The z-transform is a mathematical technique used to analyze discrete-time signals and systems in the frequency domain. Similarly to the [[discrete-time Fourier transform (DTFT)]], it converts time-domain sequences of discrete samples (a [[discrete-time signal]]) into complex numbers, which can be used to represent the frequency response of a system or signal. The z-transform is a generalization of the [[discrete-time Fourier transform (DTFT)]] that converges for a larger class of signals. The z-transform is defined as $ X(z) = \sum_{n = -\infty}^{+\infty} x(n) z^{-n} $ where $z \in \mathbb{C}$. If $x(n)$ and $X(z)$ are a z-transform pair, then: $ x(n) \xrightarrow[\cal ZT]{} X(z) $ The [[discrete-time Fourier transform (DTFT)]] can be obtained from the z-transform by restricting the variable $z$ to the unit circle $z=e^{j\omega}$ ($|z|=1$): $ X(e^{j \omega}) = \sum_{n=-\infty}^{+\infty} x(n) e^{-j \omega n} $ (see [[relation between the z-transform and the DTFT]])