Discrete-Time Fourier Transform Properties with Proofs
| Property Name | Property | Proof |
|---|---|---|
| Periodicity | $ \chi(\omega + 2\pi) = \chi(\omega) $ | $ \chi(\omega+2\pi) = \sum_{n=-\infty}^{\infty}x[n]e^{-j(\omega +2\pi)n} $ $ = \sum_{n=-\infty}^{\infty}x[n]e^{-j\omega n} e^{-j\omega 2\pi} $ |
| Linearity | $ ax_{1}[n] + bx_{2}[n] \rightarrow a\chi_{1}(\omega) + b\chi_{2}(\omega) $ | $ \mathfrak{F}(ax_{1}[n] + bx_{2}[n]) = \sum_{n=-\infty}^{\infty}[ax_{1}[n] + bx_{2}[n]]e^{-j\omega n} $ $ \sum_{n=-\infty}^{\infty}ax_{1}[n]e^{-j\omega n} + \sum_{n=-\infty}^{\infty}bx_{2}[n]e^{-j\omega n} $ |
| Time Shifting & Frequency Shifting | 1) $ x[n - n_{o}] \rightarrow e^{-j\omega n_{o}}\chi(\omega) $ 2) $ e^{-j{\omega}_{o}n}x[n] \rightarrow \chi[\omega - \omega_{o}] $ |
$ \mathfrak{F}(x[n - n_{o}]) = \sum_{n=-\infty}^{\infty}x[n - n_{o}]e^{-j\omega n} $ let $ m = n - n_{o} $ |
| Conjugate & Conjugate Symmetry | $ x[n] \rightarrow \chi^{*}(-\omega) $ | $ \mathfrak{F}(x[n]) = \sum_{n=-\infty}^{\infty}x[n]e^{-j\omega n} $ $ = \sum_{n=-\infty}^{\infty}x[n][cos(\omega n) + jsin(\omega n)] $ |
| Parversal Relation | $ \sum_{n=-\infty}^{\infty }\left | x[n] \right |^{2} = \frac{1}{2\pi }\int_{0}^{2\pi}\left | \chi (\omega) \right |^{2}d\omega $ | $ \sum_{n=-\infty}^{\infty} x[n]x[n] = \sum_{n=-\infty}^{\infty}x[n](\frac{1}{2\pi}\int_{0}^{2\pi}\chi(\omega)e^{j\omega n}d\omega) $ $ = \frac{1}{2\pi}\int_{0}^{2\pi}\chi(\omega)[\sum_{n=-\infty}^{\infty}x[n](\frac{1}{2\pi}e^{j\omega n}]d\omega $ |
| Convolution | $ x[n]*y[n] \rightarrow \chi(\omega)\gamma (\omega) $ | Recall $ x[n]*y[n] = \sum_{k=-\infty}^{\infty}x[k]*y[n-k] $ $ \mathfrak{F}(x[n]*y[n]) = \sum_{n=-\infty}^{\infty}[\sum_{k=-\infty}^{\infty}x[k]*y[n-k]]e^{-j\omega n} $ |
| Multiplication | $ x[n]y[n] \rightarrow \frac{1}{2\pi}\chi(\omega)*\gamma (\omega)^{}_{} $ | Recall for a periodic signal of period T $ x(t)y(t) = \int_{T}^{ }x(\tau)y(t-\tau)d\tau $ $ \mathfrak{F}(x[n]y[n]) = \sum_{n=-\infty}^{\infty}[x[n]y[n]]e^{-j\omega n} $ $ = \sum_{n=-\infty}^{\infty}[\frac{1}{2\pi}\int_{0}^{2\pi}\chi(j\omega ')e^{j\omega 'n}d\omega ']]y[n]e^{-j\omega n } $ $ = \frac{1}{2\pi}\int_{0}^{2\pi}\chi(j\omega ')[\sum_{n=-\infty}^{\infty}e^{j\omega 'n} y[n]e^{-j\omega n }] d\omega ' $ $ = \frac{1}{2\pi}\int_{0}^{2\pi}\chi(j\omega ')[\sum_{n=-\infty}^{\infty}e^{-jn (\omega-\omega')} y[n]] d\omega ' $ |
| Duality | NO DUALITY IN DT | NO DUALITY IN DT ________________________________ |
| Differentiation in Frequency | $ nx[n] \rightarrow j\frac{\mathrm{d} }{\mathrm{d} \omega}\chi(\omega) $ | $ j\frac{\mathrm{d} }{\mathrm{d} \omega} \chi(e^{j\omega}) = j\frac{\mathrm{d} }{\mathrm{d} \omega}\sum_{n=-\infty}^{\infty}x[n]e^{-j\omega n} $ $ = j\sum_{n=-\infty}^{\infty}x[n]\frac{\mathrm{d} }{\mathrm{d} \omega}e^{-j\omega n} $ |
List of Common Coefficients of Discrete-Time Fourier Transform
| Waveform | Digital Time Domain Signal* | Frequency Domain Signal* |
|---|---|---|
| Constant | $ 1 $ | $ \delta(k) $ |
| Unit Impulse | $ \delta(n) $ | $ \frac{1}{N} $ |
| Complex Exponential | $ e^{(j2\pi m \omega n)/N} $ | $ \delta((k-m)_{N})) $ |
| Sinusoidal | $ cos(j2\pi m \omega n)/N $ | $ \frac{1}{2}(\delta((k-m)_{N})+ \delta((k+m)_{N})) $ |
| Box* | $ \delta(n) + \sum_{m=1}^{M} \delta((n-m)_{N}))+ \delta((n+m)_{N})) $ | $ \frac{sin(\frac{(2M+1)k\pi}{N})}{N sin(\frac{k\pi}{N})} $ |
| Dsinc* | $ \frac{sin(\frac{(2M+1)n\pi}{N})}{N sin(\frac{n\pi}{N})} $ | $ \delta(k) + \sum_{m=1}^{M} \delta((k-m)_{N}))+ \delta((k+m)_{N})) $ |
| ________________________________ *$ M < \frac{N}{2} $ |
________________________________ $ n \in \mathbb{Z}[0, N - 1] $ |
________________________________ $ k \in \mathbb{Z}[0, N - 1] $ |
By --Abinhamd (talk) 01:52, 19 March 2018 (EDT)Alanoud Bin Hamdan
