Z Transform

An Integral Transform, that is the Laplace Transform but Discrete-Time.

$$X(z)=\mathcal{Z}\{x[n]\}(z)=\sum _{n=0}^{\infty }x[n]z^{-n}$$

Related to the Laplace Transform by $z=e^{sT}$, where $T$ is the sampling interval. This means the imaginary axis in the $s$-plane maps to the unit circle $|z|=1$ in the $z$-plane.

The Z Transform converts Discrete Convolution into multiplication, turning difference equations into polynomial equations in $z$, analogous to how the Laplace Transform handles differential equations.

Stability

Poles of $H(z)$ inside the unit circle ($|z|<1$) correspond to a Stable system, analogous to $\text{Re}(s)<0$ for the Laplace Transform.