ROC of z-transform
There are 5 types of ROC.
1. The ROC is bounded by two poles. ROC does not include the poles.
2. The ROC exists $ |z| < a $.
3. The ROC exists $ |z| > a $.
4. The ROC is whole plane, but not including infinity.
5. The ROC is whole plane and including infinity.
The properties of ROC
1. If the z transform is rational, i.e $ X(z) = \frac{P(z)}{Q(z)}, then the ROC does not contain any poles. X(z) can have a pole at infinity. To detect a pole at infinity, replace z by <math> \frac{1}{z} $ and consider what happens
when z goes to 0. if X(1/z) goes to infinity when z goes to 0, X(z) has one pole at infinity.
So, the ROC is finite plane but not including infinity.
2. If x[n] is right sided and if |z| = r is in the ROC, then all finite values for which |z| > r
are also in the ROC.
3. If x[n] is left sided and if |z| = r is in the ROC, then all finite values for which 0 < |z| <r
are also in the ROC.
4. If x[n] is two sided and if the cirle |z| = r is in the ROC, then the ROC will consist of a ring
in the z-plane that include the cirle |z| = r.
5. If X(z) is rational, the ROC is bounded by poles or extends to infinity.
6. If x[n] is causal, the ROC also includes z = infinity.
7. If x[n] is anticausal, the ROC also includes z = 0.
