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  "body": "\n## Definition\n\nLet $T \u003e 0$ be a sampling period.\nLet $H$ be a rational function of $s$ with nonzero leading constant $c$, [[zeros|00GC8ZQP4Q]] $q_1, \\ldots, q_M$ and [[poles|066E7PJCYE]] $p_1, \\ldots, p_N$, listed with multiplicity:\n$$H(s) = c \\, \\frac{\\prod_{i=1}^{M} (s - q_i)}{\\prod_{j=1}^{N} (s - p_j)}.$$\nThe *matched Z-transform* of $H$ is the following rational function of $z$.\n$$H_d(z) = k \\, \\frac{\\prod_{i=1}^{M} (1 - e^{q_i T} z^{-1})}{\\prod_{j=1}^{N} (1 - e^{p_j T} z^{-1})}.$$\nThe constant $k$ is chosen so that $H_d$ and $H$ agree at a chosen reference point.\nThe usual choice is $H_d(1) = H(0)$ when $H(0)$ is finite and nonzero.\n\n## Remarks\n\nExample: $H(s) = a/(s + a)$ with $a \u003e 0$ has one pole $-a$ and $H(0) = 1$.\nIts matched transform is $H_d(z) = k/(1 - e^{-aT} z^{-1})$, and $H_d(1) = 1$ gives $k = 1 - e^{-aT}$.\nNon-example: $H(s) = e^{-s}$, a pure delay, is not a rational function, so the method does not apply to it as stated.\nA pole $p$ with $\\operatorname{Re} p \u003c 0$ goes to $e^{pT}$ with $\\abs{e^{pT}} \u003c 1$, so a stable continuous filter gives a stable discrete one.\nThe map $a \\mapsto e^{aT}$ is the relation $z = e^{sT}$ between the [[Z-transform|5FY0JQGHYZ]] of samples and the Laplace transform.\nSome versions add zeros at $z = -1$ for the zeros of $H$ at infinity when $N \u003e M$.\n",
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