---
tag: 47KD8V5A7G
kind: definition
name: Volterra series
msc: [93C10, 45G10]
defines: RQ3JKGWD03
uses: [2Z29JN3VES, SASP14MA1Z, 0Z49YH6ZWY]
sources:
  - url: https://en.wikipedia.org/wiki/Volterra_series
---

## Definition

Let $h_0 \in \R$ and, for each $n \ge 1$, let $h_n \colon [0, \infty)^n \to \R$ be an integrable function, the *$n$-th Volterra kernel*.
Assume that $\sum_{n \ge 1} \norm{h_n}_{L^1} r^n < \infty$ for some $r > 0$.
For a bounded measurable input $x \colon \R \to \R$ with $\sup \abs{x} \le r$, the *Volterra series* with these kernels gives the output
$$y(t) = h_0 + \sum_{n=1}^{\infty} \int_0^\infty \!\!\cdots\! \int_0^\infty h_n(\tau_1, \ldots, \tau_n)\, x(t - \tau_1) \cdots x(t - \tau_n)\, d\tau_1 \cdots d\tau_n, \quad t \in \R.$$
The $n$-th term is bounded by $\norm{h_n}_{L^1}\, r^n$, so the series converges absolutely and uniformly in $t$.

## Remarks

The operator $x \mapsto y$ is causal, since $y(t)$ depends only on $x(s)$ for $s \le t$, and time-invariant, since shifting $x$ in time shifts $y$ by the same amount.
The kernels can be taken symmetric in their variables without changing $y$.
If only $h_1$ is nonzero, $y = h_0 + h_1 * x$ is a [[convolution|2Z29JN3VES]], the output of a linear system with impulse response $h_1$.
Some authors also allow kernels with delta functions, or integrals over all of $\R^n$ for noncausal systems.

Example: let $z(t) = \int_0^\infty e^{-\tau} x(t - \tau)\, d\tau$ and $y = z^2$.
Then $y(t) = \int_0^\infty \int_0^\infty e^{-\tau_1 - \tau_2}\, x(t - \tau_1)\, x(t - \tau_2)\, d\tau_1\, d\tau_2$ by Fubini's theorem.
So $y$ is a Volterra series with $h_2(\tau_1, \tau_2) = e^{-\tau_1 - \tau_2}$, of $L^1$ norm $1$, and all other kernels $0$.

Non-example: the system $y(t) = t\, x(t)$ is not given by any Volterra series, because it is not time-invariant.
The input $x = 1$ gives $y(t) = t$, which is not constant, while a Volterra series maps a constant input to a constant output.