| Line 4: | Line 4: | ||
| + | {| | ||
| + | ! colspan="2" style="background: #eee;" | CT Fourier transform and its Inverse | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;" | CT Fourier Transform || <math>\mathcal{X}(\omega)=\mathcal{F}(x(t))=\int_{-\infty}^{\infty} x(t) e^{-j\omega t} dt</math> | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;" | Inverse DT Fourier Transform || <math>\, x(t)=\mathcal{F}^{-1}(\mathcal{X}(\omega))=\frac{1}{2\pi} \int_{-\infty}^{\infty}\mathcal{X}(\omega)e^{j\omega t} d \omega\,</math> | ||
| + | |} | ||
| + | {| | ||
| + | |- | ||
| + | ! colspan="4" style="background: #eee;" | CT Fourier Transform Pairs | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;" | || <math>x(t)</math> || <math>\longrightarrow</math>|| <math> \mathcal{X}(\omega) </math> | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;" | CTFT of a complex exponential || <math>e^{jw_0t}</math> || || | ||
| + | |- | ||
| + | |} | ||
| − | + | {| | |
| + | |- | ||
| + | ! colspan="4" style="background: #eee;" | CT Fourier Transform Properties | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;" | || <math>x(t)</math> || <math>\longrightarrow</math>|| <math> \mathcal{X}(\omega) </math> | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;" | multiplication property|| <math>x(t)y(t) \ </math> || || <math>\frac{1}{2\pi} X(\omega)*Y(\omega) =\frac{1}{2\pi} \int_{-\infty}^{\infty} X(\theta)Y(\omega-\theta)d\theta</math> | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;" | convolution property || <math>x(t)*y(t) \!</math> || ||<math> X(\omega)Y(\omega) \!</math> | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;" | time reversal ||<math>\ x(-t) </math> || ||<math>\ X(-\omega)</math> | ||
| + | |- | ||
| + | |} | ||
| + | |||
| + | {| | ||
| + | |- | ||
| + | ! colspan="2" style="background: #eee;" | Other CT Fourier Transform Properties | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;" | Parseval's relation || | ||
| + | |} | ||
| + | ---- | ||
| + | [[ MegaCollectiveTableTrial1|Back to Collective Table]] | ||
Revision as of 07:19, 28 October 2009
CT Fourier Transform Pairs and Properties
Using $ \omega $ in radians to parametrize the Fourier transforms.
| CT Fourier transform and its Inverse | |
|---|---|
| CT Fourier Transform | $ \mathcal{X}(\omega)=\mathcal{F}(x(t))=\int_{-\infty}^{\infty} x(t) e^{-j\omega t} dt $ |
| Inverse DT Fourier Transform | $ \, x(t)=\mathcal{F}^{-1}(\mathcal{X}(\omega))=\frac{1}{2\pi} \int_{-\infty}^{\infty}\mathcal{X}(\omega)e^{j\omega t} d \omega\, $ |
| CT Fourier Transform Pairs | |||
|---|---|---|---|
| $ x(t) $ | $ \longrightarrow $ | $ \mathcal{X}(\omega) $ | |
| CTFT of a complex exponential | $ e^{jw_0t} $ | ||
| CT Fourier Transform Properties | |||
|---|---|---|---|
| $ x(t) $ | $ \longrightarrow $ | $ \mathcal{X}(\omega) $ | |
| multiplication property | $ x(t)y(t) \ $ | $ \frac{1}{2\pi} X(\omega)*Y(\omega) =\frac{1}{2\pi} \int_{-\infty}^{\infty} X(\theta)Y(\omega-\theta)d\theta $ | |
| convolution property | $ x(t)*y(t) \! $ | $ X(\omega)Y(\omega) \! $ | |
| time reversal | $ \ x(-t) $ | $ \ X(-\omega) $ | |
| Other CT Fourier Transform Properties | |
|---|---|
| Parseval's relation | |
