# Fourier transforms

• UrbanXrisis
In summary, the conversation discusses the similarities between the Fourier transform of a rectangular function and a triangular function, and how they both result in sine functions due to the even symmetry and positivity of the functions. The relationship between the power of the function and the value of n in sin^n is also discussed, with the suggestion to calculate the Fourier transform of a function bx^2 to see that it results in sin^3. The concept of convolutions is also mentioned as a way to understand the transform of a function, with the question of how to produce x^2 with a convolution and what its transform would be.

## Homework Statement

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.

## 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?

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..

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?

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?