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  "body": "\n## Definition\n\nLet $x \\colon [0, \\infty) \\to \\C$ be a function and let $T \u003e 0$ be a sampling period.\nThe impulse-sampled signal is\n$$x^*(t) = \\sum_{n = 0}^{\\infty} x(nT) \\, \\delta(t - nT),$$\nwhere $\\delta$ is the [[Dirac delta|0KPHENZ5S7]].\nThe *starred transform* of $x$ is the [[Laplace transform|0V49JEZW06]] of $x^*$:\n$$X^*(s) = \\sum_{n = 0}^{\\infty} x(nT) \\, e^{-nTs},$$\nfor the complex $s$ where this series converges absolutely.\n\n## Remarks\n\nThe impulse-sampled signal is $x$ times a [[Dirac comb|7KHGQFM8A9]] of period $T$, restricted to $t \\ge 0$.\nIf $X(z)$ is the [[unilateral Z-transform|EPG6JPVJE7]] of the samples $x_n = x(nT)$, then $X^*(s) = X(e^{sT})$.\nExample: for $x(t) = e^{-at}$ with $a \\in \\C$, $X^*(s) = \\sum_{n \\ge 0} e^{-n T (s + a)} = 1/(1 - e^{-T(s + a)})$ for $\\operatorname{Re}(s + a) \u003e 0$.\nNon-example: for $x(t) = e^{t^2}$ the terms $e^{n^2 T^2 - nTs}$ do not tend to $0$ for any $s$, so the series converges nowhere and $x$ has no starred transform.\n$X^*$ is periodic in the imaginary direction: $X^*(s + 2\\pi i/T) = X^*(s)$, since $e^{-nT \\cdot 2\\pi i/T} = 1$.\n",
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