Fourier transform step-by-step example: $f(x) = 1/2$ where $x\in[0,1]$ otherwise $f(x) = 0$

Asked Oct 21 '15 at 14:03

Active Oct 23 '15 at 14:27

Viewed 664 times

I'm trying to understand the general procedure for finding the fourier transform of a function f(x). I've seen the general theory, but feel It would help with a concrete example to see how it is applied in practice.

So wondering what a step-by-step procedure for transforming the below is?

$f(x) = 1/2$ where $x\in[0,1]$ otherwise $f(x) = 0$ How would the procedure differ if the interval for x was $x\in[-1,1]$ instead?

Added:

As I understand it after some more checking we would get, from what i've seen a procedure as follows for the first interval. Where F is the Fourier-transformation of f:

$F(x) = \frac{1}{2\pi}\int_{-\infty}^{\infty} f(x) e^{-i w x} dx = \frac{1}{2\pi}\int_{0}^{1} \frac{1}{2} e^{-i w x} dx = \frac{i}{4\pi w}(e^{-iw} - 1)$

edited Oct 23 '15 at 14:27

asked Oct 21 '15 at 14:03

jibo

What interval are you expanding over? If you are expanding over $[-\pi,\pi]$, then you use functions ${ e^{inx}}$. If you are expanding over $[-1,1]$, then you use functions ${ e^{in\pi x}}$. It makes a difference. – Disintegrating By Parts Oct 21 '15 at 21:07
I believe the interval would be the interval for x, otherwise there might be a standard interval used if that's not the right interval. – jibo Oct 22 '15 at 07:58
You have defined $f$ everywhere; that's an infinite interval. – Disintegrating By Parts Oct 22 '15 at 15:56
Your integration result does not look correct. $$\frac{1}{4\pi}\int_0^1 e^{-iwx} dx = \frac{i}{4\pi w}(e^{-iw} - 1)$$ – Oct 22 '15 at 16:40
@TrialAndError: Your comment applies to Fourier series, but the OP is evaluating a Fourier transform. – Oct 22 '15 at 16:47
@Bungo : Ah, I misread. Thank you. – Disintegrating By Parts Oct 22 '15 at 17:01
@Trial: Thx, corrected the mistake. – jibo Oct 23 '15 at 14:28

Fourier transform step-by-step example: $f(x) = 1/2$ where $x\in[0,1]$ otherwise $f(x) = 0$

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