{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $f \\colon [0, \\infty) \\to \\C$ be a function, let $T \u003e 0$ be a sampling period, and let $m$ be a real number with $0 \\le m \\le T$.\nThe *advanced Z-transform* of $f$ is\n$$F(z, m) = \\sum_{k = 0}^{\\infty} f(kT + m) \\, z^{-k},$$\nfor the complex $z$ where the series converges absolutely.\nSo $F(\\cdot, m)$ is the [[unilateral Z-transform|EPG6JPVJE7]] of the sequence $(f(kT + m))_{k \\ge 0}$.\n\n## Remarks\n\nFor $m = 0$ it is the ordinary [[Z-transform|5FY0JQGHYZ]] of the samples $f(kT)$.\nExample: for $f(t) = e^{-at}$ with $a \\in \\C$, $F(z, m) = e^{-am} \\sum_{k \\ge 0} (e^{-aT} z^{-1})^k = e^{-am} \\, z/(z - e^{-aT})$ for $\\abs{z} \u003e \\abs{e^{-aT}}$.\nFor $f(t) = t$, $F(z, m) = \\sum_{k \\ge 0} (kT + m) z^{-k} = Tz/(z - 1)^2 + m z/(z - 1)$ for $\\abs{z} \u003e 1$.\nNon-example: the sum $\\sum_k f(k T + m) z^{-k}$ with $m$ outside $[0, T]$ is not covered; for $m = T$ it is not a new transform either, since $F(z, T) = z (F(z, 0) - f(0))$.\nSome texts call a closely related transform, with a delay parameter in place of the advance, the modified Z-transform.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "K0V6T18367",
    "kind": "definition",
    "msc": [
      "44A55",
      "93C57"
    ],
    "name": "Advanced Z-transform",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Advanced_z-transform"
      }
    ],
    "tag": "759JMPAZCZ",
    "uses": [
      "EPG6JPVJE7",
      "5FY0JQGHYZ"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/759JMPAZCZ/",
  "links": [
    {
      "field": "defines",
      "target": "K0V6T18367"
    },
    {
      "field": "uses",
      "target": "EPG6JPVJE7"
    },
    {
      "field": "uses",
      "target": "5FY0JQGHYZ"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/759JMPAZCZ.md",
  "msc": [
    "44A55",
    "93C57"
  ],
  "name": "Advanced Z-transform",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/759/759JMPAZCZ.md",
  "status": null,
  "tag": "759JMPAZCZ",
  "uri": "m2://759JMPAZCZ",
  "url": "https://tamnd.github.io/math2/wiki/advanced-z-transform-definition/",
  "writing": {
    "tag": "759JMPAZCZ",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}