{
  "aliases": null,
  "backlinks": [
    {
      "field": "specializes",
      "target": "3DBAMXA80B"
    },
    {
      "field": "specializes",
      "target": "A7NF067M7N"
    },
    {
      "field": "specializes",
      "target": "EPG6JPVJE7"
    },
    {
      "field": "specializes",
      "target": "K0V6T18367"
    },
    {
      "field": "uses",
      "target": "759JMPAZCZ"
    },
    {
      "field": "uses",
      "target": "8GRA22YEMB"
    },
    {
      "field": "uses",
      "target": "FZ6KV2J5C9"
    },
    {
      "field": "uses",
      "target": "J0SV0PCDBW"
    },
    {
      "field": "uses",
      "target": "SKMQ02FP9J"
    },
    {
      "field": "uses",
      "target": "T9PRG9QP8Y"
    },
    {
      "field": "about",
      "target": "FFR8NPWG64"
    },
    {
      "field": "about",
      "target": "JWH2KYCGBQ"
    },
    {
      "field": "about",
      "target": "PH61T3B7W7"
    },
    {
      "field": "defines",
      "target": "V1WRSW2AA6"
    }
  ],
  "body": "\n## Summary\n\nThe Z-transform turns a sequence $(x_n)$ of complex numbers into the function $X(z) = \\sum_n x_n z^{-n}$ of a complex variable $z$, defined where the series converges.\nIt is the discrete analogue of the [[Laplace transform|0V49JEZW06]].\nIt turns shifts of sequences into multiplication by powers of $z$ and convolution into products.\nSo it is the main tool for linear difference equations and for discrete-time signals and control systems.\n\n## Explanations\n\n### L1 human\n\nA sequence is a list of numbers $x_0, x_1, x_2, \\ldots$, such as the samples of a signal.\nThe Z-transform packs the whole list into one function, using $z^{-n}$ as a label for position $n$.\nDelaying the list by one step then just multiplies the function by $z^{-1}$.\nFor the constant list $1, 1, 1, \\ldots$ the transform is $\\sum_{n \\ge 0} z^{-n} = z/(z - 1)$ for $\\abs{z} \u003e 1$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "44A55",
      "39A12",
      "93C55"
    ],
    "name": "Z-transform",
    "same_as": {
      "wikidata": "Q135810",
      "wikipedia": "Z-transform"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Z-transform"
      }
    ],
    "tag": "5FY0JQGHYZ"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/5FY0JQGHYZ/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/5FY0JQGHYZ.md",
  "msc": [
    "44A55",
    "39A12",
    "93C55"
  ],
  "name": "Z-transform",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/5FY/5FY0JQGHYZ.md",
  "status": null,
  "tag": "5FY0JQGHYZ",
  "uri": "m2://5FY0JQGHYZ",
  "url": "https://tamnd.github.io/math2/wiki/z-transform/",
  "writing": {
    "tag": "5FY0JQGHYZ",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}