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. 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.