Engineering Handbook/Mathematics/Fourier Transformation

Fourier Transform

$F(j\omega )=ℱ\{f(t)\}=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}f(t)e^{-j\omega t}dt$

Inverse Fourier Transform

${ℱ}^{-1}\{F(j\omega )\}=f(t)=\frac{1}{2\pi }\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}F(j\omega )e^{j\omega t}d\omega$

Table of Fourier Transforms

This table contains some of the most commonly encountered Fourier transforms.

	Time Domain	Frequency Domain
	$x(t)={ℱ}^{-1}\{X(\omega )\}$	$X(\omega )=ℱ\{x(t)\}$
1	$X(j\omega )=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}x(t)e^{-j\omega t}dt$	$x(t)=\frac{1}{2\pi }\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}X(\omega )e^{j\omega t}d\omega$
2	$1$	$2\pi \delta (\omega )$
3	$-0.5+u(t)$	$\frac{1}{j\omega }$
4	$\delta (t)$	$1$
5	$\delta (t-c)$	$e^{-j\omega c}$
6	$u(t)$	$\pi \delta (\omega )+\frac{1}{j\omega }$
7	$e^{-bt}u(t)(b>0)$	$\frac{1}{j\omega +b}$
8	$\cos {\omega }_{0}t$	$\pi [\delta (\omega +{\omega }_0)+\delta (\omega -{\omega }_0)]$
9	$\cos ({\omega }_{0}t+\theta )$	$\pi [e^{-j\theta }\delta (\omega +{\omega }_0)+e^{j\theta }\delta (\omega -{\omega }_0)]$
10	$\sin {\omega }_{0}t$	$j\pi [\delta (\omega +{\omega }_0)-\delta (\omega -{\omega }_0)]$
11	$\sin ({\omega }_{0}t+\theta )$	$j\pi [e^{-j\theta }\delta (\omega +{\omega }_0)-e^{j\theta }\delta (\omega -{\omega }_0)]$
12	$\text{rect}\left(\frac{t}{\tau }\right)$	$\tau \text{sinc}\left(\frac{\tau \omega }{2\pi }\right)$
13	$\tau \text{sinc}\left(\frac{\tau t}{2\pi }\right)$	$2\pi \text{rect}\left(\frac{\omega }{\tau }\right)$
14	$(1-\frac{2|t|}{\tau })\text{rect}\left(\frac{t}{\tau }\right)$	$\frac{\tau }{2}{\text{sinc}}^2\left(\frac{\tau \omega }{4\pi }\right)$
15	$\frac{\tau }{2}{\text{sinc}}^2\left(\frac{\tau t}{4\pi }\right)$	$2\pi (1-\frac{2|\omega |}{\tau })\text{rect}\left(\frac{\omega }{\tau }\right)$
16	$e^{-a|t|},\mathrm{\Re }\{a\}>0$	$\frac{2a}{a^2+{\omega }^2}$
Notes:	$\text{sinc}(x)=\sin (\pi x)/(\pi x)$ $\text{rect}\left(\frac{t}{\tau }\right)$ is the rectangular pulse function of width $\tau$ $u(t)$ is the Heaviside step function $\delta (t)$ is the Dirac delta function
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