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Advances in Computational Mathematics

, Volume 40, Issue 3, pp 629–650 | Cite as

A survey of uncertainty principles and some signal processing applications

  • Benjamin Ricaud
  • Bruno Torrésani
Article

Abstract

The goal of this paper is to review the main trends in the domain of uncertainty principles and localization, highlight their mutual connections and investigate practical consequences. The discussion is strongly oriented towards, and motivated by signal processing problems, from which significant advances have been made recently. Relations with sparse approximation and coding problems are emphasized.

Keywords

Uncertainty principle Concentration inequalities Frames Signal processing 

Mathematics Subject Classification (2010)

94A15 

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References

  1. 1.
    Baraniuk, R., Flandrin, P., Janssen, A.J., Michel, O.: Measuring time frequency information content using the renyi entropies. IEEE Trans. Inf. Theory 47(4), 1391–1409 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Beckner, W.: Inequalities in Fourier analysis. Ann. Math. 102(1), 159–182 (1975)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Brascamp, H.J., Lieb, E.H.: Best constants in Young’s inequality, its converse, and its generalization to more than three functions. Adv. Math. 20, 151–173 (1976)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Breitenberger, E.: Uncertainty measures and uncertainty relations for angle observables. Found. Phys. 15(3), 353–364 (1985)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Dembo, A., Cover, T.M., Thomas, J.A.: Information theoretic inequalities. IEEE Trans. Inf. Theory 37, 1501–1518 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Doerfler, M., Torrésani, B.: Representation of operators by sampling in the time-frequency domain. Sampl. Theory Sign Image Process. 10(1–2), 171–190 (2011)zbMATHGoogle Scholar
  7. 7.
    Donoho, D.L., Elad, M.: Optimally sparse representation in general (nonorthogonal) dictionaries via l1 minimization. Proc. Natl. Acad. Sci. 100, 2197–2202 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Donoho, D.L., Huo, X.: Uncertainty principles and ideal atomic decomposition. IEEE Trans. Inf. Theory 47, 2845–2862 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Donoho, D.L., Stark, P.B.: Uncertainty principles and signal recovery. SIAM J. Appl. Math. 49(3), 906–931 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Elad, M., Bruckstein, A.: A generalized uncertainty principle and sparse representation in pairs of bases. IEEE Trans. Inf. Theory 48, 2558–2567 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Erb, W.: Uncertainty Principles on Riemannian Manifolds. Logos Berlin (2011)Google Scholar
  12. 12.
    Everett, H.I.: The Many-Worlds Interpretation of Quantum Mechanics: The Theory of the Universal Wave Function. Mathematics, Princeton University, Princeton (1957)Google Scholar
  13. 13.
    Feichtinger, H.G., Onchis-Moaca, D., Ricaud, B., Torrésani, B., Wiesmeyr, C.: A method for optimizing the ambiguity function concentration. In: Proceedings of Eusipco 2012 (2012)Google Scholar
  14. 14.
    Flandrin, P.: Inequalities in mellin-fourier analysis. In: Debnath, L. (ed.) Wavelet Transforms and Time-Frequency Signal Analysis, chap. 10, pp 289–319. Birkhaüser, Cambridge (2001)CrossRefGoogle Scholar
  15. 15.
    Folland, G.B., Sitaram, A.: The uncertainty principle: a mathematical survey. J. Fourier Anal. Appl. 3(3), 207–238 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Ghobber, S., Jaming, P.: On uncertainty principles in the finite dimensional setting. Linear Algebra Appl. 435, 751–768 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Gröchenig, K.: Foundations of time-frequency analysis. In: Applied and Numerical Harmonic Analysis. Birkhäuser Boston Inc, Boston (2001)Google Scholar
  18. 18.
    Hirschman, I.: A note on entropy. Am. J. Math. 79, 152–156 (1957)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Jaillet, F., Torrésani, B.: Time-frequency jigsaw puzzle: adaptive multiwindow and multilayered gabor expansions. Int. J. Wavelets Multiresolution Inf. Process. 5(2), 293–315 (2007)CrossRefzbMATHGoogle Scholar
  20. 20.
    Jaming, P.: Nazarov’s uncertainty principle in higher dimension. J. Approx. Theory 149, 611–630 (2007)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Judge, D.: On the uncertainty relation for angle variables. Il Nuovo Cimento Ser. 10(31), 332–340 (1964)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Kutyniok, G.: Data separation by sparse representations. In: Eldar, Y. (ed.) Compressed Sensing, Theory and Applications, pp 485–514. Cambridge University Press, Cambridge (2012)CrossRefGoogle Scholar
  23. 23.
    Lieb, E.H.: Integral bounds for radar ambiguity functions and wigner distributions. J. Math. Phys. 31, 594–599 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Maass, P., Sagiv, C., Sochen, N., Stark, H.-G.: Do uncertainty minimizers attain minimal uncertainty? J. Fourier Anal. Appl. 16(3), 448–469 (2010)CrossRefzbMATHGoogle Scholar
  25. 25.
    Maassen, H., Uffink, J.: Generalized entropic uncertainty relations. Phys. Rev. Lett. 60(12), 1103–1106 (1988)CrossRefMathSciNetGoogle Scholar
  26. 26.
    Nam, S.: An uncertainty principle for discrete signals. In: Proceedings of SAMPTA’ 13. Technical report, LATP, Aix-Marseille Université, Marseille. to appear (2013)Google Scholar
  27. 27.
    Rényi, A.: On measures of information and entropy. In: Proceedings of the 4th Berkeley Symposium on Mathematics, Statistics and Probability, pp. 547–561 (1960)Google Scholar
  28. 28.
    Ricaud, B., Torrésani, B.: Refined support and entropic uncertainty inequalities. (2012) arXiv:1210.7711
  29. 29.
    Song, X., Zhou, S., Willett, P.: The role of the ambiguity function in compressed sensing. In: IEEE (ed) 2010 IEEE International Conference on Acoustics Speech and Signal Processing (ICASSP) (2010)Google Scholar
  30. 30.
    Wehner, S., Winter, A.: Entropic uncertainty relations—a survey. New J. Phys. 12(2), 025009 (2010)CrossRefMathSciNetGoogle Scholar
  31. 31.
    Woodward, P.: Probability and Information Theory with Applications to Radar. Artech House, Norwood (1980)Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Signal Processing Laboratory 2Ecole Polytechnique Fédérale de Lausanne (EPFL)LausanneSwitzerland
  2. 2.Aix-Marseille Université, CNRS, Centrale Marseille, LATP, UMR7353MarseilleFrance

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