Help understanding a Fourier Transform question

**craig** · January 5th 2011, 11:26 AM

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?

Thanks in advance.

**craig** · January 5th 2011, 11:35 AM

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

**Chris L T521** · January 5th 2011, 11:38 AM

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?

Thanks in advance.

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!

**craig** · January 5th 2011, 11:43 AM

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!

Hi thanks for the reply.

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

**Chris L T521** · January 5th 2011, 11:47 AM

Originally Posted by craig

Hi thanks for the reply.

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.

**Drexel28** · January 5th 2011, 11:52 AM

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.

**craig** · January 5th 2011, 11:55 AM

Thankyou

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