# User:Porceberkeley

## Table of important Fourier transforms

The following table records some important Fourier transforms. $G$ and $H$ denote Fourier transforms of $g(t)$ and $h(t)$, respectively. $g$ and $h$ may be integrable functions or tempered distributions. Note that the two most common unitary conventions are included.

### Functional relationships

Signal Fourier transform
unitary, angular frequency
Fourier transform
unitary, ordinary frequency
Remarks
$g(t)\!\equiv\!$

$\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty}\!\!G(\omega) e^{i \omega t} d \omega \,$
$G(\omega)\!\equiv\!$

$\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty}\!\!g(t) e^{-i \omega t} dt \,$
$G(f)\!\equiv$

$\int_{-\infty}^{\infty}\!\!g(t) e^{-i 2\pi f t} dt \,$
1 $a\cdot g(t) + b\cdot h(t)\,$ $a\cdot G(\omega) + b\cdot H(\omega)\,$ $a\cdot G(f) + b\cdot H(f)\,$ Linearity
2 $g(t - a)\,$ $e^{- i a \omega} G(\omega)\,$ $e^{- i 2\pi a f} G(f)\,$ Shift in time domain
3 $e^{ iat} g(t)\,$ $G(\omega - a)\,$ $G \left(f - \frac{a}{2\pi}\right)\,$ Shift in frequency domain, dual of 2
4 $g(a t)\,$ $\frac{1}{|a|} G \left( \frac{\omega}{a} \right)\,$ $\frac{1}{|a|} G \left( \frac{f}{a} \right)\,$ If $|a|\,$ is large, then $g(a t)\,$ is concentrated around 0 and $\frac{1}{|a|}G \left( \frac{\omega}{a} \right)\,$ spreads out and flattens
5 $G(t)\,$ $g(-\omega)\,$ $g(-f)\,$ Duality property of the Fourier transform. Results from swapping "dummy" variables of $t \,$ and $\omega \,$.
6 $\frac{d^n g(t)}{dt^n}\,$ $(i\omega)^n G(\omega)\,$ $(i 2\pi f)^n G(f)\,$ Generalized derivative property of the Fourier transform
7 $t^n g(t)\,$ $i^n \frac{d^n G(\omega)}{d\omega^n}\,$ $\left (\frac{i}{2\pi}\right)^n \frac{d^n G(f)}{df^n}\,$ This is the dual to 6
8 $(g * h)(t)\,$ $\sqrt{2\pi} G(\omega) H(\omega)\,$ $G(f) H(f)\,$ $g * h\,$ denotes the convolution of $g\,$ and $h\,$ — this rule is the convolution theorem
9 $g(t) h(t)\,$ $(G * H)(\omega) \over \sqrt{2\pi}\,$ $(G * H)(f)\,$ This is the dual of 8

### Square-integrable functions

Signal Fourier transform
unitary, angular frequency
Fourier transform
unitary, ordinary frequency
Remarks
$g(t)\!\equiv\!$

$\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty}\!\!G(\omega) e^{i \omega t} d \omega \,$
$G(\omega)\!\equiv\!$

$\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty}\!\!g(t) e^{-i \omega t} dt \,$
$G(f)\!\equiv$

$\int_{-\infty}^{\infty}\!\!g(t) e^{-i 2\pi f t} dt \,$
10 $\mathrm{rect}(a t) \,$ $\frac{1}{\sqrt{2 \pi a^2}}\cdot \mathrm{sinc}\left(\frac{\omega}{2\pi a}\right)$ $\frac{1}{|a|}\cdot \mathrm{sinc}\left(\frac{f}{a}\right)$ The rectangular pulse and the normalized sinc function
11 $\mathrm{sinc}(a t)\,$ $\frac{1}{\sqrt{2\pi a^2}}\cdot \mathrm{rect}\left(\frac{\omega}{2 \pi a}\right)$ $\frac{1}{|a|}\cdot \mathrm{rect}\left(\frac{f}{a} \right)\,$ Dual of rule 10. The rectangular function is an idealized low-pass filter, and the sinc function is the non-causal impulse response of such a filter.
12 $\mathrm{sinc}^2 (a t) \,$ $\frac{1}{\sqrt{2\pi a^2}}\cdot \mathrm{tri} \left( \frac{\omega}{2\pi a} \right)$ $\frac{1}{|a|}\cdot \mathrm{tri} \left( \frac{f}{a} \right)$ tri is the triangular function
13 $\mathrm{tri} (a t) \,$ $\frac{1}{\sqrt{2\pi a^2}} \cdot \mathrm{sinc}^2 \left( \frac{\omega}{2\pi a} \right)$ $\frac{1}{|a|}\cdot \mathrm{sinc}^2 \left( \frac{f}{a} \right) \,$ Dual of rule 12.
14 $e^{-\alpha t^2}\,$ $\frac{1}{\sqrt{2 \alpha}}\cdot e^{-\frac{\omega^2}{4 \alpha}}$ $\sqrt{\frac{\pi}{\alpha}}\cdot e^{-\frac{(\pi f)^2}{\alpha}}$ Shows that the Gaussian function $\exp(-\alpha t^2)$ is its own Fourier transform. For this to be integrable we must have $\mathrm{Re}(\alpha)>0$.
$e^{i a t^2} = \left. e^{-\alpha t^2}\right|_{\alpha = -i a} \,$ $\frac{1}{\sqrt{2 a}} \cdot e^{-i \left(\frac{\omega^2}{4 a} -\frac{\pi}{4}\right)}$ $\sqrt{\frac{\pi}{a}} \cdot e^{-i \left(\frac{\pi^2 f^2}{a} -\frac{\pi}{4}\right)}$ common in optics
$\cos ( a t^2 ) \,$ $\frac{1}{\sqrt{2 a}} \cos \left( \frac{\omega^2}{4 a} - \frac{\pi}{4} \right)$ $\sqrt{\frac{\pi}{a}} \cos \left( \frac{\pi^2 f^2}{a} - \frac{\pi}{4} \right)$
$\sin ( a t^2 ) \,$ $\frac{-1}{\sqrt{2 a}} \sin \left( \frac{\omega^2}{4 a} - \frac{\pi}{4} \right)$ $- \sqrt{\frac{\pi}{a}} \sin \left( \frac{\pi^2 f^2}{a} - \frac{\pi}{4} \right)$
$e^{-a|t|} \,$ $\sqrt{\frac{2}{\pi}} \cdot \frac{a}{a^2 + \omega^2}$ $\frac{2 a}{a^2 + 4 \pi^2 f^2}$ a>0
$\frac{1}{\sqrt{|t|}} \,$ $\frac{1}{\sqrt{|\omega|}}$ $\frac{1}{\sqrt{|f|}}$ the transform is the function itself
$J_0 (t)\,$ $\sqrt{\frac{2}{\pi}} \cdot \frac{\mathrm{rect} \left( \frac{\omega}{2} \right)}{\sqrt{1 - \omega^2}}$ $\frac{2\cdot \mathrm{rect} (\pi f)}{\sqrt{1 - 4 \pi^2 f^2}}$ J0(t) is the Bessel function of first kind of order 0, rect is the rectangular function
$J_n (t) \,$ $\sqrt{\frac{2}{\pi}} \frac{ (-i)^n T_n (\omega) \mathrm{rect} \left( \frac{\omega}{2} \right)}{\sqrt{1 - \omega^2}}$ $\frac{2 (-i)^n T_n (2 \pi f) \mathrm{rect} (\pi f)}{\sqrt{1 - 4 \pi^2 f^2}}$ it's the generalization of the previous transform; Tn (t) is the Chebyshev polynomial of the first kind.
$\frac{J_n (t)}{t} \,$ $\sqrt{\frac{2}{\pi}} \frac{i}{n} (-i)^n \cdot U_{n-1} (\omega)\,$

$\cdot \ \sqrt{1 - \omega^2} \mathrm{rect} \left( \frac{\omega}{2} \right)$

$\frac{2 i}{n} (-i)^n \cdot U_{n-1} (2 \pi f)\,$

$\cdot \ \sqrt{1 - 4 \pi^2 f^2} \mathrm{rect} ( \pi f )$

Un (t) is the Chebyshev polynomial of the second kind

### Distributions

Signal Fourier transform
unitary, angular frequency
Fourier transform
unitary, ordinary frequency
Remarks
$g(t)\!\equiv\!$

$\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty}\!\!G(\omega) e^{i \omega t} d \omega \,$
$G(\omega)\!\equiv\!$

$\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty}\!\!g(t) e^{-i \omega t} dt \,$
$G(f)\!\equiv$

$\int_{-\infty}^{\infty}\!\!g(t) e^{-i 2\pi f t} dt \,$
15 $1\,$ $\sqrt{2\pi}\cdot \delta(\omega)\,$ $\delta(f)\,$ $\delta(\omega)$ denotes the Dirac delta distribution. This rule shows why the Dirac delta is important: it shows up as the Fourier transform of a constant function.
16 $\delta(t)\,$ $\frac{1}{\sqrt{2\pi}}\,$ $1\,$ Dual of rule 15.
17 $e^{i a t}\,$ $\sqrt{2 \pi}\cdot \delta(\omega - a)\,$ $\delta(f - \frac{a}{2\pi})\,$ This follows from and 3 and 15.
18 $\cos (a t)\,$ $\sqrt{2 \pi} \frac{\delta(\omega\!-\!a)\!+\!\delta(\omega\!+\!a)}{2}\,$ $\frac{\delta(f\!-\!\begin{matrix}\frac{a}{2\pi}\end{matrix})\!+\!\delta(f\!+\!\begin{matrix}\frac{a}{2\pi}\end{matrix})}{2}\,$ Follows from rules 1 and 17 using Euler's formula: $\cos(a t) = (e^{i a t} + e^{-i a t})/2.$
19 $\sin( at)\,$ $\sqrt{2 \pi}\frac{\delta(\omega\!-\!a)\!-\!\delta(\omega\!+\!a)}{2i}\,$ $\frac{\delta(f\!-\!\begin{matrix}\frac{a}{2\pi}\end{matrix})\!-\!\delta(f\!+\!\begin{matrix}\frac{a}{2\pi}\end{matrix})}{2i}\,$ Also from 1 and 17.
20 $t^n\,$ $i^n \sqrt{2\pi} \delta^{(n)} (\omega)\,$ $\left(\frac{i}{2\pi}\right)^n \delta^{(n)} (f)\,$ Here, $n$ is a natural number. $\delta^n(\omega)$ is the $n$-th distribution derivative of the Dirac delta. This rule follows from rules 7 and 15. Combining this rule with 1, we can transform all polynomials.
21 $\frac{1}{t}\,$ $-i\sqrt{\frac{\pi}{2}}\sgn(\omega)\,$ $-i\pi\cdot \sgn(f)\,$ Here $\sgn(\omega)$ is the sign function; note that this is consistent with rules 7 and 15.
22 $\frac{1}{t^n}\,$ $-i \begin{matrix} \sqrt{\frac{\pi}{2}}\cdot \frac{(-i\omega)^{n-1}}{(n-1)!}\end{matrix} \sgn(\omega)\,$ $-i\pi \begin{matrix} \frac{(-i 2\pi f)^{n-1}}{(n-1)!}\end{matrix} \sgn(f)\,$ Generalization of rule 21.
23 $\sgn(t)\,$ $\sqrt{\frac{2}{\pi}}\cdot \frac{1}{i\ \omega }\,$ $\frac{1}{i\pi f}\,$ The dual of rule 21.
24 $u(t) \,$ $\sqrt{\frac{\pi}{2}} \left( \frac{1}{i \pi \omega} + \delta(\omega)\right)\,$ $\frac{1}{2}\left(\frac{1}{i \pi f} + \delta(f)\right)\,$ Here $u(t)$ is the Heaviside unit step function; this follows from rules 1 and 21.
$e^{- a t} u(t) \,$ $\frac{1}{\sqrt{2 \pi} (a + i \omega)}$ $\frac{1}{a + i 2 \pi f}$ $u(t)$ is the Heaviside unit step function and $a > 0$.
25 $\sum_{n=-\infty}^{\infty} \delta (t - n T) \,$ $\begin{matrix} \frac{\sqrt{2\pi }}{T}\end{matrix} \sum_{k=-\infty}^{\infty} \delta \left( \omega -k \begin{matrix} \frac{2\pi }{T}\end{matrix} \right)\,$ $\frac{1}{T} \sum_{k=-\infty}^{\infty} \delta \left( f -\frac{k }{T}\right) \,$ The Dirac comb — helpful for explaining or understanding the transition from continuous to discrete time.