(New page: Category:ECE438Fall2014Boutin Category:ECE438 Category:ECE Category:fourier transform Category:homework =Homework 1 Solution, ECE438, Fall 2014, [[user:mboutin|Pro...) |
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=Homework 1 Solution, [[ECE438]], Fall 2014, [[user:mboutin|Prof. Boutin]]= | =Homework 1 Solution, [[ECE438]], Fall 2014, [[user:mboutin|Prof. Boutin]]= | ||
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===A complex exponential=== | ===A complex exponential=== | ||
<math> x(t)=e^{j2 \pi f_0 t} </math> | <math> x(t)=e^{j2 \pi f_0 t} </math> | ||
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| + | From [https://www.projectrhea.org/rhea/index.php/CTFourierTransformPairsCollectedfromECE301withomega table], <math>e^{j\omega_0t} \leftrightarrow 2\pi \delta(\omega - \omega_0)</math>, therefore <br> | ||
| + | <math> | ||
| + | \begin{align} | ||
| + | e^{j2\pi f_0 t } \leftrightarrow &2\pi \delta(2\pi f - 2\pi f_0) \\ | ||
| + | &=\delta(f - f_0) | ||
| + | \end{align} | ||
| + | </math><br> | ||
| + | Where the last line is by the [https://www.projectrhea.org/rhea/index.php/Homework_3_ECE438F09 scaling] property of the delta function. | ||
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| − | + | ===A sine=== | |
| + | <math> x(t)=sin(t) </math> | ||
---- | ---- | ||
| − | + | ===A cosine=== | |
| + | <math>x(t)=cos(t)</math> | ||
---- | ---- | ||
| − | + | ===A periodic function=== | |
| + | <math>x(t)=x(t-T)</math> | ||
---- | ---- | ||
| − | + | ===An impulse train=== | |
| + | <math>x(t)=\sum_{n=-\infty}^{\infty} \delta (t-nT)</math> | ||
---- | ---- | ||
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== Discussion == | == Discussion == | ||
You may discuss the homework below. | You may discuss the homework below. | ||
Revision as of 15:32, 8 September 2014
Contents
Homework 1 Solution, ECE438, Fall 2014, Prof. Boutin
A complex exponential
$ x(t)=e^{j2 \pi f_0 t} $
From table, $ e^{j\omega_0t} \leftrightarrow 2\pi \delta(\omega - \omega_0) $, therefore
$ \begin{align} e^{j2\pi f_0 t } \leftrightarrow &2\pi \delta(2\pi f - 2\pi f_0) \\ &=\delta(f - f_0) \end{align} $
Where the last line is by the scaling property of the delta function.
A sine
$ x(t)=sin(t) $
A cosine
$ x(t)=cos(t) $
A periodic function
$ x(t)=x(t-T) $
An impulse train
$ x(t)=\sum_{n=-\infty}^{\infty} \delta (t-nT) $
Discussion
You may discuss the homework below.
- write comment/question here
- answer will go here
