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  "body": "\n## Summary\n\nThe matched Z-transform method turns a continuous-time filter with a rational transfer function into a discrete-time filter.\nIt sends each [[pole|066E7PJCYE]] and each zero $a$ in the $s$-plane to the point $e^{aT}$ in the $z$-plane, where $T$ is the sampling period.\nThen it fixes the overall gain.\nIt is also called the pole–zero mapping method.\nIt keeps the poles exactly where sampling puts them, and it is simple to apply, but it can distort the frequency response.\n\n## Explanations\n\n### L1 human\n\nA filter for continuous signals is described by its poles and zeros, points of the complex plane.\nTo build a digital version, one moves each of these points with the map $a \\mapsto e^{aT}$, which is how sampling moves exponentials.\nThen one scales the result so that, for example, constant signals pass with the same gain as before.\nFor $H(s) = 1/(s + 1)$ and $T = 0.1$, one gets $H(z) = k/(1 - e^{-0.1} z^{-1})$ with $k = 1 - e^{-0.1}$ for equal gain at $s = 0$, $z = 1$.\n",
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