Chapitre 5

A system is considered stable if a bounded input corresponds to a bounded output.

Let h(n) be the impulse response of a discrete system, its transfer function is given by

\(H\left(Z\right)=\sum_{n=-\infty}^{+\infty}{h(n)Z^{-n}}\) .

The system is considered stable if\( \sum_{n=-\infty}^{+\infty}{\left|h(n)Z^{-n}\right|<\infty}\)

Consider a stable system :

\(\left|H(Z)\right|=\left|\sum_{n=-\infty}^{+\infty}{h(n)Z^{-n}}\right|\le\sum_{n=-\infty}^{+\infty}\left|h(n)\right|\left|Z^{-n}\right|\) (31)

When we evaluate this inequality on the unit circle :

\(\left|Z\right|<1\) ,\(H\left(Z\right)\le\sum_{n=-\infty}^{+\infty}{\left|h(n)\right|<\infty}\)

From this result, we deduce that if a system is stable, its ROC necessarily contains the unit circle.

• For a causal system, ⇒ ROC : \(\left|Z\right|>R\) (outside of a disc)

• For a stable system, ⇒ ROC : \(R<1\)

• For a causal system to be stable, it is necessary : \(Z>R<1\)

Noted :

A stable system does not necessarily mean that it is causal.

For a system to be both stable and causal, the poles must be inside the unit circle.