Comparing Fourier Transforms of Rectangle and Triangular Functions

[UrbanXrisis]

Homework Statement

For a visual of what I am talking about, please visit: http://webhost.etc.tuiasi.ro/cin/Downloads/Fourier/Fourier.html
and scroll down to the "Examples of Fourier Transforms" part

I am ask to explain why the Fourier transform on the rectangle function was similar to the Fourier transform on the trangular function.

Homework Equations

The Attempt at a Solution

so here what I think, and I'm not totally sure about it. The FT of a rectangular function is sin and rhe FT of the trangular function is a sin^2. The FT are similar because both functions are even, symetric, and always positive. The rectangular function is a constant function, which gives the sin, while the trangular function is a linear function, which gives the sin^2. Maybe a x^2 function with bounds will give a sin^3? not really sure about that. Is my reasoning correct for why the two FTs are similar?

 

[ziad1985]

Can't you calculate the FT of x^2 function? it should be easy..
define a function bx^2 between -a and a , and see what the FT would be..

 

[UrbanXrisis]

ok i did it, and it does show that it would be sin^3

know this, why is it that the higher the power, the larger n is for sin^n?

 

[marcusl]

First notice that the transform of a square pulse is sin(aw)/(aw) which is called sinc(aw). It is not the same as a simple sine.

To answer your question, here's a different approach--think in terms of convolutions. The convolution of a square pulse with itself is what? (It should be in your book.) Therefore what is the transform of the convolution?

As for x^2, how would you produce that with a convolution and what is its transform?