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<math>= \frac{1}{2\pi }\int_{0}^{2\pi}\left | \chi (\omega) \right |^{2}d\omega</math><br />________________________________<br /> | <math>= \frac{1}{2\pi }\int_{0}^{2\pi}\left | \chi (\omega) \right |^{2}d\omega</math><br />________________________________<br /> | ||
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| − | | Convolution || <math>x[n]*y[n] \rightarrow \chi(\omega)\gamma (\omega)</math> || <math></math>________________________________<br /> | + | | Convolution || <math>x[n]*y[n] \rightarrow \chi(\omega)\gamma (\omega)</math> || Recall <math> x[n]*y[n] = \sum_{k=-\infty}^{\infty}x[k]*y[n-k] </math><br /> |
| + | <math>\mathfrak{F}(x[n]*y[n]) = \sum_{n=-\infty}^{\infty}[\sum_{k=-\infty}^{\infty}x[k]*y[n-k]]e^{-j\omega n}</math><br /> | ||
| + | We can replace <math>e^{-j\omega n}</math> by <math>e^{-j\omega n} = e^{-j\omega n + j\omega k -j\omega k }= e^{-j\omega( n - k ) - j\omega k }= e^{-j\omega( n - k )} e^{j\omega k }</math><br /> | ||
| + | So..<br /> | ||
| + | <math> = \sum_{n=-\infty}^{\infty}[\sum_{k=-\infty}^{\infty}x[k]*y[n-k]]e^{-j\omega( n - k )} e^{j\omega k }</math><br /> | ||
| + | <math> = \sum_{k=-\infty}^{\infty}e^{j\omega k }[\sum_{n=-\infty}^{\infty}x[k]*y[n-k]]e^{-j\omega( n - k )] </math><br /> | ||
| + | <math>= \chi(\omega)\gamma(\omega)</math><br />________________________________<br /> | ||
|- | |- | ||
| Multiplication || <math>x[n]y[n] \rightarrow \frac{1}{2\pi}\chi(\omega)*\gamma (\omega)^{}_{}</math> || <math></math>________________________________<br /> | | Multiplication || <math>x[n]y[n] \rightarrow \frac{1}{2\pi}\chi(\omega)*\gamma (\omega)^{}_{}</math> || <math></math>________________________________<br /> | ||
Revision as of 00:04, 19 March 2018
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)^{}_{} $ | ________________________________ |
| Duality | NO DUALITY IN DT | NO DUALITY IN DT ________________________________ |
| Differentiation in Frequency | $ nx[n] \rightarrow j\frac{\mathrm{d} }{\mathrm{d} \omega}\chi(\omega) $ | ________________________________ |
