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  "body": "\n## Summary\n\nThe starred transform is the [[Laplace transform|0V49JEZW06]] of the impulse-sampled version of a continuous-time signal.\nFor a sampling period $T$ it equals $X^*(s) = \\sum_{n \\ge 0} x(nT) e^{-nTs}$.\nIt is the [[Z-transform|5FY0JQGHYZ]] of the samples with $z = e^{sT}$, and so it links the continuous-time analysis of a system with its discrete-time model.\nIt is used in the theory of sampled-data control systems.\n\n## Explanations\n\n### L1 human\n\nA digital controller sees a continuous signal only at the instants $0, T, 2T, \\ldots$.\nOne models this by a train of spikes at those instants, each spike weighted by the value of the signal there.\nThe starred transform is the Laplace transform of that spike train.\nFor $x(t) = 1$ it is $\\sum_{n \\ge 0} e^{-nTs} = 1/(1 - e^{-Ts})$ for $\\operatorname{Re} s \u003e 0$.\n",
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