Properties of the ROC for the z-Transform :

1. X(z) converges uniformly if and only if the ROC of the z-transform X(z) of the sequence includes the unit circle. The ROC of X(z) consists of a ring in the z-plane centered about the origin. That is, the ROC of the z-transform of x(n) has values of z for which x(n) r-n is absolutely summable.
2. The ROC does not contain any poles.
3. When x(n) is of finite duration then the ROC is the entire z-plane, except POSSIBLY z=0 and/or z=infinity.
4. If x(n) is a right sided sequence, the ROC will not include infinity.
5. If x(n) is a left sided sequence, the ROC will not include z=0. HOWEVER if x(n)=0 for all n>0, the ROC will include z=0.
6. If x(n) is two sided and if the circle |z| = r0 is in the ROC, then the ROC will consist of a ring in the z-plane that includes the circle |z|=r0.
7. If X(z) is rational, then the ROC extends to infinity, i.e. the ROC is bounded by poles.
8. If x(n) is causal, then the ROC includes z=infinity.
9. If x(n) is anti- causal, trhen the ROC includes z=0.

Properties of the ROC for the z-Transform :