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− | + | [[Category:Formulas]] | |
− | + | [[Category:Fourier transform]] | |
− | + | [[Category:ECE301]] | |
− | + | [[Category:ECE438]] | |
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+ | <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> | ||
+ | |||
+ | ---- | ||
{| | {| | ||
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! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="2" | Definition 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 | ||
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| <math>\mathcal{X}(\omega)=\mathcal{F}(x(t))=\int_{-\infty}^{\infty} x(t) e^{-i\omega t} dt</math> | | <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 | + | | 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> | | <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> | ||
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! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="4" | CT Fourier Transform Pairs | ! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="4" | CT Fourier Transform Pairs | ||
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| align="right" style="padding-right: 1em;" | <br> | | align="right" style="padding-right: 1em;" | <br> | ||
| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
− | | | + | | signal (function of t) |
| <math>\longrightarrow</math> | | <math>\longrightarrow</math> | ||
− | | <math> | + | | Fourier transform (function of <math>\omega</math>) |
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| <math>\delta (t)\ </math> | | <math>\delta (t)\ </math> | ||
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− | | | + | | <math> 1 \ </math> |
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| <math>\sum^{\infty}_{n=-\infty} \delta(t-nT) \ </math> | | <math>\sum^{\infty}_{n=-\infty} \delta(t-nT) \ </math> | ||
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− | | | + | | <math>\frac{2\pi}{T}\sum^{\infty}_{k=-\infty}\delta(w-\frac{2\pi k}{T})</math> |
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| align="right" style="padding-right: 1em;" | 12 | | align="right" style="padding-right: 1em;" | 12 | ||
| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
− | | | + | | <math> 1 \ </math> |
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− | | < | + | | <math>2\pi \delta (\omega) \ </math> |
− | </ | + | |
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| align="right" style="padding-right: 1em;" | 13 | | align="right" style="padding-right: 1em;" | 13 | ||
| align="right" style="padding-right: 1em;" | CTFT of a Periodic Square Wave | | 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> | ||
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− | + | | <math>\sum^{\infty}_{k=-\infty}\frac{2 \sin(k\frac{2\pi}{T}T_1)}{k}\delta(\omega-k\frac{2\pi}{T})</math> | |
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− | | <math>\sum^{\infty}_{k=-\infty}\frac{2 \sin(k\ | + | |
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− | | align="right" style="padding-right: 1em;" | | + | | align="right" style="padding-right: 1em;" | 14 |
− | 14 | + | |
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| align="right" style="padding-right: 1em;" | CTFT of a Step Function | | align="right" style="padding-right: 1em;" | CTFT of a Step Function | ||
− | | < | + | | <math> u(t) \ </math> |
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| <math>\frac{1}{j\omega}+\pi\delta(\omega)</math> | | <math>\frac{1}{j\omega}+\pi\delta(\omega)</math> | ||
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| align="right" style="padding-right: 1em;" | 15 | | align="right" style="padding-right: 1em;" | 15 | ||
| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
− | | < | + | | <math> e^{-\alpha |t|} \ </math> |
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| <math>\frac{2\alpha}{\alpha^{2}+\omega^{2}}</math> | | <math>\frac{2\alpha}{\alpha^{2}+\omega^{2}}</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;" | |
− | | < | + | | <math> x(t) \ </math> |
| <math>\longrightarrow</math> | | <math>\longrightarrow</math> | ||
| <math> \mathcal{X}(\omega) </math> | | <math> \mathcal{X}(\omega) </math> | ||
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| align="right" style="padding-right: 1em;" | 17 | | align="right" style="padding-right: 1em;" | 17 | ||
| align="right" style="padding-right: 1em;" | convolution property | | align="right" style="padding-right: 1em;" | convolution property | ||
− | | <math>x(t)*y(t) \ | + | | <math>x(t)*y(t) \ </math> |
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| <math> \mathcal{X}(\omega)\mathcal{Y}(\omega) \!</math> | | <math> \mathcal{X}(\omega)\mathcal{Y}(\omega) \!</math> | ||
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| <math>e^{j\omega_0 t}x(t)</math> | | <math>e^{j\omega_0 t}x(t)</math> | ||
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− | | < | + | | <math> \mathcal{X} (\omega - \omega_0) </math> |
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| align="right" style="padding-right: 1em;" | 20 | | align="right" style="padding-right: 1em;" | 20 | ||
− | | align="right" style="padding-right: 1em;" | Conjugation | + | | align="right" style="padding-right: 1em;"| Conjugation |
− | | < | + | | <math> x^{*}(t) \ </math> |
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− | | < | + | | <math> \mathcal{X}^{*} (-\omega)</math> |
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| align="right" style="padding-right: 1em;" | 21 | | align="right" style="padding-right: 1em;" | 21 | ||
− | | align="right" style="padding-right: 1em;" | Time and Frequency Scaling | + | | align="right" style="padding-right: 1em;" | Time and Frequency Scaling |
− | | < | + | | <math> x(at) \ </math> |
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− | | <math>\frac{1}{|a|}X(\frac{\omega}{a})</math> | + | | <math>\frac{1}{|a|} \mathcal{X} (\frac{\omega}{a})</math> |
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| align="right" style="padding-right: 1em;" | 23 | | align="right" style="padding-right: 1em;" | 23 | ||
| align="right" style="padding-right: 1em;" | Differentiation in Frequency | | align="right" style="padding-right: 1em;" | Differentiation in Frequency | ||
− | | < | + | | <math> tx(t) \ </math> |
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− | | <math>j\frac{d}{d\omega}X(\omega)</math> | + | | <math>j\frac{d}{d\omega} \mathcal{X} (\omega)</math> |
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| align="right" style="padding-right: 1em;" | 24 | | align="right" style="padding-right: 1em;" | 24 | ||
− | | align="right" style="padding-right: 1em;" | Symmetry | + | | align="right" style="padding-right: 1em;" | Symmetry |
− | | < | + | | <math> x(t)\ \text{ real and even}</math> |
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− | | < | + | | <math> \mathcal{X} (\omega) \ \text{ real and even} </math> |
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| align="right" style="padding-right: 1em;" | 25 | | align="right" style="padding-right: 1em;" | 25 | ||
| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
− | | < | + | | <math> x(t) \ \text{ real and odd}</math> |
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− | | < | + | | <math> \mathcal{X} (\omega) \ \text{ purely imaginary and odd} </math> |
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| align="right" style="padding-right: 1em;" | 26 | | align="right" style="padding-right: 1em;" | 26 | ||
| align="right" style="padding-right: 1em;" | Duality | | align="right" style="padding-right: 1em;" | Duality | ||
− | | | + | | <math> \mathcal{X} (-t) </math> |
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− | | | + | | <math> 2 \pi x (\omega) \ </math> |
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| align="right" style="padding-right: 1em;" | 27 | | align="right" style="padding-right: 1em;" | 27 | ||
| align="right" style="padding-right: 1em;" | Differentiation | | align="right" style="padding-right: 1em;" | Differentiation | ||
− | | | + | | <math>\frac{d^{n}x(t)}{dt^{n}}</math> |
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+ | | <math> (j \omega)^{n} \mathcal{X} (\omega) </math> | ||
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| align="right" style="padding-right: 1em;" | 28 | | align="right" style="padding-right: 1em;" | 28 | ||
| align="right" style="padding-right: 1em;" | Linearity | | align="right" style="padding-right: 1em;" | Linearity | ||
− | | < | + | | <math> ax(t) + by(t) \ </math> |
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− | | < | + | | <math> a \mathcal{X}(\omega) + b \mathcal{Y} (\omega) </math> |
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| align="right" style="padding-right: 1em;" | 29 | | align="right" style="padding-right: 1em;" | 29 | ||
| align="right" style="padding-right: 1em;" | Time Shifting | | align="right" style="padding-right: 1em;" | Time Shifting | ||
− | | < | + | | <math> x(t-t_0) \ </math> |
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− | | | + | | <math>e^{-j\omega t_0}X(\omega)</math> |
− | <math>e^{-j\omega t_0}X(\omega)</math> | + | |} |
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− | |} | + | |
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---- | ---- | ||
− | + | [[Collective Table of Formulas|Back to Collective Table]] | |
− | [[Collective Table of Formulas|Back to Collective Table | + | |
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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 | |
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Parseval's relation | $ \int_{-\infty}^{\infty} |x(t)|^2 dt = \frac{1}{2\pi} \int_{-\infty}^{\infty} |\mathcal{X}(w)|^2 dw $ |