In Z transform in DSP that is a necessary and sufficient condition for the system to be BIBO stable is given as below. This article gives information about Z transform stability to better understand this topic.
n=∞
∑ |h(n)| < ∞
n=-∞
n=∞
H(Z) = ∑ h(n) z-n
n=-∞
n=∞
|H(Z)| =| ∑ h(n) z-n |
n=-∞
The magnitude of the overall sum is less than the sum of the magnitude of individual terms
n=∞
|H(Z)| < ∑ |h(n)| | z-n |
n=-∞
If H(z) is evaluated on the unit circle, then | z-n | = 1
n=∞
|H(Z)| < ∑ |h(n)|
n=-∞
If the system is BIBO stable, then
n=∞
∑ |h(n)|
n=-∞
|H(z)| < ∞
This condition requires that the unit circle should be present in the ROC full form of H(Z).