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


== '''<big> Discrete-Time Fourier Transform Properties with Proofs''' </big>''' ==
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!Property Name!! Property !! Proof
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|Periodicity|| <math>\chi(\omega + 2\pi) = \chi(\omega)</math> || <math>\chi(\omega+2\pi) = \sum_{n=-\infty}^{\infty}x[n]e^{-j(\omega +2\pi)n}</math> <br />
<math>= \sum_{n=-\infty}^{\infty}x[n]e^{-j\omega n} e^{-j\omega 2\pi}</math> <br />
<math>= e^{-j\omega 2\pi} \sum_{n=-\infty}^{\infty}x[n]e^{-j\omega n}</math> <br />
<math>= (1)\chi(\omega) = \chi(\omega)</math><br />________________________________<br />
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| Linearity || <math>ax_{1}[n] + bx_{2}[n] \rightarrow a\chi_{1}(\omega) + b\chi_{2}(\omega)</math> || <math>\mathfrak{F}(ax_{1}[n] + bx_{2}[n]) = \sum_{n=-\infty}^{\infty}[ax_{1}[n] + bx_{2}[n]]e^{-j\omega n}</math><br />
<math>\sum_{n=-\infty}^{\infty}ax_{1}[n]e^{-j\omega n} + \sum_{n=-\infty}^{\infty}bx_{2}[n]e^{-j\omega n}</math><br />
<math>=a\chi_{1}(\omega) + b\chi_{2}(\omega) </math> <br />________________________________<br />
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| 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 />
|| <math>\mathfrak{F}(x[n - n_{o}]) = \sum_{n=-\infty}^{\infty}x[n - n_{o}]e^{-j\omega n}</math><br />
let <math> m = n - n_{o} </math><br />
<math>\sum_{m=-\infty}^{\infty}x[m]e^{-j\omega m + n_{o}} </math><br />
<math>= e^{-j\omega n_{o}}\sum_{m=-\infty}^{\infty}x[m]</math><br />
<math>= e^{-j\omega n_{o}}\chi(\omega)</math> <br />
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| Conjugate & Conjugate Symmetry || <math>x[n] \rightarrow \chi^{*}(-\omega)</math> || <math>\mathfrak{F}(x[n]) = \sum_{n=-\infty}^{\infty}x[n]e^{-j\omega n}</math><br />
<math>= \sum_{n=-\infty}^{\infty}x[n][cos(\omega n) + jsin(\omega n)]</math><br />
<math>= \sum_{n=-\infty}^{\infty}x[n]cos(\omega n) + \sum_{n=-\infty}^{\infty}x[n]jsin(\omega n)</math><br />
<math>= \sum_{n=-\infty}^{\infty}x[n]\frac{1}{2}[e^{j\omega n} + e^{-j\omega n}] + \sum_{n=-\infty}^{\infty}x[n]j\frac{1}{2j}[e^{j\omega n} - e^{-j\omega n}]</math><br />
Things Cancel out and you are left with.. <br />
<math>= \sum_{n=-\infty}^{\infty}x[n]e^{j\omega n}</math> <br />
<math>= \chi^{*}(-\omega)</math> <br />________________________________<br />
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| 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>\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)</math><br />
<math>= \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</math><br />
<math>= \frac{1}{2\pi}\int_{0}^{2\pi}\chi(\omega)[\chi(-\omega)]d\omega</math><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> || 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 />
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| Multiplication || <math>x[n]y[n] \rightarrow \frac{1}{2\pi}\chi(\omega)*\gamma (\omega)^{}_{}</math> || Recall for a periodic signal of period T <math> x(t)y(t) = \int_{T}^{ }x(\tau)y(t-\tau)d\tau </math><br />
<math>\mathfrak{F}(x[n]y[n]) = \sum_{n=-\infty}^{\infty}[x[n]y[n]]e^{-j\omega n}</math><br />

<math> = \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 }</math><br />

<math> = \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 '</math><br />

<math> = \frac{1}{2\pi}\int_{0}^{2\pi}\chi(j\omega ')[\sum_{n=-\infty}^{\infty}e^{-jn (\omega-\omega')} y[n]] d\omega '</math><br />
<math>= \frac{1}{2\pi}\int_{0}^{2\pi}\chi(j\omega ')\gamma(j\omega - \omega')d\omega '</math><br />
<math>= \chi(j\omega)*\gamma(j\omega)</math><br />________________________________<br />
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| Duality || '''''NO DUALITY IN DT''''' || '''''NO DUALITY IN DT'''''<br />
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| Differentiation in Frequency || <math>nx[n] \rightarrow j\frac{\mathrm{d} }{\mathrm{d} \omega}\chi(\omega)</math> || <math>\frac{\mathrm{d} }{\mathrm{d} \omega} \chi(e^{j\omega}) = \frac{\mathrm{d} }{\mathrm{d} \omega}\sum_{n=-\infty}^{\infty}x[n]e^{-j\omega n}</math><br />

<math>= \sum_{n=-\infty}^{\infty}x[n]\frac{\mathrm{d} }{\mathrm{d} \omega}e^{-j\omega n}</math><br />

<math>= \sum_{n=-\infty}^{\infty}x[n](-jn)e^{-j\omega n}</math><br />________________________________<br />
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