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[[Category:Formulas]]
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[[Category:Fourier transform]]
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[[Category:ECE301]]
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[[Category:ECE438]]
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<center><font size= 4>
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'''[[Collective_Table_of_Formulas|Collective Table of Formulas]]'''
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</font size>
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[[Discrete-time_Fourier_transform_info|Discrete-time (DT) Fourier Transforms]] Pairs and Properties
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(used in [[ECE301]], [[ECE438]], [[ECE538]])
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</center>
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----
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{|
 
{|
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|-
 
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! colspan="2" style="background: #e4bc7e; font-size: 110%;" | Discrete-time Fourier Transform Pairs and Properties
 
 
|-
 
|-
 
! colspan="2" style="background: #eee;" | DT Fourier transform and its Inverse
 
! colspan="2" style="background: #eee;" | DT Fourier transform and its Inverse
 
|-
 
|-
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| align="right" style="padding-right: 1em;" | DT Fourier Transform  
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| align="right" style="padding-right: 1em;" | [[Discrete-time Fourier transform info|DT Fourier Transform]]
 
| <math>\,\mathcal{X}(\omega)=\mathcal{F}(x[n])=\sum_{n=-\infty}^{\infty}x[n]e^{-j\omega n}\,</math>
 
| <math>\,\mathcal{X}(\omega)=\mathcal{F}(x[n])=\sum_{n=-\infty}^{\infty}x[n]e^{-j\omega n}\,</math>
 
|-
 
|-
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{|
 
{|
 
|-
 
|-
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! colspan="4" style="background: #eee;" |
 
 
! colspan="4" style="background: #eee;" | DT Fourier Transform Pairs
 
! colspan="4" style="background: #eee;" | DT Fourier Transform Pairs
 
|-
 
|-
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| <math>e^{jw_0n} \ </math>  
 
| <math>e^{jw_0n} \ </math>  
 
|  
 
|  
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| <math>\pi\sum_{l=-\infty}^{+\infty}\delta(w-w_0-2\pi l) \ </math>
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| <math>\ 2\pi\sum_{l=-\infty}^{+\infty}\delta(w-w_0-2\pi l) \ </math>
 
|-
 
|-
 
| align="right" style="padding-right: 1em;" |  
 
| align="right" style="padding-right: 1em;" |  
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| <math>w[n]= \ </math>  
 
| <math>w[n]= \ </math>  
 
|  
 
|  
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| add formula here
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| <math> \text{add formula here} \  </math>
 
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|-
 
| align="right" style="padding-right: 1em;" |  
 
| align="right" style="padding-right: 1em;" |  
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| align="right" style="padding-right: 1em;" |  
 
| align="right" style="padding-right: 1em;" |  
 
| align="right" style="padding-right: 1em;" |  
 
| align="right" style="padding-right: 1em;" |  
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| <span class="texhtml">''c''''o''''s''(ω<sub>0</sub>''n'')</span>  
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| <math>\cos\left(\omega _0 n\right) \ </math>  
 
|  
 
|  
 
| <math>\pi \sum^{\infty}_{k=-\infty} (\delta(\omega-\omega_0 + 2\pi k)+\delta(\omega+\omega_0-2\pi k))</math>
 
| <math>\pi \sum^{\infty}_{k=-\infty} (\delta(\omega-\omega_0 + 2\pi k)+\delta(\omega+\omega_0-2\pi k))</math>
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| align="right" style="padding-right: 1em;" |  
 
| align="right" style="padding-right: 1em;" |  
 
| align="right" style="padding-right: 1em;" |  
 
| align="right" style="padding-right: 1em;" |  
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| <span class="texhtml">''s''''i''''n''(ω<sub>0</sub>''n'')</span>  
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| <math>\sin\left(\omega _0 n\right) \ </math>  
 
|  
 
|  
−
| <math>\frac{\pi}{j} \sum^{\infty}_{k=-\infty} (\delta(\omega-\omega_0 + 2\pi k)+\delta(\omega+\omega_0-2\pi k))</math>
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| <math>\frac{\pi}{j} \sum^{\infty}_{k=-\infty} (\delta(\omega-\omega_0 + 2\pi k)-\delta(\omega+\omega_0-2\pi k))</math>
 
|-
 
|-
 
| align="right" style="padding-right: 1em;" |  
 
| align="right" style="padding-right: 1em;" |  
 
| align="right" style="padding-right: 1em;" |  
 
| align="right" style="padding-right: 1em;" |  
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| 1  
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| <math> 1 \ </math>
−
|  
+
|
 
| <math>2\pi\sum^{\infty}_{k=-\infty}\delta(\omega-2\pi k)</math>
 
| <math>2\pi\sum^{\infty}_{k=-\infty}\delta(\omega-2\pi k)</math>
 
|-
 
|-
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| align="right" style="padding-right: 1em;" | DTFT of a Periodic Square Wave  
 
| align="right" style="padding-right: 1em;" | DTFT of a Periodic Square Wave  
 
|  
 
|  
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<math>\left\{\begin{array}{ll}1, &  |n|<N_1,\\ 0, & N_1<|n|<=\frac{N}{2}\end{array} \right.</math>
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<math>\left\{\begin{array}{ll}1, &  |n|<N_1,\\ 0, & N_1<|n|\leq\frac{N}{2}\end{array} \right. \text{ and }
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x[n+N]=x[n] </math>
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and  
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−
 
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<span class="texhtml">''x''[''n'' + ''N''] = ''x''[''n'']</span>  
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|  
 
|  
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| <math>\sum^{\infty}_{k=-\infty}\delta[n-kN]</math>  
 
| <math>\sum^{\infty}_{k=-\infty}\delta[n-kN]</math>  
 
|  
 
|  
−
| <math>\frac{2\pi}{N}\sum^{\infty}_{k=-\infty}\delta(\frac{2\pi k}{N})</math>
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| <math>\frac{2\pi}{N}\sum^{\infty}_{k=-\infty}\delta(\omega -\frac{2\pi k}{N})</math>
 
|-
 
|-
 
| align="right" style="padding-right: 1em;" |  
 
| align="right" style="padding-right: 1em;" |  
 
| align="right" style="padding-right: 1em;" |  
 
| align="right" style="padding-right: 1em;" |  
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| <span class="texhtml">δ[''n'']</span>  
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| <math> \delta [n] \  </math>
 
|  
 
|  
−
| 1
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| <math> 1 \  </math>
 
|-
 
|-
 
| align="right" style="padding-right: 1em;" |  
 
| align="right" style="padding-right: 1em;" |  
 
| align="right" style="padding-right: 1em;" |  
 
| align="right" style="padding-right: 1em;" |  
−
| <span class="texhtml">''u''[''n'']</span><br>  
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| <math> u[n] \  </math>  
 
|  
 
|  
 
| <math>\frac{1}{1-e^{-j\omega}}+\sum^{\infty}_{k=-\infty}\pi\delta(\omega-2\pi k)</math>
 
| <math>\frac{1}{1-e^{-j\omega}}+\sum^{\infty}_{k=-\infty}\pi\delta(\omega-2\pi k)</math>
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| align="right" style="padding-right: 1em;" |  
 
| align="right" style="padding-right: 1em;" |  
 
| align="right" style="padding-right: 1em;" |  
 
| align="right" style="padding-right: 1em;" |  
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| <math>\delta[n-n_0]</math>  
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| <math> \delta[n - n_0] \  </math>  
 
|  
 
|  
 
| <math>e^{-j\omega n_0}</math>
 
| <math>e^{-j\omega n_0}</math>
 
|-
 
|-
 
| align="right" style="padding-right: 1em;" |  
 
| align="right" style="padding-right: 1em;" |  
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| align="right" style="padding-right: 1em;" |  
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| align="right" style="padding-right: 1em;" |
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| <math>(n+1)a^{n}u[n], |a|<1</math>  
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| <math> (n + 1)a^n u[n], \quad |a| < 1 </math>
 
|  
 
|  
 
| <math>\frac{1}{(1-ae^{-j\omega})^{2}}</math>
 
| <math>\frac{1}{(1-ae^{-j\omega})^{2}}</math>
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|-
 
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| align="right" style="padding-right: 1em;" |
 
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| align="right" style="padding-right: 1em;" |
 
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|
 
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|}
 
|}
  
 
{|
 
{|
 
|-
 
|-
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! colspan="4" style="background: #eee;" |
 
 
! colspan="4" style="background: #eee;" | DT Fourier Transform Properties
 
! colspan="4" style="background: #eee;" | DT Fourier Transform Properties
 
|-
 
|-
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| <math>x[n]y[n] \ </math>  
 
| <math>x[n]y[n] \ </math>  
 
|  
 
|  
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| <math>\frac{1}{2\pi} \int_{2\pi} X(\theta)Y(\omega-\theta)d\theta</math>
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| <math>\frac{1}{2\pi} \int_{-\pi}^{\pi} X(\theta)Y(\omega-\theta)d\theta</math>
 
|-
 
|-
 
| align="right" style="padding-right: 1em;" |  
 
| align="right" style="padding-right: 1em;" |  
 
| align="right" style="padding-right: 1em;" | convolution property  
 
| align="right" style="padding-right: 1em;" | convolution property  
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| <math>x[n]*y[n] \!</math>  
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| <math>x[n]*y[n] \ </math>  
 
|  
 
|  
 
| <math> X(\omega)Y(\omega) \!</math>
 
| <math> X(\omega)Y(\omega) \!</math>
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| align="right" style="padding-right: 1em;" |  
 
| align="right" style="padding-right: 1em;" |  
 
| align="right" style="padding-right: 1em;" | Linearity  
 
| align="right" style="padding-right: 1em;" | Linearity  
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| <span class="texhtml">''a''''x'''''<b>[</b>'''''n''] + ''b''''''''y''[''n'']''</span>  
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| <math> ax[n]+by[n] \  </math>
 
|  
 
|  
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| <span class="texhtml">''a''''X'''''<b>(ω) + </b>'''''b''''''''Y''(ω)''</span>
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| <math> aX(\omega)+bY(\omega) \  </math>
 
|-
 
|-
 
| align="right" style="padding-right: 1em;" |  
 
| align="right" style="padding-right: 1em;" |  
 
| align="right" style="padding-right: 1em;" | Time Shifting  
 
| align="right" style="padding-right: 1em;" | Time Shifting  
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| <span class="texhtml">''x''[''n'' − ''n''<sub>0</sub>]</span>  
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| <math> x[n - n_0] \  </math>
 
|  
 
|  
 
| <math>e^{-j\omega n_0}X(\omega)</math>
 
| <math>e^{-j\omega n_0}X(\omega)</math>
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| <math>e^{j\omega_0 n}x[n]</math>  
 
| <math>e^{j\omega_0 n}x[n]</math>  
 
|  
 
|  
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| <span class="texhtml">''X''(ω − ω<sub>0</sub>)</span>
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| <math> X(\omega - \omega_0) \  </math>
 
|-
 
|-
 
| align="right" style="padding-right: 1em;" |  
 
| align="right" style="padding-right: 1em;" |  
 
| align="right" style="padding-right: 1em;" | Conjugation  
 
| align="right" style="padding-right: 1em;" | Conjugation  
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| <span class="texhtml">''x''<sup> * </sup>[''n'']</span>  
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| <math> x^* [n] \  </math>
 
|  
 
|  
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| <span class="texhtml">''X''<sup> * </sup>( − ω)</span>
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| <math> X^* (-\omega) \  </math>
 
|-
 
|-
 
| align="right" style="padding-right: 1em;" |  
 
| align="right" style="padding-right: 1em;" |  
 
| align="right" style="padding-right: 1em;" | Time Expansion  
 
| align="right" style="padding-right: 1em;" | Time Expansion  
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| <math>x_(k) [n]=\left\{\begin{array}{ll}x[n/k], &  \text{ if n = multiple of k},\\ 0, & \text{else.}\end{array} \right.</math>  
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| <math>x_{(k)}[n]=\left\{\begin{array}{ll}x[n/k], &  \text{ if n = multiple of k},\\ 0, & \text{else.}\end{array} \right.</math>  
 
|  
 
|  
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| <span class="texhtml">''X''(''k''ω)</span>
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| <math> X(k\omega) \  </math>
 
|-
 
|-
 
| align="right" style="padding-right: 1em;" |  
 
| align="right" style="padding-right: 1em;" |  
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| align="right" style="padding-right: 1em;" | Differentiating in Time  
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| align="right" style="padding-right: 1em;" | Differentiating in Time
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| <span class="texhtml">''x''[''n''] − ''x''[''n'' − 1]</span>  
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| <math> x[n] - x[n - 1] \  </math>
 
|  
 
|  
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| <span class="texhtml">(1 − ''e''<sup> − ''j''ω</sup>)''X''(ω)</span>
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| <math> (1 - e^{-j\omega}) X (\omega) \  </math>
 
|-
 
|-
 
| align="right" style="padding-right: 1em;" |  
 
| align="right" style="padding-right: 1em;" |  
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| <math>\sum^{n}_{k=-\infty} x[k]</math>  
 
| <math>\sum^{n}_{k=-\infty} x[k]</math>  
 
|  
 
|  
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| '''<math>\frac{1}{1-e^{-j\omega}}X(\omega)</math>'''
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| <math>\frac{1}{1-e^{-j\omega}}X(\omega)</math>
 
|-
 
|-
 
| align="right" style="padding-right: 1em;" |  
 
| align="right" style="padding-right: 1em;" |  
 
| align="right" style="padding-right: 1em;" | Symmetry  
 
| align="right" style="padding-right: 1em;" | Symmetry  
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| x[n] real and even  
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| <math> x[n] \  \text{ real and even} \ </math>
 
|  
 
|  
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| <span class="texhtml">''X''(ω)</span>&nbsp;real and even
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| <math> X(\omega) \ \text{ real and even} \  </math>
 
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|-
 
| align="right" style="padding-right: 1em;" |  
 
| align="right" style="padding-right: 1em;" |  
 
| align="right" style="padding-right: 1em;" |  
 
| align="right" style="padding-right: 1em;" |  
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| x[n] real and odd  
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| <math> x[n] \  \text{ real and odd} \  </math>
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|
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| <span class="texhtml">''X''(ω)</span>&nbsp;purely imaginary and odd
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|  
 
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| <math> X(\omega) \ \text{ purely imaginary and odd} \  </math>
 
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|-
 
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| align="right" style="padding-right: 1em;" | Parseval's relation  
 
| align="right" style="padding-right: 1em;" | Parseval's relation  
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| <math>\frac {1}{N} \sum_{n=-\infty}^{\infty}\left| x[n] \right|^2 = </math>
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| <math>\sum_{n=-\infty}^{\infty}\left| x[n] \right|^2 = \frac{1}{2\pi}\int_{-\pi}^{\pi}|X( \omega )|^2d\omega </math>
 
|}
 
|}
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[[Collective Table of Formulas|Back to Collective Table]]
 
  
−
[[Category:Formulas]]
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[[Collective Table of Formulas|Back to Collective Table]]

Latest revision as of 20:05, 4 March 2015


Collective Table of Formulas

Discrete-time (DT) Fourier Transforms Pairs and Properties

(used in ECE301, ECE438, ECE538)


DT Fourier transform and its Inverse
DT Fourier Transform $ \,\mathcal{X}(\omega)=\mathcal{F}(x[n])=\sum_{n=-\infty}^{\infty}x[n]e^{-j\omega n}\, $
Inverse DT Fourier Transform $ \,x[n]=\mathcal{F}^{-1}(\mathcal{X}(\omega))=\frac{1}{2\pi} \int_{0}^{2\pi}\mathcal{X}(\omega)e^{j\omega n} d \omega\, $
DT Fourier Transform Pairs
$ x[n] \ $ $ \longrightarrow $ $ \mathcal{X}(\omega) \ $
DTFT of a complex exponential $ e^{jw_0n} \ $ $ \ 2\pi\sum_{l=-\infty}^{+\infty}\delta(w-w_0-2\pi l) \ $
(info) DTFT of a rectangular window $ w[n]= \ $ $ \text{add formula here} \ $
$ a^{n} u[n], |a|<1 \ $ $ \frac{1}{1-ae^{-j\omega}} \ $
$ (n+1)a^{n} u[n], |a|<1 \ $ $ \frac{1}{(1-ae^{-j\omega})^2} \ $
$ \sin\left(\omega _0 n\right) u[n] \ $ $ \frac{1}{2j}\left( \frac{1}{1-e^{-j(\omega -\omega _0)}}-\frac{1}{1-e^{-j(\omega +\omega _0)}}\right) $
$ \cos\left(\omega _0 n\right) \ $ $ \pi \sum^{\infty}_{k=-\infty} (\delta(\omega-\omega_0 + 2\pi k)+\delta(\omega+\omega_0-2\pi k)) $
$ \sin\left(\omega _0 n\right) \ $ $ \frac{\pi}{j} \sum^{\infty}_{k=-\infty} (\delta(\omega-\omega_0 + 2\pi k)-\delta(\omega+\omega_0-2\pi k)) $
$ 1 \ $ $ 2\pi\sum^{\infty}_{k=-\infty}\delta(\omega-2\pi k) $
DTFT of a Periodic Square Wave

$ \left\{\begin{array}{ll}1, & |n|<N_1,\\ 0, & N_1<|n|\leq\frac{N}{2}\end{array} \right. \text{ and } x[n+N]=x[n] $

$ 2\pi\sum^{\infty}_{k=-\infty}a_k\delta(\omega-\frac{2\pi k}{N}) $
$ \sum^{\infty}_{k=-\infty}\delta[n-kN] $ $ \frac{2\pi}{N}\sum^{\infty}_{k=-\infty}\delta(\omega -\frac{2\pi k}{N}) $
$ \delta [n] \ $ $ 1 \ $
$ u[n] \ $ $ \frac{1}{1-e^{-j\omega}}+\sum^{\infty}_{k=-\infty}\pi\delta(\omega-2\pi k) $
$ \delta[n - n_0] \ $ $ e^{-j\omega n_0} $
$ (n + 1)a^n u[n], \quad |a| < 1 $ $ \frac{1}{(1-ae^{-j\omega})^{2}} $
DT Fourier Transform Properties
$ x[n] \ $ $ \longrightarrow $ $ \mathcal{X}(\omega) \ $
multiplication property $ x[n]y[n] \ $ $ \frac{1}{2\pi} \int_{-\pi}^{\pi} X(\theta)Y(\omega-\theta)d\theta $
convolution property $ x[n]*y[n] \ $ $ X(\omega)Y(\omega) \! $
time reversal $ \ x[-n] $ $ \ X(-\omega) $
Differentiation in frequency $ \ nx[n] $ $ \ j\frac{d}{d\omega}X(\omega) $
Linearity $ ax[n]+by[n] \ $ $ aX(\omega)+bY(\omega) \ $
Time Shifting $ x[n - n_0] \ $ $ e^{-j\omega n_0}X(\omega) $
Frequency Shifting $ e^{j\omega_0 n}x[n] $ $ X(\omega - \omega_0) \ $
Conjugation $ x^* [n] \ $ $ X^* (-\omega) \ $
Time Expansion $ x_{(k)}[n]=\left\{\begin{array}{ll}x[n/k], & \text{ if n = multiple of k},\\ 0, & \text{else.}\end{array} \right. $ $ X(k\omega) \ $
Differentiating in Time $ x[n] - x[n - 1] \ $ $ (1 - e^{-j\omega}) X (\omega) \ $
Accumulation $ \sum^{n}_{k=-\infty} x[k] $ $ \frac{1}{1-e^{-j\omega}}X(\omega) $
Symmetry $ x[n] \ \text{ real and even} \ $ $ X(\omega) \ \text{ real and even} \ $
$ x[n] \ \text{ real and odd} \ $ $ X(\omega) \ \text{ purely imaginary and odd} \ $
Other DT Fourier Transform Properties
Parseval's relation $ \sum_{n=-\infty}^{\infty}\left| x[n] \right|^2 = \frac{1}{2\pi}\int_{-\pi}^{\pi}|X( \omega )|^2d\omega $



Back to Collective Table

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