Absolutely summable signals Given an absolutely summable signal $x[n]$, the $z$-transform $X^z(z)$ is rational with a pole at $z=0.5$.

Given the following the statements:


$x[n]$ has a finite support in the time domain.
$x[n]$ is a left sided signal.
$x[n]$ is a right sided signal.
$x[n]$ is a two sided signal.


Can anyone please tell which statements are true and explain why or why not?
Absolute summability implies that the Fourier transform converges uniformly, so this means that the unit circle is contained in the region of convergence (ROC).

Now, the pole at 0.5 is the limit of the ROC, so the ROC is from 0.5 outwards (in order to include the unit circle). This means that the signal is right-sided.

Finite support signals cannot have poles (other than $z=0$ or $z=\infty$).