Fourier transform

The frequency-domain function $\hat f(\omega)=\int_{-\infty}^\infty f(t)e^{-i\omega t}\,dt$ built from a non-periodic $f$ .

For a function $f$ over all of $t\in\mathbb R$ , the Fourier transform of $f$ is $\hat f(\omega)=\mathcal F[f](\omega)=\int_{-\infty}^\infty f(t)e^{-i\omega t}\,dt$ , whenever the integral converges. It is the non-periodic analogue of a Fourier series' complex coefficients, the limit of a period- $T$ Fourier series as $T\to\infty$ .

Examples

A single rectangular pulse has a sinc-shaped Fourier transform, even though the pulse itself is not periodic.