Region of Convergence (RoC) Properties
1.) The RoC of a Z transform consists of rings (or washers) centered about the origin in the z-plane.
2.) If X(z) is rational, aka $ X(z) = \frac {P(z)}{Q(z)} $ ,
Then the RoC does not contain any poles of X(z). A pole is a z that would make Q(z) = 0.
3.) If x[n] is of a finite duration,
Then the RoC either DOES NOT EXIST, or is the entire Z-plane except 0 or $ \infty $
4.) If x[n] is right sided and if |z| = $ r_0 $ is in the RoC,
Then all finite values of z for which |z| > $ r_0 $ are also in the RoC.
5.) If x[n] is left sided and if |z| = $ r_0 $ is in the RoC,
Then all finite values for which 0 < |z| < $ r_0 $ are also in the RoC.
6.) If two-sided, and if the circle |z| = $ r_0 $ is in the RoC,
Then the RoC will consist of a ring (or washer) in the z-plane.
The ring will include the circle |z| = $ r_0 $
7.) If X(z) is rational, aka $ X(z) = \frac {P(z)}{Q(z)} $
Then the RoC is bounded by poles or will extend to $ \infty $
