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  "body": "\n## Summary\n\nThe bilateral Z-transform of a sequence $(x_n)_{n \\in \\Z}$ sums over all integers: $X(z) = \\sum_{n \\in \\Z} x_n z^{-n}$.\nIt is the form of the [[Z-transform|5FY0JQGHYZ]] for signals that extend into the past as well as the future.\nIts set of convergence is an annulus centred at $0$, possibly empty.\nThe same formula for $X(z)$ can come from different sequences on different annuli.\nIt is the discrete analogue of the two-sided Laplace transform.\n\n## Explanations\n\n### L1 human\n\nThe bilateral transform looks at a sequence in both directions of time.\nTerms with $n \\ge 0$ need $\\abs{z}$ large to converge, and terms with $n \u003c 0$ need $\\abs{z}$ small.\nSo the transform lives on a ring between two circles.\nFor example, $x_n = 2^{-\\abs{n}}$ has transform defined for $1/2 \u003c \\abs{z} \u003c 2$.\n",
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