Homework Help: Calculate the Fourier Transform using theorems

1. Nov 20, 2014

grandpa2390

1. The problem statement, all variables and given/known data
Using a theorem (state which theorem you are using and give the formula), Calculate the Fourier Transform of
1. rect(x)triangle(x)
2.cos(pi*x)sinc(x)
3.rect(x)exp(-pi*x^2)
4.sinc(x)sin(pi*x)
5. exp(-pi*x^2)cos(pi*x)

2. Relevant equations
not sure what theorem to use for the first one.

3. The attempt at a solution
Well I am thinkng that since the triangle function is an even function, that I could use the power theorem which states that f(x)g(-x) = F(s)G(s)
so since triangle(-x)=triangle(x) I can just take the transform of rect(x) and multiply by the transform of triangle(x)

I should be able to do the same for the rest of them. take the function that is even and make it g(-x)? or does it matter if the function is even?
maybe it is just saying that given f(x) and g(x) reverse g(x) and multiply them together to get F(s) x G(s)?

so I got for number 1 : [sinc(pi * s) / (pi * s)]^3
do I integrate that or is that the answer?

Last edited: Nov 20, 2014
2. Nov 20, 2014

BiGyElLoWhAt

I think (at least for me) some clarification would help.
What is rect(x)? Is that a square wave function? What is triangle(x)? Is that a triangular wave function? Is rect(x)triangel(x) their products? Not really sure what to do with this at this point.

Also, I should mention, I've never done fourier transforms with theorems. I didn't even know there were any. I've always justg done it the long way.

3. Nov 20, 2014

grandpa2390

I figured it out. I am supposed to use the convolution theorem.

4. Nov 20, 2014

grandpa2390

yes and yes. turned out I am supposed to do the convolution theorem which states that the fourier transform of (f(x) times g(x)) is equal to the convolution (F(s) convolved with G(s)) :)

5. Nov 20, 2014

BiGyElLoWhAt

Oh. Cool.

6. Nov 20, 2014

grandpa2390

any tips for doing the convolution of sinc(x) and sinc^2(x) ?

I converted sinc and sinc^2 into sin (pi x) / pi x

and I multiply them together but I can't do the integral... :(

7. Nov 21, 2014

BiGyElLoWhAt

for 2? so you have cos(pi*x)sin(pi*x)/(pi*x)?
u-sub u=sin(pi*x)/pi

Last edited: Nov 21, 2014
8. Nov 21, 2014

BiGyElLoWhAt

or are you using sinc for the triangle wave?

9. Nov 21, 2014

BiGyElLoWhAt

$\frac{sin^2(x)}{x}+\frac{cos^2(x)}{x} = \frac{1}{x} (sin^2 +cos^2) = \frac{1}{x}(1) = \frac{1}{x}$
still a u sub, just a different u. If you have sinc^3, that's sinc^2*sinc which is (1/(argument)-cosc^2)sinc. I already feel like I'm giving out too much, so I'm gonna stop here and let you take over.

10. Nov 21, 2014