Mostrar código-fonte de Tabelas de engenharia/Propriedades das transformadas de Fourier

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!style="background:LightSteelBlue; color:White"|ID 
!style="background:LightSteelBlue; color:White"|Sinal 
!style="background:LightSteelBlue; color:White"|Transformada de Fourier<br /> unitary, angular frequency  
!style="background:LightSteelBlue; color:White"|Fourier transform <br /> unitary, ordinary frequency 
!style="background:LightSteelBlue; color:White"|Remarks
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|align="center"|<math> g(t)\!\equiv\!</math><br /><br /><math> \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty}\!\!G(\omega) e^{i \omega t} d \omega \,</math>
|align="center"|<math> G(\omega)\!\equiv\!</math><br /><br /><math>\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty}\!\!g(t) e^{-i \omega t} dt \,</math>
|align="center"|<math> G(f)\!\equiv </math><br /><br /><math>\int_{-\infty}^{\infty}\!\!g(t) e^{-i 2\pi f t} dt \,</math>
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|-
| 1
|<math>a\cdot g(t) + b\cdot h(t)\,</math>
|<math>a\cdot G(\omega) + b\cdot H(\omega)\,</math>
|<math>a\cdot G(f) + b\cdot H(f)\,</math>
|Linearity
|-
| 2
|<math>g(t - a)\,</math>
|<math>e^{- i a \omega} G(\omega)\,</math>
|<math>e^{- i 2\pi a f} G(f)\,</math>
|Shift in time domain
|-
| 3
|<math>e^{ iat} g(t)\,</math>
|<math>G(\omega - a)\,</math>
|<math>G \left(f - \frac{a}{2\pi}\right)\,</math>
|Shift in frequency domain, dual of 2
|-
| 4
|<math>g(a t)\,</math>
|<math>\frac{1}{|a|} G \left( \frac{\omega}{a} \right)\,</math>
|<math>\frac{1}{|a|} G \left( \frac{f}{a} \right)\,</math>
|If <math>|a|\,</math> is large, then <math>g(a t)\,</math> is concentrated around 0 and <math>\frac{1}{|a|}G \left( \frac{\omega}{a} \right)\,</math> spreads out and flattens
|-
| 5
|<math>G(t)\,</math>
|<math> g(-\omega)\,</math>
|<math> g(-f)\,</math>
|Duality property of the Fourier transform.  Results from swapping "dummy" variables of <math> t \,</math> and <math> \omega \,</math>.
|-
| 6
|<math>\frac{d^n g(t)}{dt^n}\,</math>
|<math> (i\omega)^n  G(\omega)\,</math>
|<math> (i 2\pi f)^n  G(f)\,</math>
|Generalized derivative property of the Fourier transform
|-
| 7
|<math>t^n g(t)\,</math>
|<math>i^n \frac{d^n G(\omega)}{d\omega^n}\,</math>
|<math>\left (\frac{i}{2\pi}\right)^n \frac{d^n G(f)}{df^n}\,</math>
|This is the dual to 6
|-
| 8
|<math>(g * h)(t)\,</math>
|<math>\sqrt{2\pi} G(\omega) H(\omega)\,</math>
|<math>G(f) H(f)\,</ma''''''Texto em negrito''''''th>
|<math>g * h\,</math> denotes the convolution of <math>g\,</math> and <math>h\,</math> &mdash; this rule is the convolution theorem
|-
| 9
|<math>g(t) h(t)\,</math>
|<math>(G * H)(\omega) \over \sqrt{2\pi}\,</math>
|<math>(G * H)(f)\,</math>
|This is the dual of 8
|-
|10
| For a purely real even function <math>g(t)\,</math>
| <math>G(\omega)\,</math> is a purely real even function
| <math>G(f)\,</math> is a purely real even function
|-
|11
| For a purely real odd function <math>g(t)\,</math>
| <math>G(\omega)\,</math> is a purely imaginary odd function
| <math>G(f)\,</math> is a purely imaginary odd function
|}
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