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<br />
 
<br />
  
−
{| class="wikitable sortable"
+
{|  
 
|-
 
|-
 
!Property Name!! Property !! Proof
 
!Property Name!! Property !! Proof
 
|-
 
|-
−
|Periodicity|| <math>\chi(\omega + 2\pi) = \chi(\omega)</math> || Example
+
|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 2\pi}</math> <br />
 +
<math>= e^{-j 2\pi} \sum_{n=-\infty}^{\infty}x[n]e^{-j\omega n}</math> <br />
 +
<math>= (1)\chi(\omega) = \chi(\omega)</math><br />________________________________<br />
 
|-
 
|-
−
| Linearity || <math>ax_{1}[n] + bx_{2}[n] \rightarrow a\chi_{1}(\omega) + b\chi_{2}(\omega)</math> || Example
+
| 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 />
 
|-
 
|-
 
| Time Shifting & Frequency Shifting || 1) <math>x[n - n_{o}] \rightarrow e^{-j\omega n_{o}}\chi(\omega)</math><br />
 
| 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 />
 
2) <math>e^{-j{\omega}_{o}n}x[n] \rightarrow \chi[\omega - \omega_{o}]</math><br />
−
|| Example
+
|| <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 />
 +
________________________________<br />
 
|-
 
|-
−
| Conjugate & Conjugate Symmetry || <math>x[n] \rightarrow \chi^{*}(-\omega)</math> || <math></math>
+
| 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 />
 
|-
 
|-
−
| 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>\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 />
 
|-
 
|-
−
| Convolution || <math>x[n]*y[n] \rightarrow \chi(\omega)\gamma (\omega)</math> || <math></math>
+
| 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>
+
| 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 />
 
|-
 
|-
−
| Duality || '''''NO DUALITY IN DT''''' || '''''NO DUALITY IN DT'''''
+
| Duality || '''''NO DUALITY IN DT''''' || '''''NO DUALITY IN DT'''''<br />
 +
________________________________<br />
 
|-
 
|-
−
| Differentiation in Frequency || <math>nx[n] \rightarrow j\frac{\mathrm{d} }{\mathrm{d} \omega}\chi(\omega)</math> || <math></math>
+
| Differentiation in Frequency || <math>nx[n] \rightarrow j\frac{\mathrm{d} }{\mathrm{d} \omega}\chi(\omega)</math> || <math>j\frac{\mathrm{d} }{\mathrm{d} \omega} \chi(e^{j\omega}) =  j\frac{\mathrm{d} }{\mathrm{d} \omega}\sum_{n=-\infty}^{\infty}x[n]e^{-j\omega n}</math><br />
 +
 
 +
<math>= j\sum_{n=-\infty}^{\infty}x[n]\frac{\mathrm{d} }{\mathrm{d} \omega}e^{-j\omega n}</math><br />
 +
<math>= j\sum_{n=-\infty}^{\infty}x[n](-jn)e^{-j\omega n}= \sum_{n=-\infty}^{\infty}(x[n]n)e^{-j\omega n}</math><br />
 +
<math>= nx[n]\sum_{n=-\infty}^{\infty}e^{-j\omega n} = nx[n]</math><br />________________________________<br />
 
|}
 
|}
 +
== '''<big>List of Common Coefficients of Discrete-Time Fourier Transform''' </big>''' ==
 +
<br />
 +
{| class="wikitable"
 +
|-
 +
! Waveform  !! Digital Time Domain Signal* !! Frequency Domain Signal*
 +
|-
 +
| Constant || <math>1</math> || <math>\delta(k)</math>
 +
|-
 +
| Unit Impulse || <math>\delta(n)</math> || <math>\frac{1}{N}</math>
 +
|-
 +
| Complex Exponential || <math>e^{(j2\pi m \omega n)/N}</math> || <math>\delta((k-m)_{N}))</math>
 +
|-
 +
| Sinusoidal  || <math>cos(j2\pi m \omega n)/N</math> || <math>\frac{1}{2}(\delta((k-m)_{N})+ \delta((k+m)_{N}))</math>
 +
|-
 +
| Box* || <math>\delta(n) + \sum_{m=1}^{M} \delta((n-m)_{N}))+ \delta((n+m)_{N}))</math> || <math>\frac{sin(\frac{(2M+1)k\pi}{N})}{N sin(\frac{k\pi}{N})}</math>
 +
|-
 +
| Dsinc* || <math>\frac{sin(\frac{(2M+1)n\pi}{N})}{N sin(\frac{n\pi}{N})}</math> || <math>\delta(k) + \sum_{m=1}^{M} \delta((k-m)_{N}))+ \delta((k+m)_{N}))</math>
 +
|-
 +
| ________________________________<br />*<math>M < \frac{N}{2}</math><br />||________________________________<br /> <math> n \in \mathbb{Z}[0, N - 1]</math><br />||________________________________<br /> <math> k \in \mathbb{Z}[0, N - 1]</math>
 +
|}
 +
By --[[User:Abinhamd|Abinhamd]] ([[User talk:Abinhamd|talk]]) 01:52, 19 March 2018 (EDT)Alanoud Bin Hamdan

Latest revision as of 12:20, 26 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 2\pi} $
$ = e^{-j 2\pi} \sum_{n=-\infty}^{\infty}x[n]e^{-j\omega n} $
$ = (1)\chi(\omega) = \chi(\omega) $
________________________________

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} $
$ =a\chi_{1}(\omega) + b\chi_{2}(\omega) $
________________________________

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} $
$ \sum_{m=-\infty}^{\infty}x[m]e^{-j\omega m + n_{o}} $
$ = e^{-j\omega n_{o}}\sum_{m=-\infty}^{\infty}x[m] $
$ = e^{-j\omega n_{o}}\chi(\omega) $
________________________________

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)] $
$ = \sum_{n=-\infty}^{\infty}x[n]cos(\omega n) + \sum_{n=-\infty}^{\infty}x[n]jsin(\omega n) $
$ = \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}] $
Things Cancel out and you are left with..
$ = \sum_{n=-\infty}^{\infty}x[n]e^{j\omega n} $
$ = \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 $ $ \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 $
$ = \frac{1}{2\pi}\int_{0}^{2\pi}\chi(\omega)[\chi(-\omega)]d\omega $
$ = \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) $ 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} $
We can replace $ e^{-j\omega n} $ by $ 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 } $
So..
$ = \sum_{n=-\infty}^{\infty}[\sum_{k=-\infty}^{\infty}x[k]*y[n-k]]e^{-j\omega( n - k )} e^{-j\omega k} $
$ = \sum_{k=-\infty}^{\infty}e^{-j\omega k }[\sum_{n=-\infty}^{\infty}x[k]*y[n-k]]e^{-j\omega( n - k )}] $
$ = \chi(\omega)\gamma(\omega) $
________________________________

Multiplication $ x[n]y[n] \rightarrow \frac{1}{2\pi}\chi(\omega)*\gamma (\omega)^{}_{} $ Recall for a periodic signal of period T $ x(t)y(t) = \int_{T}^{ }x(\tau)y(t-\tau)d\tau $

$ \mathfrak{F}(x[n]y[n]) = \sum_{n=-\infty}^{\infty}[x[n]y[n]]e^{-j\omega n} $

$ = \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 } $

$ = \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 ' $

$ = \frac{1}{2\pi}\int_{0}^{2\pi}\chi(j\omega ')[\sum_{n=-\infty}^{\infty}e^{-jn (\omega-\omega')} y[n]] d\omega ' $
$ = \frac{1}{2\pi}\int_{0}^{2\pi}\chi(j\omega ')\gamma(j\omega - \omega')d\omega ' $
$ = \chi(j\omega)*\gamma(j\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) $ $ j\frac{\mathrm{d} }{\mathrm{d} \omega} \chi(e^{j\omega}) = j\frac{\mathrm{d} }{\mathrm{d} \omega}\sum_{n=-\infty}^{\infty}x[n]e^{-j\omega n} $

$ = j\sum_{n=-\infty}^{\infty}x[n]\frac{\mathrm{d} }{\mathrm{d} \omega}e^{-j\omega n} $
$ = j\sum_{n=-\infty}^{\infty}x[n](-jn)e^{-j\omega n}= \sum_{n=-\infty}^{\infty}(x[n]n)e^{-j\omega n} $
$ = nx[n]\sum_{n=-\infty}^{\infty}e^{-j\omega n} = nx[n] $
________________________________

List of Common Coefficients of Discrete-Time Fourier Transform


Waveform Digital Time Domain Signal* Frequency Domain Signal*
Constant $ 1 $ $ \delta(k) $
Unit Impulse $ \delta(n) $ $ \frac{1}{N} $
Complex Exponential $ e^{(j2\pi m \omega n)/N} $ $ \delta((k-m)_{N})) $
Sinusoidal $ cos(j2\pi m \omega n)/N $ $ \frac{1}{2}(\delta((k-m)_{N})+ \delta((k+m)_{N})) $
Box* $ \delta(n) + \sum_{m=1}^{M} \delta((n-m)_{N}))+ \delta((n+m)_{N})) $ $ \frac{sin(\frac{(2M+1)k\pi}{N})}{N sin(\frac{k\pi}{N})} $
Dsinc* $ \frac{sin(\frac{(2M+1)n\pi}{N})}{N sin(\frac{n\pi}{N})} $ $ \delta(k) + \sum_{m=1}^{M} \delta((k-m)_{N}))+ \delta((k+m)_{N})) $
________________________________
*$ M < \frac{N}{2} $
 ________________________________
$ n \in \mathbb{Z}[0, N - 1] $
 ________________________________
$ k \in \mathbb{Z}[0, N - 1] $

By --Abinhamd (talk) 01:52, 19 March 2018 (EDT)Alanoud Bin Hamdan

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Sean Hu, ECE PhD 2009