Inverse Fourier transform

The formula that recovers $f$ back from its Fourier transform $\hat f$ .

The inverse Fourier transform, $\mathcal F^{-1}$ , recovers $f$ from $\hat f$ : $f(t) = \frac{1}{2\pi}\int_{-\infty}^\infty \hat f(\omega)e^{i\omega t}\,d\omega$ , at every point of continuity of $f$ , provided $f$ is absolutely integrable and piecewise smooth. Applying $\mathcal F$ then $\mathcal F^{-1}$ (or the reverse order) returns the original function.

Examples

Given a filter's frequency response $H(\omega)$ , its impulse response is $h=\mathcal F^{-1}[H]$ .