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

Latest revision as of 11:45, 24 August 2016

Table of Continuous-time (CT) Fourier Transform Pairs and Properties

as a function of $\omega$ in radians per time unit

(used in ECE301)

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 CT 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
		signal (function of t)	$\longrightarrow$	Fourier transform (function of $\omega$ )
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	$x(t+T)=x(t)=\left\{\begin{array}{ll}1, & \|t\|\leq T_1,\\ 0, & T_1<\|t\|\leq T/2 \end{array} \right.$		$\sum^{\infty}_{k=-\infty}\frac{2 \sin(k\frac{2\pi}{T}T_1)}{k}\delta(\omega-k\frac{2\pi}{T})$
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	$\int_{-\infty}^{\infty} \|x(t)\|^2 dt = \frac{1}{2\pi} \int_{-\infty}^{\infty} \|\mathcal{X}(w)\|^2 dw$

Back to Collective Table

@@ 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>
+Table of Continuous-time (CT)  Fourier Transform Pairs and Properties
+as a function of <math>\omega</math> in radians per time unit
+(used in [[ECE301]])
+</center>
+----
 {|
-|-
-! style="background: none repeat scroll 0% 0% rgb(228, 188, 126); font-size: 110%;" colspan="2" | CT Fourier Transform Pairs and Properties (frequency <span class="texhtml">ω</span> in radians per time unit) [[More on CT Fourier transform|(info)]]
 |-
 ! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="2" | Definition CT Fourier Transform and its Inverse
@@ Line 8: / Line 25: @@
 | <math>\mathcal{X}(\omega)=\mathcal{F}(x(t))=\int_{-\infty}^{\infty} x(t) e^{-i\omega t} dt</math>
 |-
-| align="right" style="padding-right: 1em;" | [[More on CT Fourier transform|(info)]] Inverse DT Fourier Transform
+| align="right" style="padding-right: 1em;" | [[More on CT Fourier transform|(info)]] Inverse CT Fourier Transform
 | <math>\, x(t)=\mathcal{F}^{-1}(\mathcal{X}(\omega))=\frac{1}{2\pi} \int_{-\infty}^{\infty}\mathcal{X}(\omega)e^{i\omega t} d \omega\,</math>
 |}
@@ Line 16: / Line 33: @@
 ! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="4" | CT Fourier Transform Pairs
 |-
+| align="right" style="padding-right: 1em;" | <br>
 | align="right" style="padding-right: 1em;" |
-| <span class="texhtml">''x''(''t'')</span>
+| signal (function of t)
 | <math>\longrightarrow</math>
-| <math> \mathcal{X}(\omega) </math>
+| Fourier transform (function of <math>\omega</math>)
+|
 |-
+| align="right" style="padding-right: 1em;" | 1
 | align="right" style="padding-right: 1em;" | CTFT of a unit impulse
 | <math>\delta (t)\ </math>
 |
-| <math> 1 \! \ </math>
+| <math> 1 \ </math>
+|
 |-
+| align="right" style="padding-right: 1em;" | 2
 | align="right" style="padding-right: 1em;" | CTFT of a shifted unit impulse
 | <math>\delta (t-t_0)\ </math>
 |
-| <math>e^{iwt_0}</math>
+| <math>e^{-iwt_0}</math>
+|
 |-
+| align="right" style="padding-right: 1em;" | 3
 | align="right" style="padding-right: 1em;" | CTFT of a complex exponential
 | <math>e^{iw_0t}</math>
 |
 | <math> 2\pi \delta (\omega - \omega_0) \ </math>
+|
 |
 |-
+| align="right" style="padding-right: 1em;" | 4
 | align="right" style="padding-right: 1em;" |
-| <math>e^{-at}u(t)\ </math>, where <math>a\in {\mathbb R}, a>0 </math>
+| <math>e^{-at}u(t),\ </math> <math>a\in {\mathbb R}, a>0 </math>
 |
 | <math>\frac{1}{a+i\omega}</math>
+|
 |
 |-
+| align="right" style="padding-right: 1em;" | 5
 | align="right" style="padding-right: 1em;" |
-| <math>te^{-at}u(t)\ </math>, where <math>a\in {\mathbb R}, a>0 </math>
+| <math>te^{-at}u(t),\ </math> <math>a\in {\mathbb R}, a>0 </math>
 |
 | <math>\left( \frac{1}{a+i\omega}\right)^2</math>
+|
 |
 |-
+| align="right" style="padding-right: 1em;" | 6
 | align="right" style="padding-right: 1em;" | CTFT of a cosine
 | <math>\cos(\omega_0 t) \ </math>
 |
 | <math> \pi \left[\delta (\omega - \omega_0) + \delta (\omega + \omega_0)\right] \ </math>
+|
 |
 |-
+| align="right" style="padding-right: 1em;" | 7
 | align="right" style="padding-right: 1em;" | CTFT of a sine
 | <math>sin(\omega_0 t)  \ </math>
 |
 | <math>\frac{\pi}{i} \left[\delta (\omega - \omega_0) - \delta (\omega + \omega_0)\right]</math>
+|
 |
 |-
+| align="right" style="padding-right: 1em;" | 8
 | align="right" style="padding-right: 1em;" | CTFT of a rect
 | <math>\left\{\begin{array}{ll}1, &  \text{ if }|t|<T,\\ 0, & \text{else.}\end{array} \right. \ </math>
 |
 | <math> \frac{2 \sin \left( T \omega \right)}{\omega}  \ </math>
+|
 |
 |-
+| align="right" style="padding-right: 1em;" | 9
 | align="right" style="padding-right: 1em;" | CTFT of a sinc
-| <math>\frac{2 \sin \left( W t  \right)}{\pi t }  \ </math>
+| <math>\frac{\sin \left( W t  \right)}{\pi t }  \ </math>
 |
 | <math>\left\{\begin{array}{ll}1, &  \text{ if }|\omega| <W,\\ 0, & \text{else.}\end{array} \right.  \ </math>
+|
 |
 |-
+| align="right" style="padding-right: 1em;" | 10
 | align="right" style="padding-right: 1em;" | CTFT of a periodic function
 | <math>\sum^{\infty}_{k=-\infty} a_{k}e^{ikw_{0}t}</math>
 |
 | <math>2\pi\sum^{\infty}_{k=-\infty}a_{k}\delta(w-kw_{0}) \ </math>
+|
 |
 |-
+| align="right" style="padding-right: 1em;" | 11
 | align="right" style="padding-right: 1em;" | CTFT of an impulse train
 | <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>
+| <math>\frac{2\pi}{T}\sum^{\infty}_{k=-\infty}\delta(w-\frac{2\pi k}{T})</math>
+|
+|
+|-
+| align="right" style="padding-right: 1em;" | 12
+| align="right" style="padding-right: 1em;" |
+| <math> 1 \  </math>
+|
+| <math>2\pi \delta (\omega) \ </math>
+|
+|
+|-
+| 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|\leq T_1,\\ 0, & T_1<|t|\leq T/2 \end{array} \right.</math>
+|
+| <math>\sum^{\infty}_{k=-\infty}\frac{2 \sin(k\frac{2\pi}{T}T_1)}{k}\delta(\omega-k\frac{2\pi}{T})</math>
+|
+|
+|-
+| align="right" style="padding-right: 1em;" | 14
+| align="right" style="padding-right: 1em;" | CTFT of a Step Function
+| <math> u(t) \  </math>
+|
+| <math>\frac{1}{j\omega}+\pi\delta(\omega)</math>
+|
+|
+|-
+| align="right" style="padding-right: 1em;" | 15
+| align="right" style="padding-right: 1em;" |
+| <math> e^{-\alpha |t|}  \ </math>
+|
+| <math>\frac{2\alpha}{\alpha^{2}+\omega^{2}}</math>
+|
 |
 |}
@@ Line 88: / Line 161: @@
 {|
 |-
+! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="4" |
 ! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="4" | CT Fourier Transform Properties
 |-
 | align="right" style="padding-right: 1em;" |
-| <span class="texhtml">''x''(''t'')</span>
+| align="right" style="padding-right: 1em;" |
-| <math>\longrightarrow</math>
+| <math> x(t) \ </math>
+| <math>\longrightarrow</math>
 | <math> \mathcal{X}(\omega) </math>
 |-
-| align="right" style="padding-right: 1em;" | multiplication property
+| align="right" style="padding-right: 1em;" | 16
+| align="right" style="padding-right: 1em;" | [[Info multiplication property FT w|(info)]] multiplication property
 | <math>x(t)y(t) \ </math>
 |
-| <math>\frac{1}{2\pi} X(\omega)*Y(\omega) =\frac{1}{2\pi} \int_{-\infty}^{\infty} X(\theta)Y(\omega-\theta)d\theta</math>
+| <math>\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</math>
 |-
+| 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> X(\omega)Y(\omega) \!</math>
+| <math> \mathcal{X}(\omega)\mathcal{Y}(\omega) \!</math>
 |-
+| align="right" style="padding-right: 1em;" | 18
 | align="right" style="padding-right: 1em;" | time reversal
 | <math>\ x(-t) </math>
 |
-| <math>\ X(-\omega)</math>
+| <math>\ \mathcal{X}(-\omega)</math>
-|}
+|-
+| align="right" style="padding-right: 1em;" | 19
+| align="right" style="padding-right: 1em;" | Frequency Shifting
+| <math>e^{j\omega_0 t}x(t)</math>
+|
+| <math> \mathcal{X} (\omega - \omega_0) </math>
+|-
+| align="right" style="padding-right: 1em;" | 20
+| align="right" style="padding-right: 1em;"| Conjugation
+| <math> x^{*}(t) \  </math>
+|
+| <math> \mathcal{X}^{*} (-\omega)</math>
+|-
+| align="right" style="padding-right: 1em;" | 21
+| align="right" style="padding-right: 1em;" | Time and Frequency Scaling
+| <math> x(at) \  </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
+| <math> tx(t) \  </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
+| <math> x(t)\ \text{ real and even}</math>
+|
+| <math> \mathcal{X} (\omega) \ \text{ real and even} </math>
+|-
+| align="right" style="padding-right: 1em;" | 25
+| align="right" style="padding-right: 1em;" |
+| <math> x(t) \ \text{ real and odd}</math>
+|
+| <math> \mathcal{X} (\omega) \ \text{ purely imaginary and odd} </math>
+|-
+| align="right" style="padding-right: 1em;" | 26
+| align="right" style="padding-right: 1em;" | Duality
+| <math> \mathcal{X} (-t) </math>
+|
+| <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> (j \omega)^{n} \mathcal{X} (\omega) </math>
+|-
+| align="right" style="padding-right: 1em;" | 28
+| align="right" style="padding-right: 1em;" | Linearity
+| <math> ax(t) + by(t) \  </math>
+|
+| <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
+| <math> x(t-t_0) \  </math>
+|
+| <math>e^{-j\omega t_0}X(\omega)</math>
+|}
 {|
@@ Line 119: / Line 257: @@
 |}
+<br>
 ----
-[[Collective_Table_of_Formulas|Back to Collective Table]]
+[[Collective Table of Formulas|Back to Collective Table]]
-[[Category:Formulas]]

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

Latest revision as of 11:45, 24 August 2016

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