---
tag: YK8121V1PN
kind: definition
name: Ambiguity function
msc: [94A12, 42A38]
defines: CDVA7EJSSD
uses: [SASP14MA1Z, 3DHNP5NEWH]
sources:
  - url: https://en.wikipedia.org/wiki/Ambiguity_function
---

## Definition

Let $f \in L^2(\R)$ (see [[Lp space|SASP14MA1Z]]).
The *ambiguity function* of $f$ is the function $Af \colon \R^2 \to \C$,
$$Af(x, \omega) = \int_{\R} f\!\left(t + \tfrac{x}{2}\right) \overline{f\!\left(t - \tfrac{x}{2}\right)} \, e^{-2\pi i \omega t} \, dt,$$
for a time shift $x \in \R$ and a frequency shift $\omega \in \R$.

## Remarks

The substitution $t = s - x/2$ gives $Af(x, \omega) = e^{\pi i x \omega} V_f f(x, \omega)$, where $V_f f$ is the [[short-time Fourier transform|3DHNP5NEWH]] of $f$ with window $f$.
Hence $\abs{Af(x, \omega)} \le Af(0, 0) = \norm{f}_2^2$, by the Cauchy–Schwarz inequality.
In radar texts the ambiguity function is often written as $\chi(\tau, \nu) = \int s(t) \overline{s(t - \tau)} e^{2\pi i \nu t} \, dt$.
It differs from $Af$ by a sign convention and a phase factor, and $\abs{\chi(\tau, \nu)} = \abs{Af(\tau, -\nu)}$ for $s = f$.

Example: let $f(t) = 2^{1/4} e^{-\pi t^2}$.
Then $f(t + x/2) \overline{f(t - x/2)} = \sqrt2 \, e^{-\pi x^2/2} e^{-2\pi t^2}$, and the Fourier transform of $e^{-2\pi t^2}$ is $2^{-1/2} e^{-\pi \omega^2/2}$.
So $Af(x, \omega) = e^{-\pi (x^2 + \omega^2)/2}$, a single peak of height $1 = \norm{f}_2^2$ at the origin.

Non-example: the autocorrelation $R(x) = \int_\R f(t) \overline{f(t - x)} \, dt$ is not the ambiguity function.
The substitution $t = s + x/2$ shows $R(x) = Af(x, 0)$, so $R$ is only the slice $\omega = 0$ of $Af$ and does not see frequency shifts.