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===Answer 1=== | ===Answer 1=== | ||
− | + | There are a few things that you need to know to accomplish this problem. The two main formulas that you need are | |
+ | <math> \omega = 2 \pi f </math> and <math>\delta(cx)= \frac{1}{c} \delta(x)</math>. | ||
+ | |||
+ | PROOF | ||
+ | |||
+ | <math>\int_{-\infty}^{\infty}\delta(x)dx = 1</math> | ||
+ | |||
+ | <math> y=cx => \frac{dy}{c}=dx</math> | ||
+ | |||
+ | <math>\int_{-\infty}^\infty \delta(y)\frac{dy}{c}=\frac{1}{c}</math> | ||
+ | |||
+ | THEREFORE | ||
+ | |||
+ | <math>\delta(\omega)=\delta(2\pi f)=\frac{1}{2\pi}\delta(f)</math> | ||
+ | |||
+ | and | ||
+ | |||
+ | <math>{\mathcal X} (\omega) = \frac{1}{j \omega} + \pi \delta (\omega ) = \frac{1}{2}(\frac{1}{j\pi f} + \delta(f))</math> | ||
+ | |||
+ | -my | ||
+ | |||
===Answer 2=== | ===Answer 2=== | ||
Write it here. | Write it here. |
Revision as of 18:14, 31 August 2011
Contents
Continuous-time Fourier transform: from omega to f
In ECE301, you learned that the Fourier transform of a step function $ x(t)=u(t) $ is the following:
$ {\mathcal X} (\omega) = \frac{1}{j \omega} + \pi \delta (\omega ). $
Use this fact to obtain an expression for the Fourier transform $ X(f) $ (in terms of frequency in hertz) of the step function. (Your answer should agree with the one given in this table.) Justify all your steps.
You will receive feedback from your instructor and TA directly on this page. Other students are welcome to comment/discuss/point out mistakes/ask questions too!
Answer 1
There are a few things that you need to know to accomplish this problem. The two main formulas that you need are $ \omega = 2 \pi f $ and $ \delta(cx)= \frac{1}{c} \delta(x) $.
PROOF
$ \int_{-\infty}^{\infty}\delta(x)dx = 1 $
$ y=cx => \frac{dy}{c}=dx $
$ \int_{-\infty}^\infty \delta(y)\frac{dy}{c}=\frac{1}{c} $
THEREFORE
$ \delta(\omega)=\delta(2\pi f)=\frac{1}{2\pi}\delta(f) $
and
$ {\mathcal X} (\omega) = \frac{1}{j \omega} + \pi \delta (\omega ) = \frac{1}{2}(\frac{1}{j\pi f} + \delta(f)) $
-my
Answer 2
Write it here.
Answer 3
write it here.