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  "body": "\n## Summary\n\nThe multidimensional Z-transform sends a sequence indexed by $\\Z^n$ to a Laurent series in $n$ complex variables $z_1, \\dots, z_n$.\nIt is the discrete analogue of the [[multidimensional Laplace transform|QT3M720TCV]] and the $n$-variable form of the [[Z-transform|5FY0JQGHYZ]].\nIt is used to analyze multidimensional digital filters, for example filters acting on images.\nIts region of convergence depends only on the moduli $\\abs{z_j}$.\nIts 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.\n\n## Explanations\n\n### L1 human\n\nA digital image is a table of numbers $x[k_1, k_2]$ indexed by two integers.\nThe 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 $z_1^{-k_1} z_2^{-k_2}$.\nShifting the image becomes multiplying the function by a monomial, and filtering becomes multiplying two functions.\nFor example, the table with a single $1$ at position $(1, 2)$ has the transform $z_1^{-1} z_2^{-2}$.\n",
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