Z-transform

Summary

The Z-transform turns a sequence $(x_n)$ of complex numbers into the function $X(z) = \sum_n x_n z^{-n}$ of a complex variable $z$, defined where the series converges. It is the discrete analogue of the Laplace transform. It turns shifts of sequences into multiplication by powers of $z$ and convolution into products. So it is the main tool for linear difference equations and for discrete-time signals and control systems.