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| + | [[Category:Formulas]] | ||
| + | [[Category:Fourier transform]] | ||
| + | [[Category:ECE301]] | ||
| + | [[Category:ECE438]] | ||
| + | |||
| + | <center><font size= 4> | ||
| + | '''[[Collective_Table_of_Formulas|Collective Table of Formulas]]''' | ||
| + | </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(238, 238, 238);" colspan="2" | CT Fourier Transform and its Inverse | |
| − | + | ||
| − | ! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="2" | | + | |
|- | |- | ||
| align="right" style="padding-right: 1em;" | CT Fourier Transform | | align="right" style="padding-right: 1em;" | CT Fourier Transform | ||
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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 |
|- | |- | ||
| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
| − | | | + | | signal (function of t) |
| <math>\longrightarrow</math> | | <math>\longrightarrow</math> | ||
| − | | | + | | Fourier transform (function of f) |
|- | |- | ||
| align="right" style="padding-right: 1em;" | CTFT of a unit impulse | | align="right" style="padding-right: 1em;" | CTFT of a unit impulse | ||
| <math>\delta (t)\ </math> | | <math>\delta (t)\ </math> | ||
| | | | ||
| − | | <math> 1 | + | | <math> 1 \ </math> |
|- | |- | ||
| align="right" style="padding-right: 1em;" | CTFT of a shifted unit impulse | | align="right" style="padding-right: 1em;" | CTFT of a shifted unit impulse | ||
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|- | |- | ||
| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
| − | | <math>e^{-at}u(t) | + | | <math>e^{-at}u(t), \ \text{ where } a\in {\mathbb R}, a>0 </math> |
| | | | ||
| <math>\frac{1}{a+i2\pi f}</math> | | <math>\frac{1}{a+i2\pi f}</math> | ||
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|- | |- | ||
| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
| − | | <math>te^{-at}u(t) | + | | <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> | | <math>\left( \frac{1}{a+i2\pi f}\right)^2</math> | ||
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|- | |- | ||
| align="right" style="padding-right: 1em;" | CTFT of a sinc | | align="right" style="padding-right: 1em;" | CTFT of a sinc | ||
| − | | <math>\frac{ | + | | <math>\frac{ \sin \left( W t \right)}{\pi t } \ </math> |
| | | | ||
| <math>\left\{\begin{array}{ll}1, & \text{ if }|f| <\frac{W}{2\pi},\\ 0, & \text{else.}\end{array} \right. \ </math> | | <math>\left\{\begin{array}{ll}1, & \text{ if }|f| <\frac{W}{2\pi},\\ 0, & \text{else.}\end{array} \right. \ </math> | ||
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|- | |- | ||
| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
| − | | < | + | | <math> x(t) \ </math> |
| <math>\longrightarrow</math> | | <math>\longrightarrow</math> | ||
| − | | <math> X(f) </math> | + | | <math> X(f) \ </math> |
|- | |- | ||
| align="right" style="padding-right: 1em;" | multiplication property | | align="right" style="padding-right: 1em;" | multiplication property | ||
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| | | | ||
| <math> X(f)*Y(f) =\int_{-\infty}^{\infty} X(\theta)Y(f-\theta)d\theta</math> | | <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 | | align="right" style="padding-right: 1em;" | convolution property | ||
| − | | <math>x(t)*y(t) \ | + | | <math>x(t)*y(t) \ </math> |
| | | | ||
| − | | <math> X(f)Y(f) \ | + | | <math> X(f)Y(f) \ </math> |
|- | |- | ||
| align="right" style="padding-right: 1em;" | time reversal | | align="right" style="padding-right: 1em;" | time reversal | ||
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---- | ---- | ||
| − | + | [[Collective_Table_of_Formulas|Back to Collective Table]] | |
| − | [[Collective_Table_of_Formulas|Back to Collective Table | + | |
| − | + | ||
| − | + | ||
Latest revision as of 21:01, 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 $ |
