ROC of Z transform Doesnt include a pole on the boundary?

1. Why the ROC of z-transform can not contain any pole?
2. Does ROC have both poles and zeros?
3. What are the properties of ROC for z-transform?
4. What are poles in z-transform?

Why the ROC of z-transform can not contain any pole?

The ROC cannot contain any poles.

By definition a pole is a where X(z) is infinite. Since X(z) must be finite for all z for convergence, there cannot be a pole in the ROC.

Does ROC have both poles and zeros?

The ROC cannot contain a Pole, since at a pole H(z) is infinite by definition and hence does not converge. For a causal system (impulse response h(n) is zero for n< 0), the ROC is the exterior of a circle, including ¥. Further, for a system to be stable, its impulse response must be absolutely summable.

What are the properties of ROC for z-transform?

Properties of ROC of Z-Transform

The ROC of the Z-transform cannot contain any poles. The ROC of Z-transform of an LTI stable system contains the unit circle. The ROC of Z-transform must be connected region. When the Ztransform X(z) is a rational, then its ROC is bounded by poles or extends up to infinity.

What are poles in z-transform?

Introduction to Poles and Zeros of the Z-Transform

The two polynomials, P(z) and Q(z), allow us to find the poles and zeros of the Z-Transform. The value(s) for z where P(z)=0. The complex frequencies that make the overall gain of the filter transfer function zero.