The fractional Fourier transform corresponds to rotation by an arbitrary angle; they are the elliptic elements of SL 2 (R), represented by the matrices [ a b c d ] = [ cos ⁡ ⁡ θ θ sin ⁡ ⁡ θ θ − − sin ⁡ ⁡ θ θ cos ⁡ ⁡ θ θ ] . The Fourier transform is the fractional Fourier transform when θ θ = 90 ∘ ∘ . The inverse Fourier transform corresponds to θ θ = − − 90 ∘ ∘ .