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|Periodicity|| <math>\chi(\omega + 2\pi) = \chi(\omega)</math> || Example | |Periodicity|| <math>\chi(\omega + 2\pi) = \chi(\omega)</math> || Example | ||
|- | |- | ||
| − | | Linearity || <math>ax_{1}[n] + bx_{2}[n] | + | | Linearity || <math>ax_{1}[n] + bx_{2}[n] \rightarrow a\chi_{1}(\omega) + b\chi_{2}(\omega)</math> || Example |
|- | |- | ||
| − | | Time Shifting & Frequency Shifting || 1) <math>x[n - n_{o}] | + | | Time Shifting & Frequency Shifting || 1) <math>x[n - n_{o}] \rightarrow e^{-j\omega n_{o}}\chi(\omega)</math><br /> |
| − | 2) <math>e^{-j{\omega}_{o}n}x[n] | + | 2) <math>e^{-j{\omega}_{o}n}x[n] \rightarrow \chi[\omega - \omega_{o}]</math><br /> |
|| Example | || Example | ||
|- | |- | ||
| − | | Conjugate & Conjugate Symmetry || <math>x[n] | + | | Conjugate & Conjugate Symmetry || <math>x[n] \rightarrow \chi^{*}(-\omega)</math> || <math></math> |
|- | |- | ||
| Parversal Relation || <math>\sum_{n=-\infty}^{\infty }\left | x[n] \right |^{2} = \frac{1}{2\pi }\int_{0}^{2\pi}\left | \chi (\omega) \right |^{2}d\omega</math> || <math></math> | | Parversal Relation || <math>\sum_{n=-\infty}^{\infty }\left | x[n] \right |^{2} = \frac{1}{2\pi }\int_{0}^{2\pi}\left | \chi (\omega) \right |^{2}d\omega</math> || <math></math> | ||
Revision as of 23:12, 18 March 2018
Discrete-Time Fourier Transform Properties with Proofs
| Property Name | Property | Proof |
|---|---|---|
| Periodicity | $ \chi(\omega + 2\pi) = \chi(\omega) $ | Example |
| Linearity | $ ax_{1}[n] + bx_{2}[n] \rightarrow a\chi_{1}(\omega) + b\chi_{2}(\omega) $ | Example |
| 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}] $ |
Example |
| Conjugate & Conjugate Symmetry | $ x[n] \rightarrow \chi^{*}(-\omega) $ | |
| 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 $ | |
| Convolution | $ x[n]*y[n] \rightarrow \chi(\omega)\gamma (\omega) $ | |
| Multiplication | ||
| Duality | ||
| Differentiation in Frequency |
