Difference between revisions of "CT Fourier Transform (frequency in radians per time unit)" - Rhea

Revision as of 14:36, 14 November 2011

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CT Fourier Transform Pairs and Properties (frequency $ω$ in radians per time unit) (info)
Definition CT Fourier Transform and its Inverse
(info) CT Fourier Transform	$\mathcal{X}(\omega)=\mathcal{F}(x(t))=\int_{-\infty}^{\infty} x(t) e^{-i\omega t} dt$
(info) Inverse DT Fourier Transform	$\, x(t)=\mathcal{F}^{-1}(\mathcal{X}(\omega))=\frac{1}{2\pi} \int_{-\infty}^{\infty}\mathcal{X}(\omega)e^{i\omega t} d \omega\,$

				CT Fourier Transform Pairs
		$x(t) \$	$\longrightarrow$	$\mathcal{X}(\omega)$
1	CTFT of a unit impulse	$\delta (t)\$		$1 \$
2	CTFT of a shifted unit impulse	$\delta (t-t_0)\$		$e^{-iwt_0}$
3	CTFT of a complex exponential	$e^{iw_0t}$		$2\pi \delta (\omega - \omega_0) \$
4		$e^{-at}u(t),\$ $a\in {\mathbb R}, a>0$		$\frac{1}{a+i\omega}$
5		$te^{-at}u(t),\$ $a\in {\mathbb R}, a>0$		$\left( \frac{1}{a+i\omega}\right)^2$
6	CTFT of a cosine	$\cos(\omega_0 t) \$		$\pi \left[\delta (\omega - \omega_0) + \delta (\omega + \omega_0)\right] \$
7	CTFT of a sine	$sin(\omega_0 t) \$		$\frac{\pi}{i} \left[\delta (\omega - \omega_0) - \delta (\omega + \omega_0)\right]$
8	CTFT of a rect	$\left\{\begin{array}{ll}1, & \text{ if }\|t\|<T,\\ 0, & \text{else.}\end{array} \right. \$		$\frac{2 \sin \left( T \omega \right)}{\omega} \$
9	CTFT of a sinc	$\frac{\sin \left( W t \right)}{\pi t } \$		$\left\{\begin{array}{ll}1, & \text{ if }\|\omega\| <W,\\ 0, & \text{else.}\end{array} \right. \$
10	CTFT of a periodic function	$\sum^{\infty}_{k=-\infty} a_{k}e^{ikw_{0}t}$		$2\pi\sum^{\infty}_{k=-\infty}a_{k}\delta(w-kw_{0}) \$
11	CTFT of an impulse train	$\sum^{\infty}_{n=-\infty} \delta(t-nT) \$		$\frac{2\pi}{T}\sum^{\infty}_{k=-\infty}\delta(w-\frac{2\pi k}{T})$
12		$1 \$		$2\pi \delta (\omega) \$
13	CTFT of a Periodic Square Wave	$x(t+T)=x(t)=\left\{\begin{array}{ll}1, & \|t\|<T_1,\\ 0, & T_1<\|t\|\leq T/2 \end{array} \right.$		$\sum^{\infty}_{k=-\infty}\frac{2 \sin(k\omega_0T_1}{k}\delta(\omega-k\omega_0)$
14	CTFT of a Step Function	$u(t) \$		$\frac{1}{j\omega}+\pi\delta(\omega)$
15		$e^{-\alpha \|t\|} \$		$\frac{2\alpha}{\alpha^{2}+\omega^{2}}$

				CT Fourier Transform Properties
		$x(t) \$	$\longrightarrow$	$\mathcal{X}(\omega)$
16	(info) multiplication property	$x(t)y(t) \$		$\frac{1}{2\pi} \mathcal{X}(\omega)*\mathcal{Y}(\omega) =\frac{1}{2\pi} \int_{-\infty}^{\infty} \mathcal{X}(\theta)\mathcal{Y}(\omega-\theta)d\theta$
17	convolution property	$x(t)*y(t) \$		$\mathcal{X}(\omega)\mathcal{Y}(\omega) \!$
18	time reversal	$\ x(-t)$		$\ \mathcal{X}(-\omega)$
19	Frequency Shifting	$e^{j\omega_0 t}x(t)$		$\mathcal{X} (\omega - \omega_0)$
20	Conjugation	$x^{*}(t) \$		$\mathcal{X}^{*} (-\omega)$
21	Time and Frequency Scaling	$x(at) \$		$\frac{1}{\|a\|} \mathcal{X} (\frac{\omega}{a})$
23	Differentiation in Frequency	$tx(t) \$		$j\frac{d}{d\omega} \mathcal{X} (\omega)$
24	Symmetry	$x(t)\ \text{ real and even}$		$\mathcal{X} (\omega) \ \text{ real and even}$
25		$x(t) \ \text{ real and odd}$		$\mathcal{X} (\omega) \ \text{ purely imaginary and odd}$
26	Duality	$\mathcal{X} (-t)$		$2 \pi x (\omega) \$
27	Differentiation	$\frac{d^{n}x(t)}{dt^{n}}$		$(j \omega)^{n} \mathcal{X} (\omega)$
28	Linearity	$ax(t) + by(t) \$		$a \mathcal{X}(\omega) + b \mathcal{Y} (\omega)$
29	Time Shifting	$x(t-t_0) \$		$e^{-j\omega t_0}X(\omega)$

Other CT Fourier Transform Properties
Parseval's relation	$\int_{-\infty}^{\infty} \|x(t)\|^2 dt = \frac{1}{2\pi} \int_{-\infty}^{\infty} \|\mathcal{X}(w)\|^2 dw$

Sources:

Class Text Book

http://www1.na.infn.it/~cavalier/Download/SICSI_LAES/Lucidi_DSP/FourierTransformPairs.pdf

Back to Collective Table

@@ Line 29: / Line 29: @@
 | align="right" style="padding-right: 1em;" | <br>
 | align="right" style="padding-right: 1em;" |
-| <math> x(t) </math>
+| <math> x(t) \  </math>
 | <math>\longrightarrow</math>
 | <math> \mathcal{X}(\omega) </math>
@@ Line 38: / Line 38: @@
 | <math>\delta (t)\ </math>
 |
-| '''<span class="texhtml">1</span><br>'''
+| <math> 1 \ </math>
 |
 |-
@@ Line 116: / Line 116: @@
 | <math>\sum^{\infty}_{n=-\infty} \delta(t-nT)  \ </math>
 |
-| '''<math>\frac{2\pi}{T}\sum^{\infty}_{k=-\infty}\delta(w-\frac{2\pi k}{T})</math><br>'''
+| <math>\frac{2\pi}{T}\sum^{\infty}_{k=-\infty}\delta(w-\frac{2\pi k}{T})</math>
 |
 |
@@ Line 122: / Line 122: @@
 | align="right" style="padding-right: 1em;" | 12
 | align="right" style="padding-right: 1em;" |
-| '''1<br> '''
+| <math> 1 \  </math>
 |
-| <span class="texhtml">2πδ(ω)
+| <math>2\pi \delta (\omega) \ </math>
-</span>
 |
 |
@@ Line 131: / Line 130: @@
 | align="right" style="padding-right: 1em;" | 13
 | align="right" style="padding-right: 1em;" | CTFT of a Periodic Square Wave
-|
+| <math>x(t+T)=x(t)=\left\{\begin{array}{ll}1, &  |t|<T_1,\\ 0, & T_1<|t|\leq T/2 \end{array} \right.</math>
-<math>\left\{\begin{array}{ll}1, &  |t|<T_1,\\ 0, & T_1<|t|<=\frac{T}{2}\end{array} \right.</math>
-and
-<span class="texhtml">''x''(''t'' + ''T'') = ''x''(''t'')</span>
 |
 | <math>\sum^{\infty}_{k=-\infty}\frac{2 \sin(k\omega_0T_1}{k}\delta(\omega-k\omega_0)</math>
@@ Line 143: / Line 136: @@
 |
 |-
-| align="right" style="padding-right: 1em;" |
+| align="right" style="padding-right: 1em;" | 14
 | align="right" style="padding-right: 1em;" | CTFT of a Step Function
-| <span class="texhtml">''u''(''t'')</span>
+| <math> u(t) \  </math>
 |
 | <math>\frac{1}{j\omega}+\pi\delta(\omega)</math>
@@ Line 155: / Line 146: @@
 | align="right" style="padding-right: 1em;" | 15
 | align="right" style="padding-right: 1em;" |
-| <span class="texhtml">''e''<sup> − α &#124; ''t'' &#124; </sup></span>
+| <math> e^{-\alpha |t|}  \ </math>
 |
 | <math>\frac{2\alpha}{\alpha^{2}+\omega^{2}}</math>
@@ Line 168: / Line 159: @@
 |-
 | align="right" style="padding-right: 1em;" |
 | align="right" style="padding-right: 1em;" |
-| <span class="texhtml">''x''(''t'')</span>
+| <math> x(t) \ </math>
 | <math>\longrightarrow</math>
 | <math> \mathcal{X}(\omega) </math>
@@ Line 181: / Line 172: @@
 | align="right" style="padding-right: 1em;" | 17
 | align="right" style="padding-right: 1em;" | convolution property
-| <math>x(t)*y(t) \!</math>
+| <math>x(t)*y(t) \ </math>
 |
 | <math> \mathcal{X}(\omega)\mathcal{Y}(\omega) \!</math>
@@ Line 195: / Line 186: @@
 | <math>e^{j\omega_0 t}x(t)</math>
 |
-| <span class="texhtml">''X''(ω − ω<sub>0</sub>)</span>
+| <math> \mathcal{X} (\omega - \omega_0) </math>
 |-
 | align="right" style="padding-right: 1em;" | 20
-| align="right" style="padding-right: 1em;" | Conjugation
+| align="right" style="padding-right: 1em;"| Conjugation
-| <span class="texhtml">''x''<sup> * </sup>(''t'')</span>
+| <math> x^{*}(t) \  </math>
 |
-| <span class="texhtml">''X''<sup> * </sup>( − ω)</span>
+| <math> \mathcal{X}^{*} (-\omega)</math>
 |-
 | align="right" style="padding-right: 1em;" | 21
 | align="right" style="padding-right: 1em;" | Time and Frequency Scaling
-| <span class="texhtml">''x''(''a''''t'''''<b>)</b></span>
+| <math> x(at) \  </math>
 |
-| <math>\frac{1}{|a|}X(\frac{\omega}{a})</math>
+| <math>\frac{1}{|a|} \mathcal{X} (\frac{\omega}{a})</math>
 |-
 | align="right" style="padding-right: 1em;" | 23
 | align="right" style="padding-right: 1em;" | Differentiation in Frequency
-| <span class="texhtml">''t''''x'''''<b>(''t'')</b></span>
+| <math> tx(t) \  </math>
 |
-| <math>j\frac{d}{d\omega}X(\omega)</math>
+| <math>j\frac{d}{d\omega} \mathcal{X} (\omega)</math>
 |-
 | align="right" style="padding-right: 1em;" | 24
 | align="right" style="padding-right: 1em;" | Symmetry
-| <span class="texhtml">''x''(''t'') &nbsp; &nbsp; real and even</span>
+| <math> x(t)\ \text{ real and even}</math>
 |
-| <span class="texhtml">''X''(ω) &nbsp; &nbsp; real and even</span>
+| <math> \mathcal{X} (\omega) \ \text{ real and even} </math>
 |-
 | align="right" style="padding-right: 1em;" | 25
 | align="right" style="padding-right: 1em;" |
-| <span class="texhtml">''x''(''t'') &nbsp; &nbsp; real and odd</span>
+| <math> x(t) \ \text{ real and odd}</math>
 |
-| <span class="texhtml">''X''(ω) &nbsp; &nbsp; perely imaginary and odd</span>
+| <math> \mathcal{X} (\omega) \ \text{ purely imaginary and odd} </math>
 |-
 | align="right" style="padding-right: 1em;" | 26
 | align="right" style="padding-right: 1em;" | Duality
-| x(-t)
+| <math> \mathcal{X} (-t) </math>
 |
-| '''<span class="texhtml">2π''X''(ω)</span>'''
+| <math> 2 \pi x (\omega) \  </math>
 |-
 | align="right" style="padding-right: 1em;" | 27
 | align="right" style="padding-right: 1em;" | Differentiation
-| '''<math>\frac{d^{n}x(t)}{dt^{n}}</math>'''
+| <math>\frac{d^{n}x(t)}{dt^{n}}</math>
-|
-| '''<span class="texhtml">(''j''ω)<sup>''n''</sup>''X''(ω)</span>'''
-|-
-| align="right" style="padding-right: 1em;" |
-| align="right" style="padding-right: 1em;" | <br>
-|
-|
 |
+| <math> (j \omega)^{n} \mathcal{X} (\omega) </math>
 |-
 | align="right" style="padding-right: 1em;" | 28
 | align="right" style="padding-right: 1em;" | Linearity
-| <span class="texhtml">''a''''x'''''<b>(</b>'''''t'') + ''b''''''''y''(''t'')''</span>
+| <math> ax(t) + by(t) \  </math>
 |
-| <span class="texhtml">''a''''X'''''<b>(ω) + </b>'''''b''''''''Y''(ω)''</span>
+| <math> a \mathcal{X}(\omega) + b \mathcal{Y} (\omega) </math>
 |-
 | align="right" style="padding-right: 1em;" | 29
 | align="right" style="padding-right: 1em;" | Time Shifting
-| <span class="texhtml">''x''(''t'' − ''t''<sub>0</sub>)</span>
+| <math> x(t-t_0) \  </math>
 |
-|
+| <math>e^{-j\omega t_0}X(\omega)</math>
-<math>e^{-j\omega t_0}X(\omega)</math>
+|}
-|}
-<br>
 {|

Difference between revisions of "CT Fourier Transform (frequency in radians per time unit)" - Rhea

Revision as of 14:36, 14 November 2011

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