The z-transform extends the discrete-time Fourier transform by replacing the unit-magnitude complex exponential with a general complex variable $z$.

For a sequence $x[n]$,

$$X(z)=\sum_{n=-\infty}^{\infty}x[n]z^{-n}.$$

The transform converges only for some values of $z$. This set is the region of convergence, or ROC, and is part of the transform description.

The DTFT is obtained by evaluating the z-transform on the unit circle when the ROC includes it:

$$z=e^{i\omega}.$$

Time shifts become powers of $z$:

$$x[n-k]\quad\longleftrightarrow\quad z^{-k}X(z).$$

Convolution becomes multiplication:

$$x*h\quad\longleftrightarrow\quad X(z)H(z).$$

These properties turn linear difference equations into algebraic equations and make the z-transform the natural transform-domain tool for discrete-time LTI systems.