The sum may not converge for all values of ‘z’. The value of ‘z’ for which the sum converges is called Region of Convergence (ROC).

**PROPERTIES OF REGION OF CONVERGENCE**

1. The ROC is a concentric ring or a circle in the z-plane centered at the origin.

2. The ROC cannot contain any poles.

3. If x(n) is a finite duration causal sequence, the ROC is entire z-plane except at z=0.

If x(n) is a finite duration anti causal sequence, then the ROC is the entire z-plane except at z=∞.

If x(n) is a finite duration 2-sided sequence, then the ROC will the entire z-plane except at z=0 and z=∞.

4.If x(n) is a right sided sequence and if the circle |z|=r0 is in the ROC, then all finite values of ‘z’ for which |z|>ro will also be in ROC.

5. If x(n) is a left sided sequence and if the circle |z|=r0 is in the ROC, then all values of z for which 0<|z|<ro will be in ROC.

6. If x(n) is a 2-sided sequence and if the circle |z|=r0 is in the ROC, then the ROC will consists of a ring in the z-plane that includes the circle |z|=r0

7. If the z-Transform X(z) of x(n) is rational, then its ROC is bounded by poles or extends to ‘∞’.

8. If the z-Transform X(z) of x(n) is rational and if x(n) is right sided, then ROC is the region in the z-plane outside the outermost pole. In other words, Outside the radius of circle = the largest magnitude of pole of x(z). If x(n) is causal then ROC also includes Z= ∞.

9. If the z-Transform X(z) of x(n) is rational and if x(n) is left sided, then ROC is the region in the z-plane inside the outermost ‘non zero pole’.

In other words, inside the circle of radius = the smallest magnitude of pole of x(z) other than at z=0 and extending inwards to and possibly including z=0. If x(n) is anti-causal, ROC includes z=0.

10.If x(n) is a finite duration 2-sided sequence, then ROC will consists of a circular ring in the z-plane bounded on the interior and exterior by a pole and not containing any pole.

11. The ROC of an LTI (Linear Time Invariant) system contains the unit circle.

12. ROC must be a connected region.