# Computing Fourier Series coefficients

Hello I have to calculate the Fourier series coefficients for the following function:

$$f(t)=\sum_{n=-\infty}^{+\infty} \Pi(\dfrac{t-nT_o}{T_o/2})$$

where "$\Pi$" indicates the rectangular function.

I know that the Fourier series are in the form:

$f(t)\simeq \frac{a_o}{2}+\sum_{n=1}^{+\infty} [a_n \cos(\dfrac{2n\pi t}{T}) + b_n \sin (\dfrac{2n\pi t}{T})]$

And each coefficient is defined as:

$a_o=\dfrac{2}{T} \int_{-T/2}^{T/2} f(t)$

$a_n = \dfrac{2}{T} \int_{-T/2}^{T/2} f(t)\cos(\dfrac{2n\pi t}{T})$

$b_n= \dfrac{2}{T} \int_{-T/2}^{T/2} f(t)\sin(\dfrac{2n\pi t}{T})$

I have been given the following indications in order to compute these coefficients (if I don't want to use the formulas above):

i) $TF\{\Pi(\dfrac{t}{T/2})\}=\dfrac{2}{\omega}\sin(\dfrac{\omega T}{4})$

ii) $x(t) = \sum_{n=-\infty}^{\infty} x_b (t-nT) = x_b \star \sum_{n=-\infty}^{\infty} \delta(t-nT)$

iii) $TF\{\sum_{n=-\infty}^{\infty} \delta (t-nT) = \sum_{n=-\infty}^{\infty} e^{-jnT\omega} = \dfrac{2\pi}{T} \sum_{n=-\infty}^{\infty} \delta (\omega-n\dfrac{2 \pi n}{T})$

I know how to do it by using the coefficients equations. My question is how to compute the coefficients using indications (i),(ii),(iii).

## migrated from engineering.stackexchange.comMay 1 '15 at 20:03

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• TF= Fourier Transform? – Paul May 1 '15 at 19:54
• Yes, it is the Fourier Transform. – user3780731 May 1 '15 at 19:59