# Math Help - Fourier Transforms Help

1. ## Fourier Transforms Help

I need help with the following problems:

Question 1:

a) Find the Fourier Transform of

b) Determine the limit of this transform as and discuss the result

Question 2:

Solve for Y(x) the integral equation

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

Question 4:
Given that

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.

2. Originally Posted by atwinix
I need help with the following problems:

Question 1:

a) Find the Fourier Transform of

$
\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).

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

3. ## More help

Show that the convolution of a top-hat function with itself is the triangle function. That is

Hint: Top-Hat function and Triangle function

4. Originally Posted by atwinix
Show that the convolution of a top-hat function with itself is the triangle function. That is

Hint: Top-Hat function and Triangle function

-Dan