The z-transform exists when the infinite sum
converges.
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.