---
tag: A7NF067M7N
kind: concept
name: Multidimensional Z-transform
msc: [44A55, 44A30]
specializes: [4PVP1C3JZG, 5FY0JQGHYZ]
sources:
  - url: https://en.wikipedia.org/wiki/Multidimensional_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|QT3M720TCV]] and the $n$-variable form of the [[Z-transform|5FY0JQGHYZ]].
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.

## Explanations

### L1 human

A digital image is a table of numbers $x[k_1, k_2]$ indexed by two integers.
The two-dimensional Z-transform packs the whole table into one function of two complex variables, with $x[k_1, k_2]$ as the coefficient of $z_1^{-k_1} z_2^{-k_2}$.
Shifting the image becomes multiplying the function by a monomial, and filtering becomes multiplying two functions.
For example, the table with a single $1$ at position $(1, 2)$ has the transform $z_1^{-1} z_2^{-2}$.