A linear dynamic system where a bounded input yields a bounded zero-state response. More precisely, let be a bounded-input with as the least upper bound (i.e., there is a fixed finite constant such that for every t or k), if there exists a scalar such that for every t (or k), the output satisfies, then the system is said to be bounded-input bounded-output stable.
For a rational and discrete time system, the condition for stability is that the region of convergence (ROC) of the z-transform includes the unit circle. When the system is causal, the ROC is the open region outside a circle whose radius is the magnitude of the pole with largest magnitude. Therefore, all poles of the system must be inside the unit circle in the z-plane for BIBO stability.