View source for Discrete Time Fourior Transfrom Properties and Proofs - Rhea


== '''<big> Discrete-Time Fourier Transform Properties with Proofs''' </big>''' ==
<br />

{| class="wikitable sortable"
|-
!Property Name!! Property !! Proof
|-
|Periodicity|| <math>\chi(\omega + 2\pi) = \chi(\omega)</math> || Example
|-
| 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}] \rightarrow e^{-j\omega n_{o}}\chi(\omega)</math><br />
2) <math>e^{-j{\omega}_{o}n}x[n] \rightarrow \chi[\omega - \omega_{o}]</math><br />
|| Example
|-
| 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>
|-
| Convolution || <math>x[n]*y[n] \rightarrow \chi(\omega)\gamma (\omega)</math> || <math></math>
|-
| Multiplication || <math>x[n]y[n] \rightarrow \frac{1}{2\pi}\chi(\omega)*\gamma (\omega)^{}_{}</math> || <math></math>
|-
| Duality || '''''NO DUALITY IN DT''''' || '''''NO DUALITY IN DT'''''
|-
| Differentiation in Frequency || <math>nx[n] \rightarrow j\frac{\mathrm{d} }{\mathrm{d} \omega}\chi(\omega)</math> || <math></math>
|}