uncertainty_principle.md

Fourier Transform of Normal Distribution

$$ f(t) = \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{1}{2}\left(\frac{t-\mu}{\sigma}\right)^2} $$

Uncertainty Principle

For $ f(t) \in L^2(\mathbb{R})$, the distribution of propability density of $f(t)$ and $\hat f(\omega)$ are $$ \frac{|f(t)|^2}{|f|^2} , \quad \frac{|\hat f(\omega)|^2}{2\pi |\hat f|^2} $$