Let $x(t)$ be the sign with Fourier transformation $$X(\omega)=\delta(\omega)+ \delta(\omega-\pi)+\delta(\omega-5)$$
and let $h(t)=u(t)-u(t-2)$.
Is $x(t)$ periodic?
Is the convolution of $x(t)$ with $ h(t)$ a periodic sign?
Can convolution of two non-periodic signs be a periodic sign?
$x(t) = \frac{1}{2 \pi} \int X(\omega) e^{j \omega t} d\omega$
How can I see if it is periodic?
Since $\frac\pi5\not\in\mathbb{Q}$, $$ x(t)=\frac1{2\pi}\left(1+e^{i\pi t}+e^{i5t}\right) $$ is not periodic.
Since $h(t)=u(t)\ast(\delta(t)-\delta(t-2))$ $$ \begin{align} h(x)\ast x(t) &=\frac1{2\pi}\left(1-1+e^{i\pi t}-e^{i\pi(t-2)}+e^{i5t}-e^{i5(t-2)}\right)\ast u(t)\\ &=\frac1{2\pi}\left(1-e^{-10i}\right)e^{i5t}\ast u(t) \end{align} $$ which has period $\frac{2\pi}5$