### 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 $\infty$


## Alumni Liaison

Basic linear algebra uncovers and clarifies very important geometry and algebra.

Dr. Paul Garrett