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 | |
| DT Fourier Transform Pair_ECE301Fall2008mboutin | $ e^{jw_0n} \longrightarrow 2\pi\sum_{l=-\infty}^{+\infty}\delta(w-w_0-2\pi l) \ $ |
| DT Fourier an_ECE301Fall2008mboutin | $ a^{n} u[n], |a|<1 \longrightarrow \frac{1}{1-ae^{-j\omega}} \ $ |
