There are two sorts of transforms known as the fractional Fourier transform.
The linear fractional Fourier transform is a discrete Fourier transform in which the exponent is modified by the addition of a factor
,
 |
However, such transforms may not be consistent with their inverses unless 
is an integer relatively prime to 
so that 
. Fractional fourier transforms are implemented in the
Wolfram Language as Fourier[list,
FourierParameters -> 
a, b
], where 
is an additional scaling parameter. For example, the plots
above show 2-dimensional fractional Fourier transforms of the function 
for parameter 
ranging from 1 to 6.
The quadratic fractional Fourier transform is defined in signal processing and optics. Here, the fractional powers 
of the ordinary Fourier
transform operation 
correspond to rotation by angles 
in the time-frequency or space-frequency plane (phase
space). So-called fractional Fourier domains correspond to oblique axes in the
time-frequency plane, and thus the fractional Fourier transform (sometimes abbreviated
FRT) is directly related to the Radon transforms
of the Wigner distribution and the ambiguity
function. Of particular interest from a signal processing perspective is the
concept of filtering in fractional Fourier domains. Physically, the transform is
intimately related to Fresnel diffraction in wave and beam propagation and to the
quantum-mechanical harmonic oscillator.
