{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "47KD8V5A7G"
    }
  ],
  "body": "\n## Summary\n\nA Volterra series writes the output of a nonlinear, time-invariant system with memory as a sum of multiple integrals of the input.\nThe first term is a [[convolution|2Z29JN3VES]] with an impulse response, as for a linear system.\nThe next terms involve products of two, three or more past values of the input, weighted by kernels of several variables.\nIt plays for nonlinear operators the role that the [[Taylor series|0Z49YH6ZWY]] plays for functions.\nIt is used to model and identify nonlinear systems in engineering and biology.\n\n## Explanations\n\n### L1 human\n\nA linear system remembers its past inputs and adds them up with fixed weights.\nA nonlinear system may also multiply past inputs with each other, for example the input a second ago times the input two seconds ago.\nThe Volterra series lists all these effects: first the single inputs, then the pairs, then the triples, each with its own weights.\nFor example, squaring the output of a simple linear filter gives a system whose series has only a second order term.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "93C10",
      "45G10"
    ],
    "name": "Volterra series",
    "same_as": {
      "wikidata": "Q3957891",
      "wikipedia": "Volterra series"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Volterra_series"
      }
    ],
    "tag": "RQ3JKGWD03"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/RQ3JKGWD03/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/RQ3JKGWD03.md",
  "msc": [
    "93C10",
    "45G10"
  ],
  "name": "Volterra series",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/RQ3/RQ3JKGWD03.md",
  "status": null,
  "tag": "RQ3JKGWD03",
  "uri": "m2://RQ3JKGWD03",
  "url": "https://tamnd.github.io/math2/wiki/volterra-series/",
  "writing": {
    "tag": "RQ3JKGWD03",
    "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
      }
    ]
  }
}