Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Can I apply the Fourier Transform to a Fourier series?

share|improve this question
    
What have you tried so far?: What have you tried so far? –  draks ... Jul 1 '12 at 13:34
    
I have a Fourier series that belongs to a bipolar pulse (square wave). I have to filter that signal through an ideal low pass filter, h(f), with cutoff frequency at 4KHz. At the beginning I thought the Fourier series was the "same" as the Fourier Transform but then I realize that the Fourier series is x(t) not x(f). Then I started to think that maybe I should apply the Fourier transform but I am not sure if I should do that because as I said I thought the Fourier series was the same as Fourier transform. –  marina Jul 1 '12 at 13:43
    
Now I am confuse. What I need to know if they are equally. @dranks –  marina Jul 1 '12 at 13:47
2  
@marina: Edit the question and put that comment in. –  KennyTM Jul 1 '12 at 15:18

1 Answer 1

Let's assume your square-wave Fourier series looks like this:

$$ \begin{align} x_{\mathrm{square}}(t) & {} = \frac{4}{\pi} \sum_{k=1}^M{\sin{\left ((2k-1) 2\pi ft \right )}\over(2k-1)} \\ & {} = \frac{4}{\pi}\left (\sin(2\pi ft) + {1\over3}\sin(6\pi ft) + {1\over5}\sin(10\pi ft) + \cdots\right ) \end{align} \tag{1} $$

When you apply the Fourier Transform to this, you use the property of linearity:

For any complex numbers $a_n$, if $h(x)=\sum_n a_n\cdot f_n(x)$,  then $ \hat{h}(\xi)=\sum_n a_n\cdot \hat{f_n}(\xi) $,

so the FT of ${\sin{\left ((2k-1) 2\pi ft \right )}\over(2k-1)}$ is

$$ \mathcal{F}_t\left[{\sin{\left ((2k-1) 2\pi ft \right )}\over(2k-1)}\right](\omega)= i\frac{ \sqrt{\pi/2} \delta\left(\omega-f \pi (2k-1)\right)}{(2k-1)}-i\frac{ \sqrt{\pi/2} \delta(\omega+f \pi (2k-1))}{(2k-1)}, $$ with $\delta(\cdot)$ being the Dirac $\delta$ function. Now you can cut the frequencies above your threshold, but you might have done this in $(1)$ already, so there is actual need to transform it.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.