User:Porceberkeley

From Wikipedia, the free encyclopedia
Jump to: navigation, search

Table of important Fourier transforms[edit]

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[edit]

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[edit]

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[edit]

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.