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  "body": "\n## Summary\n\nThe advanced Z-transform is a variant of the [[Z-transform|5FY0JQGHYZ]] of sampled signals.\nIt samples a continuous-time signal at the instants $kT + m$, shifted by a part $m$ of the sampling period $T$.\nIt is $F(z, m) = \\sum_{k \\ge 0} f(kT + m) z^{-k}$, with $0 \\le m \\le T$.\nIt lets one describe delays that are not whole multiples of the sampling period, and the behaviour of a sampled system between sampling instants.\nFor fixed $m$ it has the usual properties of the Z-transform.\n\n## Explanations\n\n### L1 human\n\nSampling a signal at $0, T, 2T, \\ldots$ only shows it at those instants.\nThe advanced Z-transform samples it at $m, T + m, 2T + m, \\ldots$ instead, for a chosen offset $m$.\nLetting $m$ run from $0$ to $T$ shows the signal at every time.\nFor $f(t) = e^{-at}$ one gets $F(z, m) = e^{-am} z/(z - e^{-aT})$.\n",
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