Z-Transform

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

A delay of one sample is multiplication by $z^{-1}$. A difference equation such as

$y[n] = 0.9\,y[n-1] + 0.1\,x[n]$

becomes the transfer function $H(z) = \dfrac{0.1}{1 - 0.9z^{-1}}$. A causal system is stable when all poles lie inside the unit circle $|z| < 1$. The frequency response is $H(e^{j\omega})$.