## Signal processing/spectral uncertainties

đ see wikipedia for now.

## Entropic uncertainty

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

where the âââ indicates convergence in \(L_2\), and normalized so that (by Plancherelâs 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 BiaĹynicki-Birula and Mycielski. The equality holds in the case of Gaussian distributions.

Beckner, William. 1975. âInequalities in Fourier Analysis.â *Annals of Mathematics* 102 (1): 159â82. https://doi.org/10.2307/1970980.

Hirschman, I. I. 1957. âA Note on Entropy.â *American Journal of Mathematics* 79 (1): 152â56. https://doi.org/10.2307/2372390.

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

Ăzaydin, Murad, and Tomasz Przebinda. 2004. âAn Entropy-Based Uncertainty Principle for a Locally Compact Abelian Group.â *Journal of Functional Analysis* 215 (1): 241â52. 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.