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).