| Line 11: | Line 11: | ||
| Linearity || <math>ax_{1}[n] + bx_{2}[n] → a\chi_{1}(\omega) + b\chi_{2}(\omega)</math> || Example | | Linearity || <math>ax_{1}[n] + bx_{2}[n] → a\chi_{1}(\omega) + b\chi_{2}(\omega)</math> || Example | ||
|- | |- | ||
| − | | Time Shifting & Frequency Shifting || 1) x[n - | + | | Time Shifting & Frequency Shifting || 1) <math>x[n - n_{o}] → e^{-j\omegan_{o}}\chi(\omega)</math><br /> |
| − | 2) | + | 2) <math>e^{-j/omega_{o}n}x[n] → \chi[\omega - \omega_{o}]</math><br /> |
|| Example | || Example | ||
|- | |- | ||
| − | | Conjugate & Conjugate Symmetry || x[n] → X<sup>*</sup>(-ω) || | + | | Conjugate & Conjugate Symmetry || x[n] → X<sup>*</sup>(-ω) || <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> || | + | | 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> |
|- | |- | ||
| − | | Convolution || <math>x[n]*y[n] \rightarrow \chi(\omega)\gamma (\omega)</math> || | + | | Convolution || <math>x[n]*y[n] \rightarrow \chi(\omega)\gamma (\omega)</math> || <math></math> |
|- | |- | ||
| − | | Multiplication || | + | | Multiplication || <math></math> || <math></math> |
|- | |- | ||
| − | | Duality || | + | | Duality || <math></math> || <math></math> |
|- | |- | ||
| − | | Differentiation in Frequency || | + | | Differentiation in Frequency || <math></math> || <math></math> |
|} | |} | ||
Revision as of 23:06, 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] → a\chi_{1}(\omega) + b\chi_{2}(\omega) $ | Example |
| Time Shifting & Frequency Shifting | 1) $ x[n - n_{o}] → e^{-j\omegan_{o}}\chi(\omega) $ 2) $ e^{-j/omega_{o}n}x[n] → \chi[\omega - \omega_{o}] $ |
Example |
| Conjugate & Conjugate Symmetry | x[n] → X*(-ω) | |
| 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 |
