Pole on the unit circle

1. Can pole be on the unit circle?
2. What happens when pole is on unit circle?
3. What is pole and ROC?
4. Can poles be outside the unit circle?

Can pole be on the unit circle?

Therefore, all poles must be inside the unit circle for stability. Isolated poles on the unit circle may be called marginally stable. The impulse response component corresponding to a single pole on the unit circle never decays, but neither does it grow.

What happens when pole is on unit circle?

a pole closer to the unit circle will on one hand mean an "stronger" presence of a certain frequency it won't be decaying exponentially. But on the other hand a pole on the circle would mean a marginally stable system.

What is pole and ROC?

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. If x[n] is a finite-duration sequence, then the ROC is the entire z-plane, except possibly z=0 or |z|=∞.

Can poles be outside the unit circle?

If there is just one pole outside of the unit circle, that function in the time domain is going to exponentially increase and overpower any other exponential decays from the poles (if any) inside of the unit circle.