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A survey of uncertainty principles and some signal processing applications

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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.

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References

  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)

    Article  MATH  MathSciNet  Google Scholar 

  2. Beckner, W.: Inequalities in Fourier analysis. Ann. Math. 102(1), 159–182 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  4. Breitenberger, E.: Uncertainty measures and uncertainty relations for angle observables. Found. Phys. 15(3), 353–364 (1985)

    Article  MathSciNet  Google Scholar 

  5. Dembo, A., Cover, T.M., Thomas, J.A.: Information theoretic inequalities. IEEE Trans. Inf. Theory 37, 1501–1518 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    MATH  Google Scholar 

  7. Donoho, D.L., Elad, M.: Optimally sparse representation in general (nonorthogonal) dictionaries via l1 minimization. Proc. Natl. Acad. Sci. 100, 2197–2202 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  8. Donoho, D.L., Huo, X.: Uncertainty principles and ideal atomic decomposition. IEEE Trans. Inf. Theory 47, 2845–2862 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  9. Donoho, D.L., Stark, P.B.: Uncertainty principles and signal recovery. SIAM J. Appl. Math. 49(3), 906–931 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  10. Elad, M., Bruckstein, A.: A generalized uncertainty principle and sparse representation in pairs of bases. IEEE Trans. Inf. Theory 48, 2558–2567 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  11. Erb, W.: Uncertainty Principles on Riemannian Manifolds. Logos Berlin (2011)

  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. 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)

  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)

    Chapter  Google Scholar 

  15. Folland, G.B., Sitaram, A.: The uncertainty principle: a mathematical survey. J. Fourier Anal. Appl. 3(3), 207–238 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  16. Ghobber, S., Jaming, P.: On uncertainty principles in the finite dimensional setting. Linear Algebra Appl. 435, 751–768 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  17. Gröchenig, K.: Foundations of time-frequency analysis. In: Applied and Numerical Harmonic Analysis. Birkhäuser Boston Inc, Boston (2001)

    Google Scholar 

  18. Hirschman, I.: A note on entropy. Am. J. Math. 79, 152–156 (1957)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  20. Jaming, P.: Nazarov’s uncertainty principle in higher dimension. J. Approx. Theory 149, 611–630 (2007)

    Article  MathSciNet  Google Scholar 

  21. Judge, D.: On the uncertainty relation for angle variables. Il Nuovo Cimento Ser. 10(31), 332–340 (1964)

    Article  MathSciNet  Google Scholar 

  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)

    Chapter  Google Scholar 

  23. Lieb, E.H.: Integral bounds for radar ambiguity functions and wigner distributions. J. Math. Phys. 31, 594–599 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  25. Maassen, H., Uffink, J.: Generalized entropic uncertainty relations. Phys. Rev. Lett. 60(12), 1103–1106 (1988)

    Article  MathSciNet  Google Scholar 

  26. Nam, S.: An uncertainty principle for discrete signals. In: Proceedings of SAMPTA’ 13. Technical report, LATP, Aix-Marseille Université, Marseille. to appear (2013)

  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)

  28. Ricaud, B., Torrésani, B.: Refined support and entropic uncertainty inequalities. (2012) arXiv:1210.7711

  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)

  30. Wehner, S., Winter, A.: Entropic uncertainty relations—a survey. New J. Phys. 12(2), 025009 (2010)

    Article  MathSciNet  Google Scholar 

  31. Woodward, P.: Probability and Information Theory with Applications to Radar. Artech House, Norwood (1980)

    Google Scholar 

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Authors and Affiliations

  1. Signal Processing Laboratory 2, Ecole Polytechnique Fédérale de Lausanne (EPFL), Station 11, 1015, Lausanne, Switzerland

    Benjamin Ricaud

  2. Aix-Marseille Université, CNRS, Centrale Marseille, LATP, UMR7353, 13453, Marseille, France

    Bruno Torrésani

Authors
  1. Benjamin Ricaud
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  2. Bruno Torrésani
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Correspondence to Benjamin Ricaud.

Additional information

Communicated by: Peter Maass, Hans G. Feichtinger, Bruno Torresani, Darian M. Onchis, Benjamin Ricaud, and David Shuman

This work was supported by the European project UNLocX, grant n. 255931.

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Ricaud, B., Torrésani, B. A survey of uncertainty principles and some signal processing applications. Adv Comput Math 40, 629–650 (2014). https://doi.org/10.1007/s10444-013-9323-2

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