Difference between revisions of "CT Fourier Transform (frequency in hertz)" - Rhea

Latest revision as of 20:01, 4 March 2015

Continuous-time Fourier Transform Pairs and Properties

as a function of frequency f in hertz

(used in ECE438)

CT Fourier Transform and its Inverse
CT Fourier Transform	$X(f)=\mathcal{F}(x(t))=\int_{-\infty}^{\infty} x(t) e^{-i2\pi ft} dt$
Inverse DT Fourier Transform	$\, x(t)=\mathcal{F}^{-1}(X(f))=\int_{-\infty}^{\infty}X(f)e^{i2\pi ft} df \,$

CT Fourier Transform Pairs
	signal (function of t)	$\longrightarrow$	Fourier transform (function of f)
CTFT of a unit impulse	$\delta (t)\$		$1 \$
CTFT of a shifted unit impulse	$\delta (t-t_0)\$		$e^{-i2\pi ft_0}$
CTFT of a complex exponential	$e^{iw_0t}$		$\delta (f - \frac{\omega_0}{2\pi}) \$
	$e^{-at}u(t), \ \text{ where } a\in {\mathbb R}, a>0$		$\frac{1}{a+i2\pi f}$
	$te^{-at}u(t), \ \text{ where } a\in {\mathbb R}, a>0$		$\left( \frac{1}{a+i2\pi f}\right)^2$
CTFT of a cosine	$\cos(\omega_0 t) \$		$\frac{1}{2} \left[\delta (f - \frac{\omega_0}{2\pi}) + \delta (f + \frac{\omega_0}{2\pi})\right] \$
CTFT of a sine	$sin(\omega_0 t) \$		$\frac{1}{2i} \left[\delta (f - \frac{\omega_0}{2\pi}) - \delta (f + \frac{\omega_0}{2\pi})\right]$
CTFT of a rect	$\left\{\begin{array}{ll}1, & \text{ if }\|t\|<T,\\ 0, & \text{else.}\end{array} \right. \$		$\frac{\sin \left(2\pi Tf \right)}{\pi f} \$
CTFT of a sinc	$\frac{ \sin \left( W t \right)}{\pi t } \$		$\left\{\begin{array}{ll}1, & \text{ if }\|f\| <\frac{W}{2\pi},\\ 0, & \text{else.}\end{array} \right. \$
CTFT of a periodic function	$\sum^{\infty}_{k=-\infty} a_{k}e^{ikw_{0}t}$		$\sum^{\infty}_{k=-\infty}a_{k}\delta(f-\frac{kw_{0}}{2\pi}) \$
CTFT of an impulse train	$\sum^{\infty}_{n=-\infty} \delta(t-nT) \$		$\frac{1}{T}\sum^{\infty}_{k=-\infty}\delta(f-\frac{k}{T}) \$

CT Fourier Transform Properties
	$x(t) \$	$\longrightarrow$	$X(f) \$
multiplication property	$x(t)y(t) \$		$X(f)*Y(f) =\int_{-\infty}^{\infty} X(\theta)Y(f-\theta)d\theta$
time shifting property	$x(t-t_0) \$		$X(f)e^{-j 2 \pi f t_0} \$
frequency shifting (also called "modulation") property	$x(t) e^{j 2 \pi f_0 t} \$		$X(f-f_0) \$
scaling and shifting property	$x\left( \frac{ t- t_0}{a} \right) \$		$\|a\| X(af) e^{-j 2 \pi f t_0} \$
convolution property	$x(t)*y(t) \$		$X(f)Y(f) \$
time reversal	$\ x(-t)$		$\ X(-f)$

Other CT Fourier Transform Properties
Parseval's relation	$\int_{-\infty}^{\infty} \|x(t)\|^2 dt = \int_{-\infty}^{\infty} \|X(f)\|^2 df$

Back to Collective Table

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+[[Category:Formulas]]
-{|
+[[Category:Fourier transform]]
-|-
+[[Category:ECE301]]
-|'''If you enjoy using this [[Collective_Table_of_Formulas|collective table of formulas]], please consider  [https://donate.purdue.edu/DesignateGift.aspx?allocation=017637&appealCode=11213&amount=25&allocationDescription=RheaProjectMimiBoutin donating to Project Rhea] or [[Donations | becoming a sponsor]].'''
+[[Category:ECE438]]
-| [[Image:DonateNow.png]]
-|-
+<center><font size= 4>
-|}
+'''[[Collective_Table_of_Formulas|Collective Table of Formulas]]'''
-</div>
+</font size>
+Continuous-time Fourier Transform Pairs and Properties
+as a function of frequency f in hertz
+(used in [[ECE438]])
+</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">f</span> in hertz) [[More on CT Fourier transform|(info)]]
+! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="2" | CT Fourier Transform and its Inverse
-|-
-! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="2" | Definition CT Fourier Transform and its Inverse
 |-
 | align="right" style="padding-right: 1em;" |  CT Fourier Transform
@@ Line 23: / Line 32: @@
 {|
 |-
-! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="4" | CT Fourier Transform Pairs [[More_on_CTFTs| (info)]]
+! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="4" | CT Fourier Transform Pairs
 |-
 | align="right" style="padding-right: 1em;" |
-| <span class="texhtml">''x''(''t'')</span>
+| signal (function of t)
 | <math>\longrightarrow</math>
-| <math> X(f) </math>
+| Fourier transform (function of f)
 |-
 | 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;" | CTFT of a shifted unit impulse
@@ Line 47: / Line 56: @@
 |-
 | 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), \  \text{ where } a\in {\mathbb R}, a>0 </math>
 |
 | <math>\frac{1}{a+i2\pi f}</math>
@@ Line 53: / Line 62: @@
 |-
 | 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), \  \text{ where } a\in {\mathbb R}, a>0 </math>
 |
 | <math>\left( \frac{1}{a+i2\pi f}\right)^2</math>
@@ Line 100: / Line 109: @@
 |-
 | align="right" style="padding-right: 1em;" |
-| <span class="texhtml">''x''(''t'')</span>
+| <math> x(t) \  </math>
 | <math>\longrightarrow</math>
-| <math> X(f) </math>
+| <math> X(f) \  </math>
 |-
 | align="right" style="padding-right: 1em;" | multiplication property
@@ Line 108: / Line 117: @@
 |
 | <math> X(f)*Y(f) =\int_{-\infty}^{\infty} X(\theta)Y(f-\theta)d\theta</math>
+|-
+| align="right" style="padding-right: 1em;" | time shifting property
+| <math>x(t-t_0) \ </math>
+|
+| <math> X(f)e^{-j 2 \pi f t_0}   \ </math>
+|-
+| align="right" style="padding-right: 1em;" | frequency shifting (also called "modulation") property
+| <math>x(t) e^{j 2 \pi f_0 t}  \ </math>
+|
+| <math> X(f-f_0)   \  </math>
+|-
+| align="right" style="padding-right: 1em;" | scaling and shifting property
+| <math>x\left( \frac{ t- t_0}{a} \right) \ </math>
+|
+| <math> |a| X(af) e^{-j 2 \pi f t_0}    \  </math>
 |-
 | align="right" style="padding-right: 1em;" | convolution property
-| <math>x(t)*y(t) \!</math>
+| <math>x(t)*y(t) \  </math>
 |
-| <math> X(f)Y(f) \!</math>
+| <math> X(f)Y(f) \  </math>
 |-
 | align="right" style="padding-right: 1em;" | time reversal
@@ Line 129: / Line 153: @@
 ----
+[[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 hertz)" - Rhea

Latest revision as of 20:01, 4 March 2015

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