aj′| ≤ ∥ ν ∥ 1/k for all j with |aj| ≥ ∥ ν ∥ 1/k.

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References

  1. 1.
    Kushilevitz, E., Mansour, Y.: Learning decision trees using the Fourier spectrum. In: STOC, pp. 455–464 (1991)Google Scholar
  2. 2.
    Gilbert, A.C., Guha, S., Indyk, P., Muthukrishnan, M., Strauss, M.: Near-optimal sparse fourier representations via sampling. In: STOC, pp. 152–161 (2002)Google Scholar
  3. 3.
    Gilbert, A.C., Muthukrishnan, S., Strauss, M.: Improved time bounds for near-optimal sparse Fourier representations. In: Proceedings of Wavelets XI Conference (2005)Google Scholar
  4. 4.
    Iwen, M.: Combinatorial sublinear-time Fourier algorithms. Foundations of Computational Mathematics 10(3), 303–338 (2009)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Hassanieh, H., Indyk, P., Katabi, D., Price, E.: Simple and practical algorithm for sparse Fourier transform. In: SODA, pp. 1183–1194 (2012)Google Scholar
  6. 6.
    Hassanieh, H., Indyk, P., Katabi, D., Price, E.: Nearly optimal sparse Fourier transform. In: STOC, pp. 563–578 (2012)Google Scholar
  7. 7.
    Vetterli, M., Marziliano, P., Blu, T.: Sampling signals with finite rate of innovation. IEEE Transactions on Signal Processing 50(6), 1417–1428 (2002)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Gilbert, A.C., Li, Y., Porat, E., Strauss, M.: Approximate sparse recovery: Optimizing time and measurements. SIAM J. Comput. 41(2), 436–453 (2012)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Peter, T., Potts, D., Tasche, M.: Nonlinear approximation by sums of exponentials and translates. SIAM J. Sci. Comput. 33(4), 1920–1947 (2011)MathSciNetMATHCrossRefGoogle Scholar

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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