{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "PJN54XYB5T"
    }
  ],
  "body": "\n## Summary\n\nThe fractional Fourier transform is a family of unitary operators $\\mathcal{F}_\\alpha$ on $L^2(\\R)$, indexed by an angle $\\alpha$.\nHere $\\mathcal{F}_0$ is the identity and $\\mathcal{F}_{\\pi/2}$ is the [[Fourier transform|DPVM34ADZV]].\nThe family is a group: $\\mathcal{F}_\\alpha \\mathcal{F}_\\beta = \\mathcal{F}_{\\alpha + \\beta}$, so the Fourier transform has a square root, a cube root, and so on.\nIn the time-frequency plane, $\\mathcal{F}_\\alpha$ acts as a rotation by the angle $\\alpha$.\nIt is used in optics and signal processing to analyse chirps and other signals whose frequency changes in time.\n\n## Explanations\n\n### L1 human\n\nDoing the Fourier transform four times gives back the original function, so it behaves like a rotation by a quarter turn.\nThe fractional Fourier transform fills in the rotations by all the other angles.\nA signal is then seen from a direction between pure time and pure frequency.\nFor example, the Gaussian $e^{-x^2/2}$ is unchanged by every $\\mathcal{F}_\\alpha$, just as a round disc is unchanged by every rotation.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "42A38",
      "44A15",
      "94A12"
    ],
    "name": "Fractional Fourier transform",
    "same_as": {
      "wikidata": "Q4817582",
      "wikipedia": "Fractional Fourier transform"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Fractional_Fourier_transform"
      }
    ],
    "specializes": [
      "74BXAPVSW8"
    ],
    "tag": "28VV812XWQ"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/28VV812XWQ/",
  "links": [
    {
      "field": "specializes",
      "target": "74BXAPVSW8"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/28VV812XWQ.md",
  "msc": [
    "42A38",
    "44A15",
    "94A12"
  ],
  "name": "Fractional Fourier transform",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/28V/28VV812XWQ.md",
  "status": null,
  "tag": "28VV812XWQ",
  "uri": "m2://28VV812XWQ",
  "url": "https://tamnd.github.io/math2/wiki/fractional-fourier-transform/",
  "writing": {
    "tag": "28VV812XWQ",
    "kind": "concept",
    "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
      }
    ]
  }
}