---
tag: 28VV812XWQ
kind: concept
name: Fractional Fourier transform
msc: [42A38, 44A15, 94A12]
specializes: [74BXAPVSW8]
sources:
  - url: https://en.wikipedia.org/wiki/Fractional_Fourier_transform
same_as:
  wikidata: Q4817582
  wikipedia: Fractional Fourier transform
---

## Summary

The fractional Fourier transform is a family of unitary operators $\mathcal{F}_\alpha$ on $L^2(\R)$, indexed by an angle $\alpha$.
Here $\mathcal{F}_0$ is the identity and $\mathcal{F}_{\pi/2}$ is the [[Fourier transform|DPVM34ADZV]].
The 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.
In the time-frequency plane, $\mathcal{F}_\alpha$ acts as a rotation by the angle $\alpha$.
It is used in optics and signal processing to analyse chirps and other signals whose frequency changes in time.

## Explanations

### L1 human

Doing the Fourier transform four times gives back the original function, so it behaves like a rotation by a quarter turn.
The fractional Fourier transform fills in the rotations by all the other angles.
A signal is then seen from a direction between pure time and pure frequency.
For example, the Gaussian $e^{-x^2/2}$ is unchanged by every $\mathcal{F}_\alpha$, just as a round disc is unchanged by every rotation.