# User:David R. Keller/Sandbox

### Distributions

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

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

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