Common Fourier Transform Pairs#

	Name	$f(t)$	$F(\omega )$	Remarks
1.	DC signal	$1$	$2\pi \delta (\omega )$	Constant energy at all times.
2.	Dirac delta	$\delta (t)$	$1$	Constant energy at all frequencies.
3.	Time sample	$\delta (t-t_0)$	$e^{-j\omega t_0}$
4.	Phase shift	$e^{j{\omega }_{0}t}$	$2\pi \delta (\omega -{\omega }_0)$
5.	Signum	$\mathrm{sgn}t$	${\displaystyle \frac{2}{j\omega }}$	also known as sign function
6.	Unit step	$u_0(t)$	${\displaystyle \frac{1}{j\omega }+\pi \delta (\omega )}$
7.	Cosine	$\cos {\omega }_{0}t$	$\pi [\delta (\omega -{\omega }_0)+\delta (\omega +{\omega }_0)]$
8.	Sine	$\sin {\omega }_{0}t$	$-j\pi [\delta (\omega -{\omega }_0)-\delta (\omega +{\omega }_0)]$
9.	Single pole	$e^{-at}{u}_0(t)$	${\displaystyle \frac{1}{j\omega +a}}$	$a>0$
10.	Double pole	$t{e}^{-at}{u}_0(t)$	${\displaystyle \frac{1}{(j\omega +a{)}^2}}$	$a>0$
11.	Complex pole (cosine component)	$e^{-at}\cos {\omega }_{0}t{u}_0(t)$	${\displaystyle \frac{j\omega +a}{(j\omega +a{)}^2+{\omega }_0^2}}$	$a>0$
12.	Complex pole (sine component)	$e^{-at}\sin {\omega }_{0}t{u}_0(t)$	${\displaystyle \frac{{\omega }_0}{(j\omega +a{)}^2+{\omega }_0^2}}$	$a>0$

See also: Wikibooks: Engineering Tables/Fourier Transform Table and Fourier Transform—WolframMathworld for more complete references.