Uncertainty principles

Signal processing/spectral uncertainties

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Entropic uncertainty

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\[ g(y)\approx \int _{{-\infty }}^{\infty }\exp(-2\pi ixy)f(x)\,dx,\qquad f(x)\approx \int _{{-\infty }}^{\infty }\exp(2\pi ixy)g(y)\,dy~, \]

where the “≈” indicates convergence in \(L_2\), and normalized so that (by Plancherel’s theorem),

\[ \int _{{-\infty }}^{\infty }|f(x)|^{2}\,dx=\int _{{-\infty }}^{\infty }|g(y)|^{2}\,dy=1~. \]

He showed that for any such functions the sum of the Shannon entropies is non-negative,

\[ H(|f|^{2})+H(|g|^{2})\equiv -\int _{{-\infty }}^{\infty }|f(x)|^{2}\log |f(x)|^{2}\,dx-\int _{{-\infty }}^{\infty }|g(y)|^{2}\log |g(y)|^{2}\,dy\geq 0. \]

\[ H(|f|^{2})+H(|g|^{2})\geq \log {\frac e2}~, \]

was conjectured by Hirschman and Everett, proven in 1975 by Beckner and in the same year interpreted by as a generalized quantum mechanical uncertainty principle by Białynicki-Birula and Mycielski. The equality holds in the case of Gaussian distributions.

Beckner, William. 1975. “Inequalities in Fourier Analysis.” Annals of Mathematics 102 (1): 159–82. https://doi.org/10.2307/1970980.

Hirschman, I. I. 1957. “A Note on Entropy.” American Journal of Mathematics 79 (1): 152–56. https://doi.org/10.2307/2372390.

Massar, Serge, and Philippe Spindel. 2008. “Uncertainty Relation for the Discrete Fourier Transform.” Physical Review Letters 100 (19): 190401. https://doi.org/10.1103/PhysRevLett.100.190401.

Özaydin, Murad, and Tomasz Przebinda. 2004. “An Entropy-Based Uncertainty Principle for a Locally Compact Abelian Group.” Journal of Functional Analysis 215 (1): 241–52. https://doi.org/10.1016/j.jfa.2003.11.008.

Pinsky, Mark A. 2002. Introduction to Fourier Analysis and Wavelets. Brooks/Cole Series in Advanced Mathematics. Australia ; Pacific Grove, CA: Brooks/Cole.