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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> | | align="right" style="padding-right: 1em;" | || <math>a^{n} u[n], |a|<1 \ </math> || ||<math>\frac{1}{1-ae^{-j\omega}} \ </math> | ||
| + | |- | ||
| + | | 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> | ||
|- | |- | ||
|} | |} | ||
---- | ---- | ||
[[ MegaCollectiveTableTrial1|Back to Collective Table]] | [[ MegaCollectiveTableTrial1|Back to Collective Table]] | ||
Revision as of 06:46, 27 October 2009
Discrete-time Fourier Transform Pairs and Properties
Please feel free to add onto this table!
| 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) \ $ | |
| $ 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) $ | ||
