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  "body": "\n## Definition\n\nLet $x = (x_n)_{n \\in \\Z}$ be a [[sequence|0G9EKM0E15]] of complex numbers.\nThe *Z-transform* of $x$ is the [[Laurent series|0GXMPDNYX3]]\n$$X(z) = \\mathcal{Z}\\{x\\}(z) = \\sum_{n = -\\infty}^{\\infty} x_n z^{-n}.$$\nIt is regarded as a function of $z$ on the set of nonzero complex numbers $z$ where the series converges absolutely.\nFor a sequence $(x_n)_{n \\ge 0}$ indexed by the natural numbers, one sets $x_n = 0$ for $n \u003c 0$, and the transform is $X(z) = \\sum_{n = 0}^{\\infty} x_n z^{-n}$.\n\n## Remarks\n\nExample: for $x_n = a^n$ when $n \\ge 0$ and $x_n = 0$ when $n \u003c 0$, with $a \\in \\C$, the series is geometric with ratio $a z^{-1}$.\nIt converges absolutely exactly when $\\abs{z} \u003e \\abs{a}$, and then $X(z) = z/(z - a)$.\nNon-example: for $x_n = 1$ for all $n \\in \\Z$, the series $\\sum_{n \\in \\Z} z^{-n}$ converges for no $z$.\nThe part with $n \\ge 0$ needs $\\abs{z} \u003e 1$, and the part with $n \u003c 0$ needs $\\abs{z} \u003c 1$, so the sequence has no Z-transform as a function.\nThe sum over all $n \\in \\Z$ is the bilateral form, and the sum over $n \\ge 0$ is the unilateral form.\nA function $X(z)$ alone does not determine $x$: the set of convergence is part of the data.\nSubstituting $z = e^{i\\omega}$ gives the discrete-time Fourier transform, when the unit circle lies in the set of convergence.\n",
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