Algorithmica

, 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
Article

Abstract

We design a sublinear Fourier sampling algorithm for a case of sparse off-grid frequency recovery. These are signals with the form \(f(t) = \sum _{j=1}^k a_j \mathrm{e}^{i\omega _j t} + \int \nu (\omega )\mathrm{e}^{i\omega t}d\mu (\omega )\); 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.

Keywords

Sparse signal recovery Fourier sampling Sublinear algorithms 

Mathematics Subject Classification

94A20 68W20 

Notes

Acknowledgments

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.

References

  1. 1.
    Baraniuk, R., Cevher, V., Wakin, M.: Low-dimensional models for dimensionality reduction and signal recovery: a geometric perspective. Proc. IEEE 98(6), 959–971 (2010)CrossRefGoogle Scholar
  2. 2.
    Chahine, K., Baltazart, V., Wang, Y.: Interpolation-based matrix pencil method for parameter estimation of dispersive media in civil engineering. Signal Process. 90(8), 2567–2580 (2010)CrossRefMATHGoogle Scholar
  3. 3.
    Chui, D.: Personal communication (2013).Google Scholar
  4. 4.
    Gilbert, A., Indyk, P.: Sparse recovery using sparse matrices. Proc. IEEE 98(6), 937–947 (2010)CrossRefGoogle Scholar
  5. 5.
    Gilbert, A., Li, Y., Porat, E., Strauss, M.: Approximate sparse recovery: optimizing time and measurements. SIAM J. Comput. 41(2), 436–453 (2012)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Gilbert, A.C., Guha, S., Indyk, P., Muthukrishnan, S., Strauss, M.: Near-optimal sparse Fourier representations via sampling. In: Proceedings of the thiry-fourth annual ACM Symposium on Theory of Computing. STOC ’02, pp. 152–161. ACM, New York (2002)Google Scholar
  7. 7.
    Gilbert, A.C., Muthukrishnan, S., Strauss, M.: Improved time bounds for near-optimal sparse Fourier representations. In: Proceedings of Wavelets XI conference, pp. 398–412 (2005).Google Scholar
  8. 8.
    Hassanieh, H., Indyk, P., Katabi, D., Price, E.: Nearly optimal sparse Fourier transform. In: Proceedings of the 44th Symposium on Theory of Computing, STOC ’12, pp. 563–578. ACM, New York (2012)Google Scholar
  9. 9.
    Hassanieh, H., Indyk, P., Katabi, D., Price, E.: Simple and practical algorithm for sparse Fourier transform. In: Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA ’12, pp. 1183–1194. SIAM (2012).Google Scholar
  10. 10.
    Heider, S., Kunis, S., Potts, D., Veit, M.: A sparse Prony FFT. In: Proceedings of Tenth International Conference on Sampling Theory and Applications, SampTA 2013, pp. 572–575 (2013)Google Scholar
  11. 11.
    Helson, H.: Harmonic Analysis (2nd edn.). Hindustan Book Agency, New Delhi (1995).Google Scholar
  12. 12.
    Hua, Y., Sarkar, T.: On SVD for estimating generalized eigenvalues of singular matrix pencil in noise. In: Proceedings of the IEEE Transactions on Signal Processing, vol. 39(4), pp. 892–900 (1991)Google Scholar
  13. 13.
    Iwen, M.: Combinatorial sublinear-time Fourier algorithms. Found. Comput. Math. 10(3), 303–338 (2009)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Kushilevitz, E., Mansour, Y.: Learning decision trees using the Fourier spectrum. SIAM J. Comput. 22(6), 1331–1348 (1993)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Peter, T., Potts, D., Tasche, M.: Nonlinear approximation by sums of exponentials and translates. SIAM J. Sci. Comput. 33(4), 1920–1947 (2011)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Potts, D., Tasche, M.: Parameter estimation for exponential sums by approximate Prony method. Signal Process. 90(5), 1631–1642 (2010)CrossRefMATHGoogle Scholar
  17. 17.
    Vetterli, M., Marziliano, P., Blu, T.: Sampling signals with finite rate of innovation. In: Proceedings of the IEEE Transactions on Signal Processing, vol. 50(6), pp. 1417–1428 (2002)Google Scholar

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

Personalised recommendations