How can I calculate the DTFT of the following signal with properties? I suspect I would obtain a pulse, but I can not figure out how to calculate it.
$x[n]$ for n=0,1...,9.
$$x[n]=\frac {\sin (\frac{3\pi n}{10})} {\sin (\frac{\pi n}{10})}$$ Elsewhere $$x[n]=0$$
A step-by-step solution would be helpful.
Using the geometric series formula, you can establish the Dirichlet kernel formula: For any $x\in\mathbb R$ and $p\in\mathbb N$, $$\frac{\sin\left(\left(p+\frac 1 2\right)x\right)}{\sin\left(\frac x 2\right)}=\sum_{k=-p}^p e^{ikx}$$ So applying this with $p=1$ and $x=\frac {\pi n} 5$ $$\frac{\sin\left(\frac{3\pi n}{10}\right)}{\sin\left(\frac {\pi n}{10}\right)}=\sum_{k=-1}^1 e^{ik\frac{\pi n}5} = 1 + e^{-in\frac{\pi}5} + e^{in\frac{\pi}5}$$
Now applying the formula for the DTFT: $$\begin{split} X(\omega) &= \sum_{n=-\infty}^{+\infty}x[n]e^{-i\omega n}\\ &=\left(\sum_{n=0}^9e^{-i\omega n}\right) + \left(\sum_{n=0}^9e^{-i n(\frac {\pi}{5} +\omega)}\right) +\left(\sum_{n=0}^9e^{i n(\frac {\pi}{5} -\omega)}\right) \end{split}$$ Can you finish?