Bilateral Z-transform

Summary

The bilateral Z-transform of a sequence $(x_n)_{n \in \Z}$ sums over all integers: $X(z) = \sum_{n \in \Z} x_n z^{-n}$. It is the form of the Z-transform for signals that extend into the past as well as the future. Its set of convergence is an annulus centred at $0$, possibly empty. The same formula for $X(z)$ can come from different sequences on different annuli. It is the discrete analogue of the two-sided Laplace transform.