| 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}} \ $ | ||
| $ (n+1)a^{n} u[n], |a|<1 \ $ | $ \frac{1}{(1-ae^{-j\omega})^2} \ $ | ||
| $ \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) $ | |
| Differentiation in frequency | $ \ nx[n] $ | $ \ j\frac{d}{d\omega}X(\omega) $ | |
| Other DT Fourier Transform Properties | |
|---|---|
| Parseval's relation | $ \frac {1}{N} \sum_{n=-\infty}^{\infty}\left| x[n] \right|^2 = $ |
