, Volume 73, Issue 2, pp 261–288 | Cite as

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

  • Petros Boufounos
  • Volkan Cevher
  • Anna C. Gilbert
  • Yi Li
  • Martin J. Strauss


We design a sublinear Fourier sampling algorithm for a case of sparse off-grid frequency recovery. These are signals with the form \(f(t) = \sum _{j=1}^k a_j \mathrm{e}^{i\omega _j t} + \int \nu (\omega )\mathrm{e}^{i\omega t}d\mu (\omega )\); i.e., exponential polynomials with a noise term. The frequencies \(\{\omega _j\}\) satisfy \(\omega _j\in [\eta ,2\pi -\eta ]\) and \(\min _{i\ne j} |\omega _i-\omega _j|\ge \eta \) for some \(\eta > 0\). We design a sublinear time randomized algorithm which, for any \(\epsilon \in (0,\eta /k]\), which takes \(O(k\log k\log (1/\epsilon )(\log k+\log (\Vert a\Vert _1/\Vert \nu \Vert _1))\) samples of \(f(t)\) and runs in time proportional to number of samples, recovering \(\omega _j'\approx \omega _j\) and \(a_j'\approx a_j\) such that, with probability \(\varOmega (1)\), the approximation error satisfies \(|\omega _j'-\omega _j|\le \epsilon \) and \(|a_j-a_j'|\le \Vert \nu \Vert _1/k\) for all \(j\) with \(|a_j|\ge \Vert \nu \Vert _1/k\). We apply our model and algorithm to bearing estimation or source localization and discuss their implications for receiver array processing.


Sparse signal recovery Fourier sampling Sublinear algorithms 

Mathematics Subject Classification

94A20 68W20 



The authors would like to thank an anonymous reviewer of APPROX/RANDOM 2012 for the suggestion of improving the running time. V. Cevher was partially supported by Faculty Fellowship at Rice University, MIRG-268398, ERC Future Proof, and DARPA KeCoM program #11-DARPA-1055. A. C. Gilbert was partially supported by NSF DMS 0743372 and DARPA ONR N66001-06-1-2011. Y. Li was partially supported by NSF DMS 0743372 when the author was at University of Michigan, Ann Arbor. M. J. Strauss was partially supported by NSF DMS 0743372 and DARPA ONR N66001-06-1-2011.


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Petros Boufounos
    • 1
  • Volkan Cevher
    • 2
  • Anna C. Gilbert
    • 3
  • Yi Li
    • 4
  • Martin J. Strauss
    • 5
    • 6
  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 1Max-Planck-Institut für InformatikSaarbrückenGermany
  5. 5.Department of MathematicsUniversity of MichiganAnn ArborUSA
  6. 6.Department of Electrical Engineering and Computer ScienceUniversity of MichiganAnn ArborUSA

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