Fourier transform - application, integral calculation

I'm trying to brush the dust of my transform tinkering. (Have to use it in some electronic stuff.) I just did an exercise where I calculate the F-transform of an even function.

Let f(x) be:

$\displaystyle \displaystyle f(t)=\left\{\begin{matrix} 0 & x > 2\\ 2-x & 0 < x < 2 \end{matrix}\right.$

Find the even Fourier transform.

I calculated the transform with this formula

$\displaystyle \displaystyle f(\omega) = \int_{-\infty}^{\infty} f(t) e^{-i \omega t} dt = \int_{-\infty}^{\infty} f(t) (cos(\omega t) + i sin(\omega t))dt$

Knowing that the function is even I write:

$\displaystyle \displaystyle f(\omega) = 2 \int_{0}^{\infty} (2-t) cos(\omega t) dt$

A bit integration gives, don't hesitate to call out errors.

$\displaystyle \displaystyle f(\omega) = \frac{4sin^{2}w}{w^2}$

Now as an application to this calculation I'm supposed to determine this integral.

$\displaystyle \displaystyle I = \int_0^{\infty} \frac{sin^{2}w}{w^2}$

This is where I like to get some help... I simply don't know how to do it. Calculating the integral, in an other way, I get $\displaystyle \displaystyle \frac{\pi}{2}$

Re: Fourier transform - application, integral calculation

Quote:

Originally Posted by

**liquidFuzz** I'm trying to brush the dust of my transform tinkering. (Have to use it in some electronic stuff.) I just did an exercise where I calculate the F-transform of an even function.

Let f(x) be:

$\displaystyle \displaystyle f(t)=\left\{\begin{matrix} 0 & x > 2\\ 2-x & 0 < x < 2 \end{matrix}\right.$

Find the even Fourier transform.

I calculated the transform with this formula

$\displaystyle \displaystyle f(\omega) = \int_{-\infty}^{\infty} f(t) e^{-i \omega t} dt = \int_{-\infty}^{\infty} f(t) (cos(\omega t) + i sin(\omega t))dt$

Knowing that the function is even I write:

$\displaystyle \displaystyle f(\omega) = 2 \int_{0}^{\infty} (2-t) cos(\omega t) dt$

A bit integration gives, don't hesitate to call out errors.

$\displaystyle \displaystyle f(\omega) = \frac{4sin^{2}w}{w^2}$

Now as an application to this calculation I'm supposed to determine this integral.

$\displaystyle \displaystyle I = \int_0^{\infty} \frac{sin^{2}w}{w^2}$

This is where I like to get some help... I simply don't know how to do it. Calculating the integral, in an other way, I get $\displaystyle \displaystyle \frac{\pi}{2}$

The notation for the Fourier transform of a function should be different from the notation for the function itself. So you should write $\displaystyle \hat{f}(\omega)$ rather than $\displaystyle f(\omega).$

The Fourier inversion theorem says that you can retrieve $\displaystyle f(t)$ from $\displaystyle \hat{f}(\omega)$ by the formula $\displaystyle f(t) = \frac1{2\pi}\int_{-\infty}^\infty \hat{f}(\omega)e^{it\omega}d\omega.$ Put t=0 in that formula and you will get the result you are looking for.

Re: Fourier transform - application, integral calculation

Thanks! For slapping my fingers, and explaining how to do this.

Re: Fourier transform - application, integral calculation

How about this..?

Assume that $\displaystyle \hat{f}(w)$ converges to f(x), $\displaystyle \hat{f}(w) = f(x)$, and that the Fourier transform is correct, $\displaystyle \hat{f}(w) = \frac{4 sin^2 w}{w^2}$

Taking a arbitrary point on f(x)

$\displaystyle f(0) = 1 = \lim_{a \to \infty} \frac{1}{2 \pi} \int_0^a \frac{sin^2 x}{x^2} dx$

Gives

$\displaystyle 1 = \frac{1}{2 \pi} \int_0^\infty \frac{4sin^2 x}{x^2} dx$

$\displaystyle \drarrow \frac{\pi}{2} = \int_0^\infty \frac{sin^2 x}{x^2} dx$