Help understanding a Fourier Transform question

• Jan 5th 2011, 12:26 PM
craig
Help understanding a Fourier Transform question
Calculate the Fourier Transform of $g(x)$, where:

$g(x) =
\begin{cases}
x & 0 \le x \le 1 \\
2-x & 1 \le x \le 2 \\
0 & \text{otherwise}
\end{cases}
$

The formula that I've got is the Fourier transform $\hat{f}(s)$ of a function $f(x)$ is a function defined by:

$\hat{f}(s) := \left\int^{\infty}_{-\infty}\right f(x) e^{-isx} dx$

My question is, how would I go about calculating the Fourier Transform for the above function? Which value of $f(x)$ would I use, and which limits?

• Jan 5th 2011, 12:35 PM
craig
Just had a thought. Would I calculate the two integrals separately and then add them together?

Also if this is the case, is this something you can always do?

Thanks again :)
• Jan 5th 2011, 12:38 PM
Chris L T521
Quote:

Originally Posted by craig
Calculate the Fourier Transform of $g(x)$, where:

$g(x) =
\begin{cases}
x & 0 \le x \le 1 \\
2-x & 1 \le x \le 2 \\
0 & \text{otherwise}
\end{cases}
$

The formula that I've got is the Fourier transform $\hat{f}(s)$ of a function $f(x)$ is a function defined by:

$\hat{f}(s) := \left\int^{\infty}_{-\infty}\right f(x) e^{-isx} dx$

My question is, how would I go about calculating the Fourier Transform for the above function? Which value of $f(x)$ would I use, and which limits?

I'm pretty sure that it would be safe to say that $\hat{g}(s) = \displaystyle\int_0^1 xe^{-isx}\,dx+\int_1^2(2-x)e^{-isx}\,dx$ due to the piecewise definition of g(x).

I hope this helps!
• Jan 5th 2011, 12:43 PM
craig
Quote:

Originally Posted by Chris L T521
I'm pretty sure that it would be safe to say that $\hat{g}(s) = \displaystyle\int_0^1 xe^{-isx}\,dx+\int_1^2(2-x)e^{-isx}\,dx$ due to the piecewise definition of g(x).

I hope this helps!

So I'm guessing that this would apply for any function defined like this?
• Jan 5th 2011, 12:47 PM
Chris L T521
Quote:

Originally Posted by craig

So I'm guessing that this would apply for any function defined like this?

I want to say that the function should be piecewise continuous in order to do this; I'm pretty sure this is the case, but I'm not 100% sure.
• Jan 5th 2011, 12:52 PM
Drexel28
Quote:

Originally Posted by Chris L T521
I want to say that the function should be piecewise continuous in order to do this; I'm pretty sure this is the case, but I'm not 100% sure.

This is true for any integrable function.
• Jan 5th 2011, 12:55 PM
craig
Thankyou :)