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).