User:Porceberkeley

Table of important Fourier transforms

The following table records some important Fourier transforms. ${\displaystyle G}$ and ${\displaystyle H}$ denote Fourier transforms of ${\displaystyle g(t)}$ and ${\displaystyle h(t)}$, respectively. ${\displaystyle g}$ and ${\displaystyle h}$ 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
${\displaystyle g(t)\!\equiv \!}$

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

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

${\displaystyle \int _{-\infty }^{\infty }\!\!g(t)e^{-i2\pi ft}dt\,}$
1 ${\displaystyle a\cdot g(t)+b\cdot h(t)\,}$ ${\displaystyle a\cdot G(\omega )+b\cdot H(\omega )\,}$ ${\displaystyle a\cdot G(f)+b\cdot H(f)\,}$ Linearity
2 ${\displaystyle g(t-a)\,}$ ${\displaystyle e^{-ia\omega }G(\omega )\,}$ ${\displaystyle e^{-i2\pi af}G(f)\,}$ Shift in time domain
3 ${\displaystyle e^{iat}g(t)\,}$ ${\displaystyle G(\omega -a)\,}$ ${\displaystyle G\left(f-{\frac {a}{2\pi }}\right)\,}$ Shift in frequency domain, dual of 2
4 ${\displaystyle g(at)\,}$ ${\displaystyle {\frac {1}{|a|}}G\left({\frac {\omega }{a}}\right)\,}$ ${\displaystyle {\frac {1}{|a|}}G\left({\frac {f}{a}}\right)\,}$ If ${\displaystyle |a|\,}$ is large, then ${\displaystyle g(at)\,}$ is concentrated around 0 and ${\displaystyle {\frac {1}{|a|}}G\left({\frac {\omega }{a}}\right)\,}$ spreads out and flattens
5 ${\displaystyle G(t)\,}$ ${\displaystyle g(-\omega )\,}$ ${\displaystyle g(-f)\,}$ Duality property of the Fourier transform. Results from swapping "dummy" variables of ${\displaystyle t\,}$ and ${\displaystyle \omega \,}$.
6 ${\displaystyle {\frac {d^{n}g(t)}{dt^{n}}}\,}$ ${\displaystyle (i\omega )^{n}G(\omega )\,}$ ${\displaystyle (i2\pi f)^{n}G(f)\,}$ Generalized derivative property of the Fourier transform
7 ${\displaystyle t^{n}g(t)\,}$ ${\displaystyle i^{n}{\frac {d^{n}G(\omega )}{d\omega ^{n}}}\,}$ ${\displaystyle \left({\frac {i}{2\pi }}\right)^{n}{\frac {d^{n}G(f)}{df^{n}}}\,}$ This is the dual to 6
8 ${\displaystyle (g*h)(t)\,}$ ${\displaystyle {\sqrt {2\pi }}G(\omega )H(\omega )\,}$ ${\displaystyle G(f)H(f)\,}$ ${\displaystyle g*h\,}$ denotes the convolution of ${\displaystyle g\,}$ and ${\displaystyle h\,}$ — this rule is the convolution theorem
9 ${\displaystyle g(t)h(t)\,}$ ${\displaystyle (G*H)(\omega ) \over {\sqrt {2\pi }}\,}$ ${\displaystyle (G*H)(f)\,}$ This is the dual of 8

Square-integrable functions

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

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

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

${\displaystyle \int _{-\infty }^{\infty }\!\!g(t)e^{-i2\pi ft}dt\,}$
10 ${\displaystyle \mathrm {rect} (at)\,}$ ${\displaystyle {\frac {1}{\sqrt {2\pi a^{2}}}}\cdot \mathrm {sinc} \left({\frac {\omega }{2\pi a}}\right)}$ ${\displaystyle {\frac {1}{|a|}}\cdot \mathrm {sinc} \left({\frac {f}{a}}\right)}$ The rectangular pulse and the normalized sinc function
11 ${\displaystyle \mathrm {sinc} (at)\,}$ ${\displaystyle {\frac {1}{\sqrt {2\pi a^{2}}}}\cdot \mathrm {rect} \left({\frac {\omega }{2\pi a}}\right)}$ ${\displaystyle {\frac {1}{|a|}}\cdot \mathrm {rect} \left({\frac {f}{a}}\right)\,}$ 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 ${\displaystyle \mathrm {sinc} ^{2}(at)\,}$ ${\displaystyle {\frac {1}{\sqrt {2\pi a^{2}}}}\cdot \mathrm {tri} \left({\frac {\omega }{2\pi a}}\right)}$ ${\displaystyle {\frac {1}{|a|}}\cdot \mathrm {tri} \left({\frac {f}{a}}\right)}$ tri is the triangular function
13 ${\displaystyle \mathrm {tri} (at)\,}$ ${\displaystyle {\frac {1}{\sqrt {2\pi a^{2}}}}\cdot \mathrm {sinc} ^{2}\left({\frac {\omega }{2\pi a}}\right)}$ ${\displaystyle {\frac {1}{|a|}}\cdot \mathrm {sinc} ^{2}\left({\frac {f}{a}}\right)\,}$ Dual of rule 12.
14 ${\displaystyle e^{-\alpha t^{2}}\,}$ ${\displaystyle {\frac {1}{\sqrt {2\alpha }}}\cdot e^{-{\frac {\omega ^{2}}{4\alpha }}}}$ ${\displaystyle {\sqrt {\frac {\pi }{\alpha }}}\cdot e^{-{\frac {(\pi f)^{2}}{\alpha }}}}$ Shows that the Gaussian function ${\displaystyle \exp(-\alpha t^{2})}$ is its own Fourier transform. For this to be integrable we must have ${\displaystyle \mathrm {Re} (\alpha )>0}$.
${\displaystyle e^{iat^{2}}=\left.e^{-\alpha t^{2}}\right|_{\alpha =-ia}\,}$ ${\displaystyle {\frac {1}{\sqrt {2a}}}\cdot e^{-i\left({\frac {\omega ^{2}}{4a}}-{\frac {\pi }{4}}\right)}}$ ${\displaystyle {\sqrt {\frac {\pi }{a}}}\cdot e^{-i\left({\frac {\pi ^{2}f^{2}}{a}}-{\frac {\pi }{4}}\right)}}$ common in optics
${\displaystyle \cos(at^{2})\,}$ ${\displaystyle {\frac {1}{\sqrt {2a}}}\cos \left({\frac {\omega ^{2}}{4a}}-{\frac {\pi }{4}}\right)}$ ${\displaystyle {\sqrt {\frac {\pi }{a}}}\cos \left({\frac {\pi ^{2}f^{2}}{a}}-{\frac {\pi }{4}}\right)}$
${\displaystyle \sin(at^{2})\,}$ ${\displaystyle {\frac {-1}{\sqrt {2a}}}\sin \left({\frac {\omega ^{2}}{4a}}-{\frac {\pi }{4}}\right)}$ ${\displaystyle -{\sqrt {\frac {\pi }{a}}}\sin \left({\frac {\pi ^{2}f^{2}}{a}}-{\frac {\pi }{4}}\right)}$
${\displaystyle e^{-a|t|}\,}$ ${\displaystyle {\sqrt {\frac {2}{\pi }}}\cdot {\frac {a}{a^{2}+\omega ^{2}}}}$ ${\displaystyle {\frac {2a}{a^{2}+4\pi ^{2}f^{2}}}}$ a>0
${\displaystyle {\frac {1}{\sqrt {|t|}}}\,}$ ${\displaystyle {\frac {1}{\sqrt {|\omega |}}}}$ ${\displaystyle {\frac {1}{\sqrt {|f|}}}}$ the transform is the function itself
${\displaystyle J_{0}(t)\,}$ ${\displaystyle {\sqrt {\frac {2}{\pi }}}\cdot {\frac {\mathrm {rect} \left({\frac {\omega }{2}}\right)}{\sqrt {1-\omega ^{2}}}}}$ ${\displaystyle {\frac {2\cdot \mathrm {rect} (\pi f)}{\sqrt {1-4\pi ^{2}f^{2}}}}}$ J0(t) is the Bessel function of first kind of order 0, rect is the rectangular function
${\displaystyle J_{n}(t)\,}$ ${\displaystyle {\sqrt {\frac {2}{\pi }}}{\frac {(-i)^{n}T_{n}(\omega )\mathrm {rect} \left({\frac {\omega }{2}}\right)}{\sqrt {1-\omega ^{2}}}}}$ ${\displaystyle {\frac {2(-i)^{n}T_{n}(2\pi f)\mathrm {rect} (\pi f)}{\sqrt {1-4\pi ^{2}f^{2}}}}}$ it's the generalization of the previous transform; Tn (t) is the Chebyshev polynomial of the first kind.
${\displaystyle {\frac {J_{n}(t)}{t}}\,}$ ${\displaystyle {\sqrt {\frac {2}{\pi }}}{\frac {i}{n}}(-i)^{n}\cdot U_{n-1}(\omega )\,}$

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

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

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

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

Distributions

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

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

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

${\displaystyle \int _{-\infty }^{\infty }\!\!g(t)e^{-i2\pi ft}dt\,}$
15 ${\displaystyle 1\,}$ ${\displaystyle {\sqrt {2\pi }}\cdot \delta (\omega )\,}$ ${\displaystyle \delta (f)\,}$ ${\displaystyle \delta (\omega )}$ 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 ${\displaystyle \delta (t)\,}$ ${\displaystyle {\frac {1}{\sqrt {2\pi }}}\,}$ ${\displaystyle 1\,}$ Dual of rule 15.
17 ${\displaystyle e^{iat}\,}$ ${\displaystyle {\sqrt {2\pi }}\cdot \delta (\omega -a)\,}$ ${\displaystyle \delta (f-{\frac {a}{2\pi }})\,}$ This follows from and 3 and 15.
18 ${\displaystyle \cos(at)\,}$ ${\displaystyle {\sqrt {2\pi }}{\frac {\delta (\omega \!-\!a)\!+\!\delta (\omega \!+\!a)}{2}}\,}$ ${\displaystyle {\frac {\delta (f\!-\!{\begin{matrix}{\frac {a}{2\pi }}\end{matrix}})\!+\!\delta (f\!+\!{\begin{matrix}{\frac {a}{2\pi }}\end{matrix}})}{2}}\,}$ Follows from rules 1 and 17 using Euler's formula: ${\displaystyle \cos(at)=(e^{iat}+e^{-iat})/2.}$
19 ${\displaystyle \sin(at)\,}$ ${\displaystyle {\sqrt {2\pi }}{\frac {\delta (\omega \!-\!a)\!-\!\delta (\omega \!+\!a)}{2i}}\,}$ ${\displaystyle {\frac {\delta (f\!-\!{\begin{matrix}{\frac {a}{2\pi }}\end{matrix}})\!-\!\delta (f\!+\!{\begin{matrix}{\frac {a}{2\pi }}\end{matrix}})}{2i}}\,}$ Also from 1 and 17.
20 ${\displaystyle t^{n}\,}$ ${\displaystyle i^{n}{\sqrt {2\pi }}\delta ^{(n)}(\omega )\,}$ ${\displaystyle \left({\frac {i}{2\pi }}\right)^{n}\delta ^{(n)}(f)\,}$ Here, ${\displaystyle n}$ is a natural number. ${\displaystyle \delta ^{n}(\omega )}$ is the ${\displaystyle n}$-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 ${\displaystyle {\frac {1}{t}}\,}$ ${\displaystyle -i{\sqrt {\frac {\pi }{2}}}\operatorname {sgn}(\omega )\,}$ ${\displaystyle -i\pi \cdot \operatorname {sgn}(f)\,}$ Here ${\displaystyle \operatorname {sgn}(\omega )}$ is the sign function; note that this is consistent with rules 7 and 15.
22 ${\displaystyle {\frac {1}{t^{n}}}\,}$ ${\displaystyle -i{\begin{matrix}{\sqrt {\frac {\pi }{2}}}\cdot {\frac {(-i\omega )^{n-1}}{(n-1)!}}\end{matrix}}\operatorname {sgn}(\omega )\,}$ ${\displaystyle -i\pi {\begin{matrix}{\frac {(-i2\pi f)^{n-1}}{(n-1)!}}\end{matrix}}\operatorname {sgn}(f)\,}$ Generalization of rule 21.
23 ${\displaystyle \operatorname {sgn}(t)\,}$ ${\displaystyle {\sqrt {\frac {2}{\pi }}}\cdot {\frac {1}{i\ \omega }}\,}$ ${\displaystyle {\frac {1}{i\pi f}}\,}$ The dual of rule 21.
24 ${\displaystyle u(t)\,}$ ${\displaystyle {\sqrt {\frac {\pi }{2}}}\left({\frac {1}{i\pi \omega }}+\delta (\omega )\right)\,}$ ${\displaystyle {\frac {1}{2}}\left({\frac {1}{i\pi f}}+\delta (f)\right)\,}$ Here ${\displaystyle u(t)}$ is the Heaviside unit step function; this follows from rules 1 and 21.
${\displaystyle e^{-at}u(t)\,}$ ${\displaystyle {\frac {1}{{\sqrt {2\pi }}(a+i\omega )}}}$ ${\displaystyle {\frac {1}{a+i2\pi f}}}$ ${\displaystyle u(t)}$ is the Heaviside unit step function and ${\displaystyle a>0}$.
25 ${\displaystyle \sum _{n=-\infty }^{\infty }\delta (t-nT)\,}$ ${\displaystyle {\begin{matrix}{\frac {\sqrt {2\pi }}{T}}\end{matrix}}\sum _{k=-\infty }^{\infty }\delta \left(\omega -k{\begin{matrix}{\frac {2\pi }{T}}\end{matrix}}\right)\,}$ ${\displaystyle {\frac {1}{T}}\sum _{k=-\infty }^{\infty }\delta \left(f-{\frac {k}{T}}\right)\,}$ The Dirac comb — helpful for explaining or understanding the transition from continuous to discrete time.