User:David R. Keller/Sandbox

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Distributions[edit]

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.