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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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| − | | align="right" style="padding-right: 1em;" | [[Explain_CTFT|CT Fourier Transform | + | | align="right" style="padding-right: 1em;" | [[Explain_CTFT|Justification]] CT Fourier Transform |
| <math>X(f)=\mathcal{F}(x(t))=\int_{-\infty}^{\infty} x(t) e^{-i2\pi ft} dt</math> | | <math>X(f)=\mathcal{F}(x(t))=\int_{-\infty}^{\infty} x(t) e^{-i2\pi ft} dt</math> | ||
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| − | | align="right" style="padding-right: 1em;" | [[Explain_InverseCTFT|Inverse CT Fourier Transform | + | | align="right" style="padding-right: 1em;" | [[Explain_InverseCTFT|Justification ]] Inverse CT Fourier Transform |
| <math>\, x(t)=\mathcal{F}^{-1}(X(f))=\int_{-\infty}^{\infty}X(f)e^{i2\pi ft} df \,</math> | | <math>\, x(t)=\mathcal{F}^{-1}(X(f))=\int_{-\infty}^{\infty}X(f)e^{i2\pi ft} df \,</math> | ||
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| <math> X(f) </math> | | <math> X(f) </math> | ||
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| − | | align="right" style="padding-right: 1em;" | [[Explain_unitimpulse|CTFT of a unit impulse | + | | align="right" style="padding-right: 1em;" | [[Explain_unitimpulse|Justification]] CTFT of a unit impulse |
| <math>\delta (t)\ </math> | | <math>\delta (t)\ </math> | ||
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Latest revision as of 04:49, 9 September 2013
HW1 Solution ECE438 Fall 2013
In this homework, you were asked to start from a table of CT Fourier transforms in terms of $ \omega $ in radians and to obtain the corresponding relationships in terms of frequency f (in hertz). Below are the solutions.
To get the justification for each transform/property, click on the corresponding link.
| CT Fourier Transform Pairs and Properties (frequency f in hertz per time unit) (info) | |
|---|---|
| (Click title to see explanation on how to obtain the formula in terms of f in hertz) | |
| Definition CT Fourier Transform and its Inverse | |
| Justification CT Fourier Transform | $ X(f)=\mathcal{F}(x(t))=\int_{-\infty}^{\infty} x(t) e^{-i2\pi ft} dt $ |
| Justification Inverse CT Fourier Transform | $ \, x(t)=\mathcal{F}^{-1}(X(f))=\int_{-\infty}^{\infty}X(f)e^{i2\pi ft} df \, $ |
| CT Fourier Transform Pairs | ||||
|---|---|---|---|---|
| x(t) | $ \longrightarrow $ | $ X(f) $ | ||
| Justification 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)\ $, where $ a\in {\mathbb R}, a>0 $ | $ \frac{1}{a+i2\pi f} $ | |||
| $ te^{-at}u(t)\ $, 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{2 \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 $ | |
| 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 $ |
