Let $f(t)$ be a periodic signal of period $1$, i.e. $f(t+1)=f(t) \forall t$. Define the averaging operator depending on a fixed parameter $h>0$ as below:
\[g(x)=\frac{1}{2 h} \int_{x-h}^{x+h} f(t) d t .\]
Which of the following is $\text{TRUE}$ for the new signal $g(x)$?
- $g(x)$ is aperiodic
- $g(x)$ is periodic with period $\frac{1}{2}$
- $g(x)$ is periodic with period $1$
- The value of $h$ determines whether or not $g(x)$ is periodic
- None of the above