Calculate the Fourier Transform

Question

I'm stuck calculating the Fourier Transform of the following (periodic) signal and grateful for any help:

First, I calculated the complex Fourier Series $x_p(t) = \sum_{n=-\infty}^{+\infty} {\frac{U_{0}\sin(n\pi T_{1})}{n\pi} e^{iw_{0}nt}}$ to simplify things.

Now I start to calculate the Fourier Transform $$X_p(w)=\int_{-\infty}^{+ \infty} {x_p(t)\cdot e^{-iw_0t}dt} =$$

$$ \int_{-\infty}^{+ \infty} {\sum_{n=-\infty}^{+\infty} {\frac{U_{0}\sin(n\pi T_{1})}{n\pi} e^{iw_{0}nt}}\cdot e^{-iw_0t}dt} =$$

$$ \int_{-\infty}^{+ \infty} {\sum_{n=-\infty}^{+\infty} {\frac{U_{0}\sin(n\pi T_{1})}{n\pi} e^{iw_{0}nt-iw_0t}}dt} = $$

$$ U_{0} \cdot\int_{-\infty}^{+ \infty} {\sum_{n=-\infty}^{+\infty} {\frac{\sin(n\pi T_{1})}{n\pi}}e^{iw_0t(n-1)}dt} = ?$$

Now I think we could interchange summation and integration, but that doesn't help me much further

Answer 1

Hint: Start with computing the Fourier transform of the Rectangle function and then revisit the Sinc() function.