Multidimensional Z-transform

Summary

The multidimensional Z-transform sends a sequence indexed by $\Z^n$ to a Laurent series in $n$ complex variables $z_1, \dots, z_n$. It is the discrete analogue of the multidimensional Laplace transform and the $n$-variable form of the Z-transform. It is used to analyze multidimensional digital filters, for example filters acting on images. Its region of convergence depends only on the moduli $\abs{z_j}$. Its image under $z \mapsto (\log\abs{z_1}, \dots, \log\abs{z_n})$ is convex, and it plays the role of the annulus of the one-variable case.