{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "HM74NCW5JX"
    }
  ],
  "body": "\n## Summary\n\nProny's method recovers a sum of exponentials $f(k) = \\sum_{j=1}^{n} c_j z_j^k$ from equally spaced samples.\nIt uses the fact that such samples satisfy a [[linear recurrence|0NDVHBVBXB]] whose characteristic polynomial has the roots $z_j$.\nOne solves a [[Hankel|PQ1D8FQB3E]] linear system for the recurrence, finds the roots, and then solves a linear system for the amplitudes $c_j$.\nThe method underlies many techniques of signal processing and spectral estimation.\nIt is close to rational approximation, since the generating function $\\sum_k f(k) w^k$ of such samples is rational.\n\n## Explanations\n\n### L1 human\n\nA signal made of a few decaying or oscillating exponentials has a hidden regularity: each sample is a fixed combination of the previous few.\nProny's method first finds that combination from the data.\nThe roots of the associated polynomial reveal the rates and frequencies, and the remaining sizes come from a small linear system.\nFor example, the samples $2, 3, 5, 9$ of $2^k + 1$ satisfy $f(k+2) = 3 f(k+1) - 2 f(k)$, whose polynomial $z^2 - 3z + 2$ has roots $1$ and $2$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "65D10",
      "41A30",
      "65F20"
    ],
    "name": "Prony's method",
    "same_as": {
      "wikidata": "Q7249979",
      "wikipedia": "Prony's method"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Prony%27s_method"
      }
    ],
    "tag": "2WSV04QETW"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/2WSV04QETW/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/2WSV04QETW.md",
  "msc": [
    "65D10",
    "41A30",
    "65F20"
  ],
  "name": "Prony's method",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/2WS/2WSV04QETW.md",
  "status": null,
  "tag": "2WSV04QETW",
  "uri": "m2://2WSV04QETW",
  "url": "https://tamnd.github.io/math2/wiki/pronys-method/",
  "writing": {
    "tag": "2WSV04QETW",
    "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
      }
    ]
  }
}