Z-Plane

The Complex Plane used in Z Transform analysis. Each point $z\in \mathbb{C}$ corresponds to a complex frequency in the discrete-time domain.

Related to the S-Plane by $z=e^{sT}$, where $T$ is the sampling interval. This mapping wraps the $s$-plane onto the $z$-plane periodically.

Stability Regions

Region	Condition	Behavior
Inside unit circle	$\Vert z\Vert <1$	Stable (decaying)
Unit circle	$\Vert z\Vert =1$	Marginally stable (oscillatory)
Outside unit circle	$\Vert z\Vert >1$	Unstable (growing)

The unit circle in the $z$-plane corresponds to the imaginary axis in the S-Plane; evaluating $H(z)$ on the unit circle ($z=e^{j\omega }$) gives the frequency response. Poles of $H(z)$ inside the unit circle are required for asymptotic stability; see Infinite Impulse Response Filter.