(35 intermediate revisions by 7 users not shown)
Line 1: Line 1:
 +
[[Category:Formulas]]
 +
[[Category:Fourier transform]]
 +
[[Category:ECE301]]
 +
[[Category:ECE438]]
 +
 +
<center><font size= 4>
 +
'''[[Collective_Table_of_Formulas|Collective Table of Formulas]]'''
 +
</font size>
 +
 +
[[Discrete-time_Fourier_transform_info|Discrete-time (DT) Fourier Transforms]] Pairs and Properties
 +
 +
(used in [[ECE301]], [[ECE438]], [[ECE538]])
 +
 +
</center>
 +
 +
----
 +
 +
 
{|
 
{|
−
! 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
 
|-
 
|-
−
| align="right" style="padding-right: 1em;" | DT Fourier Transform || <math>\,\mathcal{X}(\omega)=\mathcal{F}(x[n])=\sum_{n=-\infty}^{\infty}x[n]e^{-j\omega n}\,</math>
+
| 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>
−
| align="right" style="padding-right: 1em;" | Inverse DT Fourier Transform || <math>\,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\,</math>
+
|-
 +
| align="right" style="padding-right: 1em;" | Inverse DT Fourier Transform  
 +
| <math>\,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\,</math>
 
|}
 
|}
 +
 
{|
 
{|
 
|-
 
|-
 
! colspan="4" style="background: #eee;" | DT Fourier Transform Pairs
 
! colspan="4" style="background: #eee;" | DT Fourier Transform Pairs
−
|-
 
−
| align="right" style="padding-right: 1em;" |  || <math> x[n] \ </math> || <math>\longrightarrow</math>|| <math> \mathcal{X}(\omega) \ </math>
 
−
|-
 
−
| align="right" style="padding-right: 1em;" | DTFT of a complex exponential || <math>e^{jw_0n} \ </math> || || <math>\pi\sum_{l=-\infty}^{+\infty}\delta(w-w_0-2\pi l) \ </math>
 
−
|-
 
−
| align="right" style="padding-right: 1em;" | ([[DTFT_rectangular_window|info]]) DTFT of a rectangular window || <math>w[n]= \ </math> || || add formula here
 
−
|-
 
−
| align="right" style="padding-right: 1em;" |  || <math>a^{n} u[n],  |a|<1 \ </math> || ||<math>\frac{1}{1-ae^{-j\omega}} \ </math>
 
 
|-
 
|-
−
| align="right" style="padding-right: 1em;" | || <math>(n+1)a^{n} u[n],  |a|<1 \ </math> || ||<math>\frac{1}{(1-ae^{-j\omega})^2} \ </math>
+
| align="right" style="padding-right: 1em;" |  
 +
| align="right" style="padding-right: 1em;" |
 +
| <math> x[n] \ </math>
 +
| <math>\longrightarrow</math>  
 +
| <math> \mathcal{X}(\omega) \ </math>
 
|-
 
|-
−
| align="right" style="padding-right: 1em;" | || <math>\sin\left(\omega _0 n\right) u[n] \ </math> || ||<math>\frac{1}{2j}\left( \frac{1}{1-e^{-j(\omega -\omega _0)}}-\frac{1}{1-e^{-j(\omega +\omega _0)}}\right)</math>
+
| align="right" style="padding-right: 1em;" |  
 +
| align="right" style="padding-right: 1em;" | DTFT of a complex exponential
 +
| <math>e^{jw_0n} \ </math>  
 +
|  
 +
| <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;" | ([[DTFT rectangular window|info]]) DTFT of a rectangular window
 +
| <math>w[n]= \ </math>
 +
|
 +
| <math> \text{add formula here} \  </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| align="right" style="padding-right: 1em;" |
 +
| <math>a^{n} u[n],  |a|<1 \ </math>
 +
|
 +
| <math>\frac{1}{1-ae^{-j\omega}} \ </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| align="right" style="padding-right: 1em;" |
 +
| <math>(n+1)a^{n} u[n],  |a|<1 \ </math>
 +
|
 +
| <math>\frac{1}{(1-ae^{-j\omega})^2} \ </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| align="right" style="padding-right: 1em;" |
 +
| <math>\sin\left(\omega _0 n\right) u[n] \ </math>
 +
|
 +
| <math>\frac{1}{2j}\left( \frac{1}{1-e^{-j(\omega -\omega _0)}}-\frac{1}{1-e^{-j(\omega +\omega _0)}}\right)</math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| align="right" style="padding-right: 1em;" |
 +
| <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>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| align="right" style="padding-right: 1em;" |
 +
| <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>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| align="right" style="padding-right: 1em;" |
 +
| <math> 1 \ </math>
 +
|
 +
| <math>2\pi\sum^{\infty}_{k=-\infty}\delta(\omega-2\pi k)</math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| align="right" style="padding-right: 1em;" | DTFT of a Periodic Square Wave
 +
|
 +
<math>\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] </math>
 +
 +
|
 +
| <math>2\pi\sum^{\infty}_{k=-\infty}a_k\delta(\omega-\frac{2\pi k}{N})</math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| align="right" style="padding-right: 1em;" |
 +
| <math>\sum^{\infty}_{k=-\infty}\delta[n-kN]</math>
 +
|
 +
| <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;" |
 +
| <math> \delta [n] \  </math>
 +
|
 +
| <math> 1 \  </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| align="right" style="padding-right: 1em;" |
 +
| <math> u[n] \  </math>
 +
|
 +
| <math>\frac{1}{1-e^{-j\omega}}+\sum^{\infty}_{k=-\infty}\pi\delta(\omega-2\pi k)</math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| align="right" style="padding-right: 1em;" |
 +
| <math> \delta[n - n_0] \  </math>
 +
|
 +
| <math>e^{-j\omega n_0}</math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| align="right" style="padding-right: 1em;" |
 +
| <math> (n + 1)a^n u[n], \quad |a| < 1 </math> 
 +
|
 +
| <math>\frac{1}{(1-ae^{-j\omega})^{2}}</math>
 
|}
 
|}
  
Line 29: Line 129:
 
|-
 
|-
 
! colspan="4" style="background: #eee;" | DT Fourier Transform Properties
 
! colspan="4" style="background: #eee;" | DT Fourier Transform Properties
−
|-
 
−
| align="right" style="padding-right: 1em;" |  || <math>x[n] \ </math> || <math>\longrightarrow</math>|| <math> \mathcal{X}(\omega) \ </math>
 
−
|-
 
−
| align="right" style="padding-right: 1em;" | multiplication property|| <math>x[n]y[n] \ </math> || || <math>\frac{1}{2\pi} \int_{2\pi} X(\theta)Y(\omega-\theta)d\theta</math>
 
−
|-
 
−
| align="right" style="padding-right: 1em;" |  convolution property || <math>x[n]*y[n] \!</math> || ||<math> X(\omega)Y(\omega) \!</math>
 
 
|-
 
|-
−
| align="right" style="padding-right: 1em;" | time reversal ||<math>\ x[-n] </math> || ||<math>\ X(-\omega)</math>
+
| align="right" style="padding-right: 1em;" |  
 +
| align="right" style="padding-right: 1em;" |
 +
| <math>x[n] \ </math>  
 +
| <math>\longrightarrow</math>
 +
| <math> \mathcal{X}(\omega) \ </math>
 
|-
 
|-
−
| align="right" style="padding-right: 1em;" | Differentiation in frequency ||<math>\ nx[n] </math> || ||<math>\ j\frac{d}{d\omega}X(\omega)</math>
+
| align="right" style="padding-right: 1em;" |  
 +
| align="right" style="padding-right: 1em;" | multiplication property
 +
| <math>x[n]y[n] \ </math>  
 +
|  
 +
| <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;" | convolution property
 +
| <math>x[n]*y[n] \ </math>
 +
|
 +
| <math> X(\omega)Y(\omega) \!</math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| align="right" style="padding-right: 1em;" | time reversal
 +
| <math>\ x[-n] </math>
 +
|
 +
| <math>\ X(-\omega)</math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| align="right" style="padding-right: 1em;" | Differentiation in frequency
 +
| <math>\ nx[n] </math>
 +
|
 +
| <math>\ j\frac{d}{d\omega}X(\omega)</math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| align="right" style="padding-right: 1em;" | Linearity
 +
| <math> ax[n]+by[n] \  </math>
 +
|
 +
| <math> aX(\omega)+bY(\omega) \  </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| align="right" style="padding-right: 1em;" | Time Shifting
 +
| <math> x[n - n_0] \  </math>
 +
|
 +
| <math>e^{-j\omega n_0}X(\omega)</math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| align="right" style="padding-right: 1em;" | Frequency Shifting
 +
| <math>e^{j\omega_0 n}x[n]</math>
 +
|
 +
| <math> X(\omega - \omega_0) \  </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| align="right" style="padding-right: 1em;" | Conjugation
 +
| <math> x^* [n] \  </math>
 +
|
 +
| <math> X^* (-\omega) \  </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| align="right" style="padding-right: 1em;" | Time Expansion
 +
| <math>x_{(k)}[n]=\left\{\begin{array}{ll}x[n/k], &  \text{ if n = multiple of k},\\ 0, & \text{else.}\end{array} \right.</math>
 +
|
 +
| <math> X(k\omega) \  </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| align="right" style="padding-right: 1em;" | Differentiating in Time
 +
| <math> x[n] - x[n - 1] \  </math> 
 +
|
 +
| <math> (1 - e^{-j\omega}) X (\omega) \  </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| align="right" style="padding-right: 1em;" | Accumulation
 +
| <math>\sum^{n}_{k=-\infty} x[k]</math>
 +
|
 +
| <math>\frac{1}{1-e^{-j\omega}}X(\omega)</math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| align="right" style="padding-right: 1em;" | Symmetry
 +
| <math> x[n] \  \text{ real and even} \ </math>
 +
|
 +
| <math> X(\omega) \ \text{ real and even} \  </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| align="right" style="padding-right: 1em;" |
 +
| <math> x[n] \  \text{ real and odd} \  </math>
 +
|
 +
| <math> X(\omega) \ \text{ purely imaginary and odd} \  </math>
 
|}
 
|}
  
Line 45: Line 218:
 
|-
 
|-
 
! colspan="2" style="background: #eee;" | Other DT Fourier Transform Properties
 
! colspan="2" style="background: #eee;" | Other DT Fourier Transform Properties
−
|-  
+
|-
−
| align="right" style="padding-right: 1em;" | Parseval's relation || <math>\frac {1}{N} \sum_{n=-\infty}^{\infty}\left| x[n] \right|^2 = </math>
+
| align="right" style="padding-right: 1em;" | Parseval's relation  
 +
| <math>\sum_{n=-\infty}^{\infty}\left| x[n] \right|^2 = \frac{1}{2\pi}\int_{-\pi}^{\pi}|X( \omega )|^2d\omega </math>
 
|}
 
|}
 +
 +
 
----
 
----
−
[[Collective_Table_of_Formulas|Back to Collective Table]]
+
 
−
[[Category:Formulas]]
+
 
 +
[[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

Retrieved from "https://www.projectrhea.org/rhea/index.php?title=Table_DT_Fourier_Transforms&oldid=69325"
Shortcuts
Help Main Wiki Page Random Page Special Pages Log in
Search

Alumni Liaison

Sees the importance of signal filtering in medical imaging

Dhruv Lamba, BSEE2010