# User:David R. Keller/Sandbox

### Distributions

Signal Fourier transform
unitary, angular frequency
Fourier transform
unitary, ordinary frequency
Remarks
${\displaystyle {\tilde {g}}(t)\,}$ ${\displaystyle G(\omega )\!\ {\stackrel {\mathrm {def} }{=}}\ \!}$

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

${\displaystyle G[k]={\frac {1}{T_{o}}}\int _{0}^{T_{o}}\!\!g(t)e^{-i2\pi kf_{o}t}\,dt}$
301 ${\displaystyle 1\,}$ ${\displaystyle {\sqrt {2\pi }}\cdot \delta (\omega )\,}$ ${\displaystyle \delta (f)\,}$ ${\displaystyle \displaystyle \delta (\omega )}$ denotes the Dirac delta distribution.
302 ${\displaystyle \delta (t)\,}$ ${\displaystyle {\frac {1}{\sqrt {2\pi }}}\,}$ ${\displaystyle 1\,}$ Dual of rule 301.
303 ${\displaystyle e^{iat}\,}$ ${\displaystyle {\sqrt {2\pi }}\cdot \delta (\omega -a)\,}$ ${\displaystyle \delta (f-{\frac {a}{2\pi }})\,}$ This follows from and 103 and 301.
304 ${\displaystyle \cos(at)\,}$ ${\displaystyle {\sqrt {2\pi }}{\frac {\delta (\omega \!-\!a)\!+\!\delta (\omega \!+\!a)}{2}}\,}$ ${\displaystyle {\frac {\delta (f\!-\!{\begin{matrix}{\frac {a}{2\pi }}\end{matrix}})\!+\!\delta (f\!+\!{\begin{matrix}{\frac {a}{2\pi }}\end{matrix}})}{2}}\,}$ Follows from rules 101 and 303 using Euler's formula: ${\displaystyle \displaystyle \cos(at)=(e^{iat}+e^{-iat})/2.}$
305 ${\displaystyle \sin(at)\,}$ ${\displaystyle i{\sqrt {2\pi }}{\frac {\delta (\omega \!+\!a)\!-\!\delta (\omega \!-\!a)}{2}}\,}$ ${\displaystyle i{\frac {\delta (f\!+\!{\begin{matrix}{\frac {a}{2\pi }}\end{matrix}})\!-\!\delta (f\!-\!{\begin{matrix}{\frac {a}{2\pi }}\end{matrix}})}{2}}\,}$ Also from 101 and 303 using ${\displaystyle \displaystyle \sin(at)=(e^{iat}-e^{-iat})/(2i).}$
306 ${\displaystyle t^{n}\,}$ ${\displaystyle i^{n}{\sqrt {2\pi }}\delta ^{(n)}(\omega )\,}$ ${\displaystyle \left({\frac {i}{2\pi }}\right)^{n}\delta ^{(n)}(f)\,}$ Here, ${\displaystyle \displaystyle n}$ is a natural number. ${\displaystyle \displaystyle \delta ^{n}(\omega )}$ is the ${\displaystyle \displaystyle n}$-th distribution derivative of the Dirac delta. This rule follows from rules 107 and 302. Combining this rule with 1, we can transform all polynomials.
307 ${\displaystyle {\frac {1}{t}}\,}$ ${\displaystyle -i{\sqrt {\frac {\pi }{2}}}\operatorname {sgn}(\omega )\,}$ ${\displaystyle -i\pi \cdot \operatorname {sgn}(f)\,}$ Here ${\displaystyle \displaystyle \operatorname {sgn}(\omega )}$ is the sign function; note that this is consistent with rules 107 and 302.
308 ${\displaystyle {\frac {1}{t^{n}}}\,}$ ${\displaystyle -i{\begin{matrix}{\sqrt {\frac {\pi }{2}}}\cdot {\frac {(-i\omega )^{n-1}}{(n-1)!}}\end{matrix}}\operatorname {sgn}(\omega )\,}$ ${\displaystyle -i\pi {\begin{matrix}{\frac {(-i2\pi f)^{n-1}}{(n-1)!}}\end{matrix}}\operatorname {sgn}(f)\,}$ Generalization of rule 307.
309 ${\displaystyle \operatorname {sgn}(t)\,}$ ${\displaystyle {\sqrt {\frac {2}{\pi }}}\cdot {\frac {1}{i\ \omega }}\,}$ ${\displaystyle {\frac {1}{i\pi f}}\,}$ The dual of rule 307.
310 ${\displaystyle u(t)\,}$ ${\displaystyle {\sqrt {\frac {\pi }{2}}}\left({\frac {1}{i\pi \omega }}+\delta (\omega )\right)\,}$ ${\displaystyle {\frac {1}{2}}\left({\frac {1}{i\pi f}}+\delta (f)\right)\,}$ Here ${\displaystyle u(t)}$ is the Heaviside unit step function; this follows from rules 101 and 309.
311 ${\displaystyle e^{-at}u(t)\,}$ ${\displaystyle {\frac {1}{{\sqrt {2\pi }}(a+i\omega )}}}$ ${\displaystyle {\frac {1}{a+i2\pi f}}}$ ${\displaystyle u(t)}$ is the Heaviside unit step function and ${\displaystyle a>0}$.
312 ${\displaystyle g[n]=\sum _{n=-\infty }^{\infty }g(nT)\delta (t-nT)\,}$ ${\displaystyle {\begin{matrix}{\frac {\sqrt {2\pi }}{T}}\end{matrix}}\sum _{k=-\infty }^{\infty }\delta \left(\omega -k{\begin{matrix}{\frac {2\pi }{T}}\end{matrix}}\right)\,}$ ${\displaystyle {\frac {1}{T}}\sum _{k=-\infty }^{\infty }\delta \left(f-{\frac {k}{T}}\right)\,}$ The Dirac comb — helpful for explaining or understanding the transition from continuous to discrete time.