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| + | ! colspan="2" style="background: #e4bc7e; font-size: 110%;" | Discrete-time Fourier Transform Pairs and Properties | ||
| + | |- | ||
! colspan="2" style="background: #eee;" | DT Fourier transform and its Inverse | ! colspan="2" style="background: #eee;" | DT Fourier transform and its Inverse | ||
|- | |- | ||
| align="right" style="padding-right: 1em;" | DT Fourier Transform || <math>\,\mathcal{X}(\omega)=\mathcal{F}(x[n])=\sum_{n=-\infty}^{\infty}x[n]e^{-j\omega n}\,</math> | | align="right" style="padding-right: 1em;" | DT Fourier Transform || <math>\,\mathcal{X}(\omega)=\mathcal{F}(x[n])=\sum_{n=-\infty}^{\infty}x[n]e^{-j\omega n}\,</math> | ||
|- | |- | ||
| − | | align="right" style="padding-right: 1em;" | | + | | align="right" style="padding-right: 1em;" | Inverse DT Fourier Transform || <math>\,x[n]=\mathcal{F}^{-1}(\mathcal{X}(\omega))=\frac{1}{2\pi} \int_{0}^{2\pi}\mathcal{X}(\omega)e^{j\omega n} d \omega\,</math> |
| + | |} | ||
| + | {| | ||
|- | |- | ||
| − | ! colspan=" | + | ! colspan="4" style="background: #eee;" | DT Fourier Transform Pairs |
|- | |- | ||
| − | | align="right" style="padding-right: 1em;" | [ | + | | align="right" style="padding-right: 1em;" | || <math> x[n] \ </math> || <math>\longrightarrow</math>|| <math> \mathcal{X}(\omega) \ </math> |
| + | |- | ||
| + | | align="right" style="padding-right: 1em;" | DTFT of a complex exponential || <math>e^{jw_0n} \ </math> || || <math>\pi\sum_{l=-\infty}^{+\infty}\delta(w-w_0-2\pi l) \ </math> | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;" | ([[DTFT_rectangular_window|info]]) DTFT of a rectangular window || <math>w[n]= \ </math> || || add formula here | ||
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| + | | align="right" style="padding-right: 1em;" | || <math>a^{n} u[n], |a|<1 \ </math> || ||<math>\frac{1}{1-ae^{-j\omega}} \ </math> | ||
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| + | | align="right" style="padding-right: 1em;" | || <math>\sin\left(\omega _0 n\right) u[n] \ </math> || ||<math>\frac{1}{2j}\left( \frac{1}{1-e^{-j(\omega -\omega _0)}}-\frac{1}{1-e^{-j(\omega +\omega _0)}}\right)</math> | ||
| + | |- | ||
| + | |} | ||
| + | |||
| + | {| | ||
| + | |- | ||
| + | ! colspan="4" style="background: #eee;" | DT Fourier Transform Properties | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;" | || <math>x[n] \ </math> || <math>\longrightarrow</math>|| <math> \mathcal{X}(\omega) \ </math> | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;" | multiplication property|| <math>x[n]y[n] \ </math> || || <math>\frac{1}{2\pi} \int_{2\pi} X(\theta)Y(\omega-\theta)d\theta</math> | ||
|- | |- | ||
| − | | align="right" style="padding-right: 1em;" | [[ | + | | align="right" style="padding-right: 1em;" | convolution property || <math>x[n]*y[n] \!</math> || ||<math> X(\omega)Y(\omega) \!</math> |
| + | |- | ||
| + | | align="right" style="padding-right: 1em;" | time reversal ||<math>\ x[-n] </math> || ||<math>\ X(-\omega)</math> | ||
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|} | |} | ||
| − | + | {| | |
| − | + | |- | |
| − | + | ! colspan="2" style="background: #eee;" | Other DT Fourier Transform Properties | |
| − | + | |- | |
| − | + | | align="right" style="padding-right: 1em;" | Parseval's relation || <math>\frac {1}{N} \sum_{n=-\infty}^{\infty}\left| x[n] \right|^2 = </math> | |
| + | |} | ||
---- | ---- | ||
| − | [[ | + | [[Collective_Table_of_Formulas|Back to Collective Table]] |
| + | [[Category:Formulas]] | ||
Latest revision as of 12:52, 22 October 2010
| Discrete-time Fourier Transform Pairs and Properties | |
|---|---|
| DT Fourier transform and its Inverse | |
| DT Fourier Transform | $ \,\mathcal{X}(\omega)=\mathcal{F}(x[n])=\sum_{n=-\infty}^{\infty}x[n]e^{-j\omega n}\, $ |
| Inverse DT Fourier Transform | $ \,x[n]=\mathcal{F}^{-1}(\mathcal{X}(\omega))=\frac{1}{2\pi} \int_{0}^{2\pi}\mathcal{X}(\omega)e^{j\omega n} d \omega\, $ |
| DT Fourier Transform Pairs | |||
|---|---|---|---|
| $ x[n] \ $ | $ \longrightarrow $ | $ \mathcal{X}(\omega) \ $ | |
| DTFT of a complex exponential | $ e^{jw_0n} \ $ | $ \pi\sum_{l=-\infty}^{+\infty}\delta(w-w_0-2\pi l) \ $ | |
| (info) DTFT of a rectangular window | $ w[n]= \ $ | add formula here | |
| $ a^{n} u[n], |a|<1 \ $ | $ \frac{1}{1-ae^{-j\omega}} \ $ | ||
| $ \sin\left(\omega _0 n\right) u[n] \ $ | $ \frac{1}{2j}\left( \frac{1}{1-e^{-j(\omega -\omega _0)}}-\frac{1}{1-e^{-j(\omega +\omega _0)}}\right) $ | ||
| DT Fourier Transform Properties | |||
|---|---|---|---|
| $ x[n] \ $ | $ \longrightarrow $ | $ \mathcal{X}(\omega) \ $ | |
| multiplication property | $ x[n]y[n] \ $ | $ \frac{1}{2\pi} \int_{2\pi} X(\theta)Y(\omega-\theta)d\theta $ | |
| convolution property | $ x[n]*y[n] \! $ | $ X(\omega)Y(\omega) \! $ | |
| time reversal | $ \ x[-n] $ | $ \ X(-\omega) $ | |
| Other DT Fourier Transform Properties | |
|---|---|
| Parseval's relation | $ \frac {1}{N} \sum_{n=-\infty}^{\infty}\left| x[n] \right|^2 = $ |
