Calculate the Fourier Transform using theorems

[grandpa2390]

Homework Statement

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)

Homework Equations

not sure what theorem to use for the first one.

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?

 

[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.

 

[grandpa2390]

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

 

[grandpa2390]

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.

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)) :)

 

[BiGyElLoWhAt]

Oh. Cool.

 

[grandpa2390]

Oh. Cool.

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... :(

 

[BiGyElLoWhAt]

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

 

[BiGyElLoWhAt]

or are you using sinc for the triangle wave?

 

[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 going to stop here and let you take over.

 

[BiGyElLoWhAt]

If that's not what you're looking for, either be more specific or see:
http://tutorial.math.lamar.edu/Classes/DE/ConvolutionIntegrals.aspx