Table of Fourier transforms: transform pairs and properties

The transform pairs and properties that come back in signals and systems, ODEs and physics. The function of time is on the left, its transform on the right.

Definition


$F (ω)$	$\int_{- \infty}^{\infty} f (t) e^{- iω t} d t$
$f (t)$	$\frac{1}{2 π} \int_{- \infty}^{\infty} F (ω) e^{iω t} d ω$

Transform pairs

f(t)	F(ω)
$δ (t)$	$1$
$1$	$2 π δ (ω)$
$e^{- a t} u (t), a > 0$	$\frac{1}{a + iω}$
$e^{- a ∣ t ∣}, a > 0$	$\frac{2 a}{a ^{2} + ω ^{2}}$
$e^{- a t^{2}}, a > 0$	$\frac{π}{a} e^{- ω^{2} / (4 a)}$
$rect (\frac{t}{τ})$	$τ \frac{sin ( ω τ /2 )}{ω τ /2}$
$u (t)$	$π δ (ω) + \frac{1}{iω}$
$e^{i ω_{0} t}$	$2 π δ (ω - ω_{0})$
$cos ω_{0} t$	$π [δ (ω - ω_{0}) + δ (ω + ω_{0})]$
$sin ω_{0} t$	$- iπ [δ (ω - ω_{0}) - δ (ω + ω_{0})]$

Properties

f(t)	F(ω)
$a f (t) + b g (t)$	$a F (ω) + b G (ω)$
$f (t - t_{0})$	$e^{- iω t_{0}} F (ω)$
$e^{i ω_{0} t} f (t)$	$F (ω - ω_{0})$
$f (a t), a \neq = 0$	$\frac{1}{∣ a ∣} F (\frac{ω}{a})$
$f^{'} (t)$	$iω F (ω)$
$t f (t)$	$i \frac{d F}{d ω}$
$(f * g) (t)$	$F (ω) G (ω)$
$f (t) g (t)$	$\frac{1}{2 π} (F * G) (ω)$
$F (t)$	$2 π f (- ω)$
$\int_{- \infty}^{\infty} ∣ f (t) ∣^{2} d t$	$\frac{1}{2 π} \int_{- \infty}^{\infty} ∣ F (ω) ∣^{2} d ω$

Notes

This table uses F(ω) = ∫ f(t)·e^(−iωt) dt. Some courses use frequency f in hertz (e^(−i2πft), with no 1/2π in the inverse) or 1/√(2π) in both directions. The factors of 2π in the table then change, so check which definition your course uses.

u(t) is the unit step, δ is the Dirac delta, and rect(t/τ) is a rectangular pulse of height 1 and width τ.