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− | = CTFT of periodic signals | + | = CTFT of periodic signals with properties = |
{| border="1" class="wikitable" | {| border="1" class="wikitable" | ||
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! <math> x(t) \longrightarrow \ </math> | ! <math> x(t) \longrightarrow \ </math> | ||
! <math> \mathcal{X}(\omega) </math> | ! <math> \mathcal{X}(\omega) </math> | ||
+ | ! Proof | ||
|- | |- | ||
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| <math> ax(t) + by(t) \ </math> | | <math> ax(t) + by(t) \ </math> | ||
| <math> a \mathcal{X}(\omega) + b \mathcal{Y} (\omega) </math> | | <math> a \mathcal{X}(\omega) + b \mathcal{Y} (\omega) </math> | ||
+ | | <math>\mathfrak{F}(ax(t) + by(t)) = \int_{-\infty}^{\infty}[ax(t) + by(t)]e^{-j\omega t} dt</math><br /> | ||
+ | <math>\int_{-\infty}^{\infty}ax(t)e^{-j\omega t} dt + \int_{-\infty}^{\infty}by(t)e^{-j\omega t} dt</math><br /> | ||
+ | <math>=a\mathcal{X}(\omega) + b\mathcal{Y}(\omega) </math> | ||
|- | |- | ||
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| <math> x(t-t_0) \ </math> | | <math> x(t-t_0) \ </math> | ||
| <math>e^{-j\omega t_0}X(\omega)</math> | | <math>e^{-j\omega t_0}X(\omega)</math> | ||
+ | | <math>\mathfrak{F}(x(t - t_{o})) = \int_{-\infty}^{\infty}x(t-t_{0})e^{-j\omega t} dt</math><br /> | ||
+ | let <math> t' = t - t_{o} </math><br /> | ||
+ | <math>\int_{-\infty}^{\infty}x(t')e^{-j\omega (t'+t_{o})} dt' </math><br /> | ||
+ | <math>= e^{-j\omega t_{o}}\int_{-\infty}^{\infty}x(t')e^{-j\omega t'} dt' </math><br /> | ||
+ | <math>= e^{-j\omega t_{o}}\mathcal{X}(\omega)</math> <br /> | ||
|- | |- | ||
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| <math>e^{j\omega_0 t}x(t)</math> | | <math>e^{j\omega_0 t}x(t)</math> | ||
| <math> \mathcal{X} (\omega - \omega_0) </math> | | <math> \mathcal{X} (\omega - \omega_0) </math> | ||
+ | | Refer to Time Shifting section | ||
|- | |- | ||
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| <math> x^{*}(t) \ </math> | | <math> x^{*}(t) \ </math> | ||
| <math> \mathcal{X}^{*} (-\omega)</math> | | <math> \mathcal{X}^{*} (-\omega)</math> | ||
+ | | <math> \mathcal{X}^{*}(-\omega) = \int_{-\infty}^{\infty} (x(t)e^{j\omega t}dt)^{*} </math> <br /> | ||
+ | <math>= \int_{-\infty}^{\infty} x^{*}(t)e^{-j\omega t}dt </math> <br /> | ||
+ | <math>=\mathfrak{F}(x(t)^{*}) </math> | ||
|- | |- | ||
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| <math> x(at) \ </math> | | <math> x(at) \ </math> | ||
| <math>\frac{1}{|a|} \mathcal{X} (\frac{\omega}{a})</math> | | <math>\frac{1}{|a|} \mathcal{X} (\frac{\omega}{a})</math> | ||
+ | | <math> \int_{-\infty}^{\infty} x(at)e^{-j\omega t}dt </math> <br /> | ||
+ | let <math> t' = at </math><br /> | ||
+ | <math>= \int_{-\infty}^{\infty} x(t')e^{-j\omega \frac{t'}{a}}\frac{dt'}{a}, a > 0 </math> <br /> | ||
+ | <math>= -\int_{-\infty}^{\infty} x(t')e^{-j\omega \frac{t'}{a}}\frac{dt'}{a}, a < 0 </math> <br /> | ||
+ | <math>=\frac{1}{|a|} \mathcal{X} (\frac{\omega}{a})</math> | ||
+ | |- | ||
+ | |||
+ | | Convolution | ||
+ | | <math>x(t)*y(t) \ </math> | ||
+ | | <math> \mathcal{X}(\omega)\mathcal{Y}(\omega) \!</math> | ||
+ | | Remember that <math> x(t)*y(t) =\int_{-\infty}^{\infty} x(t')y(t-t')dt' </math><br /> | ||
+ | <math>\mathfrak{F}(x(t)*y(t)) = \int_{-\infty}^{\infty}[\int_{-\infty}^{\infty} x(t')y(t-t')dt' ]e^{-j\omega t}dt </math><br /> | ||
+ | Replace <math>e^{-j\omega t}</math> by <math> e^{-j\omega( t - t' )} e^{-j\omega t' }</math><br /> | ||
+ | <math>= \int_{-\infty}^{\infty}[\int_{-\infty}^{\infty} x(t')y(t-t')dt' ]e^{-j\omega( t - t' )} e^{-j\omega t' }dt </math><br /> | ||
+ | <math>= \int_{-\infty}^{\infty}e^{-j\omega t' }dt'[\int_{-\infty}^{\infty} x(t')y(t-t')e^{-j\omega( t - t' )}dt'] </math><br /> | ||
+ | <math>= \chi(\omega)\gamma(\omega)</math><br /> | ||
|- | |- | ||
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| <math>x(t)y(t) \ </math> | | <math>x(t)y(t) \ </math> | ||
| <math>\frac{1}{2\pi} \mathcal{X}(\omega)*\mathcal{Y}(\omega) </math> | | <math>\frac{1}{2\pi} \mathcal{X}(\omega)*\mathcal{Y}(\omega) </math> | ||
+ | | Refer to Convolution section | ||
|- | |- | ||
− | | | + | | Differentiation |
− | | <math>x(t) | + | | <math> \frac{d}{dt} x(t) \ </math> |
− | + | | <math>j\omega \mathcal{X} (\omega)</math> | |
+ | | <math> x(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} \mathcal{X}(\omega)e^{j\omega t}d\omega</math> <br /> | ||
+ | <math> \frac{d}{dt} x(t)= \frac{1}{2\pi} \int_{-\infty}^{\infty} \mathcal{X}(\omega) [\frac{d}{dt}e^{j\omega t}] d\omega</math> <br /> | ||
+ | <math> \frac{d}{dt} x(t)= \frac{1}{2\pi} \int_{-\infty}^{\infty} \mathcal{X}(\omega) j\omega e^{j\omega t} d\omega</math> <br /> | ||
|- | |- | ||
− | | | + | | Duality |
− | | <math> | + | | <math> \mathcal{X} (-t) </math> |
− | | <math> | + | | <math> 2 \pi x (\omega) \ </math> |
+ | | <math> 2\pi \mathcal{X} (-\omega) = \int_{-\infty}^{\infty} \mathcal{X}(t')e^{-j\omega t'}dt'</math> <br /> | ||
+ | let t = t' <br /> | ||
+ | <math> = \int_{-\infty}^{\infty} \mathcal{X}(t)e^{-j\omega t}dt</math> <br /> | ||
+ | = CTFT[x(t)] <br /> | ||
|- | |- | ||
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| | <math>\int_{-\infty}^{\infty} |x(t)|^2 dt = </math> | | | <math>\int_{-\infty}^{\infty} |x(t)|^2 dt = </math> | ||
| <math> \frac{1}{2\pi} \int_{-\infty}^{\infty} |\mathcal{X}(w)|^2 dw</math> | | <math> \frac{1}{2\pi} \int_{-\infty}^{\infty} |\mathcal{X}(w)|^2 dw</math> | ||
+ | | <math> \int_{-\infty}^{\infty} x(t)x(t) dt = \int_{-\infty}^{\infty}x(t)dt(\frac{1}{2\pi} \int_{-\infty}^{\infty}\mathcal{X}(\omega)e^{j\omega t}d\omega)</math><br /> | ||
+ | <math>= \frac{1}{2\pi}\int_{-\infty}^{\infty}\mathcal{X}(\omega)d\omega [\int_{-\infty}^{\infty}x(t)\frac{1}{2\pi}e^{j\omega t}]dt</math><br /> | ||
+ | <math>= \frac{1}{2\pi}\int_{-\infty}^{\infty}\chi(\omega)[\chi(-\omega)]d\omega</math><br /> | ||
+ | <math>= \frac{1}{2\pi }\int_{-\infty}^{\infty} | \mathcal{X} (\omega) |^{2}d\omega</math><br /> | ||
|- | |- |
Latest revision as of 00:52, 15 November 2018
CTFT of periodic signals with properties
Function | CTFT | ||
---|---|---|---|
$ sin(\omega_0t) $ | $ \frac{\pi}{j}(\delta(\omega - \omega_0) - \delta(\omega+\omega_0)) $ | ||
$ cos(\omega_0t) $ | $ \pi(\delta(\omega - \omega_0) + \delta(\omega+\omega_0)) $ | ||
$ e^{j\omega_0t} $ | $ 2\pi\delta(\omega - \omega_0) $ | ||
$ \sum^{\infty}_{k=-\infty} a_{k}e^{ikw_{0}t} $ | $ 2\pi\sum^{\infty}_{k=-\infty}a_{k}\delta(w-kw_{0}) \ $ | ||
$ \sum^{\infty}_{n=-\infty} \delta(t-nT) \ $ | $ \frac{2\pi}{T}\sum^{\infty}_{k=-\infty}\delta(w-\frac{2\pi k}{T}) $ |
Name | $ x(t) \longrightarrow \ $ | $ \mathcal{X}(\omega) $ | Proof |
---|---|---|---|
Linearity | $ ax(t) + by(t) \ $ | $ a \mathcal{X}(\omega) + b \mathcal{Y} (\omega) $ | $ \mathfrak{F}(ax(t) + by(t)) = \int_{-\infty}^{\infty}[ax(t) + by(t)]e^{-j\omega t} dt $ $ \int_{-\infty}^{\infty}ax(t)e^{-j\omega t} dt + \int_{-\infty}^{\infty}by(t)e^{-j\omega t} dt $ |
Time Shifting | $ x(t-t_0) \ $ | $ e^{-j\omega t_0}X(\omega) $ | $ \mathfrak{F}(x(t - t_{o})) = \int_{-\infty}^{\infty}x(t-t_{0})e^{-j\omega t} dt $ let $ t' = t - t_{o} $ |
Frequency Shifting | $ e^{j\omega_0 t}x(t) $ | $ \mathcal{X} (\omega - \omega_0) $ | Refer to Time Shifting section |
Conjugation | $ x^{*}(t) \ $ | $ \mathcal{X}^{*} (-\omega) $ | $ \mathcal{X}^{*}(-\omega) = \int_{-\infty}^{\infty} (x(t)e^{j\omega t}dt)^{*} $ $ = \int_{-\infty}^{\infty} x^{*}(t)e^{-j\omega t}dt $ |
Scaling | $ x(at) \ $ | $ \frac{1}{|a|} \mathcal{X} (\frac{\omega}{a}) $ | $ \int_{-\infty}^{\infty} x(at)e^{-j\omega t}dt $ let $ t' = at $ |
Convolution | $ x(t)*y(t) \ $ | $ \mathcal{X}(\omega)\mathcal{Y}(\omega) \! $ | Remember that $ x(t)*y(t) =\int_{-\infty}^{\infty} x(t')y(t-t')dt' $ $ \mathfrak{F}(x(t)*y(t)) = \int_{-\infty}^{\infty}[\int_{-\infty}^{\infty} x(t')y(t-t')dt' ]e^{-j\omega t}dt $ |
Multiplication | $ x(t)y(t) \ $ | $ \frac{1}{2\pi} \mathcal{X}(\omega)*\mathcal{Y}(\omega) $ | Refer to Convolution section |
Differentiation | $ \frac{d}{dt} x(t) \ $ | $ j\omega \mathcal{X} (\omega) $ | $ x(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} \mathcal{X}(\omega)e^{j\omega t}d\omega $ $ \frac{d}{dt} x(t)= \frac{1}{2\pi} \int_{-\infty}^{\infty} \mathcal{X}(\omega) [\frac{d}{dt}e^{j\omega t}] d\omega $ |
Duality | $ \mathcal{X} (-t) $ | $ 2 \pi x (\omega) \ $ | $ 2\pi \mathcal{X} (-\omega) = \int_{-\infty}^{\infty} \mathcal{X}(t')e^{-j\omega t'}dt' $ let t = t' |
Parseval's Relation | $ \int_{-\infty}^{\infty} |x(t)|^2 dt = $ | $ \frac{1}{2\pi} \int_{-\infty}^{\infty} |\mathcal{X}(w)|^2 dw $ | $ \int_{-\infty}^{\infty} x(t)x(t) dt = \int_{-\infty}^{\infty}x(t)dt(\frac{1}{2\pi} \int_{-\infty}^{\infty}\mathcal{X}(\omega)e^{j\omega t}d\omega) $ $ = \frac{1}{2\pi}\int_{-\infty}^{\infty}\mathcal{X}(\omega)d\omega [\int_{-\infty}^{\infty}x(t)\frac{1}{2\pi}e^{j\omega t}]dt $ |