Fourier Transforms Help

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September 28th 2007, 04:11 AM
atwinix

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Fourier Transforms Help

I need help with the following problems:

Question 1:

a) Find the Fourier Transform of

http://i82.photobucket.com/albums/j256/atwinix/Q1.jpg

b) Determine the limit of this transform as http://i82.photobucket.com/albums/j256/atwinix/e.jpg and discuss the result

Question 2:

Solve for Y(x) the integral equation

http://i82.photobucket.com/albums/j256/atwinix/Q2.jpg

and verify the solution by direct substitution

Question 3:

Use the time-shift property to calculate the Fourier transform of the double pulse defined by

http://i82.photobucket.com/albums/j256/atwinix/Q3.jpg

Question 4:
Given that

http://i82.photobucket.com/albums/j256/atwinix/Q4.jpg

a) Draw the graph of f(t)
b) Express f(t) in terms of the Heaviside unit step function
c) Find the Fourier transform of f(t)

Thanks in advance for all the valuable help. It is most appreciated.
September 28th 2007, 06:52 AM
CaptainBlack

Quote:

Originally Posted by atwinix

I need help with the following problems:

Question 1:

a) Find the Fourier Transform of

http://i82.photobucket.com/albums/j256/atwinix/Q1.jpg

$ \mathcal{F}f(\omega)=\frac{1}{2 \varepsilon}\int_{-\varepsilon}^{\varepsilon}e^{i~\omega x} ~dx = \frac{\sin(\omega \varepsilon)}{\omega \varepsilon} $

(I will leave the last step in evaluating the integral in the above to you,
it's not difficult).

Quote:

b) Determine the limit of this transform as $\varepsilon \to 0_{+}$ and discuss the result

You are supposed to know that:

$ \lim_{\varepsilon \to 0} \frac{\sin( \varepsilon)}{\varepsilon}=1 $

So:

$ \lim_{\varepsilon \to 0} \frac{\sin(\omega \varepsilon)}{\omega \varepsilon}=1 $

RonL
September 30th 2007, 05:37 PM
atwinix

More help

Show that the convolution of a top-hat function with itself is the triangle function. That ishttp://i82.photobucket.com/albums/j256/atwinix/Q5_1.jpg

Hint: Top-Hat function http://i82.photobucket.com/albums/j256/atwinix/Q5_2.jpg and Triangle function http://i82.photobucket.com/albums/j256/atwinix/Q5_3.jpg
September 30th 2007, 05:39 PM
topsquark

Quote:

Originally Posted by atwinix

Show that the convolution of a top-hat function with itself is the triangle function. That ishttp://i82.photobucket.com/albums/j256/atwinix/Q5_1.jpg

Hint: Top-Hat function http://i82.photobucket.com/albums/j256/atwinix/Q5_2.jpg and Triangle function http://i82.photobucket.com/albums/j256/atwinix/Q5_3.jpg

Please post new questions in new threads.

-Dan